1.1

<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></div> <div> </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 1.1</B></div> <div> <B>Funciones y gráficas</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> <div>  <A NAME="top"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Top</div> <div> <B>Enunciados y soluciones de los problemas de los Ejercicios 1.1</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 1.1:</div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Georgia" ><B>E</B></FONT>n<FONT SIZE="4" COLOR="#006666" FACE="Georgia" ><B> </B></FONT>los problemas 1 a 6, encuentre los valores funcionales indicados.</div> <div> <IMG SRC="2e9a7b270.gif" border=0 width="379" height="39" 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/CFN3RRlSmYY" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          </FONT><IMG SRC="2fe490340.gif" border=0 width="400" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#1_1_2_1" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="2fe9701f0.gif" border=0 width="368" height="31" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#1_1_3" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><IMG SRC="2e9ba1340.gif" border=0 width="417" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><A HREF="https://youtu.be/CFN3RRlSmYY?t=10m9s" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY?t=10m9s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><IMG SRC="2fee7c300.gif" border=0 width="380" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="#1_1_5" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2ff194340.gif" border=0 width="404" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#1_1_6" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <B>E</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >n los problemas 7 y 8, encuentre </FONT></div> <div> <IMG SRC="2e9e971f0.gif" border=0 width="407" height="31" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> para la función dada <I>f  </I>y simplifique lo más que pueda.</div> <div> <U><IMG SRC="30edbf210.gif" border=0 width="191" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#1_1_7" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>         <U><IMG SRC="2e9cd9210.gif" border=0 width="217" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U><A HREF="https://youtu.be/CFN3RRlSmYY?t=14m31s" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY?t=14m31s"><U><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></U></A></div> <div> <U><IMG SRC="2e9d291b0.gif" border=0 width="553" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U><A HREF="https://www.youtube.com/watch?v=CFN3RRlSmYY&feature=youtu.be&t=22m32s" TARGET="_top" TITLE="https://www.youtube.com/watch?v=CFN3RRlS"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="30f3d01d0.gif" border=0 width="464" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#1_1_10" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="4" COLOR="#006666" FACE="Georgia" ><B>E</B></FONT>n los problemas 11 a 26, encuentre el dominio de la función <I>f </I>dada.</div> <div> <FONT SIZE="4" COLOR="#330033" FACE="Georgia" ><IMG SRC="30f69b1d0.gif" border=0 width="155" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="#1_1_11" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>         <IMG SRC="30f8a31d0.gif" border=0 width="163" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#1_1_12" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>         <IMG SRC="30fb91330.gif" border=0 width="145" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#1_1_13" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><A HREF="https://youtu.be/CFN3RRlSmYY?t=23m30s" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY?t=23m30s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          </FONT><FONT SIZE="4" COLOR="#330033" FACE="Georgia" ><IMG SRC="310299330.gif" border=0 width="153" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="#1_1_14" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="31079f350.gif" border=0 width="159" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#1_1_15" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          </FONT><IMG SRC="310a8d300.gif" border=0 width="141" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#1_1_16" TITLE="1.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>          <IMG SRC="2ea0bc300.gif" border=0 width="188" height="48" 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/CFN3RRlSmYY?t=23m30s" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY?t=23m30s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>          <IMG SRC="2ea1a51f0.gif" border=0 width="165" height="31" 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/CFN3RRlSmYY?t=23m30s" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY?t=23m30s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="2ea295350.gif" border=0 width="149" 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/CFN3RRlSmYY?t=23m30s" TARGET="_top" TITLE="https://youtu.be/CFN3RRlSmYY?t=23m30s"><U></U></A></FONT></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>Solución de los Ejercicios 1.1</B></div> <div>  <A NAME="1_1_2_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="1" COLOR="#FFFFFF" FACE="Georgia" >1.1_2</FONT></div> <div> <IMG SRC="2fe6905d0.gif" border=0 width="400" height="93" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para hallar los valores funcionales respectivos, se sustituye la variable independiente <I>x</I> en la fórmula de la función dada en (<FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >1</FONT>), se efectúan las operaciones aritméticas indicadas, se reduce y se simplifica:</div> <div> <IMG SRC="2fe8aca10.gif" border=0 width="684" height="161" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_3</div> <div> <IMG SRC="2feb705c0.gif" border=0 width="368" height="92" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para hallar los valores funcionales respectivos, se sustituye la variable independiente <I>x</I> en la fórmula de la función dada en (<FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >1</FONT>), se efectúan las operaciones aritméticas indicadas, se reduce y se simplifica:</div> <div> <IMG SRC="2fec58870.gif" border=0 width="344" height="135" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_5</div> <div> <IMG SRC="2fef7c7d0.gif" border=0 width="380" height="125" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para hallar los valores funcionales respectivos, se sustituye la variable independiente <I>x</I> en la fórmula de la función dada en (<FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >1</FONT>), se efectúan las operaciones aritméticas indicadas, se reduce y se simplifica:</div> <div> <IMG SRC="2ff0c0e90.gif" border=0 width="448" height="233" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_6</div> <div> <IMG SRC="2ff294830.gif" border=0 width="404" height="131" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para hallar los valores funcionales respectivos, se sustituye la variable independiente <I>x</I> en la fórmula de la función dada en (<FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >1</FONT>), se efectúan las operaciones aritméticas indicadas, se reduce y se simplifica:</div> <div> <IMG SRC="2ff3cd3c0.gif" border=0 width="973" height="316" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_7</div> <div> <IMG SRC="30eae03b0.gif" border=0 width="992" height="315" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_10</div> <div> <IMG SRC="30f1d0340.gif" border=0 width="464" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Se debe resolver la siguiente ecuación:</div> <div> <IMG SRC="30f03f710.gif" border=0 width="575" height="113" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_11</div> <div> <IMG SRC="30f4d3340.gif" border=0 width="211" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Debido a que las raíces cuadradas de números negativos no son números reales, debemos resolver la siguiente inecuación:</div> <div> <IMG SRC="30f58c850.gif" border=0 width="396" height="133" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_12</div> <div> <IMG SRC="30f7d3340.gif" border=0 width="211" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Debido a que las raíces cuadradas de números negativos no existen en los reales, debemos resolver la siguiente inecuación:</div> <div> <IMG SRC="30f9e7690.gif" border=0 width="487" height="105" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_13</div> <div> <IMG SRC="30fad3440.gif" border=0 width="211" height="68" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Debido a que las raíces cuadradas de números negativos no existen en los reales y que la división por 0 no tiene sentido, la fórmula que involucra la variable independiente <I>x</I> debe cumplir las siguientes dos condiciones:</div> <div> <IMG SRC="30fce51d0.gif" border=0 width="229" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> que se resumen en esta única condición:</div> <div> <IMG SRC="30fd1f570.gif" border=0 width="287" height="87" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_14</div> <div> <IMG SRC="30fed3490.gif" border=0 width="211" height="73" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Debido a que las raíces cuadradas de números negativos no existen en los reales y que la división por 0 no tiene sentido, la fórmula que involucra la variable independiente <I>x</I> debe cumplir las siguientes dos condiciones:</div> <div> <IMG SRC="3103f91d0.gif" border=0 width="249" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> que se resumen en esta única condición:</div> <div> <IMG SRC="310437850.gif" border=0 width="311" height="133" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_15</div> <div> <IMG SRC="3106d34d0.gif" border=0 width="211" height="77" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para evitar la división por cero se debe satisfacer la siguiente condición:</div> <div> <IMG SRC="310840570.gif" border=0 width="576" height="87" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_1_16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1.1_16</div> <div> <IMG SRC="3109d3470.gif" border=0 width="211" height="71" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para evitar la división por cero se debe satisfacer la siguiente condición:</div> <div> <IMG SRC="310b60570.gif" border=0 width="608" height="87" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c128c8c0.gif" border=0 width="140" height="140" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <A HREF="http://twitter.com/Calculo21" TARGET="_top" TITLE="http://twitter.com/Calculo21"><U><IMG SRC="248328280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.facebook.com/juan.beltran.58511" TARGET="_top" TITLE="http://www.facebook.com/juan.beltran.585"><U><IMG SRC="248428280.gif" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" 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