A.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> <B>C á l c u l o  2 1</B><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="id488.htm" TARGET="_top" TITLE="Leithold EC7"><U>Leithold</U></A> </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> Ejercicios A.1</div> <div> <B>Números reales y desigualdades</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> <bR> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >1</FONT> y <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >2</FONT>, acomode los elementos del subconjunto dado de R en el mismo orden que sus puntos correspondientes de la recta numérica real de izquierda a derecha.</div> <div> <IMG SRC="27c3ab2b0.gif" border=0 width="427" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#BF0000" FACE="Bell MT" ><A HREF="#1" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <B><IMG SRC="27c216280.gif" border=0 width="534" height="40" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <IMG SRC="27c6ca270.gif" border=0 width="714" height="39" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#BF0000" FACE="Bell MT" ><A HREF="#3" TITLE="A.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="27caca380.gif" border=0 width="714" height="56" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#BF0000" FACE="Bell MT" ><A HREF="#6" TITLE="A.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="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >7</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >14</FONT>, haga lo siguiente: (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><I>i</I></FONT>) muestre el conjunto sobre la recta real; (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><I>ii</I></FONT>) represente el conjunto por medio de notación de intervalos; (<FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><I>iii</I></FONT>) describa el conjunto en palabras.</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="27cc53150.gif" border=0 width="339" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#9" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="27cf8b150.gif" border=0 width="395" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#12" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >15</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >18</FONT>, muestre el intervalo sobre la recta numérica real y utilice la notación de conjuntos y símbolos de desigualdad para denotar el intervalo indicado.</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="27d280150.gif" border=0 width="384" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#15" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="27d58f150.gif" border=0 width="399" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#18" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >21</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >24</FONT>, muestre los puntos que corresponden a <I>u</I> y <I>v</I> sobre la recta real y después calcule la distancia entre ellos.</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="27d8fd150.gif" border=0 width="509" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#21" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="27db7b250.gif" border=0 width="379" height="37" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#24" TITLE="A.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >25</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >30</FONT>, demuestre que las dos desigualdades son equivalentes.</div> <div> <IMG SRC="27ded9250.gif" border=0 width="217" height="37" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#BF0000" FACE="Bell MT" ><A HREF="#27" TITLE="A.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="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >33</FONT> y <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >34</FONT>, utilice la desigualdad del triángulo para demostrar la proposición indicada.</div> <div> <IMG SRC="27e17b250.gif" border=0 width="379" height="37" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#BF0000" FACE="Bell MT" ><A HREF="#33" TITLE="A.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="27e21b1a0.gif" border=0 width="539" height="26" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#36" TITLE="A.1"><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> Soluciones</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="27c4a8a50.gif" border=0 width="680" height="421" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="27c7ca270.gif" border=0 width="714" height="39" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27c8a7300.gif" border=0 width="423" height="48" ALT="MathType 6.0 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>6</div> <div> <IMG SRC="27c9ca380.gif" border=0 width="714" height="56" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27cbb3300.gif" border=0 width="435" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>9</div> <div> <IMG SRC="27cda8f20.gif" border=0 width="936" height="242" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="27ce8b2b0.gif" border=0 width="395" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27d08f620.gif" border=0 width="911" height="98" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="27d1802b0.gif" border=0 width="384" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27d361fd0.gif" border=0 width="865" height="253" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> <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="27d48f2b0.gif" border=0 width="399" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27d685f30.gif" border=0 width="901" height="243" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="27d7fd2b0.gif" border=0 width="509" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27d9dd440.gif" border=0 width="733" height="324" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="24"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>24</div> <div> <IMG SRC="27da7b3b0.gif" border=0 width="379" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="27dcb2c00.gif" border=0 width="690" height="192" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="27dd97650.gif" border=0 width="663" height="101" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="30"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>30</div> <div> <IMG SRC="27df9cd10.gif" border=0 width="668" height="209" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="33"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></div> <div> <IMG SRC="27e03f540.gif" border=0 width="575" height="340" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">33</div> <bR> <div>  <A NAME="36"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>36</div> <div> <IMG SRC="27e4ddf90.gif" border=0 width="733" 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