Interpretación geométrica

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B>C á l c u l o   2 1</B><FONT SIZE="4" COLOR="#FFCC66" FACE="Bell MT" ><B> </B></FONT><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" >   </FONT><B>  </B>|   <A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U>Home</U></A>  |  <A HREF="id98.htm" TARGET="_top" TITLE="Contenido"><U>Contenido</U></A>  |  <A HREF="id307.htm" TARGET="_top" TITLE="Introducción al cálculo diferencial"><U>Cálculo diferencial</U></A>  |</div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Interpretación geométrica</div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3926688482" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B><IMG BORDER="0" SRC="Wing0a73007700f000cc0000.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">G</B></FONT>eométricamente la derivada de una función  <I>f </I>en un punto determinado se interpreta como el valor de la pendiente de la recta tangente a la gráfica de <I>f</I> en dicho punto.</div> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="636" BORDERCOLORLIGHT="#006600" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFCC99 HEIGHT= 191 WIDTH="624"><div>  <A NAME="ejercicios_resueltos"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>Ejercicios resueltos</B></FONT></div> <div> <FONT SIZE="3" COLOR="#33FFFF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >1</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >3</FONT>, encuentre la pendiente de la recta tangente a la gráfica en el punto (<I>x</I><FONT SIZE="1" COLOR="#000000" FACE="Times New Roman" ><I>1</I></FONT><I>, y</I><FONT SIZE="1" COLOR="#000000" FACE="Times New Roman" ><I>1</I></FONT>). Elabore una tabla de valores de <I>x, y, m</I> en el intervalo [<I>a, b</I>], e incluya en ella todos los puntos donde la gráfica tiene una pendiente horizontal. Trace la gráfica y muestre un segmento de la tangente en cada uno de los puntos localizados.</div> <div> En los ejercicios <FONT SIZE="3" COLOR="#CC0000" FACE="Times New Roman" >4</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >6</FONT>, determine la pendiente de la tangente a la gráfica de la función  <I>f</I> en el punto (<I>x</I><FONT SIZE="1" COLOR="#000000" FACE="Times New Roman" ><I>1</I></FONT><I>, f(x</I><FONT SIZE="1" COLOR="#000000" FACE="Times New Roman" ><I>1</I></FONT><I>)</I>). Elabore una tabla de valores de <I>x, f(x) </I>y<I> m</I> en diversos puntos de la gráfica e incluya en dicha tabla todos los puntos donde la gráfica tiene un tangente horizontal. Trace la gráfica de la función.</div> <div> En los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >7</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >11</FONT>, obtenga las ecuaciones de las rectas tangente y normal de la curva en el punto indicado. Trace una gráfica de la curva junto con la tangente y la normal.</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 31 ><NOBR><div> <A HREF="#1" TITLE="Interpretación geométrica"><U><IMG SRC="1dc0bd180.gif" border=0 width="189" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#2" TITLE="Interpretación geométrica"><U><IMG SRC="1dc1d3180.gif" border=0 width="211" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#3" TITLE="Interpretación geométrica"><U><IMG SRC="1dc2bd180.gif" border=0 width="189" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 32 WIDTH="198"><div> <A HREF="#4" TITLE="Interpretación geométrica"><U><IMG SRC="1dc35f190.gif" border=0 width="95" height="25" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#5_1" TITLE="Interpretación geométrica"><U><IMG SRC="1dc49d180.gif" border=0 width="157" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#6" TITLE="Interpretación geométrica"><U><IMG SRC="1dc5b5180.gif" border=0 width="181" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 ><NOBR><div> <A HREF="#7" TITLE="Interpretación geométrica"><U><IMG SRC="1dc6b4180.gif" border=0 width="180" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#8" TITLE="Interpretación geométrica"><U><IMG SRC="1dc77f290.gif" border=0 width="127" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="222"><div> <IMG SRC="1dc871290.gif" border=0 width="113" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 79 ><NOBR><div> <IMG SRC="1dc99c180.gif" border=0 width="156" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1dca9b2c0.gif" border=0 width="155" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#12" TITLE="Interpretación geométrica"><U><IMG SRC="1dcbdd480.gif" border=0 width="221" height="72" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 63 ><NOBR><div> <A HREF="#13" TITLE="Interpretación geométrica"><U><IMG SRC="1dcc78380.gif" border=0 width="376" height="56" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="222"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="63" ALIGN="bottom" WIDTH="222" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <div> <B>S o l u c i o n e s</B></div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1dcdbd180.gif" border=0 width="189" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="1" CELLSPACING="1" WIDTH="653" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 533 ><NOBR><div> <IMG SRC="1dce4d2d0.gif" border=0 width="333" height="301" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="237" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>y</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>m</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -3</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 6</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 5</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 4</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 8</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 2</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 9</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 0</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 8</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -2</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 5</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -4</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 3</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -6</div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1dcf38730.gif" border=0 width="312" height="371" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1dd0d3180.gif" border=0 width="211" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="1" CELLSPACING="1" WIDTH="672" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 404 ><NOBR><div> <IMG SRC="1dd1cb1b0.gif" border=0 width="459" height="283" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1dd2d0de0.gif" border=0 width="208" height="222" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="187" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="52"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="61"><div> <I><B>y</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="56"><div> <I><B>m</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="52"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="61"><div> 4</div> </tD> <tD VALIGN=TOP WIDTH="56"><div> -4</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="52"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="61"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="56"><div> -2</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="52"><div> 3</div> </tD> <tD VALIGN=TOP WIDTH="61"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="56"><div> 0</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="52"><div> 4</div> </tD> <tD VALIGN=TOP WIDTH="61"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="56"><div> 2</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="52"><div> 5</div> </tD> <tD VALIGN=TOP WIDTH="61"><div> 4</div> </tD> <tD VALIGN=TOP WIDTH="56"><div> 4</div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1dd3bd180.gif" border=0 width="189" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="1" CELLSPACING="1" WIDTH="671" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 492 ><NOBR><div> <IMG SRC="1dd4a04c0.gif" border=0 width="416" height="332" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1dd5fa360.gif" border=0 width="250" height="310" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="237" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>y</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>m</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -7</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 12</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> -1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 3</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 0</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 0</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 3</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> 2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 9</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 12</div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1dd65f190.gif" border=0 width="95" height="25" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B> </B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="644" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 143 ><NOBR><div> <IMG SRC="1dd7d0880.gif" border=0 width="464" height="136" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 351 ><NOBR><div> <IMG SRC="1dd840540.gif" border=0 width="320" height="340" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1dd9fbc20.gif" border=0 width="251" height="194" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="237" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>y</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>m</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 76 WIDTH="73"><div> -5</div> <div> 0</div> <div> 3</div> <div> 4</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 3</div> <div> 2</div> <div> 1</div> <div> 0</div> </tD> <tD VALIGN=TOP><NOBR><div> -0.167</div> <div> -0.25</div> <div> -0.5</div> <div> No existe</div> </NOBR></tD> </tR> </TABLE></div> <bR> </NOBR></tD> </tR> </TABLE></div> <bR> <div>  <A NAME="5_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1dda99180.gif" border=0 width="153" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="658" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 338 ><NOBR><div> <IMG SRC="1ddb8fb50.gif" border=0 width="399" height="181" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="237" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>y</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>m</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 95 WIDTH="73"><div> 0</div> <div> 1</div> <div> 2</div> <div> 3</div> <div> 4</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 8</div> <div> -1</div> <div> -4</div> <div> -1</div> <div> 9</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -12</div> <div> -6</div> <div> 0</div> <div> 6</div> <div> 12</div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1ddcf4f30.gif" border=0 width="244" height="243" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1dddb3180.gif" border=0 width="179" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1ddead710.gif" border=0 width="685" height="113" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="658" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 301 ><NOBR><div> <IMG SRC="1ddf807d0.gif" border=0 width="384" height="125" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="237" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP BGCOLOR=#FFCC99 HEIGHT= 20 WIDTH="73"><div> <I><B>x</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>y</B></I></div> </tD> <tD VALIGN=TOP BGCOLOR=#FFCC99 WIDTH="73"><div> <I><B>m</B></I></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 95 WIDTH="73"><div> 0</div> <div> 1</div> <div> 2</div> <div> 3</div> <div> 4</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> -2</div> <div> 2</div> <div> 0</div> <div> -2</div> <div> 2</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> 9</div> <div> 0</div> <div> -3</div> <div> 0</div> <div> 9</div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1de0e30b0.gif" border=0 width="227" height="267" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1de18c180.gif" border=0 width="140" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <U>Solución</U>:</div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="656" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 330 ><NOBR><div> <IMG SRC="1de2a4430.gif" border=0 width="420" height="323" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1de3e22a0.gif" border=0 width="226" height="298" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 254 ><NOBR><div> <IMG SRC="1de468f70.gif" border=0 width="616" height="247" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <bR> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1de557170.gif" border=0 width="87" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="1" CELLSPACING="1" WIDTH="671" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 306 ><NOBR><div> <IMG SRC="1de68f230.gif" border=0 width="399" height="291" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1de7f92b0.gif" border=0 width="249" height="299" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 410 ><NOBR><div> <IMG SRC="1de859930.gif" border=0 width="601" height="403" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <bR> <div>  <A NAME="12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1de95f170.gif" border=0 width="95" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1dea69330.gif" border=0 width="617" height="307" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B><IMG SRC="1deb5f170.gif" border=0 width="95" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1dec8bd50.gif" border=0 width="651" height="469" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <bR> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B> <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:4577786115" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="100" /> <input type="submit" name="sa" value="Buscar" /> </div> </form> <script type="text/javascript" src="http://www.google.com.co/coop/cse/brand?form=cse-search-box&lang=es"></script>  </B></div> </td> </tr></table></body>