5.3

<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> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>M</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>atemáticas de la U</B></FONT><B> (MatU)  | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U><B>Cálculo de una variable</B></U></A><B> | </B><A HREF="id543.htm" TARGET="_top" TITLE="D.G. Zill"><U><B>D. G. Zill</B></U></A><B> | </B><A HREF="id551_m.htm#apuntes" TARGET="TRLX_Middle" TITLE="5.3"><U><B>Top</B></U></A><B> |</B></div> </td> </tr><tr> <td valign="top" rowspan="2" width=165 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 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios 5.3</B></div> <div> <B>El problema de área</B></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> <TABLE BORDER="0" CELLPADDING="4" CELLSPACING="1" WIDTH="903" BORDERCOLORLIGHT="#006666" BORDERCOLORDARK="#006666" FRAME="VOID" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 23 WIDTH="885"><div> <B> <A NAME="apuntes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Apuntes: Definiciones, t</B><B>e</B><B>oremas.</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="442"><div> <A HREF="#notación_sigma" TITLE="5.3"><U><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Notación sigma</U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_5_3_1" TITLE="5.3"><U><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.3.1</U></A>:<B>  </B>Propiedades de la notación sigma</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 ><NOBR><div> <A HREF="#teorema_5_3_2" TITLE="5.3"><U><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.3.2</U></A>: Fórmulas de sumas especiales</div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#definición_5_3_1" TITLE="5.3"><U><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición 5.3.1:</U></A> Área bajo una gráfca</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 45 WIDTH="442"><div> <A HREF="#ejercicios_5_3" TITLE="5.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 5.3</U></A>: Enunciados y soluciones de los problemas</div> </tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC WIDTH="442"><div> <A HREF="#lista_5_3" TITLE="5.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 5.3</U></A>: Lista de reproduccción de los videos en You Tube</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 20 ><NOBR><div> <U>Nota</U>: Las explicaciones magistrales de esta teoría las puede leer en el libro "Cálculo", Trascendentes tempranas, Edi. 4, del Profesor D.G. Zill </div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="notación_sigma"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Notación sigma</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Notación sigma</B></div> <div> <IMG SRC="2d99adec0.gif" border=0 width="941" height="236" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <B> <A NAME="teorema_5_3_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema 5.3.1</B></div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.3.1:  </B>Propiedades de la notación sigma</div> <div> <IMG SRC="2d9ab4bc0.gif" border=0 width="436" height="188" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <B> <A NAME="teorema_5_3_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema 5.3.2</B></div> <div> <B>Teorema 5.3.2:</B> Fórmulas de sumas especiales</div> <div> Para <I>n</I> un entero positivo y <I>c</I> cualquier constante,</div> <div> <TABLE BORDER="0" CELLPADDING="0" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="VOID" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 62 WIDTH="400"><div> <IMG SRC="2da16f340.gif" border=0 width="111" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="400"><div> <IMG SRC="2d9e9b370.gif" border=0 width="155" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 64 WIDTH="400"><div> <IMG SRC="2d9fdd370.gif" border=0 width="221" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="400"><div> <IMG SRC="2da0af390.gif" border=0 width="175" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> </TABLE></div> <div>  <A NAME="definición_5_3_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Definición 5.3.1</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición 5.3.1:</B> Área bajo una gráfca</div> <div> <IMG SRC="2da25bec0.gif" border=0 width="859" height="236" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <B> <A NAME="ejercicios_5_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Ejercicios 5.3</B></div> <div> <B>Enunciados y soluciones de los problemas de los Ejercicios 5.3</B></div> <bR> <div> <FONT SIZE="5" COLOR="#006600" FACE="Bell MT" ><B>S</B></FONT>olución en imagen <IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT> de los problemas de los Ejercicios 5.3:</div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT><B>1</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>10</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, desarolle la suma indicada.</FONT></div> <div> <IMG SRC="2da341340.gif" border=0 width="65" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#1" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2da66c340.gif" border=0 width="108" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2da845350.gif" border=0 width="69" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2daa5f390.gif" border=0 width="95" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dac673f0.gif" border=0 width="103" height="63" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dae703f0.gif" border=0 width="112" height="63" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2db077370.gif" border=0 width="119" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#7" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2db270350.gif" border=0 width="112" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2db463340.gif" border=0 width="99" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#9" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2db694370.gif" border=0 width="148" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#10" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT><B>11</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>20</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, use notación sigma para escribir la suma dada.</FONT></div> <div> <IMG SRC="2db8e3140.gif" border=0 width="227" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#11" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dbacf140.gif" border=0 width="207" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#12" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dbcc8140.gif" border=0 width="200" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#13" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2dbfd4140.gif" border=0 width="212" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#14" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dc19d300.gif" border=0 width="157" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#15" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dc3b7300.gif" border=0 width="183" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#16" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dc5e5140.gif" border=0 width="229" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#17" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2dc7f71b0.gif" border=0 width="247" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#18" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <IMG SRC="2dc9852f0.gif" border=0 width="389" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#19" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2dcb5d2d0.gif" border=0 width="605" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#20" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT><B>21</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>28</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, encuentre el valor numérico de la suma dada.</FONT></div> <div> <IMG SRC="2dcd4f350.gif" border=0 width="79" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#21" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dcf69350.gif" border=0 width="105" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#22" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dd16b350.gif" border=0 width="107" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#23" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2dd27b350.gif" border=0 width="123" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><A HREF="https://youtu.be/aQclkf4CRDM" TARGET="_top" TITLE="https://youtu.be/aQclkf4CRDM"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       </FONT><IMG SRC="2dda75350.gif" border=0 width="117" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#25" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <IMG SRC="2dd380340.gif" border=0 width="128" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><A HREF="https://youtu.be/aQclkf4CRDM?t=4m18s" TARGET="_top" TITLE="https://youtu.be/aQclkf4CRDM?t=4m18s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="2e167c380.gif" border=0 width="124" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">        <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#27" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><IMG SRC="2dd498350.gif" border=0 width="152" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="https://youtu.be/aQclkf4CRDM?t=6m44s" TARGET="_top" TITLE="https://youtu.be/aQclkf4CRDM?t=6m44s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      </div> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT><B>29</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>42</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, use la Definición 5.3.1 y el Teorema 5.3.2 para hallar el área bajo la gráfica de la función función dada y el intervalo indicado.</FONT></div> <div> <IMG SRC="2e17a01c0.gif" border=0 width="160" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#29" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e1ead1c0.gif" border=0 width="173" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#34" TITLE="5.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <DIV ALIGN="LEFT"></DIV> <div> <B> <A NAME="lista_5_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Lista 5.3</B></div> <div> <B>Lista de reproducción de los Ejercicios 5.3</B></div> <bR> <div> <B> <iframe width="853" height="480" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo1VQ3kwWEi4oJQS5u3RgQ52" frameborder="0" allowfullscreen></iframe>  </B></div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1</div> <div> <IMG SRC="2da400880.gif" border=0 width="512" height="136" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2</div> <div> <IMG SRC="2da5f9880.gif" border=0 width="761" height="136" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3</div> <div> <IMG SRC="2da78d890.gif" border=0 width="653" height="137" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>4</div> <div> <IMG SRC="2da910cf0.gif" border=0 width="784" height="207" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>5</div> <div> <IMG SRC="2dab7b5f0.gif" border=0 width="891" height="351" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>6</div> <div> <IMG SRC="2dad945f0.gif" border=0 width="916" height="351" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>7</div> <div> <IMG SRC="2daf738d0.gif" border=0 width="883" height="141" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>8</div> <div> <IMG SRC="2db15c880.gif" border=0 width="860" height="136" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#FFFFFF" FACE="Times New Roman" >9</FONT></div> <div> <IMG SRC="2db383880.gif" border=0 width="643" height="136" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#FFFFFF" FACE="Times New Roman" >10</FONT></div> <div> <IMG SRC="2db5c8fc0.gif" border=0 width="712" height="252" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <IMG SRC="2db793ff0.gif" border=0 width="659" height="255" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>12</div> <div> <IMG SRC="2db993010.gif" border=0 width="659" height="257" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>13</div> <div> <IMG SRC="2dbb17250.gif" border=0 width="791" height="293" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>14</div> <div> <IMG SRC="2dbd1b250.gif" border=0 width="795" height="293" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>15</div> <div> <IMG SRC="2dc093270.gif" border=0 width="659" height="295" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>16</div> <div> <IMG SRC="2dc29c240.gif" border=0 width="668" height="292" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>17</div> <div> <IMG SRC="2dc485cc0.gif" border=0 width="645" height="204" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="18"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>18</div> <div> <IMG SRC="2dc685f50.gif" border=0 width="645" height="245" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="19"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>19</div> <div> <IMG SRC="2dc8bfd80.gif" border=0 width="703" height="472" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="20"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>20</div> <div> <IMG SRC="2dca4ddf0.gif" border=0 width="845" height="479" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>21</div> <div> <IMG SRC="2dccbd170.gif" border=0 width="445" height="279" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="22"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>22</div> <div> <IMG SRC="2dce9b170.gif" border=0 width="667" height="279" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="23"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>23</div> <div> <IMG SRC="2dd0a3170.gif" border=0 width="931" height="279" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>25</div> <div> <IMG SRC="2dd907890.gif" border=0 width="775" height="137" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>27</div> <div> <IMG SRC="2ddbd4930.gif" border=0 width="980" height="147" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="29"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>29</div> <div> <IMG SRC="2e1915530.gif" border=0 width="789" height="595" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <B><IMG SRC="2e1bd0b50.gif" border=0 width="464" height="437" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="2e1c2c2c0.gif" border=0 width="556" height="300" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="34"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>34</div> <div> <B><IMG SRC="2e1d15540.gif" border=0 width="789" height="596" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <DIV ALIGN="LEFT"></DIV> <div> <B><IMG SRC="2e1f82190.gif" border=0 width="386" height="281" 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