tag:blogger.com,1999:blog-34247280600589082372024-03-12T18:04:27.061-07:00MatematicasDavid Manuelhttp://www.blogger.com/profile/12554619794128383540noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3424728060058908237.post-31117337082188903512010-08-25T16:42:00.001-07:002010-08-25T16:42:33.910-07:00Pares ordenados<meta content="text/html; charset=utf-8" http-equiv="Content-Type"></meta><meta content="Word.Document" name="ProgId"></meta><meta content="Microsoft Word 12" name="Generator"></meta><meta content="Microsoft Word 12" name="Originator"></meta><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml" rel="File-List"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_editdata.mso" rel="Edit-Time-Data"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx" rel="themeData"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml" rel="colorSchemeMapping"></link> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><style>
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<h1 align="center" style="text-align: center;"><span style="color: black; font-size: 18pt; line-height: 115%;">Par ordenado</span><span style="color: black;"><o:p></o:p></span></h1><div style="text-align: justify;"><br />
</div><div style="text-align: justify;"><span style="color: black;">Un <b>par ordenado</b> es una <span style="color: black; text-decoration: none;">tupla</span> de dos elementos, tal que uno puede ser distinguido como el <i>primero</i> y el otro como el <i>segundo</i>. Un par ordenado con primer elemento <i>a</i> y con segundo elemento <i>b</i> es escrito usualmente como (<i>a</i>, <i>b</i>).</span></div><div style="text-align: justify;"><br />
<span style="color: black;">Dos pares ordenados cumplen:<o:p></o:p></span></div><br />
<div></div><div style="background: none repeat scroll 0% 0% white; text-align: justify;"><span style="color: black;">(<i>a</i>, <i>b</i>) = (<i>c</i>, <i>d</i>) si y sólo si <i>a = c</i> y <i>b = d</i><o:p></o:p></span></div><br />
<div style="text-align: justify;"></div><div style="text-align: justify;"><span style="color: black;">El <span style="color: black; text-decoration: none;">conjunto</span> de todos los pares ordenados en los cuales el primer elemento se toma de un conjunto <i>X</i> determinado y el segundo de un conjunto <i>Y</i> se llama <b><span style="color: black; text-decoration: none;">producto cartesiano</span></b> de <i>X</i> e <i>Y</i>, escrito .<o:p></o:p></span></div>David Manuelhttp://www.blogger.com/profile/12554619794128383540noreply@blogger.com1tag:blogger.com,1999:blog-3424728060058908237.post-60090112929032957272010-08-25T16:37:00.000-07:002010-08-25T16:37:25.025-07:00Como ubicar un punto en el plano cartesiano<div class="separator" style="clear: both; text-align: center;"></div><div style="margin-left: 1em; margin-right: 1em;"><meta content="text/html; charset=utf-8" http-equiv="Content-Type"></meta><meta content="Word.Document" name="ProgId"></meta><meta content="Microsoft Word 12" name="Generator"></meta><meta content="Microsoft Word 12" name="Originator"></meta><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml" rel="File-List"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx" rel="themeData"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml" rel="colorSchemeMapping"></link> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><style>
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</style> </m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div align="center" class="MsoNormal" style="line-height: normal; text-align: center;"><span style="color: black; font-size: 18pt;">Plano cartesiano<o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; text-align: justify;"><span style="color: black; font-size: 12pt;">El <b>plano cartesiano</b> es un <span style="color: black; text-decoration: none;">sistema de referencia</span> respecto ya sea a un solo eje (<span style="color: black; text-decoration: none;">línea recta</span>), respecto a dos ejes (un <span style="color: black; text-decoration: none;">plano</span>) o respecto a tres ejes (en el <span style="color: black; text-decoration: none;">espacio</span>), perpendiculares entre sí (plano y espacio), que se cortan en un punto llamado <i><span style="color: black; text-decoration: none;">origen de coordenadas</span></i>. En el plano, las coordenadas cartesianas (o rectangulares) <b>x</b> e <b>y</b> se denominan <b>abscisa</b> y <b>ordenada</b>, respectivamente.<o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; text-align: justify;"><span style="color: black; font-size: 12pt;">Los cuatro espacios que conforman el plano cartesiano se denominan “cuadrantes” y son los espacios en los que se ubican los puntos.<o:p></o:p></span></div><h3 align="center" style="line-height: normal; text-align: center;"><span class="mw-headline"><span style="color: black; font-size: 14pt;">Localización de un punto en el plano cartesiano</span></span><span style="color: black; font-size: 14pt;"><o:p></o:p></span></h3><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">En un plano traza dos rectas orientadas perpendiculares entre sí (ejes) —que por convenio se trazan de manera que una de ellas sea horizontal y la otra vertical—, y cada punto del plano queda unívocamente determinado por las distancias de dicho punto a cada uno de los ejes, siempre y cuando se dé también un criterio para determinar sobre qué <span style="color: black; text-decoration: none;">semiplano</span> determinado por cada una de las rectas hay que tomar esa distancia, criterio que viene dado por un signo. Ese par de números, las <span style="color: black; text-decoration: none;">coordenadas</span>, quedará representado por un par ordenado <span class="texhtml">(<i>x</i>,<i>y</i>)</span>, siendo <span class="texhtml"><i>x</i></span> la distancia a uno de los ejes (por convenio será la distancia al eje vertical) e <span class="texhtml"><i>y</i></span> la distancia al otro eje (al horizontal).<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">En la coordenada <span class="texhtml"><i>x</i></span>, el signo positivo (que suele omitirse) significa que la distancia se toma hacia la derecha del eje horizontal (<b>eje de las abscisas</b>), y el signo negativo (nunca se omite) indica que la distancia se toma hacia la izquierda. Para la coordenada <span class="texhtml"><i>y</i></span>, el signo positivo (también se suele omitir) indica que la distancia se toma hacia arriba del eje vertical (<b>eje de ordenadas</b>), tomándose hacia abajo si el signo es negativo (tampoco se omite nunca en este caso).<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">A la coordenada <span class="texhtml"><i>x</i></span> se la suele denominar <i>abscisa</i> del punto, mientras que a la <span class="texhtml"><i>y</i></span> se la denomina <i>ordenada</i> del punto.<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">Los puntos del eje de abscisas tienen por lo tanto ordenada igual a <span class="texhtml">0</span>, así que serán de la forma <span class="texhtml">(<i>x</i>,0)</span>, mientras que los del eje de ordenadas tendrán abscisa igual a <span class="texhtml">0</span>, por lo que serán de la forma <span class="texhtml">(0,<i>y</i>)</span>.<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">El punto donde ambos ejes se cruzan tendrá por lo tanto distancia <span class="texhtml">0</span> a cada uno de los ejes, luego su abscisa será <span class="texhtml">0</span> y su ordenada también será <span class="texhtml">0</span>. A este punto —el <span class="texhtml">(0,0)</span>— se le denomina <b>origen de coordenadas</b>.<o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; text-align: justify;"><br />
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David Manuelhttp://www.blogger.com/profile/12554619794128383540noreply@blogger.com0tag:blogger.com,1999:blog-3424728060058908237.post-50512374550380354702010-08-25T16:16:00.000-07:002010-08-25T16:16:38.568-07:00Geometria Analitica<meta content="text/html; charset=utf-8" http-equiv="Content-Type"></meta><meta content="Word.Document" name="ProgId"></meta><meta content="Microsoft Word 12" name="Generator"></meta><meta content="Microsoft Word 12" name="Originator"></meta><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml" rel="File-List"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx" rel="themeData"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml" rel="colorSchemeMapping"></link> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><style>
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<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">pitch</span>:variable;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">signature</span>:-1610611985 1107304683 0 0 159 0;}
@<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">face</span>
{<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Calibri</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">panose</span>-1:2 15 5 2 2 2 4 3 2 4;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">charset</span>:0;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">generic</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">swiss</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">pitch</span>:variable;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">signature</span>:-1610611985 1073750139 0 0 159 0;}
/* <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Style</span> <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Definitions</span> */
p.<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">MsoNormal</span>, <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">li</span>.<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">MsoNormal</span>, <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">div</span>.<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">MsoNormal</span>
{<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">unhide</span>:no;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">qformat</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">yes</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">parent</span>:"";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">top</span>:0cm;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">right</span>:0cm;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bottom</span>:10.0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">left</span>:0cm;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">line</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">height</span>:115%;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">pagination</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">widow</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">orphan</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">size</span>:11.0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:"<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Calibri</span>","<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">sans</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">serif</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">ascii</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Calibri</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">ascii</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">minor</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">latin</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Calibri</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">minor</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">latin</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">hansi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Calibri</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">hansi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">minor</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">latin</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:"Times <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">New</span> <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Roman</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">minor</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">language</span>:EN-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">US</span>;}
h1
{<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">priority</span>:9;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">unhide</span>:no;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">qformat</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">yes</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">link</span>:"Título 1 <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Car</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">next</span>:Normal;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">top</span>:24.0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">right</span>:0cm;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bottom</span>:0cm;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">left</span>:0cm;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">margin</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bottom</span>:.0001pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">line</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">height</span>:115%;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">pagination</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">widow</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">orphan</span> <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">lines</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">together</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">page</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">break</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">after</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">avoid</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">outline</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">level</span>:1;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">size</span>:14.0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:"<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Cambria</span>","<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">serif</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">ascii</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Cambria</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">ascii</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">major</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">latin</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:"Times <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">New</span> <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Roman</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">major</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">hansi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Cambria</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">hansi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">major</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">latin</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:"Times <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">New</span> <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Roman</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">theme</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">major</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>;
color:#365F91;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">themecolor</span>:accent1;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">themeshade</span>:191;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">kerning</span>:0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">fareast</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">language</span>:EN-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">US</span>;}
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">span</span>.Ttulo1Car
{<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">name</span>:"Título 1 <span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Car</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">priority</span>:9;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">unhide</span>:no;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">locked</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">yes</span>;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">style</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">link</span>:"Título 1";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">ansi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">size</span>:14.0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">bidi</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">size</span>:14.0pt;
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:"<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Cambria</span>","<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">serif</span>";
<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">mso</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">ascii</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">font</span>-<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">family</span>:<span style="background: none repeat scroll 0% 0% yellow;" class="goog-spellcheck-word">Cambria</span>;
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<h1 align="center" style="text-align: center;"><span style="color: black; font-family: "Calibri","sans-serif"; font-size: 18pt; line-height: 115%;">Geometría analítica<o:p></o:p></span></h1><div class="MsoNormal" style="line-height: normal;"><span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt;">Se conoce como <b>geometría analítica</b> al estudio de ciertos objetos geométricos mediante técnicas básicas del <span style="color: black; text-decoration: none;">análisis matemático</span> y del <span style="color: black; text-decoration: none;">álgebra</span><span style="color: black; text-decoration: none;">cartesiana</span> y concluye con la aparición de la <span style="color: black; text-decoration: none;">geometría diferencial</span> con <span style="color: black; text-decoration: none;">Carl Friedrich Gauss</span> y más tarde con el desarrollo de la <span style="color: black; text-decoration: none;">geometría algebraica</span>.<o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal;"><span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt;">Los dos problemas fundamentales de la geometría analítica son:<o:p></o:p></span></div><ol start="1" type="1"><li class="MsoNormal" style="color: black; line-height: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Dado el <span style="color: black; text-decoration: none;">lugar geométrico</span> en un sistema de <span style="color: black; text-decoration: none;">coordenadas</span>, obtener su <span style="color: black; text-decoration: none;">ecuación</span>.<o:p></o:p></span></li>
<li class="MsoNormal" style="color: black; line-height: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Dada la <span style="color: black; text-decoration: none;">ecuación</span> en un sistema de coordenadas, determinar la gráfica o <span style="color: black; text-decoration: none;">lugar geométrico</span> de los puntos que verifican # dicha <span style="color: black; text-decoration: none;">ecuación</span>.<o:p></o:p></span></li>
</ol><div class="MsoNormal" style="line-height: normal;"><span style="color: black; font-family: "Times New Roman","serif"; font-size: 12pt;">Lo novedoso de la geometría analítica es que permite representar figuras geométricas mediante fórmulas del tipo <i>f</i>(<i>x</i>,<i>y</i>) = 0, donde <i>f</i> representa una <span style="color: black; text-decoration: none;">función</span> u otro tipo de expresión matemática. En particular, las <span style="color: black; text-decoration: none;">rectas</span> pueden expresarse como <span style="color: black; text-decoration: none;">ecuaciones</span> polinómicas de grado 1 (por ejemplo, 2<i>x</i> + 6<i>y</i> = 0) y las <span style="color: black; text-decoration: none;">circunferencias</span> y el resto de <span style="color: black; text-decoration: none;">cónicas</span> como ecuaciones polinómicas de grado 2 (la circunferencia <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 4, la hipérbola <i>xy</i> = 1)<o:p></o:p></span></div>David Manuelhttp://www.blogger.com/profile/12554619794128383540noreply@blogger.com0tag:blogger.com,1999:blog-3424728060058908237.post-79337018864962176652010-08-25T16:05:00.000-07:002010-08-25T16:05:35.328-07:00Historia de la geometria analitica<meta content="text/html; charset=utf-8" http-equiv="Content-Type"></meta><meta content="Word.Document" name="ProgId"></meta><meta content="Microsoft Word 12" name="Generator"></meta><meta content="Microsoft Word 12" name="Originator"></meta><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_filelist.xml" rel="File-List"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_themedata.thmx" rel="themeData"></link><link href="file:///C:%5CDOCUME%7E1%5CADMINI%7E1%5CCONFIG%7E1%5CTemp%5Cmsohtmlclip1%5C01%5Cclip_colorschememapping.xml" rel="colorSchemeMapping"></link> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><style>
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<div align="center" class="MsoNormal" style="line-height: normal; text-align: center;"><b><span style="font-family: "Times New Roman","serif"; font-size: 18pt;">Historia de la geometría analítica<o:p></o:p></span></b></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">Existe una cierta controversia sobre la verdadera paternidad de este método. Lo único cierto es que se publica por primera vez como "Geometría analítica", apéndice al <i><span style="color: black; text-decoration: none;">Discurso del método</span></i>, de <span style="color: black; text-decoration: none;">Descartes</span>, si bien se sabe que <span style="color: black; text-decoration: none;">Pierre de Fermat</span> conocía y utilizaba el método antes de su publicación por Descartes. Aunque <span style="color: black; text-decoration: none;">Omar Khayyam</span> ya en el siglo XI utilizara un método muy parecido para determinar ciertas intersecciones entre curvas, es imposible que alguno de los citados matemáticos franceses tuviera acceso a su obra.<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">El nombre de <i>geometría analítica</i> corrió parejo al de <i>geometría cartesiana</i>, y ambos son indistinguibles. Hoy en día, paradójicamente, se prefiere denominar <i>geometría cartesiana</i> al apéndice del <i>Discurso del método</i>, mientras que se entiende que <i>geometría analítica</i> comprende no sólo a la geometría cartesiana (en el sentido que acabamos de citar, es decir, al texto apéndice del <i>Discurso del método</i>), sino también todo el desarrollo posterior de la geometría que se base en la construcción de ejes coordenados y la descripción de las figuras mediante funciones —algebraicas o no— hasta la aparición de la <span style="color: black; text-decoration: none;">geometría diferencial</span> de <span style="color: black; text-decoration: none;">Gauss</span> (decimos "paradójicamente" porque se usa precisamente el término "geometría cartesiana" para aquello que el propio Descartes bautizó como "geometría analítica"). El problema es que durante ese periodo no existe una diferencia clara entre geometría analítica y análisis matemático —esta falta de diferencia se debe precisamente a la identificación hecha en la época entre los conceptos de <span style="color: black; text-decoration: none;">función</span> y <span style="color: black; text-decoration: none;">curva</span>—, por lo que resulta a veces muy difícil intentar determinar si el estudio que se está realizando corresponde a una u otra rama.<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">La geometría diferencial de curvas sí que permite un estudio mediante un sistema de coordenadas, ya sea en el plano o en el espacio tridimensional. Pero en el estudio de las <span style="color: black; text-decoration: none;">superficies</span>, en general, aparecen serios obstáculos. Gauss salva dichos obstáculos creando la geometría diferencial, y marcando con ello el fin de la geometría analítica como disciplina. Es con el desarrollo de la geometría algebraica cuando se puede certificar totalmente la superación de la geometría analítica.<o:p></o:p></span></div><div style="text-align: justify;"><span style="color: black; font-family: "Calibri","sans-serif";">Es de puntualizar que la denominación de <i>analítica</i> dada a esta forma de estudiar la geometría provocó que la anterior manera de estudiarla (sea es decir, la manera axiomático-deductiva, sin la intervención de coordenadas)<o:p></o:p></span></div><div class="MsoNormal" style="text-align: justify;"><br />
</div>David Manuelhttp://www.blogger.com/profile/12554619794128383540noreply@blogger.com0