Propiedades de la diferenciación de funciones

<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> <bR> <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> <div> <A HREF="id441_m.htm#teorema_d9" TARGET="TRLX_Middle" TITLE="Propiedades de la diferenciación de func"><U><B><IMG BORDER="0" SRC="Wing0a73007700c8ffffff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema D9</B></U></A></div> <div> <A HREF="id441_m.htm#teorema_d10" TARGET="TRLX_Middle" TITLE="Propiedades de la diferenciación de func"><U><B><IMG BORDER="0" SRC="Wing0a73007700c8ffffff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema D10</B></U></A></div> <div> <A HREF="id441_m.htm#teorema_d11" TARGET="TRLX_Middle" TITLE="Propiedades de la diferenciación de func"><U><B><IMG BORDER="0" SRC="Wing0a73007700c8ffffff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema D11</B></U></A></div> <div> <A HREF="id441_m.htm#teorema_d12" TARGET="TRLX_Middle" TITLE="Propiedades de la diferenciación de func"><U><B><IMG BORDER="0" SRC="Wing0a73007700c8ffffff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema D12</B></U></A></div> <div> <A HREF="id441_m.htm#teorema_d13" TARGET="TRLX_Middle" TITLE="Propiedades de la diferenciación de func"><U><B><IMG BORDER="0" SRC="Wing0a73007700c8ffffff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema D13</B></U></A></div> <div> <A HREF="id441_m.htm#teorema_d14" TARGET="TRLX_Middle" TITLE="Propiedades de la diferenciación de func"><U><B><IMG BORDER="0" SRC="Wing0a73007700c8ffffff00.gif" HEIGHT="13" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema D14</B></U></A></div> <bR> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Propiedades de la diferenciación de funciones</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>  H</B></FONT>allar la derivada de una función aplicando la <I>definición de derivada</I> es un proceso largo y la mayor de las veces bastante tedioso. Afortunadamente existen varias propiedades en la derivación de funciones que los matemáticos han descubierto y establecido como teoremas. Algunos de estos teoremas son generales, aplicables a cualquier función, y otros sólo se aplican a funciones particulares. A continuación se enuncian algunos de los teoremas más importantes (se nombran enumerándolos consecutivamente para facilitar una futura referencia a ellos):</div> <div> Nota:<FONT SIZE="2" COLOR="#000000" FACE="Times New Roman" > se supone, obviamente, que las funciones a las que hacen referencia los teoremas son diferenciables, esto es, que tienen derivada.</FONT></div> <bR> <div>  <A NAME="teorema_d1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D1:</B></FONT></div> <div> <IMG SRC="1e1b002d0.gif" border=0 width="256" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>la </I></FONT><I>derivada de una constante por una función </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>es igual a la constante multiplicada por la derivada de la función</I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div>  <A NAME="teorema_d2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D2:</B></FONT></div> <div> <B><IMG SRC="1e1c31300.gif" border=0 width="305" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>la </I></FONT><I>derivada de la suma de un número finito, n, de funciones </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>(términos), positivas o negativas, es igual a la suma de las derivadas de cada función y con su respectivo signo</I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div>  <A NAME="teorema_d3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D3:</B></FONT></div> <div> <B><IMG SRC="1e1d182d0.gif" border=0 width="280" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>la </I></FONT><I>derivada del producto de dos funciones </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>es igual a la primera función por la derivada de la segunda más la segunda función por la derivada de la primera</I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div>  <A NAME="teorema_d4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D4:</B></FONT></div> <div> <B><IMG SRC="1e1e1b5d0.gif" border=0 width="283" height="93" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>la </I></FONT><I>derivada del cociente de dos funciones </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>es igual a una fracción cuyo denominador es el cuadrado de la función del dividendo y cuyo numerador es la diferencia entre la función del dividendo por la derivada de la función del divisor y la función del divisor por la derivda de la función del dividendo</I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div>  <A NAME="teorema_d5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D5:</B></FONT></div> <div> <B><IMG SRC="1e1f88330.gif" border=0 width="136" height="51" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>para hallar la </I></FONT><I>derivada de la función potencia </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>se multiplica la función por un coeficiente igual al exponente y el exponente se disminuye en la unidad </I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div>  <A NAME="teorema_d6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D6:</B></FONT></div> <div> <B><IMG SRC="1e20d42d0.gif" border=0 width="212" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>la </I></FONT><I>derivada de la función constante </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>es cero </I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div>  <A NAME="teorema_d7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D7:</B></FONT></div> <div> <B><IMG SRC="1e215d2d0.gif" border=0 width="93" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >En palabras: </FONT><B> </B>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>la </I></FONT><I>derivada de la función identidad </I><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>es uno </I></FONT>"<FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >.</FONT></div> <bR> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="604" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFFFCC HEIGHT= 20 WIDTH="594"><div> <B>Derivadas de las funciones trigonométricas</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 71 WIDTH="196"><div>  <A NAME="teorema_d8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D8:</B></FONT></div> <div> <B><IMG SRC="1e22792d0.gif" border=0 width="121" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> </tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="teorema_d9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D9:</B></FONT></div> <div> <B><IMG SRC="1e23842d0.gif" border=0 width="132" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="196"><div>  <A NAME="teorema_d10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D10:</B></FONT></div> <div> <IMG SRC="1e2480300.gif" border=0 width="128" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 71 ><NOBR><div>  <A NAME="teorema_d11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D11:</B></FONT></div> <div> <IMG SRC="1e258b300.gif" border=0 width="139" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="teorema_d12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D12:</B></FONT></div> <div> <B><IMG SRC="1e26992d0.gif" border=0 width="153" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="teorema_d13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D13:</B></FONT></div> <div> <B><IMG SRC="1e27a42d0.gif" border=0 width="164" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> </NOBR></tD> </tR> </TABLE></div> <bR> <div>  <A NAME="teorema_d14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B>Teorema D14:</B></FONT></div> <div> <B><IMG SRC="1e282b990.gif" border=0 width="299" height="153" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="612" BORDERCOLORLIGHT="#006600" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFCC99 HEIGHT= 39 ><NOBR><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="#CC0000" FACE="Times New Roman" ><B>Ejercicios resueltos</B></FONT></div> <div> <FONT SIZE="3" COLOR="#CC0000" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0000" FACE="Times New Roman" >1</FONT> a <FONT SIZE="3" COLOR="#CC0000" FACE="Times New Roman" >12 </FONT>halle la derivada de la función dada aplicando los teoremas <FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><B>TD1</B></FONT> a <FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><B>TD14</B></FONT>:</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 WIDTH="198"><div> <A HREF="#1" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e296d150.gif" border=0 width="109" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#2" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e2a98180.gif" border=0 width="152" 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="Propiedades de la diferenciación de func"><U><IMG SRC="1e2b97290.gif" border=0 width="151" height="41" 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="#4" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e2cb0180.gif" border=0 width="176" 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="#5" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e2d691c0.gif" border=0 width="105" height="28" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#6" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e2e65290.gif" border=0 width="101" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 ><NOBR><div> <A HREF="#7" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e2f9f290.gif" border=0 width="159" height="41" 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="Propiedades de la diferenciación de func"><U><IMG SRC="1e3070150.gif" border=0 width="112" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#9" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e31a7150.gif" border=0 width="167" height="21" 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="#10" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e3297150.gif" border=0 width="151" 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="#11" TITLE="Propiedades de la diferenciación de func"><U><IMG SRC="1e3388290.gif" border=0 width="136" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="1e3485290.gif" border=0 width="133" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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><IMG SRC="1e356d150.gif" border=0 width="109" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1e3638790.gif" border=0 width="312" height="121" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1e3798180.gif" border=0 width="152" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1e387b7f0.gif" border=0 width="379" height="127" ALT="MathType 5.0 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><IMG SRC="1e3997290.gif" border=0 width="151" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1e3a119f0.gif" border=0 width="529" height="159" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e3bb0180.gif" border=0 width="176" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e3c3f8d0.gif" border=0 width="575" height="141" 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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e3d691c0.gif" border=0 width="105" height="28" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e3e808b0.gif" border=0 width="384" height="139" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e3f65290.gif" border=0 width="101" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e40c9d50.gif" border=0 width="457" height="213" 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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e419f290.gif" border=0 width="159" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e42519d0.gif" border=0 width="593" height="413" 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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e4370150.gif" border=0 width="112" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e441b480.gif" border=0 width="283" height="72" 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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e45a7150.gif" border=0 width="167" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e46ff790.gif" border=0 width="511" height="121" 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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e4797150.gif" border=0 width="151" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e48686b0.gif" border=0 width="616" height="107" 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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="1e4988290.gif" border=0 width="136" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="1e4a54a80.gif" border=0 width="596" height="168" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <bR> <bR> <bR> <bR> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=24 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