Rapideces de variación relacionadas

<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 = 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value="always"></param><embed src="http://www.youtube.com/v/ldnKNhU1F0Q&hl=es&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object>  </div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#FFCC99 HEIGHT= 39 ><NOBR><div>  <A NAME="e_n_u_n_c_i_a_d_o_s"><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>E n u n c i a d o s</B></FONT></div> <div> <FONT SIZE="3" COLOR="#CCFF00" 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" >4</FONT>, <I>x</I>  e  <I>y</I>  son funciones de una tercera variable <I>t</I>.</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 42 ><NOBR><div> <A HREF="#1" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fcc3b150.gif" border=0 width="315" height="21" 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="Rapideces de variación relacionadas"><U><IMG SRC="1fcd1d230.gif" border=0 width="285" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 87 ><NOBR><div> <A HREF="#3" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fce39500.gif" border=0 width="313" height="80" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#4" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fcff04d0.gif" border=0 width="240" height="77" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 473 ><NOBR><div> <A HREF="#5" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd08c350.gif" border=0 width="652" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#6" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd180280.gif" border=0 width="640" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#7" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd281330.gif" border=0 width="641" height="51" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#8" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd385350.gif" border=0 width="645" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#9" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd45b370.gif" border=0 width="603" height="55" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#10" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd57c450.gif" border=0 width="636" height="69" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#11" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd678350.gif" border=0 width="632" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="#12" TITLE="Rapideces de variación relacionadas"><U><IMG SRC="1fd781450.gif" border=0 width="641" height="69" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> </TABLE></div> <bR> <div> <B>S o l u c i o n e s</B></div> <bR> <div>  <A NAME="1"><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><IMG SRC="1fd83b150.gif" border=0 width="315" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1fd91b1d0.gif" border=0 width="539" height="285" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <bR> <div>  <A NAME="2"><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><IMG SRC="1fda73150.gif" border=0 width="371" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1fdb61270.gif" border=0 width="609" height="295" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="3"><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><IMG SRC="1fdce82d0.gif" border=0 width="488" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1fdd5f9c0.gif" border=0 width="607" height="412" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="4"><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><IMG SRC="1fde5f290.gif" border=0 width="351" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1fdf373c0.gif" border=0 width="567" height="316" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="5"><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><IMG SRC="1fe08c350.gif" border=0 width="652" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="649" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 188 ><NOBR><div> <IMG SRC="1fe141b50.gif" border=0 width="321" height="181" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1fe237b30.gif" border=0 width="311" height="179" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> Para calcular la distancia horizontal, <I>x</I>, de la cometa, respecto al niño, cuando la longitud de la cuerda es de 50 pies, vamos a utilizar el recíproco del Teorema de Pitágoras:</div> <div> <IMG SRC="1fe378300.gif" border=0 width="632" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> De acuerdo con el Teorema de Pitágoras, se tiene:</div> <div> <IMG SRC="1fe480d80.gif" border=0 width="640" height="216" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1fe580280.gif" border=0 width="640" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="623" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 167 ><NOBR><div> <IMG SRC="1fe6d0a00.gif" border=0 width="464" height="160" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1fe77a910.gif" border=0 width="122" height="145" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <B><IMG SRC="1fe835e70.gif" border=0 width="565" height="231" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <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="#CC0033" FACE="Times New Roman" ><B><IMG SRC="1fe981330.gif" border=0 width="641" height="51" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="672" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 486 ><NOBR><div> <IMG SRC="1fea71d90.gif" border=0 width="369" height="473" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1feb09ee0.gif" border=0 width="265" height="238" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <IMG SRC="1fecdfdb0.gif" border=0 width="223" height="219" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <B><IMG SRC="1fed41e80.gif" border=0 width="577" height="232" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></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="#CC0033" FACE="Times New Roman" ><B><IMG SRC="1fee85350.gif" border=0 width="645" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="648" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 146 WIDTH="397"><div> <IMG SRC="1fefa58b0.gif" border=0 width="165" height="139" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="240"><bR> <div> <IMG SRC="1ff07e770.gif" border=0 width="126" height="119" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> </TABLE></div> <div> <B><IMG SRC="1ff16c1c0.gif" border=0 width="620" height="284" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <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="#CC0033" FACE="Times New Roman" ><B><IMG SRC="1ff25b370.gif" border=0 width="603" height="55" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="662" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 299 ><NOBR><div> <IMG SRC="1ff3d3240.gif" border=0 width="467" height="292" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1ff4b5f80.gif" border=0 width="181" height="248" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <B><IMG SRC="1ff589a80.gif" border=0 width="649" height="424" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <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="#CC0033" FACE="Times New Roman" ><B><IMG SRC="1ff67d450.gif" border=0 width="637" height="69" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1ff70db30.gif" border=0 width="525" height="435" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="11"><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><IMG SRC="1ff878350.gif" border=0 width="632" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="657" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 210 ><NOBR><div> <IMG SRC="1ff94fcb0.gif" border=0 width="335" height="203" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1ffa12c40.gif" border=0 width="274" height="196" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <B><IMG SRC="1ffb836d0.gif" border=0 width="643" height="109" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></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="#CC0033" FACE="Times New Roman" ><B><IMG SRC="1ffc81450.gif" border=0 width="641" height="69" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="673" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 218 ><NOBR><div> <IMG SRC="1ffd7cd30.gif" border=0 width="380" height="211" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1ffe18ca0.gif" border=0 width="280" height="202" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <B><IMG SRC="1fffab9d0.gif" border=0 width="683" height="413" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <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" 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