1MA 09. DERIVADES/1.MA.09.0 TVM 1MA.09.0.10DT TVM

Taxa de variació mitjana

La taxa de variació mitjana TVM d'una funció entre els punts d'abscissa a i b es calcular amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»T«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»VM«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.09.0.11Q CàlculTVMPolinG2 De la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«/mstyle»«/math»,

a) Calcula la taxa de variació mitjana en l’interval «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced mathcolor=¨#003300¨ open=¨[¨ close=¨]¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»p«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»,«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math»

b) Calcula l'equació de la recta secant al gràfic que passa pels punts d’abscisses «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»p«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#003300¨»p«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»2«/mn»«/mstyle»«/math»

]]> Mira el gràfic #G1

i constata si la funció és creixent o decreixent en tots els punts de l'interval o no.

]]> 1.0000000 0.3333333 0 0 a. =#sol1b. =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>p1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>p2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c1</mi></mtd></mtr><mtr><mtd><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>q1</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>p1</mi><mn>2</mn></msup><mo>+</mo><mi>b1</mi><mo>*</mo><mi>p1</mi><mo>+</mo><mi>c1</mi></mtd></mtr><mtr><mtd><mi>q2</mi><mo>=</mo><mi>a1</mi><mo>*</mo><msup><mi>p2</mi><mn>2</mn></msup><mo>+</mo><mi>b1</mi><mo>*</mo><mi>p2</mi><mo>+</mo><mi>c1</mi></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><msup><mi>b1</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a1</mi><mo>*</mo><mi>c1</mi><mo>></mo><mn>0</mn></mrow></mfenced><mo>&and;</mo><mo>(</mo><mi>p1</mi><mo><</mo><mi>p2</mi><mo>)</mo><mo>&and;</mo><mfenced><mrow><mfenced close="|" open="|"><mi>q1</mi></mfenced><mo>&les;</mo><mn>10</mn></mrow></mfenced><mo>&and;</mo><mfenced><mrow><mfenced close="|" open="|"><mi>q2</mi></mfenced><mo>&les;</mo><mn>10</mn></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfrac><mrow><mi>q2</mi><mo>-</mo><mi>q1</mi></mrow><mrow><mi>p2</mi><mo>-</mo><mi>p1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>p1</mi><mo>,</mo><mi>q1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>p2</mi><mo>,</mo><mi>q2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>recta</mi><mfenced><mrow><mi>R1</mi><mo>,</mo><mi>R2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Gràfics</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p1</mi><mo>,</mo><mi>p2</mi><mo>,</mo><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q1</mi><mo>,</mo><mi>q2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>,</mo><mo>-</mo><mn>3</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option><option name="answer_parameter">true</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Només cal aplicar la definició de la TVM:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»TVM«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mfrac mathcolor=¨#00007F¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»q«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»(«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»p«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«/mstyle»«/math»

]]> L'equació d'una recta que passa per dos punts es pot calcular a partir d'un dels punts (a,b) i del seu pendent que és la TVM: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«/mrow»«/mstyle»«/math»

y = TVM · (x - a) + b on b = f(a)

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»sol«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»p«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#00007F¨»q«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»)«/mo»«/mrow»«/mstyle»«/math»

]]> 1MA.09.0.21Q CàlculTVMRacG1/G1 De la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«/mstyle»«/math»,

]]> Mira el gràfic #G1

i constata si la funció és creixent o decreixent en tots els punts de l'interval o no.

]]> 1.0000000 0.3333333 0 0 a. =#sol1b. =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>p1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>p2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mi>a1</mi></mrow><mrow><mi>x</mi><mo>-</mo><mi>b1</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>p1</mi><mo>≠</mo><mi>b1</mi></mrow></mfenced><mo>&and;</mo><mfenced><mrow><mi>p2</mi><mo>≠</mo><mi>b1</mi></mrow></mfenced><mo>&and;</mo><mfenced><mrow><mi>a1</mi><mo>≠</mo><mi>b1</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>q1</mi><mo>=</mo><mfrac><mrow><mi>p1</mi><mo>-</mo><mi>a1</mi></mrow><mrow><mi>p1</mi><mo>-</mo><mi>b1</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mi>q2</mi><mo>=</mo><mfrac><mrow><mi>p2</mi><mo>-</mo><mi>a1</mi></mrow><mrow><mi>p2</mi><mo>-</mo><mi>b1</mi></mrow></mfrac></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfrac><mrow><mi>q2</mi><mo>-</mo><mi>q1</mi></mrow><mrow><mi>p2</mi><mo>-</mo><mi>p1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>p1</mi><mo>,</mo><mi>q1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>p2</mi><mo>,</mo><mi>q2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>recta</mi><mfenced><mrow><mi>R1</mi><mo>,</mo><mi>R2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Gràfics</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>sol2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>x</mi><mo>=</mo><mi>b1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p1</mi><mo>,</mo><mi>p2</mi><mo>,</mo><mi>q1</mi><mo>,</mo><mi>q2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option><option name="answer_parameter">true</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Només cal aplicar la definició de la TVM:

y = TVM · (x - a) + b on b = f(a)

]]> 1MA.09.0.22Q CàlculTVMLog De la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«/mstyle»«/math»,

Format de resposta: als centèsims

]]> Mira el gràfic #G1

i constata si la funció és creixent o decreixent en tots els punts de l'interval o no.

]]> 1.0000000 0.3333333 0 0 a. =#sol1b. =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>p1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>p2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>p1</mi><mo><</mo><mi>p2</mi></mrow></mfenced><mo>&and;</mo><mfenced><mrow><mi>p1</mi><mo>></mo><mo>-</mo><mfrac><mi>b1</mi><mi>a1</mi></mfrac></mrow></mfenced><mo>&and;</mo><mfenced><mrow><mi>a1</mi><mo>*</mo><mi>p1</mi><mo>+</mo><mi>b1</mi><mo>></mo><mn>0</mn></mrow></mfenced><mo>&and;</mo><mfenced><mrow><mi>a1</mi><mo>*</mo><mi>p2</mi><mo>+</mo><mi>b1</mi><mo>></mo><mn>0</mn></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b1</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>f</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q1</mi><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a1</mi><mo>*</mo><mi>p1</mi><mo>+</mo><mi>b1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q2</mi><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a1</mi><mo>*</mo><mi>p2</mi><mo>+</mo><mi>b1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfrac><mrow><mi>q2</mi><mo>-</mo><mi>q1</mi></mrow><mrow><mi>p2</mi><mo>-</mo><mi>p1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>p1</mi><mo>,</mo><mi>q1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>p2</mi><mo>,</mo><mi>q2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>recta</mi><mfenced><mrow><mi>R1</mi><mo>,</mo><mi>R2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Gràfics</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>10</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>5</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>f</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>sol2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mn>5</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.2695</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p1</mi><mo>,</mo><mi>p2</mi><mo>,</mo><mi>q1</mi><mo>,</mo><mi>q2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>2.6391</mn><mo>,</mo><mn>3.1781</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0.2695</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1.8306</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-1)</option><option name="relative_tolerance">true</option><option name="precision">2</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option><option name="answer_parameter">true</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Només cal aplicar la definició de la TVM:

y = TVM · (x - a) + b on b = f(a)

]]> 1MA 09. DERIVADES/1MA.09.1 DerivadaRectaTgNormal 1MA.09.1.10DT EN UN PUNT TEORIA

Derivada en un punt

La derivada de la funció f(x) en el punt d'abscissa x_o es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»`«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«munder mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»l§#237;m«/mi»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mn mathvariant=¨bold¨»0«/mn»«/mrow»«/munder»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

i és el pendent de la recta tangent en aquest punt.

]]> 0.0000000 0.0000000 0 1MA.09.1.11Q Definició de derivada Considera la funció f(x)=#funcio. Considera el punt x=#a i el paràmetre h=#h.
1.- Calcula la recta secant que passa pel punt x = #a i x = #a+#h.
2.- Calcula la recta tangent que passa pel punt x =#a.
3.- Calcula el pendent de la recta tangent.
4.- Calcula la derivada de la funció f(x) en el punt x=#a.

Nota: introduïu la solució en forma de llista.
Així si la recta secant és s(x)=Ax+B, la tangent t(x)=Cx+D, el pendent P i la derivada E, la solució que heu d'introduir és

{Ax+B,Cx+D,P,E}

]]> Recorda que si tenim una funció f(x) i un punt x=a,

- la recta secant que passa pels punts x=a i x=a+h és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»f«/mi»«mo»(«/mo»«mi»a«/mi»«mo»+«/mo»«mi»h«/mi»«mo»)«/mo»«mo»-«/mo»«mi»f«/mi»«mo»(«/mo»«mi»a«/mi»«mo»)«/mo»«/mrow»«mi»h«/mi»«/mfrac»«mo»(«/mo»«mi»x«/mi»«mo»-«/mo»«mi»a«/mi»«mo»)«/mo»«mo»+«/mo»«mi»f«/mi»«mo»(«/mo»«mi»a«/mi»«mo»)«/mo»«/math»
- la recta tangent en el punt x=a és y=f'(a)(x-a)+f(a).

Ax+B

]]> Recorda que si tenim una funció f(x) i un punt x=a,

- la recta secant que passa pels punts x=a i x=a+h és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«/mrow»«/mstyle»«/math»
- la recta tangent en el punt x=a és y=f'(a)(x-a)+f(a).

]]> 1.0000000 0.5000000 0 0 #L <question><wirisCasSession><session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>10</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>2</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r2</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>+</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>*</mo><mi>x</mi><mo>+</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r15</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>5</mn><mo>.</mo><mo>.</mo><mn>15</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mrow><mi>r15</mi><mo>(</mo><mo>)</mo></mrow><mn>10</mn></mfrac></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>derivada</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>a</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pendent</mi><mo>=</mo><mi>derivada</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>seccant</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>tanggent</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mi>s</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>10</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>2</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r2</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>+</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>*</mo><mi>x</mi><mo>+</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mo>-</mo><mn>10</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>10</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r15</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>5</mn><mo>.</mo><mo>.</mo><mn>15</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mrow><mi>r15</mi><mo>(</mo><mo>)</mo></mrow><mn>10</mn></mfrac></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi><mo>(</mo><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>seccant</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mi>h</mi></mfrac><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mo>-</mo><mn>29</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>17</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tanggent</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mo>-</mo><mn>21</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>derivada</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pendent</mi><mo>=</mo><mi>derivada</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>seccant</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>29</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>17</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>tanggent</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>s</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>pendent</mi><mo>,</mo><mi>derivada</mi></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>29</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>17</mn><mo>,</mo><mo>-</mo><mn>21</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn><mo>,</mo><mo>-</mo><mn>21</mn><mo>,</mo><mo>-</mo><mn>21</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #L </correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">inlineEditor</data><data name="inputCompound">false</data><data name="cas">false</data></localData></question> 1MA.09.1.13Q Definició de derivada (recta tangent) Considera la funció f(x)=#funció.
Calcula la recta tangent que passa pel punt x =#a.

Nota: si la tangent t(x)=Cx+D has d'introduir estrictament

Cx+D

]]> Recorda que si tens una funció f(x) i un punt x=a,

- la recta secant que passa pels punts x=a i x=a+h és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«/mrow»«/mstyle»«/math»
- la recta tangent en el punt x=a és y=f'(a)(x-a)+f(a).

Calcula el pendent de la recta tangent en x=#a.

]]> Recorda que si tens una funció f(x) i un punt x=a,

- la recta secant que passa pels punts x=a i x=a+h és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»y«/mi»«mo mathvariant=¨bold¨»=«/mo»«mfrac»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mstyle»«/math»
- la recta tangent en el punt x=a és y=f'(a)(x-a)+f(a).

]]> 1.0000000 0.5000000 0 0 #L <question><wirisCasSession><session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>10</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>2</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r2</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>+</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>*</mo><mi>x</mi><mo>+</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r15</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>5</mn><mo>.</mo><mo>.</mo><mn>15</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mrow><mi>r15</mi><mo>(</mo><mo>)</mo></mrow><mn>10</mn></mfrac></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mo>-</mo><mn>10</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>10</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r15</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>5</mn><mo>.</mo><mo>.</mo><mn>15</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mrow><mi>r15</mi><mo>(</mo><mo>)</mo></mrow><mn>10</mn></mfrac></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi><mo>(</mo><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>seccant</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mi>h</mi></mfrac><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mo>-</mo><mn>29</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>17</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tanggent</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mo>-</mo><mn>21</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>derivada</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pendent</mi><mo>=</mo><mi>derivada</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>seccant</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>29</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>17</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>tanggent</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>s</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>pendent</mi><mo>,</mo><mi>derivada</mi></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>29</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>17</mn><mo>,</mo><mo>-</mo><mn>21</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>9</mn><mo>,</mo><mo>-</mo><mn>21</mn><mo>,</mo><mo>-</mo><mn>21</mn></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #L </correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">inlineEditor</data><data name="inputCompound">false</data><data name="cas">false</data></localData></question> 1MA.09.1.15Q Definició de derivada (derivada en x=a) Considera la funció f(x)=#funcio.
Calcula la derivada de la funció f(x) en el punt x=#a.

]]> Recorda que si tens una funció f(x) i un punt x=a,

- la recta secant que passa pels punts x=a i x=a+h és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«/mrow»«/mstyle»«/math»
- la recta tangent en el punt x=a és y=f'(a)(x-a)+f(a).

Derivabilitat d'una funció

Una funció és derivable en el punt d'abscissa x₀ si i només si les derivades laterals existeixen i són iguals:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«munder mathcolor=¨#003300¨»«mo largeop=¨true¨ mathvariant=¨bold¨»l§#237;m«/mo»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mn mathvariant=¨bold¨»0«/mn»«/mrow»«/munder»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«munder mathcolor=¨#003300¨»«mo largeop=¨true¨ mathvariant=¨bold¨»l§#237;m«/mo»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mn mathvariant=¨bold¨»0«/mn»«/mrow»«/munder»«mfrac mathcolor=¨#003300¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

que també es pot escriure f'_-(x) = f'₊(x)

]]> 0.0000000 0.0000000 0 1MA.09.1.21Q Derivada lateral esquerra d'una funció en un punt Considera la funció: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨¨»«mtable»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#10877;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#62;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mstyle»«/math».
Calcula «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«msub mathcolor=¨#003300¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»`«/mo»«mo mathvariant=¨bold¨»-«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math».

]]> Recorda que:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«msub mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»`«/mo»«mo mathvariant=¨bold¨»+«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«munder mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»lim«/mi»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«msup»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨»+«/mo»«/msup»«/mrow»«/munder»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«msub mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»`«/mo»«mo mathvariant=¨bold¨»-«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«munder mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»lim«/mi»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«msup»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨»-«/mo»«/msup»«/mrow»«/munder»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 1.0000000 0.1000000 0 true true abc #sol1

]]> #sol2

]]> #sol3

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definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>sol2</mi><mo>==</mo><mi>sol1</mi></mrow><mrow><mi>sol1</mi><mo>=</mo><cn>+∞</cn></mrow></apply></mtd></mtr><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>sol2</mi><mo>==</mo><mi>sol3</mi></mrow><mrow><mi>sol3</mi><mo>=</mo><cn>+∞</cn></mrow></apply></mtd></mtr></mtable></apply><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>opcio</mi><mo>(</mo><mi>suma</mi><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>suma</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>false</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><cn>+∞</cn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>122</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.22Q Derivada lateral dreta d'una funció en un punt Considera la funció: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨¨»«mtable»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#10877;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»f«/mi»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#62;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mstyle»«/math».
Calcula «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«msub mathcolor=¨#003300¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»`«/mo»«mo mathvariant=¨bold¨»+«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math»

]]> Recorda que: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«msub mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»`«/mo»«mo mathvariant=¨bold¨»+«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«munder mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»lim«/mi»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«msup»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨»+«/mo»«/msup»«/mrow»«/munder»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«msub mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»`«/mo»«mo mathvariant=¨bold¨»-«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«munder mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»lim«/mi»«mrow»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«msup»«mn mathvariant=¨bold¨»0«/mn»«mo mathvariant=¨bold¨»-«/mo»«/msup»«/mrow»«/munder»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«mi mathvariant=¨bold¨»h«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 1.0000000 0.5000000 0 true true abc #sol1

]]> #sol2

]]> #sol3

Determina els valors d'a i de b que fan que f sigui contínua i derivable (amb continuitat) a tot «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8477;«/mo»«/math».

Nota: si la solució és a=3 i b=4, has d'introduir {3,4}.

]]> Una funció és contínua en un punt si

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»s«/mi»«mo»)«/mo»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mi»s«/mi»«mo»-«/mo»«/msup»«/mrow»«/munder»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«munder»«mi»lim«/mi»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«msup»«mi»s«/mi»«mo»+«/mo»«/msup»«/mrow»«/munder»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math».

Una funció és contínua i derivable en un punt s si «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»f«/mi»«mo»+«/mo»«/msub»«mo»§apos;«/mo»«mo»(«/mo»«mi»s«/mi»«mo»)«/mo»«mo»=«/mo»«msub»«mi»f«/mi»«mo»-«/mo»«/msub»«mo»§apos;«/mo»«mo»(«/mo»«mi»s«/mi»«mo»)«/mo»«/math».

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xmlns="http://www.w3.org/1998/Math/MathML"><mi>p1prima</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>p1</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></apply></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>A</mi></mrow></mfenced><mo>&map;</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p2prima</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>p2</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></apply></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>B</mi></mrow></mfenced><mo>&map;</mo><mo>-</mo><mn>18</mn><mo>*</mo><mfenced><mrow><mi>B</mi><mo>*</mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>30</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L1</mi><mo>=</mo><mi>solve</mi><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>p1</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>p2</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>B</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>p1prima</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>p2prima</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>B</mi><mo>)</mo></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>A</mi><mo>=</mo><mo>-</mo><mfrac><mn>3974404</mn><mn>243</mn></mfrac><mo>,</mo><mi>B</mi><mo>=</mo><mfrac><mn>21881</mn><mn>486</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi><mo>=</mo><msub><mi>L1</mi><mn>1</mn></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>3974404</mn><mn>243</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>=</mo><msub><mi>L1</mi><mn>1</mn></msub><mo>(</mo><mi>B</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>21881</mn><mn>486</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>a1</mi><mo>,</mo><mi>b1</mi></mtd></mtr></mtable></mfenced></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mfrac><mn>3974404</mn><mn>243</mn></mfrac><mo>,</mo><mfrac><mn>21881</mn><mn>486</mn></mfrac></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f1</mi><mo>=</mo><mi>p1</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>*</mo><mi>a</mi><mo>+</mo><mn>3</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f2</mi><mo>=</mo><mi>p2</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mo>*</mo><mi>b</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>*</mo><mi>b</mi><mo>-</mo><mn>10</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #L </correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">inlineEditor</data><data name="inputCompound">false</data><data name="cas">false</data></localData></question> 1MA.09.1.25Q 2005S Paràmetres per contínua i derivable Considera la funció: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨¨»«mtable columnspacing=¨1.4ex¨ columnalign=¨left¨»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»g«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«mtd»«mi mathvariant=¨bold¨»si«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#60;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»g«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mtd»«mtd»«mi mathvariant=¨bold¨»si«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#8805;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mstyle»«/math»

Determina per quins valors de a i b, la funció és contínua i derivable en tots els nombres reals.

Setembre 2005 MA

Format: {{a=2,b=3},{a=-1,b=-2}}

]]> Una funció és contínua en un punt d'abscissa #s si

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«munder mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»lim«/mi»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mo mathvariant=¨bold¨»#«/mo»«msup»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»-«/mo»«/msup»«/mrow»«/munder»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«munder mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»lim«/mi»«mrow»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#8594;«/mo»«mo mathvariant=¨bold¨»#«/mo»«msup»«mi mathvariant=¨bold¨»s«/mi»«mo mathvariant=¨bold¨»+«/mo»«/msup»«/mrow»«/munder»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«/math».

Una funció és derivable en un punt d'abscissa #s si les derivades laterals són iguals «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨»+«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»`«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨»-«/mo»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»`«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»s«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«/math».

]]> 1MA.09.1.26Q 2008J Paràmetres per contínua i derivable Considera la funció: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨¨»«mtable columnspacing=¨1.4ex¨ columnalign=¨left¨»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»g«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«mtd»«mi mathvariant=¨bold¨»si«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#60;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»g«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mtd»«mtd»«mi mathvariant=¨bold¨»si«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»§#8805;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mstyle»«/math»

Troba els valors dels paràmetres a i b per tal que la funció sigui contínua i derivable en x = #s.

Juny 2008MA

Format: {{a=2,b=3}}

]]> Una funció és contínua en un punt d'abscissa #s si

]]> 1MA.09.1.30DT EQ RECTA TANGENT/PERP

Equació de la recta tangent
i de la recta perpendicular

L'equació de la recta tangent en el punt d'abscissa x₀ es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfenced mathcolor=¨#003300¨ open=¨[¨ close=¨]¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»`«/mo»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfenced»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math»

L'equació de la recta perpendicular en el punt d'abscissa x₀ es calcula amb:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»y«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mfrac mathcolor=¨#003300¨»«mn mathvariant=¨bold¨»1«/mn»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»`«/mo»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«msub mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«/mrow»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.09.1.31T RectaTgNormal L'equació de la recta tangent al gràfic de la funció f(x) en el punt d'abscissa a es calcula amb:

y - {1:SA: ~=f(a)} =[{1:SA: ~=f'(a)}] (x - {1:SA: ~=a}).

El pendent de la recta tangent al gràfic de la funció f(x) en el punt d'abscissa a és:{1:SA: ~=f'(a)}

Si dues rectes són paral·leles els seus pendents són {1:SA: ~=iguals}

Si dues rectes són perpendiculars,

els seus pendents m i m' són tal que m·m' = {1:SA: ~=-1}

]]> 0.5000000 0 1MA.09.1.32T Trobar PendentDeRectaAmb2punts Quin és pendent de la recta que passa pels punts A(#x_1,#y_1) i B(#x_2,#y_2)? {#1}.

Quin és el pendent d'una recta paral·lela a la recta que passa per aquests punts A i B?{#2}.

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>x_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>y_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>y_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>x_2</mi><mo>-</mo><mi>x_1</mi></mrow></mfenced><mo>≠</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>y_2</mi><mo>-</mo><mi>y_1</mi></mrow><mrow><mi>x_2</mi><mo>-</mo><mi>x_1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>m</mi></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x_1</mi><mo>,</mo><mi>y_1</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mo>-</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x_2</mi><mo>,</mo><mi>y_2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>8</mn><mo>,</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>7</mn><mn>12</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>12</mn><mn>7</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> 1MA.09.1.33T Trobar PendentDeRecta (cartesiana) Quin és el pendent de la recta #e ? {#1}.

Quin és el pendent d'una recta paral·lela a la recta #e ? {#2}.

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>*</mo><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>m</mi></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>,</mo><mo>-</mo><mn>4</mn><mo>,</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn><mo>*</mo><mi>y</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>4</mn><mn>7</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> 1MA.09.1.34T Trobar PendentDeRecta Explícita COMPLETA:
El pendent de la recta #e és {#1}.

El pendent d'una recta paral·lela a la recta #e és {#2}.

]]> 2.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>m</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>m</mi><mo>≠</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>m</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>m</mi></mfrac></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>7</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> 1MA.09.1.41APREN Recta tangent en un punt Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

Calcula primer f(#x_0) = {#1}
Deriva f(x): f'(x)= {#2} Format de la resposta : 3*x^2-2*x+4
Calcula f'(#x_0)= {#3}
L'equació de la recta tangent es calcula doncs amb:
y - {#4} = {#5}·(x - {#6})

En forma explícita l'equació és: {#7}

]]> 7.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a_3</mi><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfenced><mi>f_2</mi></mfenced><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_0</mi><mo>)</mo><mo>+</mo><mi>f_0</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_0</mi><mo>,</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>x_0</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>446</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>254</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>254</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>824</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.42APREN RectaTangentPuntRacionalG1G1 Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

Calcula primer f(#x_0) = {#1}
Deriva f(x): f'(x)= {#2} Format de la resposta : -3*x/(2*x^2-12*x+9)
Calcula f'(#x_0)= {#3}
L'equació de la recta tangent es calcula doncs amb:
y - {#4} = {#5}·(x - {#6})

En forma explícita l'equació és: {#7}

]]> 7.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>x_0</mi><mo>≠</mo><mfenced><mrow><mo>-</mo><mfrac><mi>a_3</mi><mi>a_2</mi></mfrac></mrow></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><mfrac><mi>a_1</mi><mi>a_0</mi></mfrac><mo>≠</mo><mfrac><mi>a_3</mi><mi>a_2</mi></mfrac></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>a_0</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_1</mi></mrow><mrow><mi>a_2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_3</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfenced><mi>f_2</mi></mfenced><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_0</mi><mo>)</mo><mo>+</mo><mi>f_0</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_0</mi><mo>,</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>x_0</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>6</mn><mn>11</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mn>13</mn><mrow><mo>-</mo><mn>16</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>40</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>25</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>13</mn></mrow><mrow><mn>16</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>40</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>13</mn><mn>121</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>13</mn><mn>121</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>118</mn><mn>121</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.43APREN RectaTangentPuntLogaritme Considereu la funció f(x) = #f_5.
Trobeu l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.
Format dels logaritmes: arrodoniu als cent mil·lèsims
Calculeu primer f(#x_0) = {#1}
Deriveu f(x): f'(x)= {#2} Format de la resposta : -3*x/(2*x^2-12*x+9)
Calculeu f'(#x_0)= {#3}
L'equació de la recta tangent es calcula doncs amb:
y - {#4} = {#5}·(x - {#6})

En forma explícita l'equació és: {#7}

]]> 7.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a_1</mi><mo>*</mo><mi>x_0</mi><mo>+</mo><mi>a_2</mi><mo>&gt;</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfenced><mi>f_2</mi></mfenced><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_0</mi><mo>)</mo><mo>+</mo><mi>f_0</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_0</mi><mo>,</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>x_0</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.9444</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mn>5</mn><mrow><mn>5</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mrow><mn>5</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>19</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0.26316</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2.155</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.44APREN RectaTangentPunt Polinomi Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

Calcula primer f(#x_0) = {#1}
Deriva f(x): f'(x)= {#2} Format de la resposta : 3*x^2-2*x+4
Calcula f'(#x_0)= {#3}
L'equació de la recta tangent es calcula doncs amb:
y = {#4}·(x - {#5}) + {#6}

En forma explícita l'equació és: {#7}

]]> 7.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a_3</mi><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>446</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>254</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>254</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>824</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.45APREN Recta tangent en un punt racionalG2G1 Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

Calculeu primer f(#x_0) = {#1}
Deriveu f(x): f'(x)= {#2} Format de la resposta : (x^2-3*x)/(2*x^2-12*x+9)
Calculeu f'(#x_0)= {#3}
L'equació de la recta tangent es calcula doncs amb:
y - {#4} = {#5}·(x - {#6})

En forma explícita l'equació és: {#7}

]]> 7.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>x_0</mi><mo>≠</mo><mfenced><mrow><mo>-</mo><mfrac><mi>a_3</mi><mi>a_2</mi></mfrac></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>a_0</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_1</mi></mrow><mrow><mi>a_2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_3</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfenced><mi>f_2</mi></mfenced><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_0</mi><mo>)</mo><mo>+</mo><mi>f_0</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_0</mi><mo>,</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>x_0</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mrow><mn>3</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mrow><mn>3</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>12</mn><mn>17</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mn>2</mn><mrow><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>289</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>289</mn></mfrac><mo>*</mo><mi>x</mi><mo>-</mo><mfrac><mn>214</mn><mn>289</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.51Q RTangent PolinG3 Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

En forma explícita l'equació és: {#1}

]]> 1.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_3</mi><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi><mo>=</mo><mi>y</mi><mo>=</mo><mfenced><mi>f_2</mi></mfenced><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_0</mi><mo>)</mo><mo>+</mo><mi>f_0</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_0</mi><mo>,</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>x_0</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_0</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>446</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>254</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>254</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>824</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.52Q RTangent RacionalG1G1 Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

En forma explícita l'equació és: {#1}

]]> 1.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>x_0</mi><mo>≠</mo><mfenced><mrow><mo>-</mo><mfrac><mi>a_3</mi><mi>a_2</mi></mfrac></mrow></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><mfrac><mi>a_1</mi><mi>a_0</mi></mfrac><mo>≠</mo><mfrac><mi>a_3</mi><mi>a_2</mi></mfrac></mrow></mfenced></mrow></apply></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_0</mi><mo>,</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>x_0</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi></math></input><output><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>13</mn></mrow><mrow><mn>16</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>40</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>13</mn><mn>121</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>13</mn><mn>121</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>118</mn><mn>121</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.53Q RTangent RacionalG1G1 Considereu la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/math»
Trobeu l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x0.

]]> Per determinar l'equació de la recta tangent:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨»y«/mi»«mo mathvariant=¨bold¨»=«/mo»«mfenced mathcolor=¨#FF0000¨ open=¨[¨ close=¨]¨»«mrow»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»`«/mo»«mo mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/mfenced»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mn mathvariant=¨bold¨»0«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«/math»

Calculem f'(#x0) = #df0 i f(#x0) = #f0

El gràfic és #g1.

Calculem f'(#x0) = #df0 i f(#x0) = #f0

y = #df0 (#w0) + (#f0)

]]> 1MA.09.1.54.1Q RTangent RacionalG2G1 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math»
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x0.

]]> Per determinar l'equació de la recta tangent:

Calculem f'(#x0) = #df0 i f(#x0) = #f0

El gràfic és #g1.

Calculem f'(#x0) = #df0 i f(#x0) = #f0

y = #df0 (#w0) + (#f0)

]]> 1MA.09.1.54.2Q RTangent RacionalG2G1 Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.

En forma explícita l'equació és: {#1}

]]> 1.0000000 0.1000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mn>2</mn><mrow><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>289</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>289</mn></mfrac><mo>*</mo><mi>x</mi><mo>-</mo><mfrac><mn>214</mn><mn>289</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.1.55Q RTangent RacionalG2G2 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math»
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x0.

]]> Per determinar l'equació de la recta tangent:

Calculo f'(#x0) = #df0 i f(#x0) = #f0

El gràfic és #g1.

Calculem f'(#x0) = #df0 i f(#x0) = #f0

y = #df0 (#w0) + (#f0)

]]> 1MA.09.1.56Q RTangent Logaritme Considera la funció f(x) = #f_5.
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x_0.
Format dels logaritmes: arrodoniu als cent mil·lèsims

En forma explícita l'equació és: {#1}

]]> 1.0000000 0.1000000 0 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_5</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>f_3</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0.35714</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0.35714</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1.5676</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="precision">8</option><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="casSession"/></localData></question> 1MA.09.1.57Q RTangent IrracionalG1 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math»
Troba l'equació de la recta tangent al gràfic de f(x) en el punt d'abscissa x₀ = #x0.

Format:
arrodoneix als dècims les arrels, i els resultats de les operacions que en derivin.

]]> Per determinar l'equació de la recta tangent:

Calculem f'(#x0) = #df0 i f(#x0) = #f0

El gràfic és #g1.

Calculem f'(#x0) = #df0 i f(#x0) = #f0

y = #df0 (#w0) + (#f0)

]]> 1MA.09.1.58Q RTangent Irracional G2 Considera la funció f(x) = #f_1
Quina és l'equació de la recta tangent en el punt d'abscissa x₀ = #x_0?

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«mo»*«/mo»«mi»x«/mi»«mo»-«/mo»«mfrac»«mrow»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mrow»«mn»2«/mn»«/mfrac»«/math»

Format de la resposta: {[2/3,-5],[3/4,1/4]}

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definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_A</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>y_A</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_B</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>y_B</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>x_A</mi><mo>,</mo><mi>y_A</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>x_B</mi><mo>,</mo><mi>y_B</mi></mrow></mfenced></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>x_B</mi><mo>≠</mo><mi>x_A</mi></mrow></apply><mrow><mi>a_1</mi><mo>≠</mo><mi>a_2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>y_B</mi><mo>-</mo><mi>y_A</mi></mrow><mrow><mi>x_B</mi><mo>-</mo><mi>x_A</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>6</mn><mo>*</mo><mi>a_2</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfenced><mrow><mo>∫</mo><mrow><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>ⅆ</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mi>a_3</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mi>m</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_1</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_2</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>r_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>r_1</mi><mo>,</mo><mi>r_2</mi></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>a_4</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>,</mo><mo>-</mo><mn>7</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x_A</mi><mo>,</mo><mi>y_A</mi><mo>,</mo><mi>x_B</mi><mo>,</mo><mi>y_B</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>1</mn><mo>,</mo><mn>7</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>42</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>336</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>336</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>21</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>336</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>42</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>336</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>167</mn><mn>21</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_01</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mfrac><mn>167</mn><mn>21</mn></mfrac><mo>,</mo><mfrac><mn>28244</mn><mn>21</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.1.62Q RTangent//Explícita PolinG2 Considera la funció f(x) = #f_1.
En quin punt, la recta tangent al gràfic de f(x) és paral·lela a la recta d'equació #t

Format de la resposta: {[2/3,-5],[3/4,1/4]}

]]> 1.0000000 0.5000000 0 0 #e_01 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_A</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>y_A</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_B</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>y_B</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>x_A</mi><mo>,</mo><mi>y_A</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>x_B</mi><mo>,</mo><mi>y_B</mi></mrow></mfenced></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>x_B</mi><mo>≠</mo><mi>x_A</mi></mrow></apply><mrow><mi>a_1</mi><mo>≠</mo><mi>a_2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mi>y_B</mi><mo>-</mo><mi>y_A</mi></mrow><mrow><mi>x_B</mi><mo>-</mo><mi>x_A</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>y_A</mi><mo>-</mo><mi>m</mi><mo>*</mo><mi>x_A</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>m</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>n</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>6</mn><mo>*</mo><mi>a_2</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfenced><mrow><mo>∫</mo><mrow><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>&DifferentialD;</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mi>a_3</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mi>m</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_1</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_2</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>r_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_01</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>r_1</mi><mo>,</mo><mi>r_2</mi></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>a_3</mi><mo>,</mo><mi>a_4</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x_A</mi><mo>,</mo><mi>y_A</mi><mo>,</mo><mi>x_B</mi><mo>,</mo><mi>y_B</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>8</mn><mo>,</mo><mn>6</mn><mo>,</mo><mo>-</mo><mn>3</mn><mo>,</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mn>6</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>48</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>288</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>288</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>288</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>48</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>288</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>721</mn><mn>120</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>46</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_01</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mfrac><mn>721</mn><mn>120</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>517199</mn><mn>600</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #e_01 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.1.63Q RTangent//rectaAB PolinG3 Considera la funció f(x) = #f_1.
En quin(s) punt(s), la recta tangent al gràfic de f(x) és paral·lela a la recta que passa pels punts #A i #B?

Format de la resposta: {[2/3,-5],[3/4,1/4]}

]]> 1.0000000 0.5000000 0 0 #e_01 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_02</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mn>5</mn><mo>,</mo><mo>-</mo><mn>447</mn></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>309</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #e_01 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> Per una banda, el pendent de la recta tangent és la derivada en el punt.
Per altra banda, amb dos punts podem trobar el pendent de la recta AB.
Si són paral·leles, la derivada ha de ser igual al pendent de la recta AB.
Igualant, es troba l'abscissa (x) del punt.
Substituint, la seva ordenada (y).]]> 1MA.09.1.64Q RTangent//rectaAB PolinG3 Considera la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math».
En quin(s) punt(s), la recta tangent al gràfic de f(x) és paral·lela a la recta que passa pels punts «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«/mrow»«/mstyle»«/math»?

Format de la resposta: {[2/3,-5],[3/4,1/4]}

]]> Per una banda, el pendent de la recta tangent és la derivada en el punt.
Per altra banda, amb dos punts podem trobar el pendent de la recta AB.
Si són paral·leles, la derivada ha de ser igual al pendent de la recta AB.
Igualant, es troba l'abscissa (x) del punt.
Substituint, la seva ordenada (y).]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mfrac><mn>681</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mfrac><mn>303</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mfrac><mn>681</mn><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mfrac><mn>303</mn><mn>2</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per una banda, el pendent de la recta tangent és la derivada en el punt, #f_2.
Per altra banda, amb dos punts podem trobar el pendent #m de la recta AB.
Si són paral·leles, la derivada ha de ser igual al pendent de la recta AB.
Igualant #f_2 = #m, es troba l'abscissa (x) del punt.
Substituint, la seva ordenada (y).]]> 1MA.09.1.71Q RTangentPerpAB PolinG3 Considera la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»1«/mn»«/mrow»«/mstyle»«/math».
En quin(s) punt(s), la recta tangent al gràfic de f(x) és perpendicular a la recta que passa pels punts «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»A«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»B«/mi»«/mrow»«/mstyle»«/math»?

Format de la resposta: {[2/3,-5],[3/4,1/4]}

]]> 1.0000000 0.5000000 0 0 #e_01 #e_02 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mo>-</mo><mn>9</mn><mo>,</mo><mn>2468</mn></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>7</mn><mo>,</mo><mo>-</mo><mn>1708</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_02</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mn>7</mn><mo>,</mo><mo>-</mo><mn>1708</mn></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mo>-</mo><mn>9</mn><mo>,</mo><mn>2468</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #e_01 </correctAnswer><correctAnswer type="mathml"> #e_02 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> El pendent de la recta tangent és la derivada f'(x).
La recta AB té per vector director #v i per pendent #m.
Si són perpendiculars el producte dels pendents és m · m' = -1; cal doncs resoldre f '(x) = -1/m.

]]> 1MA.09.1.72Q RTangentPerp PolinG3 2Punts Considera la funció f(x) = #f_1.
En quin(s) punt(s), la recta tangent al gràfic de f(x) és perpendicular a la recta que passa pels punts #A i #B?

Format de la resposta: {[2/3,-5],[3/4,1/4]}

]]> Si són perpendiculars, el producte dels seus pendents és (-1):

#m · (#f_2) = -1

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#m · (#f_2) = -1

]]> 1MA 09. DERIVADES/1MA.09.2 CàlculDerivades 1MA.09.2.10DT TAULA: DERIVADES SIMPLES

]]> 0.0000000 0.0000000 0 1MA.09.2.11Q Polinomi G5 Calcula la derivada de f(x) = #g

Format de la resposta: asteriscs pels coeficients i circumflex pel grau: 3*x^3.

]]> És una suma de monomis: axⁿ

]]> 1.0000000 0.5000000 0 0 #r_28 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>c_5</mi><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mi>c_4</mi><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>c_3</mi><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>c_2</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_28</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>35</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>20</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>18</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_28</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>35</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>20</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>18</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_28 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.2.12Q Polinomi G5 paràmetre Calcula la derivada de f(x) = #g

Format de la resposta: asteriscs pels coeficients i circumflex pel grau: 3*x^3.

]]> Deriva cada monomi amb la taula.

]]> 1.0000000 0.5000000 0 0 #r_28 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>c_5</mi><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mi>c_4</mi><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>c_3</mi><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>m</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_28</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>-</mo><mn>7</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>-</mo><mn>7</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><mi>m</mi><mo>*</mo><mi>x</mi><mo>-</mo><mn>45</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>28</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_28</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><mi>m</mi><mo>*</mo><mi>x</mi><mo>-</mo><mn>45</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>28</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_28 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.2.13Q Arrel Simple Calcula la derivada de f(x) = #g

]]> N'hi ha prou amb fer servir la taula. La derivada de #c_5·f(x) és #c_5·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de a·f(x) és a·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de #c_5·f(x) és #c_5·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de m·f(x) és m·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de #c_0·f(x) és #c_0·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de m·f(x) és m·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de #c_0·f(x) és #c_0·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de (a+#c_0)·f(x) és (a+#c_0)·f'(x)

]]> N'hi ha prou amb fer servir la taula. La derivada de (#a_5)·f(x) és (#a_5)·f'(x)]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_5</mi><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>c_5</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>(</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>c_5</mi><mo>)</mo><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.22Q Arcsinus(paràmetre) Calcula la derivada de f(x) = #g

]]> N'hi ha prou amb fer servir la taula. La derivada de (#a_5)·f(x) és (#a_5)·f'(x)]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_5</mi><mo>=</mo><mfrac><mi>a</mi><mn>4</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_5</mi><mo>*</mo><mi>asin</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mo>*</mo><mi>arcsin</mi><mfenced><mi>x</mi></mfenced></mrow><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mo>*</mo><mi>arcsin</mi><mfenced><mi>x</mi></mfenced></mrow><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mrow><mn>4</mn><mo>*</mo><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mrow><mn>4</mn><mo>*</mo><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.23Q Arctangent(paràmetre) Calcula la derivada de f(x) = #g

]]> N'hi ha prou amb fer servir la taula. La derivada de (#a_5)·f(x) és (#a_5)·f'(x)]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_5</mi><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_5</mi><mo>*</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>arctan</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><mi>arctan</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.31Q SumaAleat_1 Calcula la derivada de f(x) = #g

Format de la resposta: redueix a denominador comú si s'escau.

]]> La derivada és la suma de les derivades.
]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_4</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_5</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≠</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>(</mo><mi>f_1</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>f_2</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn><mo>*</mo><mi>ln</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>8</mn><mo>*</mo><msqrt><mi>x</mi></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn><mo>*</mo><mi>ln</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>8</mn><mo>*</mo><msqrt><mi>x</mi></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>7</mn><mo>*</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi></mrow><mrow><mi>x</mi><mo>*</mo><msqrt><mi>x</mi></msqrt></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>7</mn><mo>*</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi></mrow><mrow><mi>x</mi><mo>*</mo><msqrt><mi>x</mi></msqrt></mrow></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.32Q SumaAleat_2 Calcula la derivada de f(x) = #g

Format de la resposta: redueix a denominador comú si s'escau.

]]> La derivada és la suma de les derivades.
]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>(</mo><mi>f_1</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>f_2</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msqrt><mi>x</mi></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>40</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>*</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>*</mo><msqrt><mi>x</mi></msqrt></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>40</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>*</mo><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>3</mn></mrow><mrow><mn>2</mn><mo>*</mo><msqrt><mi>x</mi></msqrt></mrow></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.33Q SumaAleat_3 Calcula la derivada de f(x) = #g

Format de la resposta: redueix a denominador comú si s'escau.

]]> La derivada és la suma de les derivades.
]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_4</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_5</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≠</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>(</mo><mi>f_1</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>f_2</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><mi>ln</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup><mo>-</mo><mn>6</mn><mo>*</mo><mi>ln</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup><mo>+</mo><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mi>x</mi></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup><mo>+</mo><mfrac><mrow><mo>-</mo><mn>6</mn></mrow><mi>x</mi></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.34Q RestaAleat Calcula la derivada de f(x) = #g

Format de la resposta: redueix a denominador comú si s'escau.

]]> La derivada és la diferència de les derivades.
]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_4</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_5</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≠</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>(</mo><mi>f_1</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mo>(</mo><mi>f_2</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><mi>sin</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup><mo>-</mo><mn>3</mn><mo>*</mo><mi>sin</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup><mo>-</mo><mn>3</mn><mo>*</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><exponentiale/><mi>x</mi></msup><mo>-</mo><mn>3</mn><mo>*</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.35Q RestaAleat_2 Calcula la derivada de f(x) = #g

Format de la resposta: redueix a denominador comú si s'escau.

]]> La derivada és la diferència de les derivades.
]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_5</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_4</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>c_5</mi><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mi>c_1</mi></msup><mo>,</mo><msqrt><mi>x</mi></msqrt><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mi>atan</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≠</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>(</mo><mi>f_1</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mo>(</mo><mi>f_2</mi><mo>)</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>*</mo><mi>sin</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>*</mo><mi>sin</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>*</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>20</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>*</mo><mi>cos</mi><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>20</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.41Q ProducteSimple Pol*Ln Quina és la derivada de: f(x) = (#f) · #g?

]]> 1MA.09.2.42Q ProducteSimple Pol*Arrel Quina és la derivada de: f(x) = (#f) · #g?

]]> És la derivada d'un producte

]]> És una funció quocient

]]> 0.0000000 0.0000000 0 1MA.09.2.51.1Q CompostaPotènciaPolinomi Calcula la derivada de «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨bold¨»f«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨»=«/mo»«mfenced»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»t«/mi»«/mrow»«/mfenced»«msup»«mo mathvariant=¨bold¨»§#160;«/mo»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold-italic¨»p«/mi»«/mrow»«/msup»«/mstyle»«/math»

Format de la resposta: 9*(x²-3x+9)⁸*(2x-3)

]]> És una potència de polinomi: a·uⁿ, on #t és el polinomi.

]]> 1.0000000 0.5000000 0 0 #sol2 <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>c</mi></mrow></mfenced><mo><</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>(</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c</mi><msup><mo>)</mo><mi>p</mi></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>h</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p2</mi><mo>=</mo><mi>p</mi><mo>-</mo><mn>1</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mi>p</mi><mo>*</mo><mfenced><mrow><mi>h</mi><mo>(</mo><mi>x</mi><msup><mo>)</mo><mi>p2</mi></msup></mrow></mfenced><mo>·</mo><mi>i</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>factoritza</mi><mo>(</mo><mi>sol</mi><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>p</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>,</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo>*</mo><msup><mi>x</mi><mn>15</mn></msup><mo>-</mo><mn>120</mn><mo>*</mo><msup><mi>x</mi><mn>14</mn></msup><mo>+</mo><mn>504</mn><mo>*</mo><msup><mi>x</mi><mn>13</mn></msup><mo>-</mo><mn>1456</mn><mo>*</mo><msup><mi>x</mi><mn>12</mn></msup><mo>+</mo><mn>3192</mn><mo>*</mo><msup><mi>x</mi><mn>11</mn></msup><mo>-</mo><mn>5544</mn><mo>*</mo><msup><mi>x</mi><mn>10</mn></msup><mo>+</mo><mn>7840</mn><mo>*</mo><msup><mi>x</mi><mn>9</mn></msup><mo>-</mo><mn>9144</mn><mo>*</mo><msup><mi>x</mi><mn>8</mn></msup><mo>+</mo><mn>8856</mn><mo>*</mo><msup><mi>x</mi><mn>7</mn></msup><mo>-</mo><mn>7112</mn><mo>*</mo><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>4704</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>-</mo><mn>2520</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>1064</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>336</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>72</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>*</mo><mfenced><mrow><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>*</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>7</mn></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol2</correctAnswer></correctAnswers><assertions><assertion name="check_factorized"/><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.2.51.2Q Composta: PotLog Calcula la derivada de la funció: f(x) = #f_1

ln(x)⁶ vol dir [ln(x)]⁶

]]> És una funció composta de la taula

]]> És una potència a·uⁿ

]]> És una funció irracional

]]> És del tipus «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»u«/mi»«/msqrt»«/mstyle»«/math»

]]> És del tipus ln(u)

]]> És del tipus e^u

]]> És del tipus sin(u)

]]> És una funció composta de la taula

]]> És del tipus asin(u)

]]> És una funció composta de la taula

]]> És una funció composta de l'arc tangent.

]]> És una funció quocient

]]> És una funció irracional

]]> És un producte: (uv)' = u'v + uv'

]]> És la derivada d'un producte

]]> Cal derivar la derivada primera que és: #p

]]> Cal derivar la derivada primera que és #p

]]> Cal aplicar la taula

]]> Cal derivar la derivada primera que és: #p

Monotonia i concavitat

f'(x)>0 → f(x) creixent;

f'(x)<0 → f(x) decreixent

f''(x) > 0 → f(x) convexa;

f''(x) < 0 → f(x) còncava

]]> 0.0000000 0.0000000 0 1MA.09.3.1.11Q CreixPolG3 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

La funció és creixent quan la derivada és positiva

]]> 1MA.09.3.1.12Q CreixPolG4 Escriu els intervals de creixement i decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

La funció és creixent quan la derivada és positiva

]]> 1MA.09.3.1.16Q CreixRacG2G1 Escriu el domini i els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Domini=#domb) Creixent en =#sol1c) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><msqrt><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup></mrow></msqrt><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></mfenced></mrow></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup><mo>)</mo></mrow><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura</mi><mo>=</mo><mn>200</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>y</mi><mo>=</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>*</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>48</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>48</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mfrac><mrow><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>48</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>D</mi><mi>o</mi><mi>min</mi><mi mathvariant="normal">i</mi><mo>=</mo><mo>#</mo><mi mathvariant="normal">d</mi><mi>o</mi><mi>m</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.3.1.17Q CreixRacG1G2 Escriu el domini i els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Domini =#domb) Creixent en =#sol1c) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mfenced close="|" open="|"><mi>s1</mi></mfenced><mo>≠</mo><mfenced close="|" open="|"><mi>s2</mi></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><msqrt><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup></mrow></msqrt><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mfenced close="|" open="|"><mi>s1</mi></mfenced><mo>≠</mo><mfenced close="|" open="|"><mi>s2</mi></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></mfenced></mrow></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo></mrow><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfenced><mo>*</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mi>s2</mi></mrow></mfenced></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>punt</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>=</mo><mi>numerador</mi><mo>(</mo><mi>d1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>d1</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>50</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>altura_finestra</mi><mo>=</mo><mn>550</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>550</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>49</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>7</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>7</mn><mo>,</mo><mn>7</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>7</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>49</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>7</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>7</mn><mo>,</mo><mn>7</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>7</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mfrac><mrow><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>147</mn></mrow><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>98</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2401</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>D</mi><mi>o</mi><mi>min</mi><mi mathvariant="normal">i</mi><mo> </mo><mo>=</mo><mo>#</mo><mi mathvariant="normal">d</mi><mi>o</mi><mi>m</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.3.1.18Q CreixRacG2G2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mfenced close="|" open="|"><mi>s1</mi></mfenced><mo>≠</mo><mfenced close="|" open="|"><mi>s2</mi></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><msqrt><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup></mrow></msqrt><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mfenced close="|" open="|"><mi>s1</mi></mfenced><mo>≠</mo><mfenced close="|" open="|"><mi>s2</mi></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></mfenced></mrow></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><msup><mo>)</mo><mn>2</mn></msup></mrow><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfenced><mo>*</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mi>s2</mi></mrow></mfenced></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>y</mi><mo>=</mo><mi>a</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>altura_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>20</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>50</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>d1</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mn>3</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>3</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mfrac><mrow><mo>-</mo><mn>8</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>52</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>72</mn></mrow><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>18</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>81</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador de 2n grau:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.3.1.21Q CreixArrelG1 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent en cas contrari.

]]> 1MA.09.3.1.22Q CreixArrelG2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan és negativa.

]]> 1MA.09.3.1.23Q CreixArrelRacG1G1 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msqrt><mfrac><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo></mrow></mfrac></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mfrac><mrow><mi>s1</mi><mo>-</mo><mi>s2</mi></mrow><mn>2</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>d2</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>-</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>s1</mi><mo>,</mo><mi>s2</mi></mtd></mtr></mtable></mfenced></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></mtd></mtr></mtable><mtable><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>-</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>s1</mi><mo>,</mo><mi>s2</mi></mtd></mtr></mtable></mfenced></mtd></mtr><mtr><mtd><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></mtd></mtr></mtable></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>450</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>,</mo><mi>n1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>3</mn></mrow><mrow><mfenced><mrow><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>*</mo><msqrt><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></msqrt></mrow></mfrac><mo>,</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>,</mo><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»; aquest signe depèn exclusivament del numerador.

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan és negativa.

]]> 1MA.09.3.1.25Q CreixLnG1 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»; s'ha simplificat per (#a)

El signe de f'(x) depèn exclusivament del numerador.

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan f'(x) és negativa.

]]> 1MA.09.3.1.26Q CreixLnG2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>+</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>∩</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>∩</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>20</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>24</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>24</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>,</mo><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>4</mn></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»; s'ha simplificat per (#a)

El signe de f'(x) depèn exclusivament del numerador.

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan f'(x) és negativa.

]]> 1MA.09.3.1.31Q CreixExpG1 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan f'(x) és negativa.

]]> 1MA.09.3.1.32Q CreixExpG2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msup><exponentiale/><mrow><mo>(</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>)</mo></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>+</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfenced><mrow><mi>f</mi><mfenced><mfrac><mrow><mi>s1</mi><mo>+</mo><mi>s2</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>2</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>50</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>55</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>20</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>100</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>110</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>50</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>500</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>550</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>220</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>4000</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1950</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable></apply></apply></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><exponentiale/><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><reals/></math></output></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>,</mo><mi>k1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.7788</mn><mo>,</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

Com que una exponencial és sempre positiva, el signe de f'(x) depèn exclusivament del polinomi #d2

La funció és creixent quan la derivada és positiva i decreixent quan f'(x) és negativa.

]]> 1MA 09. DERIVADES/1MA.09.3 Variacions/1MA.09.3.2 Extrems relatius 1MA.09.3.2.10DT MONOTONIA CONCAVITAT (còpia)

Derivades i extrems

màxim per f(x): f'(x) = 0 i passa de + a - (o f'(x) = 0 i f''(x) < 0).

mínim per f(x): f'(x) = 0 i passa de - a + (o f'(x) = 0 i f''(x) > 0)

f''(x) = 0 i canvia de signe → Inflexió per f(x) (o f''(x) = 0 i l'ordre de la primera derivada que no s'anul·la és imparell)

]]> 0.0000000 0.0000000 0 1MA.09.3.2.11Q f'→variacions (GRÀFIC) Si el gràfic de la derivada f'(x) d'una funció f(x) és el següent:

#g1

se'n dedueix que:

]]> 1.0000000 0.3333333 0 false true abc La teva resposta és correcta.

]]> La teva resposta és parcialment correcta.

]]> La teva resposta és incorrecta.

]]> La funció és sempre #c1

]]> Si la funció fós #c1, la derivada seria sempre #r1

]]> La funció presenta un #m1 quan x = #a1

]]> Si la funció presenta un #m1 la derivada s'anul·la i passa de #r2 a #r3

]]> La funció presenta un #m2 quan x = #a1

]]> Si la funció presenta un #m2 la derivada s'anul·la i passa de #r4 a #r5

]]> La funció és #c2 si x<#a1

]]> La funció és #c2 si la derivada primera és #r6

]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a2</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>a2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>b1</mi><mo>,</mo><mi>b2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>recta</mi><mfenced><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>10</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T1</mi><mo>,</mo><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mrow><mi>b2</mi><mo>-</mo><mi>b1</mi></mrow><mrow><mi>a2</mi><mo>-</mo><mi>a1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>p</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>creixent</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r1</mi><mo>=</mo><mo>"</mo><mi>positiva</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mo>"</mo><mi>decreixent</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>m1</mi><mo>=</mo><mo>"</mo><mi>mínim</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r2</mi><mo>=</mo><mo>"</mo><mi>negativa</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r3</mi><mo>=</mo><mo>"</mo><mi>positiva</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>m2</mi><mo>=</mo><mo>"</mo><mi>màxim</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r5</mi><mo>=</mo><mo>"</mo><mi>negativa</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r4</mi><mo>=</mo><mo>"</mo><mi>positiva</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r6</mi><mo>=</mo><mo>"</mo><mi>negativa</mi><mo>"</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mtable><mtr><mtd><mi>c1</mi><mo>=</mo><mo>"</mo><mi>decreixent</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r1</mi><mo>=</mo><mo>"</mo><mi>negativa</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>c2</mi><mo>=</mo><mo>"</mo><mi>creixent</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>m1</mi><mo>=</mo><mo>"</mo><mi>màxim</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r3</mi><mo>=</mo><mo>"</mo><mi>negativa</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r2</mi><mo>=</mo><mo>"</mo><mi>positiva</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>m2</mi><mo>=</mo><mo>"</mo><mi>mínim</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r4</mi><mo>=</mo><mo>"</mo><mi>negativa</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r5</mi><mo>=</mo><mo>"</mo><mi>positiva</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>r6</mi><mo>=</mo><mo>"</mo><mi>positiva</mi><mo>"</mo></mtd></mtr></mtable></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><ms>creixent</ms></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><ms>màxim</ms></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><options><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="casSession"/></localData></question> 1MA.09.3.2.12Q Màx. mín o res? Considera la següent funció: f(x)=#f i el punt x=#punt.
Determina si és un màxim, un mínim o res.

]]> 1.0000000 1.0000000 0 true true abc #sol1

]]> #sol2

]]> #sol3

]]> <question><wirisCasSession><session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r10</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r4</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r0</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>0</mn><mo>.</mo><mo>.</mo><mn>1</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>r10</mi><mo>(</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r10</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>+</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r10</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>*</mo><mi>x</mi><mo>+</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r10</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>r10</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><mi>r0</mi><mo>(</mo><mo>)</mo><mo>*</mo><mi>r</mi><mo>(</mo><mo>)</mo><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>llista</mi><mo>=</mo><mo>{</mo><mi>x</mi><mo>*</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><mi>x</mi><mrow><mi>r4</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>+</mo><mi>x</mi><mo>,</mo><mroot><mi>x</mi><mrow><mi>r4</mi><mo>(</mo><mo>)</mo></mrow></mroot><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msup><exponentiale/><mi>x</mi></msup><mo>,</mo><mfrac><mn>1</mn><msup><mi>x</mi><mrow><mi>r4</mi><mo>(</mo><mo>)</mo></mrow></msup></mfrac><mo>,</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mrow><mi>r4</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>-</mo><mi>r4</mi><mo>(</mo><mo>)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>x</mi></mfrac><mo>,</mo><mi>sin</mi><mo>(</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>)</mo><mo>,</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>*</mo><mo>(</mo><mn>6</mn><mo>-</mo><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup><msup><mo>)</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>,</mo><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced 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xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1.6053</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1.6074</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.2.13Q Màx, mín o res? Considera la següent funció: f(x)=#f i el punt x=#punt.
Determina si és un màxim, un mínim o res.

]]> 1.0000000 1.0000000 0 true true abc #sol1

]]> #sol2

]]> #sol3

]]> <question><wirisCasSession><session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r10</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>10</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r4</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>1</mn><mo>.</mo><mo>.</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r0</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>random</mi><mo>(</mo><mn>0</mn><mo>.</mo><mo>.</mo><mn>1</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>(</mo><mo>)</mo><mo>:</mo><mo>=</mo><mi>r10</mi><mo>(</mo><mo>)</mo></math></input></command><command><input><math 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definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>)</mo><mo>&lt;</mo><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>)</mo><mo>&amp;</mo><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>)</mo><mo>&lt;</mo><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>)</mo></mrow><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>maxim</mi></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>minim</mi></mtd></mtr><mtr><mtd><mi>sol3</mi><mo>=</mo><mi>res</mi></mtd></mtr><mtr><mtd><mi>seguir</mi><mo>=</mo><mi>false</mi></mtd></mtr></mtable></apply></mtd></mtr><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>seguir</mi><mo>==</mo><mi>true</mi></mrow><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>res</mi></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>maxim</mi></mtd></mtr><mtr><mtd><mi>sol3</mi><mo>=</mo><mi>minim</mi></mtd></mtr></mtable></apply></mtd></mtr></mtable></apply></apply><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cutrillo</mi><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>minim</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>maxim</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>res</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>punt</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1.6053</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>funcio</mi><mo>(</mo><mi>punt</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1.6074</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.2.15Q GrafCreixPolG3 Escriu els intervals de creixement i de decreixement de la funció f(x) si el gràfic de la seva derivada és el següent:

#G1.

Determina l'abscissa dels seus extrems relatius

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

Abscisses: 3

]]> La funció és:

#G2

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Abscissa del màxim =#sol3d) Abscissa del mínim =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>∫</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m11</mi><mo>=</mo><mi>d</mi><mo>(</mo><mfrac><mrow><mi>s1</mi><mo>+</mo><mi>s2</mi></mrow><mn>2</mn></mfrac><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m12</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>*</mo><mfenced><mfenced close="|" open="|"><mi>m11</mi></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>sol3</mi><mo>=</mo><mi>s1</mi></mtd></mtr><mtr><mtd><mi>sol4</mi><mo>=</mo><mi>s2</mi></mtd></mtr></mtable><mtable><mtr><mtd><mi>sol3</mi><mo>=</mo><mi>s2</mi></mtd></mtr><mtr><mtd><mi>sol4</mi><mo>=</mo><mi>s1</mi></mtd></mtr></mtable></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mi>m12</mi><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>10</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mfrac><mrow><mi>s1</mi><mo>+</mo><mi>s2</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T2</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>20</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>representa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G2</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T2</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>15</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>15</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>18</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>15</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>3</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>,</mo><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>A</mi><mi>b</mi><mi>s</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>s</mi><mi>s</mi><mi>a</mi><mo> </mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi>l</mi><mo> </mo><mi>m</mi><mi>à</mi><mi>x</mi><mi mathvariant="normal">i</mi><mi>m</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>A</mi><mi>b</mi><mi>s</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>s</mi><mi>s</mi><mi>a</mi><mo> </mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi><mi>l</mi><mo> </mo><mi>m</mi><mi>í</mi><mi>n</mi><mi mathvariant="normal">i</mi><mi>m</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La funció és creixent quan la derivada és positiva i decreixent quan és negativa.

Té extrems quan la derivada s'anul·la i canvia de signe

]]> 1MA.09.3.2.21.1Q Extrems PolinomiG2 Determina, si s'escau, les coordenades de l'extrem relatiu de la funció f(x) = #g. Indica si és un màxim o un mínim.

Formats:

Extrem=[3/4,1/4]

Tipus(M/m)=m

]]> La derivada primera és #d1 que s'anul·la per #s1 . El seu signe és:

x	-∞	#s1	+∞
#d1	#w1	0	#w2

Per això la funció presenta un #z1

El gràfic és #D

]]> 1.0000000 0.5000000 0 0 Extrem/s =#solTipus(M/m)=#t1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>x1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>x2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>x1</mi><mo><</mo><mi>x2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a0</mi><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>x1</mi></mrow></mfenced><mo>·</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>x2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>d1</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mo>(</mo><mi>d1</mi><mo>></mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mfenced><mrow><mi>d1</mi><mo><</mo><mn>0</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>=</mo><mi>f</mi><mo>(</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="]" open="["><mrow><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>y1</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>punt</mi><mo>(</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>f</mi><mo>(</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a0</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>z1</mi><mo>=</mo><mo>"</mo><mi>mínim</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>t1</mi><mo>=</mo><mi>m</mi></mtd></mtr></mtable><mtable><mtr><mtd><mi>z1</mi><mo>=</mo><mo>"</mo><mi>màxim</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>t1</mi><mo>=</mo><mi>M</mi></mtd></mtr></mtable></apply></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Taula</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a0</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>w1</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mtable><mtr><mtd><mi>w1</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd/></mtr></mtable></apply></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Dibuix</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>30</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>250</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>P</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>10</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>16</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo><</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>,</mo><mi>w2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mo>,</mo><mo>+</mo></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mn>18</mn></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mo>/</mo><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>m</mi><mo>)</mo><mo>=</mo><mo>#</mo><mi>t</mi><mn>1</mn><mspace linebreak="newline"/><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="nobracketslist">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La derivada primera és #d1 que s'anul·la per #s1 . El seu signe és:

x	-∞	#s1	+∞
#d1	#w1	0	#w2

]]> 1MA.09.3.2.21.2Q ExtremsPolinomiG3 Calcula, si s'escau, les coordenades i els tipus dels extrems relatius de la funció f(x) = #g.

Format:

Extrems={[1,2/3],[-2,3]}

Tipus={M,m} en el mateix ordre: M,m si 1 correspon a un màxim i -2 a un mínim

]]> La derivada primera és #d1 que s'anul·la per #s1 i #s2. El seu signe és:

x	-∞	#s1		#s2	+∞
#d1	#w1	0	#w2	0	#w3

El gràfic és #D

]]> 1.0000000 0.5000000 0 0 Extrems=#solTipus(M/m)=#t1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a0</mi><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s1</mi></mrow></mfenced><mo>·</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>∫</mo><mfenced><mi>d1</mi></mfenced><mo>+</mo><mi>a3</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mo>(</mo><mi>d1</mi><mo>></mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mfenced><mrow><mi>d1</mi><mo><</mo><mn>0</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>s1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y2</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mi>s1</mi><mo>,</mo><mi>y1</mi></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mi>s2</mi><mo>,</mo><mi>y2</mi></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>t11</mi><mo>,</mo><mi>t12</mi></mtd></mtr></mtable></mfenced></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Taula</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a0</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>w1</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w3</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>t11</mi><mo>=</mo><mi>M</mi></mtd></mtr><mtr><mtd><mi>t12</mi><mo>=</mo><mi>m</mi></mtd></mtr></mtable><mtable><mtr><mtd><mi>w1</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w3</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>t11</mi><mo>=</mo><mi>m</mi></mtd></mtr><mtr><mtd><mi>t12</mi><mo>=</mo><mi>M</mi></mtd></mtr></mtable></apply></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Dibuix</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>16</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>250</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mn>2</mn><mo>∨</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mo>-</mo><mn>3</mn><mo>∧</mo><mi>x</mi><mo><</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>,</mo><mi>w2</mi><mo>,</mo><mi>w3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mo>,</mo><mo>-</mo><mo>,</mo><mo>+</mo></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>77</mn><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>2</mn><mo>,</mo><mo>-</mo><mn>24</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>M</mi><mo>,</mo><mi>m</mi></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>m</mi><mo>)</mo><mo>=</mo><mo>#</mo><mi>t</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La derivada primera és #d1 que s'anul·la per #s1 i #s2. El seu signe és:

x	-∞	#s1		#s2	+∞
#d1	#w1	0	#w2	0	#w3

]]> 1MA.09.3.2.21.3Q Extrems PolinomiG4 Determina, si s'escau, les coordenades i els tipus dels extrems relatius de la funció f(x) = #g. Classifiqueu-los.

Format:

Extrems={[1,2],[21],[5,2]}

Tipus={m,m,M} si x=1 i x=2 corresponen a mínims i x=5 correspon a un màxim

]]> La derivada primera és #d1 que s'anul·la per #s1, #s2 i #s3.

El gràfic és #D

]]> 1.0000000 0.5000000 0 0 Extrems=#sol3Tipus=#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>s3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a0</mi><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s1</mi></mrow></mfenced><mo>·</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfenced><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s3</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>d1</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>∫</mo><mfenced><mrow><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced><mo>+</mo><mi>a3</mi></mtd></mtr><mtr><mtd><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>S</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>d1</mi><mo>=</mo><mn>0</mn></mrow></mfenced></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>s2</mi><mo><</mo><mi>s3</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>s1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y2</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y3</mi><mo>=</mo><mi>k</mi><mo>(</mo><mi>s3</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mo>(</mo><mi>d1</mi><mo>></mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mfenced><mrow><mi>d1</mi><mo><</mo><mn>0</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mi>s1</mi><mo>,</mo><mi>y1</mi></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mi>s2</mi><mo>,</mo><mi>y2</mi></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mi>s3</mi><mo>,</mo><mi>y3</mi></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Taula</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a0</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>w1</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w3</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w4</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>e1</mi><mo>=</mo><mi>m</mi></mtd></mtr><mtr><mtd><mi>e2</mi><mo>=</mo><mi>M</mi></mtd></mtr><mtr><mtd><mi>e3</mi><mo>=</mo><mi>m</mi></mtd></mtr></mtable><mtable><mtr><mtd><mi>w1</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w3</mi><mo>=</mo><mo>"</mo><mo>+</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>w4</mi><mo>=</mo><mo>"</mo><mo>-</mo><mo>"</mo></mtd></mtr><mtr><mtd><mi>e1</mi><mo>=</mo><mi>M</mi></mtd></mtr><mtr><mtd><mi>e2</mi><mo>=</mo><mi>m</mi></mtd></mtr><mtr><mtd><mi>e3</mi><mo>=</mo><mi>M</mi></mtd></mtr></mtable></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>e1</mi><mo>,</mo><mi>e2</mi><mo>,</mo><mi>e3</mi></mtd></mtr></mtable></mfenced></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Dibuix</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>40</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>500</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>24</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>24</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>,</mo><mi>s2</mi><mo>,</mo><mi>s3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>98</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo><</mo><mo>-</mo><mn>2</mn><mo>∨</mo><mi>x</mi><mo>></mo><mn>3</mn><mo>∧</mo><mi>x</mi><mo><</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mn>4</mn><mo>∨</mo><mi>x</mi><mo>></mo><mo>-</mo><mn>2</mn><mo>∧</mo><mi>x</mi><mo><</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>,</mo><mi>w2</mi><mo>,</mo><mi>w3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mo>,</mo><mo>-</mo><mo>,</mo><mo>+</mo></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mfrac><mn>98</mn><mn>3</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mfrac><mn>161</mn><mn>4</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>4</mn><mo>,</mo><mo>-</mo><mfrac><mn>118</mn><mn>3</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>M</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>M</mi></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La derivada primera és #d1 que s'anul·la per #s1, #s2 i #s3.

El gràfic és #D

]]> 1MA.09.3.2.21.5Q Abscissa Extrem FQuadràtica Per quin valor de x la funció f(x) = #g presenta un extrem relatiu?

]]> Cal trobar el punt on la derivada primera s'anul·la i canvia de signe.

]]> 1.0000000 0.5000000 0 0 #r_83 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_83 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.3.2.21.6Q Abscisses Extrems PolG3 Per quins valors de x la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math» presenta un extrem relatiu?

Format de la resposta: ordenats: [-3,5]

]]> Cal trobar els punts on la derivada primera s'anul·la i canvia de signe.

]]> 1.0000000 0.5000000 0 0 #r_83 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>a_2</mi><mo>≠</mo><mi>a_0</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_2</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_0</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>∫</mo><mi>e</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>a_3</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_1</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_2</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi><mo>=</mo><mfenced close="]" open="["><mrow><mi>r_1</mi><mo>,</mo><mi>r_2</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>12</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_83 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.3.2.22.1Q Extrems RacionalG1G2 Determina si s'escau els extrems relatius de la funció f(x) = #g.

Classifica'ls (m per mínim i M per màxim)

Format:

Extrems={[1,3/3],[2,5]}

Tipus={M,m} si x=1 correspon a un màxim i x=2 correspon a un mínim.

]]> La derivada primera és #d1 que canvia de signe en x = #b2 (que no està en el domini) i quan #s1 .

El gràfic és #D

]]> 1.0000000 0.5000000 0 0 Extrem=#sol3Tipus=#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>mcd</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>b1</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>a2</mi><mo>≠</mo><mi>b2</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a1</mi><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a2</mi></mrow></mfenced></mrow><mrow><mi>b1</mi><mo>*</mo><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>b2</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s11</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>=</mo><mi>k</mi><mo>(</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mo>(</mo><mi>d1</mi><mo>></mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mfenced><mrow><mi>d1</mi><mo><</mo><mn>0</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>y1</mi></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mfrac><mrow><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>+</mo><mi>b2</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d21</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>s11</mi></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>d21</mi><mo>></mo><mn>0</mn></mrow><mrow><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m</mi></mtd></mtr></mtable></mfenced></mrow><mrow><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>M</mi></mtd></mtr></mtable></mfenced></mrow></apply></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Dibuix</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mfrac><mrow><mi>b2</mi><mo>+</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></mfrac><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>60</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>x</mi><mo>=</mo><mi>b2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>32</mn></mrow><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>64</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mo>-</mo><mn>8</mn><mo>∧</mo><mi>x</mi><mo><</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mn>4</mn><mo>∨</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>8</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d21</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>432</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La derivada primera és #d1 que canvia de signe en x = #b2 (que no està en el domini) i quan #s1 .

El gràfic és #D

]]> 1MA.09.3.2.22.2Q Extrems RacionalG2G1 Determina si s'escau els extrems relatius de la funció f(x) = #g.

Classifica'ls (m per mínim i M per màxim)

Format:

Extrems={[1,3/3],[2,5]}

Tipus={M,m} si x=1 correspon a un màxim i x=2 correspon a un mínim

]]> La derivada primera és #d1 que canvia de signe en x = #b2 (que no està en el domini) i quan #s1 .

El gràfic és #D

]]> 1.0000000 0.5000000 0 0 Extrems=#sol3Tipus=#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>mcd</mi><mo>(</mo><mi>a1</mi><mo>,</mo><mi>b1</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>a2</mi><mo>≠</mo><mi>b2</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a1</mi><mo>*</mo><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a2</mi></mrow></mfenced><mn>2</mn></msup></mrow><mrow><mi>b1</mi><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>b2</mi></mrow></mfenced></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s2</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>s1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mo>(</mo><mi>d1</mi><mo>></mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>resol_inequació</mi><mfenced><mrow><mi>d1</mi><mo><</mo><mn>0</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mi>s1</mi><mo>,</mo><mi>y1</mi></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mi>s2</mi><mo>,</mo><mi>y2</mi></mrow></mfenced></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>(</mo><mi>s1</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow><mrow><mi>e1</mi><mo>=</mo><mi>m</mi></mrow><mrow><mi>e1</mi><mo>=</mo><mi>M</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>(</mo><mi>s2</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow><mrow><mi>e2</mi><mo>=</mo><mi>m</mi></mrow><mrow><mi>e2</mi><mo>=</mo><mi>M</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>e1</mi><mo>,</mo><mi>e2</mi></mtd></mtr></mtable></mfenced></math></input></command><maction actiontype="comment"><comment><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Dibuix</mi></math></comment></maction><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>amplada</mi><mo>=</mo><mn>40</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>200</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>x</mi><mo>=</mo><mi>b2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>32</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>64</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>,</mo><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>,</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi><mo>,</mo><mi>y2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mo>-</mo><mn>48</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>≠</mo><mn>2</mn><mo>∧</mo><mi>x</mi><mo>></mo><mo>-</mo><mn>4</mn><mo>∧</mo><mi>x</mi><mo><</mo><mn>8</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>></mo><mn>8</mn><mo>∨</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="]" open="["><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>8</mn><mo>,</mo><mo>-</mo><mn>48</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m</mi><mo>,</mo><mi>M</mi></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La derivada primera és #d1 que canvia de signe en x = #b2 (que no està en el domini) i quan #s1 .

El gràfic és #D

]]> 1MA.09.3.2.23.1Q Abscisses Extrems ln Determina si s'escau les abscisses dels extrems relatius de la funció f(x) = #g.

Classifica'ls (m per mínim i M per màxim)

Format:

Abscissa={1/3,2}

Tipus={M,m} si x=1/3 és un màxim i x=2 correspon a un mínim

]]> La funció només està definida si #a > 0.

La derivada és #i.

El gràfic és #D

]]> 1.0000000 0.5000000 0 0 Abscissa=#sol1Tipus=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mi>a1</mi></mrow></mfenced><mo>*</mo><mi>ln</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>a1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>s1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>(</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo><mo>></mo><mn>0</mn></mrow><mrow><mi>e1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m</mi></mtd></mtr></mtable></mfenced></mrow><mrow><mi>e1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>M</mi></mtd></mtr></mtable></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>e1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mo>*</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mo>*</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mo>+</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>8</mn><mo>*</mo><exponentiale/><mo>+</mo><mn>1</mn></mrow><exponentiale/></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfrac><mrow><mo>-</mo><mn>8</mn><mo>*</mo><exponentiale/><mo>+</mo><mn>1</mn></mrow><exponentiale/></mfrac></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfrac><mrow><mo>-</mo><mn>8</mn><mo>*</mo><exponentiale/><mo>+</mo><mn>1</mn></mrow><exponentiale/></mfrac></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m</mi></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>b</mi><mi>s</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>s</mi><mi>s</mi><mi>a</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> La funció només està definida si #a > 0.

La derivada és #i.

El gràfic és D

]]> 1MA.09.3.2.24.1Q AbscissaExtremExponencial Per quin valor de x la funció f(x) = #g presenta un extrem relatiu?

]]> Cal trobar el punt on la derivada primera s'anul·la i canvia de signe.

]]> 1.0000000 0.5000000 0 0 #r_83 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msup><exponentiale/><mrow><mi>a_1</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>x</mi></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>x</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><exponentiale/><mrow><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><exponentiale/><mrow><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_83 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.3.2.24.2Q Abscisses Extrems Exponencial Determina si s'escau les abscisses dels extrems relatius de la funció f(x) = #g.

Classifica'ls (m per mínim i M per màxim)en el mateix ordre.

Format:

Abscisses dels extrems={3,5}

Tipus={M,m}

]]> Estudia el signe de la derivada #i

El gràfic de la funció és #D

]]> 1.0000000 0.5000000 0 0 Abscisses=#sol1Tipus=#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>*</mo><msup><exponentiale/><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a1</mi><mo>)</mo></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s2</mi><mo>=</mo><msub><mi>S</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>s1</mi><mo>,</mo><mi>s2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>(</mo><mi>s1</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow><mrow><mi>e1</mi><mo>=</mo><mi>m</mi></mrow><mrow><mi>e1</mi><mo>=</mo><mi>M</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>(</mo><mi>s2</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow><mrow><mi>e2</mi><mo>=</mo><mi>m</mi></mrow><mrow><mi>e2</mi><mo>=</mo><mi>M</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>e1</mi><mo>,</mo><mi>e2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>7</mn><mo>)</mo><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>10</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>16</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>*</mo><msup><exponentiale/><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>*</mo><msup><exponentiale/><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><mi>x</mi></mrow></mfenced><mo>*</mo><msup><exponentiale/><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>*</mo><msup><exponentiale/><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msup></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>,</mo><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>b</mi><mi>s</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>s</mi><mi>s</mi><mi mathvariant="normal">e</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>T</mi><mi mathvariant="normal">i</mi><mi>p</mi><mi>u</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_literal"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de #i

El gràfic de la funció és #D

]]> 1MA.09.3.2.24.5Q ExtremExponencial Determina l'extrem relatiu de la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«/mrow»«/mstyle»«/math»

Format: (1,2)

]]>

]]> 1MA.09.3.2.25.1Q AbscissaExtremArrel(G2) Per quin valor de x la funció f(x) = #g presenta un extrem relatiu?

Format de la resposta:fracció (3/4) o N, si no hi ha cap extrem en el domini.

]]> Cal trobar el punt on la derivada primera s'anul·la i canvia de signe, sempre i quan sigui un punt del domini.

]]> 1.0000000 0.5000000 0 0 #r_83 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msqrt><mrow><mi>a_1</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_0</mi></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a_1</mi><mo>&lt;</mo><mn>0</mn></mrow><mrow><mi>r_83</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow><mrow><mi>r_83</mi><mo>=</mo><mi>N</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mo>-</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>3</mn></mrow></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mo>-</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>3</mn></mrow></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_83 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.3.2.25.5Q ExtremArrelPolG2 Determina l'extrem relatiu de la funció f(x) = «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Format de la resposta: (2,3)

]]>

La derivada primera és #d1

]]> 1MA.09.3.2.26.2Q ExtremsRacG2G1 Escriu el domini i els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Troba els seus extrems relatius, si s'escau

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

Format dels extrems: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»,«/mo»«mo»(«/mo»«mn»3«/mn»«mo»,«/mo»«mn»4«/mn»«mo»)«/mo»«/mrow»«/mfenced»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Domini=#domb) Creixent en =#sol1c) Decreixent en =#sol2d) Extrems relatius =#sol3]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>7</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>7</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><msqrt><mrow><msup><mi>s2</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup></mrow></msqrt><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s1</mi><mn>2</mn></msup><mo>)</mo></mrow><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>d1</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e11</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e12</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mi>e11</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>e11</mi><mo>)</mo></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>e12</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>e12</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>e1</mi><mo>,</mo><mi>e2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>50</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>50</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>y</mi><mo>=</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>*</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>e1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>e2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>5</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>5</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>8</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>16</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>,</mo><mi>e11</mi><mo>,</mo><mi>e12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced><mo>,</mo><mn>2</mn><mo>,</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>2</mn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>8</mn><mo>,</mo><mo>-</mo><mn>16</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>D</mi><mi>o</mi><mi>min</mi><mi mathvariant="normal">i</mi><mo>=</mo><mo>#</mo><mi mathvariant="normal">d</mi><mi>o</mi><mi>m</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo> </mo><mi>r</mi><mi mathvariant="normal">e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi mathvariant="normal">i</mi><mi>u</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.3.2.51Q ExtremsInflexPolG3 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Determina els seus extrems relatius, i el seu punt d'inflexió

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

Format pels extrems «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»,«/mo»«mn»5«/mn»«/mrow»«/mfenced»«mo»,«/mo»«mfenced»«mrow»«mn»2«/mn»«mo»,«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/math» i del punt d'inflexió: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mn»3«/mn»«mo»,«/mo»«mn»1«/mn»«mo»)«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Extrems =#sol3d) Inflexió =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>+</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>∫</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>s1</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>s1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>s2</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>s2</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>m1</mi><mo>,</mo><mi>m2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>d2</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mi>i1</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>i1</mi><mo>)</mo></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>250</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>m1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>m2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>sol4</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>verd</mi><mo>,</mo><mi>mida_punt</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>18</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m1</mi><mo>,</mo><mi>m2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>73</mn><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>2</mn><mo>,</mo><mo>-</mo><mn>26</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>3</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>2</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mfrac><mn>73</mn><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>2</mn><mo>,</mo><mo>-</mo><mn>26</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>21</mn><mn>4</mn></mfrac></mrow></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi mathvariant="normal">e</mi><mi>x</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

La funció és creixent quan la derivada és positiva.

Pels extrems, cal que la derivada s'anul·li i canviï de signe (de positiva a negativa en un màxim, i viceversa en un mínim).

Pel punt d'inflexió, és la derivada segona f''(x) = #d2 que s'ha d'anul·lar i canviar de signe.

]]> 1MA.09.3.2.52Q ExtremsInflexPolG4 Escriu els intervals de creixement i decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

Determina els seus extrems i punts d'inflexió

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

Format dels extrems: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»,«/mo»«mfenced»«mrow»«mn»3«/mn»«mo»,«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/math»

Format dels punts d'inflexió: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»,«/mo»«mfenced»«mrow»«mn»3«/mn»«mo»,«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Extrems =#sol3d) Inflexió =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>2</mn><mo>,</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>D</mi><mo>=</mo><mn>4</mn><mo>*</mo><mo>(</mo><mi>s1</mi><mo>+</mo><mi>s2</mi><mo>+</mo><mi>s3</mi><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>12</mn><mo>*</mo><mo>(</mo><mi>s1</mi><mo>*</mo><mi>s2</mi><mo>+</mo><mi>s1</mi><mo>*</mo><mi>s3</mi><mo>+</mo><mi>s2</mi><mo>*</mo><mi>s3</mi><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>s2</mi><mo><</mo><mi>s3</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>D</mi><mo>></mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><msqrt><mi>D</mi></msqrt><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s3</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s3</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>∫</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>s1</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>s1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>s2</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>s2</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e3</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>s3</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>s3</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi><mo>=</mo><mi>d</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>i1</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>e1</mi><mo>,</mo><mi>e2</mi><mo>,</mo><mi>e3</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i11</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i12</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>punt</mi><mo>(</mo><mi>i11</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>i11</mi><mo>)</mo><mo>)</mo><mo>,</mo><mi>punt</mi><mo>(</mo><mi>i12</mi><mo>,</mo><mi>f</mi><mo>(</mo><mi>i12</mi><mo>)</mo><mo>)</mo></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>500</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>22</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>4000</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>110</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>112</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>55</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>112</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>55</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>112</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>8</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>7</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mfrac><mn>3782</mn><mn>3</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo>-</mo><mfrac><mn>325</mn><mn>6</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>7</mn><mo>,</mo><mfrac><mn>10939</mn><mn>6</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i1</mi><mo>,</mo><mi>i11</mi><mo>,</mo><mi>i12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>110</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mfrac><mn>11</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mfrac><mn>4027</mn><mn>6</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mfrac><mn>11</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>161353</mn><mn>162</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi mathvariant="normal">e</mi><mi>x</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math» (Ruffini?)

La funció és creixent quan la derivada és positiva

]]> 1MA 09. DERIVADES/1MA.09.3 Variacions/1MA.09.3.3 Curvatura 1MA.09.3.3.11.1Q AbscissaInflexióPolG3 Per quin valor de x la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math» presenta un punt d'inflexió?

]]> Cal trobar el punt on la derivada primera s'anul·la i canvia de signe.

]]> 1.0000000 0.5000000 0 0 #r_83 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_3</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>∫</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>a_2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>∫</mo><mi>c</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>a_0</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>&apos;</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>6</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>6</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_83</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"> #r_83 </correctAnswer></correctAnswers><localData><data name="cas">false</data><data name="inputField">textField</data></localData></question> 1MA.09.3.3.11.5Q InflexióPolG3 Determina, si s'escau, en quins punts la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»t«/mi»«/mstyle»«/math» presenta un punt d'inflexió.

]]> Cal trobar el punt on la derivada segona s'anul·la i canvia de signe.

Format: {(1,2);(3,4)}

]]> Cal trobar el punt on la derivada segona s'anul·la i canvia de signe.

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a0</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a3</mi><mo>=</mo><mn>3</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a1</mi><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a3</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>∫</mo><mfenced><mrow><mn>6</mn><mo>*</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a3</mi></mrow></mfenced><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a1</mi></mrow></mfenced><mo>)</mo><mo>+</mo><mi>a2</mi></mtd></mtr><mtr><mtd/></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>∫</mo><mi>c</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>a0</mi></mtd></mtr><mtr><mtd><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>s1</mi><mo>=</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>y1</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>a1</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>y2</mi><mo>=</mo><mi>g</mi><mfenced><mi>a3</mi></mfenced></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mfenced 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xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>48</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>72</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>6</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>85</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>197</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>85</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>,</mo><mn>197</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>2</mn><mo>,</mo><mn>85</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>6</mn><mo>,</mo><mn>197</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-2)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> 1MA.09.3.3.51Q Extrems+Inflexió→CoefPolG3 Troba a, b, c i d de la funció f(x) = ax³ + bx² + cx + d
si la funció presenta un extrem relatiu en el punt (#m_1,#m_2) i un punt d'inflexió en el punt (#i_1,#i_2)?

]]> Si la funció presenta un extrem relatiu:

passa pel punt indicat: f(#m_1) = #m_2
per aquest valor de x, la derivada primera s'anul·la i canvia de signe: f'(#m_1) = 0

Si la funció presenta un punt d'inflexió:

passa pel punt indicat: f(#i_1) = (#i_2)
per aquest valor de x, la derivada segona s'anul·la i canvia de signe: f''(#i_1) = 0

Resolent el sistema de 4 equacions, es calculen a, b, c i d.

]]> 1.0000000 0.3333333 0 0 a=#a_1b=#b_1c=#c_1d=#d_1]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mn>3</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mn>6</mn><mo>*</mo><mi>a_1</mi></mtd></mtr><mtr><mtd><mi>d</mi><mo>=</mo><msup><mfenced><mi>a_1</mi></mfenced><mn>2</mn></msup><mo>*</mo><mfenced><mrow><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mfenced><mrow><mi>a_2</mi><mo>+</mo><mi>a_3</mi><mo>+</mo><mi>a_4</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>12</mn><mo>*</mo><mfenced><mrow><mi>a_2</mi><mo>*</mo><mi>a_3</mi><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>a_4</mi><mo>+</mo><mi>a_3</mi><mo>*</mo><mi>a_4</mi></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mi>d_1</mi><mo>=</mo><msqrt><mi>d</mi></msqrt></mtd></mtr></mtable><mrow><mi>d</mi><mo>></mo><mn>0</mn></mrow></apply><mrow><mi>d_1</mi><mo> </mo><mo>∈</mo><naturalnumbers/></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi><mo>=</mo><mo>-</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mi>a_2</mi><mo>+</mo><mi>a_3</mi><mo>+</mo><mi>a_4</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mi>a_1</mi><mo>*</mo><mfenced><mrow><mi>a_2</mi><mo>*</mo><mi>a_3</mi><mo>+</mo><mi>a_2</mi><mo>*</mo><mi>a_4</mi><mo>+</mo><mi>a_3</mi><mo>*</mo><mi>a_4</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>d_1</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>a_1</mi><mo>*</mo><mi>a_2</mi><mo>*</mo><mi>a_3</mi><mo>*</mo><mi>a_4</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>2</mn><mo>*</mo><mo>(</mo><mi>a_2</mi><mo>+</mo><mi>a_3</mi><mo>+</mo><mi>a_4</mi><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>t</mi><mo>=</mo><mi>f_2</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_2</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_3</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_2</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_4</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_3</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_4</mi><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>s</mi><mo>=</mo><mi>f_1</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_2</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_3</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_4</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>r</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>s</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q_1</mi><mo>=</mo><mi>longitud</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><msub><mi>Q</mi><mn>2</mn></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w_1</mi><mo>=</mo><mi>longitud</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m_1</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mo>*</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mi>a_2</mi><mo>+</mo><mi>a_3</mi><mo>+</mo><mi>a_4</mi><mo>)</mo><mo>+</mo><msqrt><mi>d</mi></msqrt></mrow><mrow><mn>6</mn><mo>*</mo><mi>a_1</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>m_1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi><mo>=</mo><mfrac><mrow><mi>a_2</mi><mo>+</mo><mi>a_3</mi><mo>+</mo><mi>a_4</mi></mrow><mn>3</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>i_1</mi><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>#</mo><mi>a</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>=</mo><mo>#</mo><mi>b</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi>c</mi><mo>=</mo><mo>#</mo><mi>c</mi><mi>_</mi><mn>1</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>=</mo><mo>#</mo><mi mathvariant="normal">d</mi><mi>_</mi><mn>1</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution">25 25 25 25</data><data name="inputField">popupEditor</data><data name="casSession"/></localData></question> Si la funció presenta un extrem relatiu:

passa pel punt indicat: f(#m_1) = #m_2
per aquest valor de x, la derivada primera s'anul·la i canvia de signe: f'(#m_1) = 0

Si la funció presenta un punt d'inflexió:

passa pel punt indicat: f(#i_1) = (#i_2)
per aquest valor de x, la derivada segona s'anul·la i canvia de signe: f''(#i_1) = 0

Resolent el sistema de 4 equacions, es calculen a, b, c i d.

]]> 1MA 09. DERIVADES/1MA.09.3 Variacions/1MA.09.3.4 Taula de variacions 1MA.09.3.4.11Q TVExtrems PolG3 Feu el quadre de variacions per determinar la MONOTONIA de la funció f(x) = #e

La primera derivada és {#1}

x	-oo	{#2}		{#3}	+oo
f'	{#4}	{#5}	{#6}	{#7}	{#8}
creixement	{#9}	{#10}	{#11}	{#12}	{#13}

La funció presenta extrems en els punt
[{#14},{#15}]
i [{#16},{#17}]

]]> Escala vertical: 1:100
#G

]]> 17.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_4</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_5</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>f_1</mi><mo>=</mo><mi>a_5</mi><mo>*</mo><mn>6</mn><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>a_2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfenced><mrow><mo>∫</mo><mi>f_1</mi></mrow></mfenced><mo>+</mo><mi>a_3</mi></mtd></mtr><mtr><mtd><mi>e</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mn>2</mn><mo>*</mo><mi>a_5</mi><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><mi>a_5</mi><mo>*</mo><mo>(</mo><mi>a_1</mi><mo>+</mo><mi>a_2</mi><mo>)</mo><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>6</mn><mo>*</mo><mi>a_5</mi><mo>*</mo><mi>a_1</mi><mo>*</mo><mi>a_2</mi><mo>)</mo><mo>*</mo><mi>x</mi><mo>+</mo><mi>a_3</mi></mtd></mtr><mtr><mtd><mi>G</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mfrac><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mn>100</mn></mfrac><mo>)</mo></mtd></mtr><mtr><mtd><mi>b_1</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>a_1</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>b_2</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>a_2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>d_1</mi><mo>=</mo><mi>f_1</mi></mtd></mtr><mtr><mtd><mi>d_2</mi><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>d_1</mi></apply></mtd></mtr><mtr><mtd><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>d_2</mi><mo>=</mo><mn>0</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>s_2</mi><mo>=</mo><mfrac><mrow><mi>a_1</mi><mo>+</mo><mi>a_2</mi></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>b_3</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>s_2</mi><mo>)</mo></mtd></mtr></mtable><mrow><mi>a_1</mi><mo>&lt;</mo><mi>a_2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a_5</mi><mo>&gt;</mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>r_11</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_12</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_13</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_14</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_15</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_16</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_17</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_18</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_21</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_22</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_23</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_24</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_25</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_26</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_27</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_28</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_31</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_32</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_33</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_34</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_35</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_36</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_37</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_38</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_41</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_42</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_43</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_44</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_45</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_46</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_47</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_48</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>m_11</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_12</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_13</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_21</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_22</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_23</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_31</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_32</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_33</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mtable><mtr><mtd><mi>r_11</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_12</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_13</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_14</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_15</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_16</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_17</mi><mo>=</mo><mi>postiva</mi></mtd></mtr><mtr><mtd><mi>r_18</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_21</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_22</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_23</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_24</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_25</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_26</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_27</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_28</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_31</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_32</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_33</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_34</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_35</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_36</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_37</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_38</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_41</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_42</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_43</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_44</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_45</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_46</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_47</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_48</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>m_11</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_12</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_13</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_21</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_22</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_23</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_31</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_32</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_33</mi><mo>=</mo><mi>màxim</mi></mtd></mtr></mtable></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>s_2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>34</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.15Q TVInflexió PolG3 Estudia els punts d'inflexió i la concavitat de f(x) = #e

La primera derivada és {#1}
La segona derivada és {#2}

x	-oo	{#3}	+oo
f''	{#4}	{#5}	{#6}
concavitat	{#7}		{#8}

La funció presenta un punt d'inflexió: [{#9},{#10}]

]]> Escala vertical: 1:100
#G

]]> 10.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" 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close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>s_2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>72</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>72</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>129</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>214</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>85</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.16Q TVExtremsInflexióPolG3 Fes el quadre de variacions que correspon a la funció f(x) = #e

La primera derivada és {#1}
La segona derivada és {#2}

x	-oo	{#3}		{#4}		{#5}	+oo
f''	{#6}		{#7}	{#8}	{#9}		{#10}
concavitat	{#11}		{#12}		{#13}		{#14}
f'	{#15}	{#16}	{#17}		{#18}	{#19}	{#20}
creixement	{#21}	{#22}	{#23}	{#24}	{#25}	{#26}	{#27}

La funció presenta extrems en els punt
[{#28},{#29}]
i [{#30},{#31}]
La funció presenta un punt d'inflexió: [{#32},{#33}]

]]> Escala vertical: 1:100
#G

]]> 33.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" 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definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a_5</mi><mo>&gt;</mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>r_11</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_12</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_13</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_14</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_15</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_16</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_17</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_18</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_21</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_22</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_23</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_24</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_25</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_26</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_27</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_28</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_31</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_32</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_33</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_34</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_35</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_36</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_37</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_38</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_41</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_42</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_43</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_44</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_45</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_46</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_47</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_48</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>m_11</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_12</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_13</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_21</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_22</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_23</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_31</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_32</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_33</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mtable><mtr><mtd><mi>r_11</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_12</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_13</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_14</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_15</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_16</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_17</mi><mo>=</mo><mi>postiva</mi></mtd></mtr><mtr><mtd><mi>r_18</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_21</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_22</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_23</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_24</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_25</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_26</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_27</mi><mo>=</mo><mi>convexa</mi></mtd></mtr><mtr><mtd><mi>r_28</mi><mo>=</mo><mi>còncava</mi></mtd></mtr><mtr><mtd><mi>r_31</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_32</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_33</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_34</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_35</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_36</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_37</mi><mo>=</mo><mi>positiva</mi></mtd></mtr><mtr><mtd><mi>r_38</mi><mo>=</mo><mi>negativa</mi></mtd></mtr><mtr><mtd><mi>r_41</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_42</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_43</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_44</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_45</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>r_46</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_47</mi><mo>=</mo><mi>creixent</mi></mtd></mtr><mtr><mtd><mi>r_48</mi><mo>=</mo><mi>decreixent</mi></mtd></mtr><mtr><mtd><mi>m_11</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_12</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_13</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_21</mi><mo>=</mo><mi>màxim</mi></mtd></mtr><mtr><mtd><mi>m_22</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_23</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_31</mi><mo>=</mo><mi>inflexió</mi></mtd></mtr><mtr><mtd><mi>m_32</mi><mo>=</mo><mi>mínim</mi></mtd></mtr><mtr><mtd><mi>m_33</mi><mo>=</mo><mi>màxim</mi></mtd></mtr></mtable></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>24</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>90</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>48</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>90</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>48</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a_1</mi><mo>,</mo><mi>a_2</mi><mo>,</mo><mi>s_2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>24</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>90</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>24</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>90</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>107</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>99</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>103</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.21Q TVExtrems PolG4 Fes el quadre de variacions que correspon a la funció f(x) = #e

La primera derivada és {#1}

x	-oo	{#2}		{#3}		{#4}	+oo
f''	{#5}	{#6}	{#7}	{#8}	{#9}		{#10}
concavitat	{#11}		{#12}		{#13}		{#14}
f'	{#15}	{#16}	{#17}		{#18}	{#19}	{#20}
creixement	{#21}	{#22}	{#23}	{#24}	{#25}	{#26}	{#27}

La funció presenta un extrem en el punt
[{#28},{#29}]

]]> Escala vertical: 1:50
#G

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xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.25Q TVInflexióPolG4 Fes el quadre de variacions respecte a f'' que correspon a la funció f(x) = #e

La primera derivada és {#1}
La segona derivada és {#2}

x	-oo	{#3}		{#4}	+oo
f''	{#5}	{#6}	{#7}	{#8}	{#9}
concavitat	{#10}		{#11}		{#12}

La funció presenta dos punts d'inflexió:
[{#13},{#14}]
[{#15},{#16}]

]]> Escala vertical: 1:50
#G

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xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.26Q TVExtremsInflexióPolG4 Completa el quadre de variacions que correspon a la funció f(x) = #e

La primera derivada és {#1}
La segona derivada és {#2}

x	-oo	{#3}		{#4}		{#5}	+oo
f''	{#6}	{#7}	{#8}	{#9}	{#10}		{#11}
concavitat	{#12}		{#13}		{#14}		{#15}
f'	{#16}	{#17}	{#18}		{#19}	{#20}	{#21}
creixement	{#22}	{#23}	{#24}	{#25}	{#26}	{#27}	{#28}

La funció presenta un extrem en el punt
[{#29},{#30}]
i dos punts d'inflexió:
[{#31},{#32}]
[{#33},{#34}]

]]> Escala vertical: 1:50
#G

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xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>48</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.31Q TV RacionalG1/G1 Completa el quadre de variacions que correspon a la funció f(x) = #e

x	-oo	{#1}	+oo
f''	{#2}		{#3}
concavitat	{#4}		{#5}
f'	{#6}		{#7}
f: creixement	{#8}		{#9}

]]> #G

]]> 9.0000000 0.5000000 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_2</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_3</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>a_4</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfrac><mi>a_2</mi><mi>a_1</mi></mfrac><mo>≠</mo><mfrac><mi>a_4</mi><mi>a_3</mi></mfrac></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>264</mn><mrow><mn>64</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>144</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>108</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>27</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>264</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_12</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>negativa</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_14</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>positiva</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_22</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>còncava</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_24</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>convexa</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_32</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>negativa</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_34</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>negativa</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_42</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>decreixent</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r_44</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>decreixent</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><localData><data name="cas">false</data></localData></question> 1MA.09.3.4.91Q EstudiPolG3 Per representar la funció f(x) = #g

1) La seva derivada és

f'(x) = {#1}

2) La seva taula de variacions és

x	-oo	{#2}		{#3}	+oo
f'	{#4}	{#5}	{#6}	{#7}	{#8}
creixement	{#9}	{#10}	{#11}	{#12}	{#13}

3) Els seus punts d'intersecció amb els eixos són:

(0, {#14} )
i (ordenats) ( {#15} ,0), ({#16},0), ({#17},0)

4) Dibuixa la funció i compara-la amb el gràfic solució

]]> #G

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xmlns="http://www.w3.org/1998/Math/MathML"><cn>-∞</cn><mo>,</mo><cn>+∞</cn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="casSession"/></localData></question> 1MA 09. DERIVADES/1MA.09.4 Estudi de funcions 1MA.09.4.10DT ESTUDI FUNCIÓ

Estudi i representació d'una funció

1. Domini
Cal tenir en compte si té denominadors, arrels, tangents, logaritmes o exponencials.
2. Límits i asímptotes:
Es calculen els límits a les vores obertes del domini, i s'esbrina si corresponen a asímptotes horitzontals, verticals o obliqües.
3. Derivades
Es calcula la derivada primera (i si cal, la segona). Es calculen les seves arrels i es fa la seva taula de signes.
4. Taula de variacions
Es centralitza tota la informació:

COMENÇANT PEL DOMINI
escrivint el signe de la o les derivades

Se'n dedueix la monotonia, els extrems, i (si s'escau) la concavitat i punts d'inflexió.

Es comprova que "quadri" quan s'afegeixen els límits.

5. Punts de tall amb els eixos

Es calculen amb x = 0 i f(x) = 0, si és que aquests valors són possibles.

6. Gràfic

Es col·loquen les asímptotes, tots els punts disponibles. Si cal es calcula algun punt més i es fa un esquema senzill de la funció.

]]> 0.0000000 0.0000000 0 1MA.09.4.11Q EstudiPolG3 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/mrow»«/mstyle»«/math»

a) Escriu els seus intervals de creixement. Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

b) Escriu els seus intervals de decreixement.

c) Determina els seus extrems relatius {(1,2),(3,4)}

d) Determina el seus punts d'inflexió {(1,2),(3,4)}

e) Dibuixa la funció i compara-la amb la solució

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Extrems =#sol3d) Inflexió =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable 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xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>3</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>3</mn><mo>,</mo><mo>-</mo><mn>25</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mn>9</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi mathvariant="normal">e</mi><mi>x</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

La funció és creixent quan la derivada és positiva

]]> 1MA.09.4.12Q EStudiPolG4 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

a) Escriu els seus intervals de creixement. Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

b) Escriu els seus intervals de decreixement.

c) Determina els seus extrems relatius {(1,2),(3,4)}

d) Determina el seus punts d'inflexió {(1,2),(3,4)}

e) Dibuixa la funció i compara-la amb la solució

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Extrems =#sol3d) Inflexió =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mn>6</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>s2</mi><mo><</mo><mi>s3</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><msup><mo>)</mo><mn>2</mn></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mi>d</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d3</mi><mo>=</mo><mi>d</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f1</mi><mo>=</mo><mi>factoritza</mi><mo>(</mo><mi>d1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f2</mi><mo>=</mo><mi>factoritza</mi><mo>(</mo><mi>d2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>∫</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mn>1</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>2</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>1</mn><mo>,</mo><mfrac><mn>57</mn><mn>2</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>2</mn><mo>,</mo><mn>27</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>251</mn><mn>9</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi mathvariant="normal">e</mi><mi>x</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math» (Ruffini?)

La funció és creixent quan la derivada és positiva

]]> 1MA.09.4.21Q EstudiRacG1G1 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

a) Escriu els seus intervals de creixement. Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

b) Escriu els seus intervals de decreixement.

c) Determina les seves asímptotes {x=2,y=2}

d) Determina el seus punts de tall amb els eixos {(1,2),(3,4)}

e) Dibuixa la funció i compara-la amb la que et surti quan comprovis

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Asímptotes =#sol3d) Punts d'intersecció amb els eixos =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mi>s1</mi><mo><</mo><mi>s2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo></mrow><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>-</mo><mi>s2</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><msup><mo>)</mo><mn>2</mn></msup></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>y</mi><mo>=</mo><mi>a</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>a</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mi>y</mi><mo>=</mo><mi>a</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I1</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mi>s1</mi></mrow><mi>s2</mi></mfrac></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I2</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mi>s1</mi><mo>,</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>I1</mi><mo>,</mo><mi>I2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura</mi><mo>=</mo><mn>200</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>r1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>></mo><mn>0</mn></mrow><mrow><mi>w1</mi><mo>=</mo><mo>"</mo><mi>positiu</mi><mo>"</mo></mrow><mrow><mi>w1</mi><mo>=</mo><mo>"</mo><mi>negatiu</mi><mo>"</mo></mrow></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>2</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>2</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>0</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>,</mo><mi>r1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><ms>positiu</ms><mo>,</mo><mi>y3</mi><mo>=</mo><mo>-</mo><mn>3</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>A</mi><mi>s</mi><mi>í</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>o</mi><mi>t</mi><mi mathvariant="normal">e</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>P</mi><mi>u</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo> </mo><mi mathvariant="normal">d</mi><mo>'</mo><mi mathvariant="normal">i</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">e</mi><mi>r</mi><mi>sec</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo> </mo><mi>a</mi><mi>m</mi><mi>b</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>l</mi><mi>s</mi><mo> </mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi>o</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

La funció és creixent quan la derivada és positiva

]]> 1MA.09.4.22Q EstudiRacG2G1 Escriu el domini i els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.4.23Q EstudiRacG1G2 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

a) Escriu els seus intervals de creixement. Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

b) Escriu els seus intervals de decreixement.

c) Determina les seves asímptotes {x=2,y=2}

d) Determina el seus punts de tall amb els eixos {(1,2),(3,4)}

e) Dibuixa la funció i compara-la amb la solució.

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Asímptotes =#sol3d) Intersecció amb els eixos =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mfenced close="|" open="|"><mi>s1</mi></mfenced><mo>≠</mo><mfenced close="|" open="|"><mi>s2</mi></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><msqrt><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup></mrow></msqrt><mo>∈</mo><rationals/></mrow></mfenced></mrow></apply><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mfenced close="|" open="|"><mi>s1</mi></mfenced><mo>≠</mo><mfenced close="|" open="|"><mi>s2</mi></mfenced></mrow></mfenced><mo>∧</mo><mfenced><mrow><msup><mi>s1</mi><mn>2</mn></msup><mo>-</mo><msup><mi>s2</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></mrow></mfenced></mrow></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo></mrow><mrow><mfenced><mrow><mi>x</mi><mo>-</mo><mi>s2</mi></mrow></mfenced><mo>*</mo><mfenced><mrow><mi>x</mi><mo>+</mo><mi>s2</mi></mrow></mfenced></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>recta</mi><mo>(</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>punt</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n1</mi><mo>=</mo><mi>numerador</mi><mo>(</mo><mi>d1</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>d1</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d1</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>s2</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I1</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mi>s1</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I2</mi><mo>=</mo><mi>punt</mi><mfenced><mrow><mn>0</mn><mo>,</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mi>s1</mi></mrow><msup><mi>s2</mi><mn>2</mn></msup></mfrac></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>I1</mi><mo>,</mo><mi>I2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>50</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>altura_finestra</mi><mo>=</mo><mn>550</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>550</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>r</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mi>x</mi></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>4</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>4</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>4</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><cn>-∞</cn><mo>,</mo><mo>-</mo><mn>4</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>2</mn></mrow></mfenced><mo>∪</mo><mfenced><mrow><mn>8</mn><mo>,</mo><cn>+∞</cn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced><mrow><mn>5</mn><mo>,</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mfrac><mn>5</mn><mn>16</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mfrac><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>16</mn></mrow><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>32</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>256</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>A</mi><mi>s</mi><mi>í</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>o</mi><mi>t</mi><mi mathvariant="normal">e</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>I</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">e</mi><mi>r</mi><mi>sec</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo> </mo><mi>a</mi><mi>m</mi><mi>b</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>l</mi><mi>s</mi><mo> </mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi>o</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.4.23Q EstudiRacG2G2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

El denominador és positiu en el domini. El signe de la derivada depèn exclusivament del numerador de 2n grau:

Si no té solució, té el signe de a

Si té solucions, té el signe contrari de a entre les solucions...

]]> 1MA.09.4.31Q EstudiArrelG2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan és negativa.

]]> 1MA.09.4.41Q EstudiLnG2 Considera la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

a) Escriu els seus intervals de creixement. Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

b) Escriu els seus intervals de decreixement.

c) Determina les seves asímptotes {x=2,y=2}

d) Determina l'abscissa del seu extrem 3/4

e) Dibuixa la funció i compara-la amb la solució

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2c) Asímptotes =#sol3d) Abscissa extrem =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>w1</mi><mo>=</mo><mfrac><mrow><mi>s1</mi><mo>+</mo><mi>s2</mi></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>w2</mi><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>w1</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>w1</mi><mo>-</mo><mi>s2</mi><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>w1</mi><mo>∈</mo><rationals/></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>w2</mi><mo>></mo><mn>0</mn></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>ln</mi><mo>(</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>+</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>∩</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>∩</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mi>s1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow><mrow><mi>sol4</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd/></mtr></mtable></mfenced></mrow><mrow><mi>sol4</mi><mo>=</mo><mi>w1</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>450</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>18</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mfenced><mrow><mi>T1</mi><mo>,</mo><mi>g</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>blau</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s1</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G1</mi><mo>=</mo><mi>dibuixa</mi><mo>(</mo><mi>T1</mi><mo>,</mo><mi>x</mi><mo>=</mo><mi>s2</mi><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>color</mi><mo>=</mo><mi>vermell</mi><mo>,</mo><mi>amplada_línia</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>14</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>24</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>3</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>14</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>24</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>12</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s1</mi><mo>,</mo><mi>s2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>14</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mn>3</mn></mrow></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>,</mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><mi>A</mi><mi>s</mi><mi>í</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>o</mi><mi>t</mi><mi mathvariant="normal">e</mi><mi>s</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </mo><mi>A</mi><mi>b</mi><mi>s</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>s</mi><mi>s</mi><mi>a</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi>m</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»; s'ha simplificat per (#a)

El signe de f'(x) depèn exclusivament del numerador.

EN EL DOMINI, la funció és creixent quan la derivada és positiva i decreixent quan f'(x) és negativa.

]]> 1MA.09.4.45Q EstudiExpG2 Escriu els intervals de creixement i de decreixement de la funció «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»g«/mi»«/math»

Format: Si un dels intervals és buit, escriu «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8709;«/mo»«/math»

]]> El gràfic és #G1

]]> 1.0000000 0.5000000 0 0 a) Creixent en =#sol1b) Decreixent en =#sol2]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>s1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><mfenced><mrow><mi>s1</mi><mo>≠</mo><mi>s2</mi></mrow></mfenced></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msup><exponentiale/><mrow><mo>(</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>s2</mi><mo>)</mo><mo>)</mo></mrow></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d1</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d2</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>s1</mi><mo>+</mo><mi>s2</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfenced><mrow><mi>f</mi><mfenced><mfrac><mrow><mi>s1</mi><mo>+</mo><mi>s2</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dom</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>sol1</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>sol2</mi><mo>=</mo><mi>conjunt_domini</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mo>-</mo><mi>d2</mi><mo>)</mo><mo>)</mo></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>2</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>50</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>55</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>20</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>100</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>altura_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>amplada_finestra</mi><mo>=</mo><mn>600</mn><mo>,</mo><mi>altura</mi><mo>=</mo><mn>110</mn><mo>,</mo><mi>amplada</mi><mo>=</mo><mn>9</mn><mo>,</mo><mi>centre</mi><mo>=</mo><mi>punt</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>50</mn><mo>)</mo></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>m</mi><mo><</mo><mn>500</mn></mrow><mtable><mtr><mtd><mi>k1</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mi>T1</mi><mo>=</mo><mi>tauler</mi><mfenced><mfenced close="}" open="{"><mtable 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xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mi>C</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mi>D</mi><mi mathvariant="normal">e</mi><mi>c</mi><mi>r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi>x</mi><mi mathvariant="normal">e</mi><mi>n</mi><mi>t</mi><mo> </mo><mi mathvariant="normal">e</mi><mi>n</mi><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"><param name="intervals">true</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> Cal estudiar el signe de la derivada «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#191919¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#191919¨»d«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#191919¨»1«/mn»«/math»

Com que una exponencial és sempre positiva, el signe de f'(x) depèn exclusivament del polinomi #d2

La funció és creixent quan la derivada és positiva i decreixent quan f'(x) és negativa.

]]> 1MA.09.4.51Q FPolG3 NOPUNTUA Estudia la funció f(x) = #g

Envia per veure la correcció.

]]> El domini és (-oo,+oo)
Talla els eixos en els punts (0,#c_1) i (#x_1,0), (#x_2,0) i (#x_3,0)
La derivada primera és #h
La funció presenta extrems relatius en #t_1 i #t_2 ja que la derivada primera s'hi anul·la i canvia de signe.Els extrems són #e_1 i #e_2
Amb la derivada segona f''(x) = #w es pot trobar el punt d'inflexió #i_11.
El gràfic és #G

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align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>11</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>11</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>3</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml">envia</correctAnswer></correctAnswers><localData><data name="inputField">textField</data></localData></question> 1MA.09.4.52Q FPolG4 NOPUNTUA Estudia la funció f(x) = #g

Envia per veure la correcció

]]> El domini és (-oo, +oo)
Si els punts de tall amb els eixos no són evidents, calculeu algun punt per situar la corba.
La derivada primera és #h
La funció presenta extrems relatius en #t_1, #t_2 i #t_3 ja que la derivada primera s'hi anul·la i canvia de signe.
Amb la derivada segona #m es poden trobar les abscisses dels punts d'inflexió on la derivada segona s'anul·la i canvia de signe #R.
Els punts que corresponen als extrems són (#t_1, #e_1), (#t_2,#e_2), (#t_3,#e_3)
El gràfic és #G

]]> 1.0000000 0.0000000 0 0 #r_1 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd/></mtr><mtr><mtd><mi>x_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>x_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable 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align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>h</mi><mo>=</mo><mn>12</mn><mo>*</mo><mi>x</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>g</mi><mo>=</mo><mo>∫</mo><mi>h</mi><mo>+</mo><mi>x_3</mi></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi></mtd></mtr><mtr><mtd><mi>t_1</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>t_2</mi><mo>=</mo><mi>x_1</mi></mtd></mtr><mtr><mtd><mi>t_3</mi><mo>=</mo><mi>x_2</mi></mtd></mtr><mtr><mtd><mi>m</mi><mo>=</mo><mi>h</mi><mo>&apos;</mo></mtd></mtr><mtr><mtd><mi>R</mi><mo>=</mo><mi>resol</mi><mo>(</mo><mi>m</mi><mo>=</mo><mn>0</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>t_1</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_2</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>t_2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_3</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>t_3</mi><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mfenced><mrow><mi>x_1</mi><mo>&lt;</mo><mi>x_2</mi></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>x_2</mi><mo>&lt;</mo><mi>x_3</mi></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mi>representa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>28</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>60</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>84</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>120</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>168</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>120</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>123</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>66</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><msqrt><mn>19</mn></msqrt><mn>3</mn></mfrac><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>19</mn></msqrt><mn>3</mn></mfrac><mo>-</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml">#r_1</correctAnswer></correctAnswers><localData><data name="inputField">textField</data></localData></question> 1MA.09.4.61Q FRacionalG1/G1 NOPUNTUA Estudia la funció f(x) = #g

Envia per veure la correcció

]]> El domini és (-oo,#x_2)U(#x_2, +oo)
Talla els eixos en els punts (0,#c_1) i (#x_1,0).
Té asímptota horitzontal y = #a_1 i asímptota vertical x = #x_2
La derivada primera és #j i la derivada segona és #i
El gràfic és #G

]]> 1.0000000 0.0000000 0 0 #r_1 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>x_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>x_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>x_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mi>a_1</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_1</mi><mo>)</mo></mrow><mrow><mi>x</mi><mo>-</mo><mi>x_2</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mi>c_1</mi><mo>=</mo><mfrac><mrow><mi>a_1</mi><mo>*</mo><mi>x_1</mi></mrow><mi>x_2</mi></mfrac></mtd></mtr><mtr><mtd><mi>g</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>j</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo>=</mo><mi>f</mi><mo>&apos;</mo><mo>&apos;</mo></mtd></mtr><mtr><mtd><mi>t</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a_1</mi><mo>*</mo><mo>(</mo><mi>x_2</mi><mo>-</mo><mi>x_1</mi><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>x_2</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>i</mi><mo>=</mo><mfrac><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>t</mi></mtd></mtr></mtable></mfenced><msup><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>-</mo><mi>x_2</mi></mtd></mtr></mtable></mfenced><mn>4</mn></msup></mfrac></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mi>x_1</mi><mo>≠</mo><mi>x_2</mi></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mi>representa</mi><mfenced><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mn>2</mn><mrow><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>2</mn></mtd></mtr></mtable></mfenced><msup><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mn>4</mn></msup></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml">#r_1</correctAnswer></correctAnswers><localData><data name="inputField">textField</data></localData></question> 1MA.09.4.62Q FRacionalG1/G2 NOPUNTUA Estudia la funció f(x) = #g
No estudiïs la inflexió

Envia per veure la correcció

]]> El domini és (-oo, 0)U(0,#x_2)U(#x_2,+oo)
Talla els eixos en el punt (#x_1,0)
Té asímptota horitzontal y = 0 i dues asímptotes verticals: x = 0 i x = #x_2
La derivada primera és #i
La funció presenta extrems relatius si #t té arrels.
El gràfic és #G

]]> 1.0000000 0.0000000 0 0 #r_1 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a_1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>x_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>x_3</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></mtd></mtr><mtr><mtd><mi>x_4</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable 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xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>30</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>15</mn></mrow><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mo>*</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></mrow><msup><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mtd></mtr></mtable></mfenced><mn>2</mn></msup></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&map;</mo><mfrac><mrow><mn>6</mn><mo>*</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>90</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>90</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>30</mn></mrow><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tauler1</mi></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo></math></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml">#r_1</correctAnswer></correctAnswers><localData><data name="inputField">textField</data></localData></question> 1MA 09. DERIVADES/1MA.09.5 Optimització 1MA.09.5.11Q Funció donada Una empresa produeix dos tipus de productes A i B. De productes del tipus A en produeix a kg, mentre que de productes del tipus B en produeix b kg. La quantitat de producte total que produeix l'empresa cada dia són #prod.

La funció que dona els costos de l'empresa diaris en funció del nombre de productes produïts és:«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»f«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»,«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»P«/mi»«msup mathcolor=¨#003300¨»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»Q«/mi»«msup mathcolor=¨#003300¨»«mo mathvariant=¨bold¨»)«/mo»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mstyle»«/math».

Mantenint la mateixa producció, quines quantitats de A i de B cal fabricar per tal que els costos siguin mínims.

Escriu la funció de a, f(a) que cal optimitzar, el valor de a que determina un cost mínim i aquest cost mínim (a i costos amb decimals si s'escau)

]]> 1.0000000 1.0000000 0 0 funció=#sol1a=#sol2costos=#sol3]]> <question><wirisCasSession><![CDATA[<session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mi>g</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><mi>m2</mi><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><mi>G</mi><mo>(</mo><mi>m2</mi><mo>)</mo><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>prod</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>,</mo><mn>117</mn><mo>,</mo><mn>141</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>108</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>1476</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>,</mo><mi>m</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>108</mn><mo>*</mo><mi>a</mi><mo>+</mo><mn>1476</mn><mo>,</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>-</mo><mn>108</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>,</mo><mi>m2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>a</mi><mo>=</mo><mn>27</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced><mo>,</mo><mn>27</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mn>2</mn><mo>*</mo><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>108</mn><mo>*</mo><mi>a</mi><mo>+</mo><mn>1476</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27.</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18.</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mi>u</mi><mi>n</mi><mi>c</mi><mi mathvariant="normal">i</mi><mi>ó</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>a</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>cos</mi><mi>t</mi><mi>o</mi><mi>s</mi><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="answer_parameter">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/><data name="inputCompound">true</data></localData></question> 1MA.09.5.12 Trobar funció: AugmentPreu El bar d'un parlament subministra gintònics d'alt preu a baix cost.
Si el preu de cada gintònic és de p = #p_1 €, en consumeixen d= #d_1 diputats/des. S'ha comprovat que per cada augment de preu de #p_2 €, #d_2 diputats/des comencen a fer "vida sana".

a) Escriu la funció que descriu el nombre de diputats/des que beuen gintònics en funció del seu preu: f(p) = {#1}
b) Escriu la funció que relaciona el preu del gintònic amb el nombre de diputats/des que en beuen: f(d) = {#2}
c) Escriu la funció que relaciona els ingressos del bar en concepte de gintònic en funció del seu preu i(p) = {#3}
d) Per quin preu, els ingressos són màxims? {#4}

]]> 4.0000000 0.3333333 0 <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p_1</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>40</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>150</mn><mo>,</mo><mn>365</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>4</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p_2</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>*</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mfrac><mrow><mi>d_1</mi><mo>*</mo><mi>p_2</mi><mo>+</mo><mi>d_2</mi><mo>*</mo><mi>p_1</mi><mo>-</mo><mi>d_2</mi><mo>*</mo><mi>p</mi></mrow><mi>p_2</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mfenced><mfrac><mrow><mi>d_1</mi><mo>*</mo><mi>p_2</mi><mo>+</mo><mi>d_2</mi><mo>*</mo><mi>p_1</mi></mrow><mi>d_2</mi></mfrac></mfenced><mo>-</mo><mfrac><mi>p_2</mi><mi>d_2</mi></mfrac><mo>*</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi><mo>=</mo><mi>p</mi><mo>*</mo><mi>f_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>p</mi><mo>*</mo><mi>f_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>(</mo><mi>i_1</mi><mo>)</mo><mo>&apos;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi><mo>=</mo><mfrac><mrow><mi>arrodoneix</mi><mo>(</mo><mn>100</mn><mo>*</mo><msub><mi>R</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></mrow><mn>100</mn></mfrac><mo>*</mo><mn>1</mn><mo>.</mo><mn>0</mn></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p_1</mi><mo>,</mo><mi>d_1</mi><mo>,</mo><mi>d_2</mi><mo>,</mo><mi>p_2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.</mn><mo>,</mo><mn>169</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>0.2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>40.</mn><mo>*</mo><mi>p</mi><mo>+</mo><mn>289.</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0.025</mn><mo>*</mo><mi>d</mi><mo>+</mo><mn>7.225</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>40.</mn><mo>*</mo><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mn>289.</mn><mo>*</mo><mi>p</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>40.</mn><mo>*</mo><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mn>289.</mn><mo>*</mo><mi>p</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>80.</mn><mo>*</mo><mi>p</mi><mo>+</mo><mn>289.</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>p</mi><mo>=</mo><mn>3.6125</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i_2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.61</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession></question> a i b) Si el preu augment x cops de #p_2 €, el preu serà p i el nombre de consumidors serà d, tal que:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd/»«/mtr»«mtr»«mtd»«mfenced close=¨¨ open=¨{¨»«mtable»«mtr»«mtd»«mi»p«/mi»«mo»=«/mo»«mi»#p_1«/mi»«mo»+«/mo»«mi»#p_2«/mi»«mo»§nbsp;«/mo»«mo»·«/mo»«mo»§nbsp;«/mo»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»d«/mi»«mo»=«/mo»«mi»#d_1«/mi»«mo»-«/mo»«mi»#d_2«/mi»«mo»§nbsp;«/mo»«mo»·«/mo»«mo»§nbsp;«/mo»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»
Aïllant p o d es pot respondre als apartats a i b.

]]> c)N'hi ha prou amb multiplicar el preu p pel nombre de diputats corresponent al preu p: #f_1

]]> d) Només cal derivar i maximitzar la funció #i_1 que hem trobat a l'apartat c

]]> 1MA.09.5.22Q Geometria: Perímetre Volem construir un marc que tingui una superfície de #a dm² . Cada dm de marc horitzontal ens costa #e_1 € i cada dm de marc vertical, #e_2. Quines dimensions té el marc de cost mínim?

Format de la resposta (fraccions amb un sol denominador):
A=funció a optimitzar
B= derivada
C= valor de x que fa mínima la funció
D=altura que correspon a aquest mínim.]]>

]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#CD=#D]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>11</mn><mo>,</mo><mn>100</mn><mo>)</mo></mtd></mtr><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>e_1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>e_2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr></mtable><mrow><mi>e_2</mi><mo>></mo><mi>e_1</mi></mrow></apply></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mfrac><mi>a</mi><mi>x</mi></mfrac></mtd></mtr><mtr><mtd><mi>A</mi><mo>:</mo><mo>=</mo><mi>e_1</mi><mo>*</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mi>e_2</mi><mo>*</mo><mn>2</mn><mo>*</mo><mfrac><mi>a</mi><mi>x</mi></mfrac></mtd></mtr><mtr><mtd><mi>f_1</mi><mo>=</mo><mi>A</mi></mtd></mtr><mtr><mtd><mi>B</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></mtd></mtr><mtr><mtd><mi>f_2</mi><mo>=</mo><mi>B</mi></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><msqrt><mfrac><mrow><mi>a</mi><mo>·</mo><mi>e_2</mi></mrow><mi>e_1</mi></mfrac></msqrt></mtd></mtr><mtr><mtd><mi>D</mi><mo>=</mo><mfrac><mi>a</mi><mi>C</mi></mfrac></mtd></mtr><mtr><mtd/></mtr><mtr><mtd/></mtr></mtable><mrow><mi>C</mi><mo>∈</mo><rationals/></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>e_1</mi><mo>,</mo><mi>e_2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>,</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>192</mn></mrow><mi>x</mi></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>12</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>192</mn></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>#</mo><mi>A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mo>#</mo><mi>B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mo>#</mo><mi>C</mi><mspace linebreak="newline"/><mi>D</mi><mo>=</mo><mo>#</mo><mi>D</mi></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data><data name="casSession"/></localData></question> La funció a optimitzar f(x) ve donada per #e_1·2·x (bases) + #e_2·2·y (altures) .
Cal doncs relacionar la base amb l'altura amb l'àrea:

#a = x ·altura «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8660;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»altura«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«mi mathvariant=¨bold¨»x«/mi»«/mfrac»«/math»

La funció a optimitzar és doncs: #A

]]> 1MA.09.5.23Q Geometria RectangleÀreaMàxima Volem delimitar un recinte rectangular amb una tanca de longitud #p. Quines són les dimensions d'aquest recinte si es vol que la seva àrea sigui màxima? Anomena x la base del rectangle.

Format de la resposta:
A=expressió de la funció a optimitzar
B= expressió de la seva derivada
C= valor de x que fa màxima l'àrea
D=altura que correspon a aquesta àrea màxima.]]>

]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#CD=#D]]> <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mrow><mi>p</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>88</mn><mo>)</mo></mrow><mrow><mi>p</mi><mo> </mo><mi>mod</mi><mo> </mo><mn>4</mn><mo> </mo><mo>≠</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>p</mi><mn>2</mn></mfrac><mo>-</mo><mi>x</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mi>x</mi><mo>*</mo><mfenced><mrow><mfrac><mi>p</mi><mn>2</mn></mfrac><mo>-</mo><mi>x</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>A</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>B</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mi>p</mi><mn>4</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mfrac><mi>p</mi><mn>4</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>59</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>59</mn><mn>2</mn></mfrac><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>59</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>59</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>59</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>59</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>#A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mi>#B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mi>#C</mi><mspace linebreak="newline"/><mi>D</mi><mo>=</mo><mi>#D</mi></math></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">popupEditor</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data></localData></question> La funció a optimitzar f(x) ve donada per x ·y.
Cal doncs relacionar la base x amb l'altura y, gràcies al perímetre P:

#p = 2x + 2y per tant: y = #p/2 -x
La funció a optimitzar és doncs: #A

]]> 1MA.09.5.24Q Geometria TriangleÀreaMàxima De tots els triangles d'hipotenusa #a, trobeu la base x del que té àrea màxima.

Format de la resposta:
A= funció a optimitzar
B=derivada d'aquesta funció
C=valor de x que la fa màxima.

]]> La funció a optimitzar és #A

]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#C]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo>,</mo><mn>15</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mo>*</mo><mi>y</mi></mrow><mn>2</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>f_1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mi>f_2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mi>a</mi><msqrt><mn>2</mn></msqrt></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>*</mo><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>49</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>49</mn></mrow><mrow><mn>2</mn><mo>*</mo><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>49</mn></mrow></msqrt></mrow></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>#</mo><mi>A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mo>#</mo><mi>B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mo>#</mo><mi>C</mi></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_equations"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data><data name="casSession"/></localData></question> L'área a maximitzar es calcula amb «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»b«/mi»«mi»a«/mi»«mi»s«/mi»«mi»e«/mi»«mo»§#183;«/mo»«mi»a«/mi»«mi»l«/mi»«mi»t«/mi»«mi»u«/mi»«mi»r«/mi»«mi»a«/mi»«/mrow»«mn»2«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»x«/mi»«mi»y«/mi»«/mrow»«mn»2«/mn»«/mfrac»«/math»

Ara cal aïllar y, emprant el teorema de Pitàgores

]]> 1MA.09.5.25Q Geometria RectanglePerímetreMínim De tots els rectangles d'àrea #a troba la base x del que té perímetre mínim.

Format de la resposta (fraccions amb un sol denominador):
A=funció a optimitzar
B= derivada
C= valor de x que fa mínima la funció
D=altura que correspon a aquest perímetre mínim.]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#CD=#D]]> <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>12</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo>,</mo><mn>15</mn><mo>,</mo><mn>17</mn><mo>,</mo><mn>18</mn><mo>,</mo><mn>19</mn><mo>,</mo><mn>20</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>23</mn><mo>,</mo><mn>24</mn><mo>,</mo><mn>25</mn><mo>,</mo><mn>26</mn><mo>,</mo><mn>27</mn><mo>,</mo><mn>28</mn><mo>,</mo><mn>29</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>31</mn><mo>,</mo><mn>32</mn><mo>,</mo><mn>33</mn><mo>,</mo><mn>34</mn><mo>,</mo><mn>35</mn><mo>,</mo><mn>37</mn><mo>,</mo><mn>38</mn><mo>,</mo><mn>39</mn><mo>,</mo><mn>40</mn></mtd></mtr></mtable></mfenced><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>a</mi><mi>x</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mn>2</mn><mfenced><mrow><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mi>x</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>A</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>B</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><msqrt><mi>a</mi></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>C</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>34</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>68</mn></mrow><mi>x</mi></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>68</mn></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>34</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><mn>34</mn></msqrt></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><msqrt><mn>34</mn></msqrt></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>#A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mi>#B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mi>#C</mi><mspace linebreak="newline"/><mi>D</mi><mo>=</mo><mi>#D</mi></math></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">popupEditor</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data><data name="cas">false</data></localData></question> La funció a optimitzar és el perímetre del rectangle = 2x+2y si x és la base i y l'altura.
Cal doncs relacionar la base amb l'altura, gràcies a l'àrea A = x ·y;
per tant: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»y«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mi mathvariant=¨normal¨»A«/mi»«mi mathvariant=¨normal¨»x«/mi»«/mfrac»«/math»
La funció a optimitzar és doncs: #A

]]> 1MA.09.5.26Q Geometria RectangleÀreaMàxima De tots els rectangles de perímetre #p troba la base x del que té àrea màxima

Format de la resposta:
A=expressió de la funció a optimitzar
B= expressió de la seva derivada
C= valor de x que fa màxima l'àrea
D=altura que correspon a aquesta àrea màxima.]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#CD=#D]]> <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mrow><mi>p</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>12</mn><mo>,</mo><mn>88</mn><mo>)</mo></mrow><mrow><mi>p</mi><mo> </mo><mi>mod</mi><mo> </mo><mn>4</mn><mo> </mo><mo>≠</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>p</mi><mn>2</mn></mfrac><mo>-</mo><mi>x</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mi>x</mi><mo>*</mo><mfenced><mrow><mfrac><mi>p</mi><mn>2</mn></mfrac><mo>-</mo><mi>x</mi></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>A</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>B</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mi>p</mi><mn>4</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mfrac><mi>p</mi><mn>4</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>59</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>59</mn><mn>2</mn></mfrac><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mfrac><mn>59</mn><mn>2</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>59</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>59</mn><mn>4</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>59</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>#A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mi>#B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mi>#C</mi><mspace linebreak="newline"/><mi>D</mi><mo>=</mo><mi>#D</mi></math></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">popupEditor</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data><data name="cas">false</data></localData></question> La funció a optimitzar f(x) ve donada per x ·y.
Cal doncs relacionar la base x amb l'altura y, gràcies al perímetre P:

#p = 2x + 2y per tant: y = #p/2 -x
La funció a optimitzar és doncs: #A

]]> 1MA.09.5.27Q RectangIncritTriangleAmàx De tots els rectangles inscrits en un triangle rectangle de base #b i d'altura #c,troba la base x del que té àrea màxima.

Format de la resposta:
A=expressió de la funció a optimitzar
B= expressió de la seva derivada
C= valor de x que fa màxima l'àrea
D=altura que correspon a aquesta àrea màxima.]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#CD=#D]]> <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>100</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>100</mn><mo>)</mo></mtd></mtr></mtable><mrow><mfenced><mrow><mi>b</mi><mo> </mo><mi>mod</mi><mo> </mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></mfenced><mo>∧</mo><mfenced><mrow><mi>c</mi><mo> </mo><mi>mod</mi><mo> </mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></mfenced></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>c</mi><mo>*</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>x</mi><mo>)</mo></mrow><mi>b</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mo>*</mo><mi>c</mi><mo>*</mo><mfenced><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow><mi>b</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>A</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>B</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mi>b</mi><mn>2</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mfrac><mi>c</mi><mn>2</mn></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>58</mn><mo>,</mo><mn>84</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>42</mn><mn>29</mn></mfrac><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>84</mn><mo>*</mo><mi>x</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>84</mn><mn>29</mn></mfrac><mo>*</mo><mi>x</mi><mo>+</mo><mn>84</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>29</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>42</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>29</mn></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>#A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mi>#B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mi>#C</mi><mspace linebreak="newline"/><mi>D</mi><mo>=</mo><mi>#D</mi></math></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">popupEditor</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data><data name="cas">false</data></localData></question> Si es dibuixa el triangle, ja es veu que hi ha proporcionalitat entre el triangle gran i el petit.

Se'n pot deduir la relació entre x i y: y = #c · (#b - x)
La funció a optimitzar és doncs: #A

]]> 1MA.09.5.28Q RectangleInscritCircumfAmax De tots els rectangles inscrits en una circumferència de radi #r troba la base x del que té l'àrea màxima.

Format de la resposta:
A=expressió de la funció a optimitzar
B=expressió de la seva derivada
C=valor de la base x que correspon a una àrea màxima
D=valor de l'altura y que correspon a una àrea màxima

]]> 1.0000000 0.5000000 0 0 A=#AB=#BC=#CD=#D]]> <question><wirisCasSession><session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>10</mn><mo>,</mo><mn>50</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>r</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mrow><msup><mi>d</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>:</mo><mo>=</mo><mi>x</mi><mo>*</mo><msqrt><mrow><msup><mi>d</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_1</mi><mo>=</mo><mi>A</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>:</mo><mo>=</mo><apply><diff/><bvar><mi>x</mi></bvar><mi>f_1</mi></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f_2</mi><mo>=</mo><mi>B</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mi>d</mi><msqrt><mn>2</mn></msqrt></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mi>C</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>f_2</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2304</mn></mrow></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>*</mo><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2304</mn></mrow></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>2</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2304</mn></mrow><msqrt><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2304</mn></mrow></msqrt></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mn>24</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>24</mn><mo>*</mo><msqrt><mn>2</mn></msqrt></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session></wirisCasSession><correctAnswers><correctAnswer type="mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>#A</mi><mspace linebreak="newline"/><mi>B</mi><mo>=</mo><mi>#B</mi><mspace linebreak="newline"/><mi>C</mi><mo>=</mo><mi>#C</mi><mspace linebreak="newline"/><mi>D</mi><mo>=</mo><mi>#D</mi></math></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression" correctAnswer="0"/><assertion name="equivalent_symbolic" correctAnswer="0"/></assertions><localData><data name="inputField">popupEditor</data><data name="gradeCompound">distribute</data><data name="gradeCompoundDistribution"></data><data name="inputCompound">true</data><data name="cas">false</data></localData></question> Aplica el teorema de Pitàgores

]]> 1MA.09.5.51Q PendentMínimRectaTangent En quin punt el pendent de la recta tangent a la funció f(x) = #E és mínim?

Format de la resposta:
A=expressió de la funció a optimitzar
B= expressió de la seva derivada
C= valor de x que fa mínim el pendent

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name="inputCompound">true</data><data name="cas">false</data></localData></question> La funció a optimitzar #A és la derivada de la funció f(x), ja que la derivada de la funció és el pendent de la recta tangent.
La seva derivada és la segona derivada de f(x) i permet trobar el valor de x.]]>