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<i><font size="4" face="times new roman,times,serif">Encuentre la ecuación principal de la recta tangente a la curva definida por:<br /><br /></font></i> <div align="center"><i><font size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#f1«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /></font></i></div><i><font size="4" face="times new roman,times,serif"><br />en el punto de la abscisa <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»#p«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>.</font></i><br /> <div align="center">#C1</div> <div align="center"></div> <div align="center"></div> <div align="center"></div> <div align="justify"></div> <div align="justify"><font color="#990000" size="4" face="times new roman,times,serif"><em>Obs.: La ecuación principal de la recta es de la forma <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></em></font><br /></div><br /> <div align="justify"><br /></div>
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<i><font size="4" face="times new roman,times,serif"><span style="font-family: times new roman,times,serif">Solución:<br /><br /></span></font></i> <div style="text-align: justify; font-style: italic"><i><font size="4" face="times new roman,times,serif"><span style="font-family: times new roman,times,serif"></span></font><font size="4" face="times new roman,times,serif"><span style="font-family: times new roman,times,serif"></span></font><font size="4" face="times new roman,times,serif"><span style="font-family: times new roman,times,serif"></span></font><font size="4" face="times new roman,times,serif"></font><font style="font-style: italic" size="4" face="times new roman,times,serif"><span style="font-family: times new roman,times,serif"></span></font><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif">Lo que buscamos es una recta de la forma:<br /><br /></font></i> <div align="center"><i><font size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></i> <br /><br /> <div align="left"><font size="4" face="times new roman,times,serif">La cual debe ser tangente a la función en el punto #q, ya que cuando <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mi»#p«/mi»«/math»</span></span></span></span></span></span></span></span></span></span>, <span style="font-family: times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#fun«/mi»«mo»(«/mo»«mi»#p«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#p2«/mi»«/math»</span></span></span></span></span></span></span></span></span></span>.</span></font></div> <div align="left"><font size="4" face="times new roman,times,serif">Por otra parte, la interpretación geométrica de la derivada es que al ser evaluada en un punto, representa la pendiente de la recta tangente a la función en dicho punto; es decir, </font><br /></div> <div align="center"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»m«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»#fun«/mi»«mo»(«/mo»«mi»#p«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /></div></div><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif">Entonces, lo primero que haremos <i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif">es calcular la derivada de <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>, </font></i></font></i><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><br />Así:<br /><br /></font></i> <div align="center"><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#f1«/mi»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#d«/mi»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font><font size="4" face="times new roman,times,serif"><br /></font></i> <div align="justify"><font size="4" face="times new roman,times,serif">Notación : </font><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»§apos;«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /></div><i><font size="4" face="times new roman,times,serif"><br /></font></i> <div align="justify"><i><font size="4" face="times new roman,times,serif">Ahora evaluamos el punto </font><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#q«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> en <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»§apos;«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mi»#d«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>.<br /><br />Así:<br /></font></i> <div align="center"><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»m«/mi»«mo»=«/mo»«mi»y«/mi»«mo»§apos;«/mo»«mfenced»«mrow»«mi»#p«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mi»#pen«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></font></i><br /></div><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><br /><br />Por lo tanto, tenemos que </font><font size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»#pen«/mi»«mi»#x«/mi»«mo»+«/mo»«mi»b«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /><br />Ahora, faltaría encontrar <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»b«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> para tener la ecuación de la recta. Para esto, reemplazamos el punto </font><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif">#q en la ecuación anterior.<br /><br /></font></i> <div align="center"><i><font size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable»«mtr»«mtd»«mi»#p2«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#pen«/mi»«mo»·«/mo»«mi»#p11«/mi»«mo»+«/mo»«mi»b«/mi»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#p2«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»#r3«/mi»«mo»+«/mo»«mi»b«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»/«/mo»«mo»+«/mo»«mfenced»«mi»#r4«/mi»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»#p2«/mi»«mo»+«/mo»«mi»#r6«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»b«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi»#r5«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»b«/mi»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«mtd»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /></font></i></div><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><br /><br />En consecuencia, la ecuación de la recta que es tangente a la curva definida por la función </font><font style="font-style: italic" size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#f1«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> y que pasa por el punto </font><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#q«/mi»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> es:<br /></font></i> <div align="center"><i><font size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»#pen«/mi»«mi»#x«/mi»«mo»+«/mo»«mo»(«/mo»«mi»#r5«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /></font></i><font size="4" face="times new roman,times,serif"><i><br /></i></font> <div align="justify"><font size="4" face="times new roman,times,serif"><br />Gráficamente:<br /></font> <div align="center"><font size="4" face="times new roman,times,serif">#C3</font><br /></div></div></div></div></div><i><font style="font-style: italic; font-family: times new roman,times,serif" size="4" face="times new roman,times,serif"><br /><br /><br /><br /></font><font size="4" face="times new roman,times,serif"><br /><br /><br /><br /></font></i><br /><font size="4"><span style="font-family: times new roman,times,serif"></span></font><font size="4"><span style="font-family: times new roman,times,serif"></span></font></div>
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<i><font size="4" face="times new roman,times,serif">¡Muy bien! Sigue así.</font></i>
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<i><font size="4" face="times new roman,times,serif"><font size="4" face="times new roman,times,serif"><em>Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</em></font></font></i>
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