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<i>Encuentre la ecuación principal de la recta tangente a la curva definida por:<br><br></i> <div align="center"><i><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><br></i></div><i><br>que pasa por el punto <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>.</i><br> <div align="center">#C1<br></div> <p></p> <p align="justify"><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">«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></em><br></p> <div align="center"><br></div>
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<p><i><span style="font-family: times new roman,times,serif">Solución:</span></i></p> <p><i><span style="font-family: times new roman,times,serif"><br></span></i></p> <div style="text-align: justify; font-style: italic"><i></i></div> <div style="text-align: justify; font-style: italic"><i>Lo que buscamos es una recta de la forma:<br><br></i> <div style="text-align: justify; font-style: italic" align="center"><i><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><br><br></i></div> <div style="text-align: justify; font-style: italic"><br><i>La cual debe ser tangente a la función en el punto #q.</i></div> <div style="text-align: justify; font-style: italic"><i>Además tenemos que saber que el valor de la derivada de una función en un punto puede interpretarse geométricamente, y corresponde a la pendiente de la recta tangente a la gráfica de la función en dicho punto.<br><br>Así que primero calcularemos 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">«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>, ya que <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¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#fun«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span></span> evaluada en el punto <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> es la pendiente <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»«/math»</span></span></span></span></span></span></span></span></span> de la recta buscada.<br>Así:<br><br></i> <div style="text-align: justify; font-style: italic" align="center"><i><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><br></i> <div align="justify">Notación : <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><br></div><i><br></i> <div align="justify"><i>Ahora evaluamos el punto <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> 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">«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>.<br><br>Así:<br></i> <div align="center"><i><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></i><br></div><i><br><br>Por lo tanto tenemos que <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><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">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»b«/mi»«/math»</span></span></span></span></span></span></span></span></span> para tener la ecuación de la recta. Para esto, reemplazamos el punto #q en la ecuación anterior.<br><br></i> <div align="center"><i><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»«mi»#c2«/mi»«mi»#r4«/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><br></i></div><i><br><br>Por lo tanto la ecuación de la recta que es tangente a la curva deifinida por la función <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>y que pasa por el punto <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> es:<br></i> <div align="center"><i><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><br></i><i><br></i> <div align="justify"><br>Gráficamente:<br> <div align="center">#C3<br></div></div></div></div></div><i> <div style="text-align: justify; font-style: italic"><br><br><br><br></div><span style="font-family: times new roman,times,serif"></span><span style="font-family: times new roman,times,serif"></span><br><br><br><br></i><br></div></div> <p></p>
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<i>¡Muy bien! Sigue así.</i>
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