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).

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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 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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>pendent</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.09.1.15Q 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

]]> <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>posneg</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>signe</mi><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>posneg</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>suma</mi><mo>=</mo><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>signe</mi><mo>=</mo><mn>1</mn></mrow><mi>true</mi><mi>false</mi></apply><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>resta</mi><mo>=</mo><apply><csymbol 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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

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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.09.1.23Q Derivabilitat de funcions 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¨»§#60;«/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¨»§#10878;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»s«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/mrow»«/mstyle»«/math».

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></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></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></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></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></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></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></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></command></group></library><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"><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>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><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>p1</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>A</mi><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><msup><mi>A</mi><mrow><mi>r0</mi><mo>(</mo><mo>)</mo></mrow></msup></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>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>p2</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>B</mi><mo>)</mo><mo>=</mo><mi>B</mi><mo>*</mo><mi>R</mi><mo>(</mo><mo>)</mo><mo>+</mo><msup><mi>B</mi><mrow><mi>r0</mi><mo>(</mo><mo>)</mo></mrow></msup><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>B</mi><mrow><mi>r0</mi><mo>(</mo><mo>)</mo></mrow></msup><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>R</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><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>9</mn><mo>*</mo><mfenced><mrow><mi>B</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><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><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>rr</mi><mo>(</mo><mo>)</mo><mo>=</mo><mi>random</mi><mo>(</mo><mo>-</mo><mn>50</mn><mo>.</mo><mo>.</mo><mn>50</mn><mo>)</mo><mo>;</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>rr</mi><mo>(</mo><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>27</mn></math></output></command><command><input><math 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 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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.09.1.42APREN 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 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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 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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_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 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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 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.09.1.45APREN 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 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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"><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 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.09.1.53Q 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"/></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.09.1.55Q 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 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>></mo><mn>0</mn></mrow></apply></math></input></command><command><input><math 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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>1</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>2.6391</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>↦</mo><mfrac><mn>5</mn><mrow><mn>5</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>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>1</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><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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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 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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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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

]]> 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"><mfrac><mn>2</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><mfrac><mn>281</mn><mn>7</mn></mfrac><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>f_2</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>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mfrac><mn>281</mn><mn>7</mn></mfrac></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><mn>5</mn><mo>,</mo><mn>2</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>4</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>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>5</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>4</mn><mo>,</mo><mfrac><mn>2035</mn><mn>21</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>5</mn><mo>,</mo><mo>-</mo><mfrac><mn>3095</mn><mn>21</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>4</mn><mo>,</mo><mfrac><mn>2035</mn><mn>21</mn></mfrac></mrow></mfenced><mo>,</mo><mfenced close="]" open="["><mrow><mn>5</mn><mo>,</mo><mo>-</mo><mfrac><mn>3095</mn><mn>21</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">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Si són perpendiculars, el producte dels seus pendents és (-1):

#m · (#f_2) = -1

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