tag:blogger.com,1999:blog-26198813172925764752024-03-13T13:05:59.657-05:00Aportes MatemáticosCompartiendo mis conocimientos en Física y Matemáticas.salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-2619881317292576475.post-4382481169115471692012-04-20T16:42:00.000-05:002017-05-16T06:07:47.273-05:00Rozamiento Mecánico (Video 1h:43min)<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-rWT3-e-k5T8/T5GAAsn2wMI/AAAAAAAABWg/6jaP35k3DCw/s1600/snapcap998.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="154" src="https://4.bp.blogspot.com/-rWT3-e-k5T8/T5GAAsn2wMI/AAAAAAAABWg/6jaP35k3DCw/s200/snapcap998.jpg" width="200" /></a></div>
<span style="text-align: justify;">Cuando un cuerpo se pone en contacto con otro y se desliza o intenta resbalar respecto a él, se generan fuerzas de oposición a estos movimientos, a los que llamamos fuerzas de fricción o de rozamiento. La naturaleza de estas fuerzas es electromagnética y se generan por el hecho de que las superficies en contacto tienen irregularidades (deformaciones), las mismas que al ponerse en contacto y pretender deslizar producen fuerzas predominantemente repulsivas. La fuerza de rozamiento es una componente de la resultante de estas fuerzas, su línea de acción es paralela a las superficies, y su sentido es opuesto al del movimiento relativo de los cuerpos. </span><br />
<div style="text-align: justify;">
</div>
<div style="text-align: justify;">
Debido a su compleja naturaleza, el cálculo de la fuerza de rozamiento es hasta cierto punto empírico. Sin embargo, cuando los cuerpos son sólidos, las superficies en contacto son planas y secas, se puede comprobar que estas fuerzas dependen básicamente de la normal (N), y son aproximadamente independientes del área de contacto y de la velocidad relativa del deslizamiento</div>
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<br /></div>
<div style="text-align: justify;">
Esta serie de vídeos presenta 8 ejercicios tipo del nivel secundario-preuniversitario de la asignatura de física elemental sección dinámica y estática, documentación disponible más abajo.</div>
<br />
<div style="text-align: center;">
<iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/videoseries?list=PL1127B982E654248C" width="560"></iframe></div>
<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com5tag:blogger.com,1999:blog-2619881317292576475.post-24773689270285781062012-04-17T20:32:00.002-05:002017-05-16T06:08:16.672-05:00Análisis Dimensional (Video: 42 minutos)<a href="http://3.bp.blogspot.com/-znHFQhEAqCw/T44amUxLujI/AAAAAAAABWY/ThV6eb1lTSc/s1600/snapcap993.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://3.bp.blogspot.com/-znHFQhEAqCw/T44amUxLujI/AAAAAAAABWY/ThV6eb1lTSc/s320/snapcap993.jpg" width="230" /></a><span style="font-family: "arial" , "helvetica" , sans-serif;">Las fórmulas dimensionales de las magnitudes físicas fundamentales (L,M,T,..etc.) permiten mostrar su relación con sus magnitudes derivadas mediante operaciones de multiplicación, potenciacion y radicación. Estas operaciones matemáticas junto con el operador de dimensión [ ] forman propiedades muy similares a las de las leyes de los exponentes. Este tema es un capítulo previo al de vectores, y es un complemento de la notación de unidades y notación científica. Aquí vemos un par de ejercicios tipo para la práctica usando el principio de homogeneidad.</span><br />
<div align="center">
<iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/videoseries?list=PL09BFF0AF0D9A17D7" width="560"></iframe> </div>
<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com0tag:blogger.com,1999:blog-2619881317292576475.post-77086949939436902992011-11-14T07:58:00.002-05:002011-11-14T13:39:04.600-05:00Analizando más Inferencias Lógicas<a href="http://es.scribd.com/doc/72676431/Demostrando-y-Validando-Mas-Inferencias" style="display: block; font-family: Helvetica, Arial, sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin-bottom: 6px; margin-left: auto; margin-right: auto; margin-top: 12px; text-align: center; text-decoration: underline;" title="View Demostrando y Validando Mas Inferencias on Scribd">Demostrando y Validando Mas Inferencias</a> <br />
<div style="text-align: center;"><object data="http://d1.scribdassets.com/ScribdViewer.swf" height="600" id="doc_45985" name="doc_45985" style="outline: none;" type="application/x-shockwave-flash" width="97%"> <param name="movie" value="http://d1.scribdassets.com/ScribdViewer.swf"><param name="wmode" value="opaque"><param name="bgcolor" value="#ffffff"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><param name="FlashVars" value="document_id=72676431&access_key=key-15715ixortxx7ix9e00d&page=1&viewMode=list"><embed id="doc_45985" name="doc_45985" src="http://d1.scribdassets.com/ScribdViewer.swf?document_id=72676431&access_key=key-15715ixortxx7ix9e00d&page=1&viewMode=list" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="600" width="97%" wmode="opaque" bgcolor="#ffffff"></embed> </object></div><div style="text-align: center;"><div style="text-align: -webkit-auto;"><br />
</div></div><div style="text-align: center;"><span class="Apple-style-span" style="color: blue;"><b>Video Explicativo</b></span></div><div class="separator" style="clear: both; text-align: center;"><br />
<object width="320" height="266" class="BLOGGER-youtube-video" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,40,0" data-thumbnail-src="http://i.ytimg.com/vi/WNwfin_aSBc/0.jpg"><param name="movie" value="http://www.youtube.com/v/WNwfin_aSBc?version=3&f=user_uploads&c=google-webdrive-0&app=youtube_gdata" /><param name="bgcolor" value="#FFFFFF" /><embed width="320" height="266" src="http://www.youtube.com/v/WNwfin_aSBc?version=3&f=user_uploads&c=google-webdrive-0&app=youtube_gdata" type="application/x-shockwave-flash"></embed></object></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/--l7QdT3CwGw/TsER2WYCTLI/AAAAAAAABQ4/MCuhT2Ep0LI/s1600/Hypsnap011.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/--l7QdT3CwGw/TsER2WYCTLI/AAAAAAAABQ4/MCuhT2Ep0LI/s1600/Hypsnap011.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com1tag:blogger.com,1999:blog-2619881317292576475.post-83057517162410176122011-05-14T01:07:00.009-05:002011-08-17T06:08:19.169-05:00Números Reales - Ecuaciones Irracionales o ecuaciones con radicales (√a = b)Esta vez observamos como resolver una ecuación sobre los números reales que contiene un término irracional (radical irreductible), esto implica el conocimiento de las inecuaciones y el de las ecuaciones cuadráticas, así como la formula del cuadrado de un trinomio en los productos notables. Más adelante colgaré las propiedades usadas en este problema.<br />
<br />
Detalles del problema.<br />
Institución: USAT. Chiclayo - Perú<br />
Especialidad: Arquitectura.<br />
Asignatura: Matemática para Ingenieros II<br />
Tipo: Pregunta de Práctica calificada<br />
<br />
<span class="Apple-style-span" style="color: blue;"><b>Resolver la ecuación</b> \[\boldsymbol{{x}^{2}+6\,x-24+2\,\sqrt {{x}^{2}+6\,x}=0}\]</span><span class="Apple-style-span" style="color: #990000;"><b><i><u>Solución</u>.</i></b></span><br />
\[{x}^{2}+6\,x-24+2\,\sqrt {{x}^{2}+6\,x}=0\] Para que exista \(\sqrt {x^2 + 6x}\) , la parte subradical tendrá que ser mayor o igual que cero \[0\leq {x}^{2}+6\,x\] factorizando y hallando puntos críticos:<br />
\[0\leq x \left( x+6 \right)\] El conjunto universo de la ecuación es: \[U=\left\langle { - \infty ,-6} \right\rangle \cup \left\langle {0,\infty } \right\rangle \] Despejando el término radical: \[2\,\sqrt {{x}^{2}+6\,x}=-{x}^{2}-6\,x+24\] Elevando al cuadrado, se cancela el radical en el lado izquierdo: \[4\,{x}^{2}+24\,x= \left( -{x}^{2}-6\,x+24 \right) ^{2}\] y se desarrolla el trinomio en el lado derecho: \[4\,{x}^{2}+24\,x={x}^{4}+12\,{x}^{3}-12\,{x}^{2}-288\,x+576\] Si pasamos el lado derecho al lado izquierdo y multiplicamos por -1 se tiene, ordenando el polinomio: \[{x}^{4}+12\,{x}^{3}-16\,{x}^{2}-312\,x+576=0\] Si factorizamos el polinomio del lado izquierdo por el método Paolo Ruffini, se tendrá:<br />
\[\left( x-2 \right) \left( x+8 \right) \left( {x}^{2}+6\,x-36 \right) =0\] el lector puede comprobarlo en lápiz y papel que el polinomio se puede factorizar de ese modo por dicho método.<br />
<br />
Igualando a cero cada factor vemos que: \[x = 8\,\quad \vee \,\quad\,x = - 2\,\quad \vee \,\quad x = - 3 + \sqrt 5 \,\quad \vee \,\quad x = - 3 - \sqrt 5 \] pero el único valor de \(x\) que están el universo \(U=\left\langle { - \infty ,-6} \right\rangle \cup \left\langle {0,\infty } \right\rangle \) es \[x = 8\] luego el conjunto solución es: \[CS = \left\{ {8} \right\}\]En maple puede comprobarse graficando la función del lado izquierdo de la ecuación. Las raíces de dicha ecuación señalarán la ubicación de los puntos de corte de <b>f</b> con el eje <b style="font-style: italic;">x </b>, la instrucción es:<br />
<b><span class="Apple-style-span" style="color: #cc0000; font-family: 'Courier New', Courier, monospace;"><b><span class="Apple-style-span" style="color: #cc0000; font-family: 'Courier New', Courier, monospace;"><b><span class="Apple-style-span" style="color: #cc0000; font-family: 'Courier New', Courier, monospace;">> </span></b></span></b></span></b><b><span class="Apple-style-span" style="color: #cc0000; font-family: 'Courier New', Courier, monospace;">a1:=x^2+6*x-24+2*sqrt(x^2+6*x):</span></b><br />
<b><span class="Apple-style-span" style="color: #cc0000; font-family: 'Courier New', Courier, monospace;"> > plot(a1,x=-14..10,y=-28..100, thickness=2,color=magenta);</span></b><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-Cr6GpJ9Ct0c/Tc4apMHbPLI/AAAAAAAABOU/klRG0rHUpYA/s1600/snap680.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="198" src="http://2.bp.blogspot.com/-Cr6GpJ9Ct0c/Tc4apMHbPLI/AAAAAAAABOU/klRG0rHUpYA/s200/snap680.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Gráfica de las raíces de <br />
\(y={x}^{2}+6\,x-24+2\,\sqrt {{x}^{2}+6\,x}\)</span></td></tr>
</tbody></table><b><span class="Apple-style-span" style="color: blue;">Bibliografía:</span></b><br />
Espinoza Ramos E, <i>Matemática Básica</i>. Editorial <i>Serv. Graf. J.J</i>. Lima - Perú. 2002.<br />
Figueroa García R, <i>Matemática Basica</i>. Editorial <i>América.</i> Lima - Perú. 1992.<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com2tag:blogger.com,1999:blog-2619881317292576475.post-34098777007657453752011-05-13T19:43:00.001-05:002011-05-13T19:45:41.845-05:00Ecuaciones cuadráticas de 2 variables - Graficando un Hiperbolide<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-gC6_JtHS7Bs/TcwBx2hGHjI/AAAAAAAABOQ/xjlBSGIX7ys/s1600/snap677.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"><img border="0" height="146" src="http://2.bp.blogspot.com/-gC6_JtHS7Bs/TcwBx2hGHjI/AAAAAAAABOQ/xjlBSGIX7ys/s200/snap677.jpg" width="200" /></a></div>Ahora enfocamos nuestra atención al cálculo multivariable. El criterio para graficar ecuaciones cuadráticas es esencial para el cálculo de integrales dobles y triples a la hora de determinar los dominios de integración y el integrando. Damos aquí una de las superficies obtenidas por una ecuación cuadrática. El Hiperboloide de dos hojas.<br />
<br />
<b>Datos del ejercicio</b><br />
Area: <span class="Apple-style-span" style="color: #0c343d;">Cálculo</span><br />
Especialidad: <span class="Apple-style-span" style="color: #0c343d;">Ingeniería Civil</span><br />
Asignatura: <span class="Apple-style-span" style="color: #0c343d;">Matemática para Ingenieros II</span><br />
Tipo: <span class="Apple-style-span" style="color: #073763;">Practica Calificada</span><br />
Institución: <span class="Apple-style-span" style="color: #7f6000;">USAT - Chiclayo Perú</span><br />
<br />
<span class="Apple-style-span" style="color: blue;">Graficar la superficie: \({x}^{2}-{y}^{2}+3\,{z}^{2}-2\,y=0\), indicando sus</span><br />
<span class="Apple-style-span" style="color: blue;">secciones transversales solo para dos valores de \(k\) en cada plano coordenado, y la intersección con los ejes coordenados.</span><br />
<br />
<b><span class="Apple-style-span" style="color: #660000;"><i><u>Solución</u></i>.</span></b><br />
\[{x}^{2}-{y}^{2}+3\,{z}^{2}-2\,y=0\] Agrupando la variable \(y\) y completando sus cuadrados:<br />
\[{x}^{2}- \left( y+1 \right) ^{2}+3\,{z}^{2}+1=0\] entonces, despejando la unidad: \[-{x}^{2}+\left( y+1 \right) ^{2}-3\,{z}^{2}=1\] esto puede expresarse: \[ - {x^2} + {\left( {y + 1} \right)^2} - \frac{{{\mkern 1mu} {z^2}}}{{\frac{1}{3}}} = 1\] \[ - {x^2} + {\left( {y + 1} \right)^2} - \frac{{{\mkern 1mu} z^2}}{{{{\left( {\sqrt {\frac{1}{3}} } \right)}^2}}} = 1\quad\ldots\,(1)\]Esta ecuación tiene corresponde a la gráfica de un hiperboloide de dos hojas (por tener 2 términos negativos) que se extiende a lo largo del eje de la variable corrspondiente a \((y+1)^2\), por ser éste un término positivo, luego el hiperboloide se extiende a lo largo de y. Además como ésta ecuación puede escribirse:<br />
\[ - {(x - 0)^2} + {\left( {y + 1} \right)^2} - \frac{{{\mkern 1mu} {{(z - 0)}^2}}}{{{{\left( {\sqrt {\frac{1}{3}} } \right)}^2}}} = 1\] entonces el centro del hiperboloide es \((0,-1,0)\)<br />
Entonces, el gráfico solicitado es:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-3Ai57gBy_ZE/Tcv5gWrQXLI/AAAAAAAABOA/lYtydcb_glc/s1600/hiperboloide+de+2+hojas..png" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="258" src="http://3.bp.blogspot.com/-3Ai57gBy_ZE/Tcv5gWrQXLI/AAAAAAAABOA/lYtydcb_glc/s320/hiperboloide+de+2+hojas..png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Gráfica del hiperboloide <br />
\(\left( y+1 \right) ^{2}-{x}^{2}-3\,{z}^{2}=1\)</span></td></tr>
</tbody></table>Graficando ahora las <span class="Apple-style-span" style="color: #660000;">secciones transversales</span><br />
<br />
<b>1)</b> Plano \(xy\):<br />
valor: \(k=0\) \(\to\) \(z=0\), \[\left( y+1 \right) ^{2}-{x}^{2}-3\,{z}^{2}=1\] \[\left( y+1 \right) ^{2}-{x}^{2}=1\] hipérbola en el plano \(xy\)<br />
<br />
valor: \(k=-1\) \(\to\) \(z=-1\), \[\left( y+1 \right) ^{2}-{x}^{2}-3=1\] \[\left( y+1 \right) ^{2}-{x}^{2}=4\] hipérbola en el plano \(xy\)<br />
<br />
Graficando:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-L0O3YqtIt2s/Tcv92yPSLjI/AAAAAAAABOE/d4IuvVQnnZ4/s1600/snap674.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="200" src="http://2.bp.blogspot.com/-L0O3YqtIt2s/Tcv92yPSLjI/AAAAAAAABOE/d4IuvVQnnZ4/s200/snap674.jpg" width="188" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Sección transversal del <br />
hiperboloide de dos hojas<br />
para \(x=0\,,\,x=-1.\)<br />
en el plano \(xy\)</span></td></tr>
</tbody></table><b>2)</b> Plano \(xz\):<br />
valor: \(k=0\) \(\to\) \(y=0\), \[\left( y+1 \right) ^{2}-{x}^{2}-3\,{z}^{2}=1\] \[1-{x}^{2}-3\,{z}^{2}=1\]<br />
\[{x}^{2}+3\,{z}^{2}=0\] elipse degradada al punto \((0,0)\), en el plano \(xz\)<br />
<br />
valor: \(k=-3\) \(\to\) \(y=-3\), \[4-{x}^{2}-3\,{z}^{2}=1\] \[{x}^{2}+3\,{z}^{2}=3\] elipse en el plano \(xy\)<br />
<br />
valor: \(k=-4\) \(\to\) \(y=-4\), \[9-{x}^{2}-3\,{z}^{2}=1\] \[{x}^{2}+3\,{z}^{2}=8\] elipse en el plano \(xz\). Vemos que el hiperboloide de dos hojas es elíptico.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-aW-J26FtcCM/Tcv-ugK7eBI/AAAAAAAABOI/bG60TI8PCVk/s1600/snap675.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="106" src="http://1.bp.blogspot.com/-aW-J26FtcCM/Tcv-ugK7eBI/AAAAAAAABOI/bG60TI8PCVk/s200/snap675.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Sección transversal del<br />
hiperboloide elíptico de dos hojas<br />
para \(y=0\,,\,y=-3\,,\,y=-4.\)<br />
en el plano \(xz\)</span></td></tr>
</tbody></table><b>3)</b> Plano \(yz\):<br />
valor: \(k=0\) \(\to\) \(x=0\), \[\left( y+1 \right) ^{2}-{x}^{2}-3\,{z}^{2}=1\] \[\left( y+1 \right) ^{2}-3\,{z}^{2}=1\] hipérbola en el plano \(yz\)<br />
<br />
valor: \(k=1\) \(\to\) \(x=1\), \[\left( y+1 \right) ^{2}-1-3\,{z}^{2}=1\] \[\left( y+1 \right) ^{2}-3\,{z}^{2}=2\] hipérbola en el plano \(yz.\)<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-G-_W38x3Z8Q/Tcv_oJdiimI/AAAAAAAABOM/rDipmUdLCTc/s1600/snap676.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="142" src="http://4.bp.blogspot.com/-G-_W38x3Z8Q/Tcv_oJdiimI/AAAAAAAABOM/rDipmUdLCTc/s200/snap676.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Sección transversal del<br />
hiperboloide elíptico para \(x=0\,,\,x=1.\)<br />
en el plano \(yz\)</span></td></tr>
</tbody></table><b><span class="Apple-style-span" style="color: #660000; font-family: Times, 'Times New Roman', serif;">Comprobación</span></b><br />
En una hoja de cálculo Maple podemos escribir la orden<br />
<span class="Apple-style-span" style="color: red; font-family: 'Courier New', Courier, monospace;"><b>> with(plots):</b></span><br />
<span class="Apple-style-span" style="color: red; font-family: 'Courier New', Courier, monospace;"><b>> implicitplot3d((y+1)^2-x^2-3*z^2 = 1,x=-7..7,y=-6..6,z=-5..5, numpoints=10000);</b></span><br />
y así coprobar la validés del resultado obtenido<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-gC6_JtHS7Bs/TcwBx2hGHjI/AAAAAAAABOQ/xjlBSGIX7ys/s1600/snap677.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="146" src="http://2.bp.blogspot.com/-gC6_JtHS7Bs/TcwBx2hGHjI/AAAAAAAABOQ/xjlBSGIX7ys/s200/snap677.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Graficando el hiperboloide de 2 hojas<br />
\(\left( y+1 \right) ^{2}-{x}^{2}-3\,{z}^{2}=1\)<br />
en maple con el comando <i>implicitplo3d</i><br />
de la librería <i>plots</i></span></td></tr>
</tbody></table><b><span class="Apple-style-span" style="color: #660000;">Bibliografía</span></b><br />
1. Stewart, James. 2008. <a href="http://bit.ly/mQW8Kw" target="_blank">Calculus Early Trascendentals</a>. 6th ed. Belmont, CA. USA : Thomson Learning Inc, 2008. pág. 808 de 1336 pp. ISBN 0-495-01166-5.<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com1tag:blogger.com,1999:blog-2619881317292576475.post-20871478633239425302011-05-09T00:21:00.004-05:002017-04-28T04:42:13.083-05:00Propiedades del Máximo entero o Mayor Entero [[x]]<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
Enumero aquí los teoremas más escenciales el operador máximo entero de un número real \(x\). El cual está definido por \[\boldsymbol{\left[\kern-0.17em\left[x \right]\kern-0.17em\right] = n\qquad \leftrightarrow \qquad n = \text{máx} \left\{ {m \in {\Bbb Z}\;|\; m \leq x} \right\}}\]</div>
<br />
<b><span class="Apple-style-span" style="color: #660000;">Propiedades</span></b><br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">1. </span></b> \(\boldsymbol{\left[\kern-0.17em\left[{ x }\right]\kern-0.17em\right]\in\mathbb{Z}\,,\quad \forall x\in\mathbb{R}}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">2. </span></b> \(\boldsymbol{\left[\kern-0.17em\left[{ x }\right]\kern-0.17em\right]=x\quad \leftrightarrow \quad x\in\mathbb{Z}}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">3.</span></b> \(\boldsymbol{\left[\kern-0.15em\left[ x
\right]\kern-0.15em\right] \le x < \left[\kern-0.15em\left[ x
\right]\kern-0.15em\right] + 1\,,\quad\forall x \in \mathbb{R}}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">5.</span></b> \(\boldsymbol{\left[\kern-0.17em\left[ x<br />
\right]\kern-0.17em\right] = n \quad\leftrightarrow\quad n \leq x \leqslant n + 1\;,\quad n \in \mathbb{Z}}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">6.</span></b> Si \(\boldsymbol{a \in \mathbb{Z}\;,\quad\left[\kern-0.17em\left[ x<br />
\right]\kern-0.17em\right] \geq a \quad\leftrightarrow\quad x \geqslant a}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">7. </span></b>Si \(\boldsymbol{a \in \mathbb{Z}\;,\quad\left[\kern-0.17em\left[ x<br />
\right]\kern-0.17em\right] < a \quad\leftrightarrow\quad x < a}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">8.</span></b> Si \(\boldsymbol{a \in \mathbb{Z}\;,\quad\left[\kern-0.17em\left[ x<br />
\right]\kern-0.17em\right] \leq a\quad \leftrightarrow\quad x < a + 1}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">9.</span></b> Si \(\boldsymbol{m \in \mathbb{Z}\quad\to\quad\left[\kern-0.17em\left[ {x + m}<br />
\right]\kern-0.17em\right] = \left[\kern-0.17em\left[ x<br />
\right]\kern-0.17em\right] + m}\)<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">10.</span></b> \(\boldsymbol{\forall\;x,y\in\mathbb{R}},\) Si \(\boldsymbol{x\leq y\quad\to\quad\left[\kern-0.17em\left[ x \right]\kern-0.17em\right] \leq \left[\kern-0.17em\left[ y \right]\kern-0.17em\right]}\)<br />
<br />
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<b><span class="Apple-style-span" style="color: #660000;"><br /></span></b>
<b><span class="Apple-style-span" style="color: #660000;">Bibliografía.</span></b><br />
Figueroa G, Ricardo. <i>Matemática Básica</i>. Editorial América S.R.L., Lima-Perú, 1995.<br />
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<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com6tag:blogger.com,1999:blog-2619881317292576475.post-24073045614941771962011-05-08T22:46:00.025-05:002017-04-28T04:46:40.303-05:00Rango de funciones compuestas: máximo entero y valor absolutoEl último mensaje recibido es una consulta de un estudiante de Lima-Perú, gracias al <a href="http://www.youtube.com/user/mathsalomon" target="_blank">Canal AporteMath</a>.<br />
<br />
Materia: <b><span class="Apple-style-span" style="color: #274e13;">Matemática Básica</span></b><br />
Tema: <b><span class="Apple-style-span" style="color: #0c343d;">Funciones Reales de Variable Real</span></b><br />
Universidad Inca Garcilazo de la Vega<br />
<br />
<span class="Apple-style-span" style="color: blue;">Calcule la intersección del rango de las funciones: \[\begin{array}{lcl}<br />
f(x)&=&\left[\kern-0.17em\left[ {x + 3} \right]\kern-0.17em\right] + \left[\kern-0.17em\left[ {1 - x}\right]\kern-0.17em\right] \\<br />
g(x)&=&\left| {x + 3} \right| - \left| {1 - x} \right| \\<br />
\end{array} \] </span><b><span class="Apple-style-span" style="color: #660000;">Solución:</span></b><br />
<span class="Apple-style-span" style="color: #990000;"><b>1º)</b> Calculando el rango de \(f\quad\) [ \({\rm{ran}}(f)\) ]</span><br />
Por <a href="http://aportemath.blogspot.com/2011/05/propiedades-del-maximo-entero-o-mayor.html" target="_blank">propiedad Nº 9 del máximo entero</a> se tiene: \[<br />
\left[\kern-0.17em\left[{x + 3}\right]\kern-0.17em\right]=<br />
\left[\kern-0.17em\left[{x}\right]\kern-0.17em\right]+3\] y \[<br />
\left[\kern-0.17em\left[{1-x} \right]\kern-0.17em\right]=<br />
1+\left[\kern-0.17em\left[{-x} \right]\kern-0.17em\right]\] como ejercicio para el lector, dejo que verifique que: \[\left[\kern-0.17em\left[x<br />
\right]\kern-0.17em\right] + \left[\kern-0.17em\left[ { - x}<br />
\right]\kern-0.17em\right] = - 1\;,\quad\forall x\in\mathbb{R}\] entonces, reemplazando estos resultados en \(f\) se tiene: \[ \begin{array}{r@{\,}c@{\,}c@{\,}c@{\,}l}<br />
f(x) &=& \left[\kern-0.17em\left[{x+3} \right]\kern-0.17em\right] &+& \left[\kern-0.17em\left[{1-x}\right]\kern-0.17em\right] \\<br />
&=& \left(\left[\kern-0.17em\left[{x} \right]\kern-0.17em\right] + 3\right)&+& \left(1+\left[\kern-0.17em\left[{-x}\right]\kern-0.17em\right]\right) \\<br />
&=&\left[\kern-0.17em\left[{x} \right]\kern-0.17em\right]&+& \left[\kern-0.17em\left[{-x}\right]\kern-0.17em\right]+4 \\<br />
&=&-1&+&4\\<br />
&=& &3&<br />
\end{array} \] \(f\) tiene un valor constante e igual a 3 para toda \(x\) en \(\mathbb{R}\). Por tanto \[{\rm{ran}}(f)=\{3\}\]<br />
<div>
</div>
<span class="Apple-style-span" style="color: #990000;"><b>2º)</b> Calculando el \({\rm{ran}}(g)\)</span><br />
Calculando los puntos críticos: \[ \begin{array}{rccl}<br />
x + 3=0 & \wedge & 1 - x=0 \\<br />
x =-3 & \wedge & x = 1 \\<br />
\end{array} \] éstos puntos originan 3 intervalos en la recta real (casos 1º - 3º), en los que los términos \(\;\;\left| {1-x} \right|\;\;\) y \(\;\;\left| {x + 3} \right|\), que conforman \(g\), asumirán valores distintos: \[ \begin{array}{rccl}<br />
\rm{1º)} & x < -3 & \rightarrow & \left|{x + 3}\right|=-x-3 \quad \rm{,} \\<br />
& & & \left|{x-1}\right|= -x+1 \\ <br />
\rm{2º)} & -3\leq x <1 & \rightarrow & \left|{x + 3}\right|=x+3 \quad \rm{,} \\<br />
& & & \left|{x-1}\right|= -x+1 \\<br />
\rm{3º)} & 1 \leq x & \rightarrow & \left|{x + 3}\right|=x+3 \quad \rm{,} \\<br />
& & & \left|{x-1}\right|= x-1 \\<br />
\end{array} \] nótese que: \(\left| {1 - x} \right|\) puede escribirse como \(\left| {x - 1} \right|\)<br />
Restando los términos del lado derecho en cada caso se obtiene:<br />
<br />
\[ \begin{array}{rccl}<br />
\rm{1º)} & x<-3 & \rightarrow & g(x)=(-x-3)-(-x+1)=-4 \\<br />
\rm{2º)} & -3\leq x <1 & \rightarrow & g(x)=(x+3)-(-x+1)=2x+2 \\<br />
\rm{3º)} & 1 \leq x & \rightarrow & g(x)=(x+3)-(x-1)=4\\<br />
\end{array} \] Es decir la función \(g\) es la función seccionada: \[g(x)=\left\{ {\begin{array}{*{20}{c}}<br />
{ - 4}&,&{x < - 3} \\ {2x + 2}&,&{ - 3 \leq x < 1} \\ 4&,&{1 \leq x} \end{array}} \right.\] En la Segunda sección de esta función se observa que si \[ \begin{array}{rcl}<br />
& -3\leq x<1\\<br />
\rightarrow & -6\leq 2x<2\\<br />
\rightarrow & -4\leq 2x+2<4\\<br />
\rightarrow & -4\leq g(x)<4\\<br />
\rightarrow & g(x)\in\langle-4,4]<br />
\end{array} \] En la primera sección se ve que \(g(x)=-4\) cuando \(x<-3\), luego \[g(x)\in[-4,4] \;,\quad \forall x\in\mathbb{R}\] es decir se ha deducido que el \[{\rm{ran}}(g)=[-4,4]\] Con el software maple, para graficar puede usarse la orden<br />
<span class="Apple-style-span" style="color: red; font-family: "courier new" , "courier" , monospace;"><b>> plot(abs(x+3)-abs(1-x)); </b></span><br />
<span class="Apple-style-span" style="color: red; font-family: "courier new" , "courier" , monospace;"><b><span class="Apple-style-span" style="color: black; font-family: "times new roman"; font-weight: normal;">(úsese solo para verificar, no para omitir el paso del desarrollo de l ejercicio)</span></b></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-_ZHUhCUp1YI/TcgRRG4f38I/AAAAAAAABN4/vI1jpKGDNb0/s1600/grafica-funcion-valor-absoluto.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="320" src="https://3.bp.blogspot.com/-_ZHUhCUp1YI/TcgRRG4f38I/AAAAAAAABN4/vI1jpKGDNb0/s320/grafica-funcion-valor-absoluto.jpg" target="_blank" width="304" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Gráfica en maple de la función valor absoluto<br />
compuesta \(g(x)=\left|{x+3}\right|-\left|{1-x}\right|\)</span></td></tr>
</tbody></table>
Finalmente, lo que nos solicitaba el ejercicio es: \[ \begin{array}{rcl}<br />
{\rm{ran}}(f) \cap {\rm{ran}}(g) &=& [ - 4,4] \cap \{3\} \\<br />
&=& \left\{ \,3 \right\} \\<br />
\end{array} \]<br />
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<b><b><span class="Apple-style-span" style="color: #660000;"><br /></span></b></b>
<b><b><span class="Apple-style-span" style="color: #660000;">Bibliografía.</span></b></b><br />
<div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;">
Figueroa G, Ricardo. <i>Matemática Básica</i>. Editorial América S.R.L., Lima-Perú, 1995.</div>
<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com3tag:blogger.com,1999:blog-2619881317292576475.post-56470160734461282592011-05-05T16:58:00.004-05:002011-05-06T16:49:59.649-05:00Validando una Inferencia Lógica (VIDEO)Últimamente me plantearon el ejercicio de demostrar una inferencia. Pero aquí veremos que no se debe demostrar la inferencia hasta que se tenga la seguridad de que se haya validado dicha inferencia con la Prueba Formal de Invalidez o Método Abreviado, el mismo que se ha mencionado en el vídeo <a href="http://www.youtube.com/watch?v=o1szEhVNXvE" target="_blank">parte 7</a>.<br />
Queda abierta cualquier duda al respecto, pueden formular más preguntas si desean.<br />
<br />
- Mat. Salomón Ching -<br />
APORTES MATEMÁTICOS<br />
<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="http://www.youtube.com/watch?v=tfF6WpBBwLY" target="_blank"><img border="0" height="150" src="http://3.bp.blogspot.com/-n668S0TsjKI/TcMgr4vC-JI/AAAAAAAABNw/x6UZtLE1DM0/s200/snap665.jpg" width="200" /></a></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com2tag:blogger.com,1999:blog-2619881317292576475.post-5753039935677414532011-05-02T01:07:00.015-05:002011-05-02T14:54:23.218-05:00¿Cómo demuestro: √(│y-x│) ≥ │√y-√x│ ?Dirigiéndonos a la teoría de los números reales, a petición de uno de mis lectores de Sincelejo Colombia, desmostraré la desigualdad: <span class="Apple-style-span" style="color: #660000;">\[\sqrt {\left| {y - x} \right|} \geqslant \left| {\sqrt y - \sqrt x } \right|\] para todo \(x,y \in \mathbb{R}_0^{+}\)</span><br />
<br />
Enumeraré cada paso para dejar lugar a los comentarios, si es que hay algo que no se entienda, fácilmente se podrá ubicar la parte con el número de paso correspondiente.<br />
<br />
Además la dividiré en 2 partes. En la primera, transformo la desigualdad en otra equivalente a través de una serie de pasos válidos en los teoremas de números reales. En la segunda efectuaré la demostración formal de la desigualdad equivalente.<br />
<br />
Empezamos:<br />
<br />
Obsérvese que la igualdad \(\sqrt {\left| {y - x} \right|} = \left| {\sqrt y - \sqrt x } \right|\) solo se satisface para \(x = 0\,\,\,,\,\,\,\,y = 0\) así que solo se trabajará con la desigualdad estricta: \[\sqrt {\left| {y - x} \right|} > \left| {\sqrt y - \sqrt x } \right|\quad\ldots(1)\] <b><span class="Apple-style-span" style="color: red;">Parte I: </span></b><br />
<b><span class="Apple-style-span" style="color: red;"></span></b><b><span class="Apple-style-span" style="color: #660000;">1.</span></b> Elevando al cuadrado \[\begin{gathered}<br />
{\sqrt {\left| {y - x} \right|} ^2} > {\left| {\sqrt y - \sqrt x } \right|^2}\\<br />
\left| {y - x} \right| > {\left| {\sqrt y - \sqrt x } \right|^2}\\<br />
\left| {y - x} \right| > {\left( {\sqrt y - \sqrt x } \right)^2}\\<br />
\end{gathered} \] <b><span class="Apple-style-span" style="color: #660000;">2.</span></b> Multiplicando por \({\left( {\sqrt y + \sqrt x } \right)^2}\) a cada lado de la desigualdad \[ \begin{array}{rcl}<br />
\left| {y - x} \right|{\left( {\sqrt y + \sqrt x } \right)^2} &>&{\left( {\sqrt y - \sqrt x } \right)^2}{\left( {\sqrt y + \sqrt x } \right)^2} \\<br />
\left| {y - x} \right|{\left( {\sqrt y + \sqrt x } \right)^2} &>&{\left[ {\left( {\sqrt y - \sqrt x } \right)\left( {\sqrt y + \sqrt x } \right)} \right]^2} \\<br />
\left| {y - x} \right|{\left( {\sqrt y + \sqrt x } \right)^2} &>& {\left( {{{\sqrt y }^2} - {{\sqrt x }^2}} \right)^2}\\<br />
\left| {y - x} \right|{\left( {\sqrt y + \sqrt x } \right)^2} &>& {\left( {y - x} \right)^2}\\<br />
\left| {y - x} \right|{\left( {\sqrt y + \sqrt x } \right)^2} &>& {\left| {y - x} \right|^2}\\<br />
\end{array} \] <b><span class="Apple-style-span" style="color: #660000;">3.</span></b> Cancelando el factor \(\left| {y - x} \right|\) \[{\left( {\sqrt y + \sqrt x } \right)^2} > \left| {y - x} \right|\] <b><span class="Apple-style-span" style="color: #660000;">4.</span></b> Esta desigualdad la podemos escribir con el símbolo de menor que cambiando de lado sus miembros \[\left| {y - x} \right| < {\left( {\sqrt y + \sqrt x } \right)^2}\] <b><span class="Apple-style-span" style="color: #660000;">5.</span></b> Recordemos la propiedad \(\left| x \right| < b \Leftrightarrow - b < x < b\), en donde podemos considerar \(b = \scriptstyle{{\left( {\sqrt y + \sqrt x } \right)^2}}\), entonces: \[ - {\left( {\sqrt y + \sqrt x } \right)^2} < y - x < {\left( {\sqrt y + \sqrt x } \right)^2}.\] <b><span class="Apple-style-span" style="color: red;">Parte II</span></b><br />
<b><span class="Apple-style-span" style="color: #660000;">1. </span></b>Acabamos de transformar la desigualdad inicial (1) en: \[ - {\left( {\sqrt y + \sqrt x } \right)^2} < y - x < {\left( {\sqrt y + \sqrt x } \right)^2}\] que es la intersección de \[ - {\left( {\sqrt y + \sqrt x } \right)^2} < y - x\quad\ldots(2)\] con \[y - x < {\left( {\sqrt y + \sqrt x } \right)^2}\quad\ldots(3)\] <b><span class="Apple-style-span" style="color: #660000;">2. </span></b>Para demostrar la desigualdad (2) tenemos: \[ \begin{array}{lcl}<br />
- {\left({\sqrt y +\sqrt x } \right)^2} &=& - \left( {y + 2\sqrt {xy} + x} \right) \\<br />
&<& - \left( {y + x} \right) \\<br />
&=& - y - x \\<br />
&<& - y + x\\<br />
&=& - y + x\\<br />
\end{array} \] <b><span class="Apple-style-span" style="color: #660000;">3. </span></b>Uniendo los extremos se ve claramente que \[ - {\left( {\sqrt y + \sqrt x } \right)^2} < y - x\] <span class="Apple-style-span" style="color: #660000;">Dejo como ejercicio al lector la demostración de la desigualdad (3).</span><br />
<br />
Habiendo demostrado las desigualdades estrictas (2) y (3), y la igualdad con \(x = 0\; , \quad y = 0\) puede afirmarse que se cumple \[{\left( {\sqrt y + \sqrt x } \right)^2} > \left| {y - x} \right|\] \[\forall\; x,y \in {\mathbb{R}^+}\] entonces multiplicando esta última desigualdad por \(\left| {y - x} \right|\) y realizando el proceso visto en el <span class="Apple-style-span" style="color: #274e13;"><i><b>paso 2</b></i> y <i><b>paso 1</b></i> de la <b><i>parte I</i></b></span> de forma inversa, quedará demostrado que: \[\sqrt {\left| {y - x} \right|} \geqslant \left| {\sqrt y - \sqrt x } \right|\] \[\forall\; x,y \in \mathbb{R}_0^{+}\] <span class="Apple-style-span" style="color: #660000;">l.q.q.d.</span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-m3QfFLayyRY/Tb5XptgXsiI/AAAAAAAABNs/cBh0QniRrAo/s1600/chiste_ecuaciones.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="popupwindow"><img border="0" height="150" src="http://2.bp.blogspot.com/-m3QfFLayyRY/Tb5XptgXsiI/AAAAAAAABNs/cBh0QniRrAo/s200/chiste_ecuaciones.jpg" width="200" /></a></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com5tag:blogger.com,1999:blog-2619881317292576475.post-21828205482577782762011-05-01T00:24:00.001-05:002011-05-01T00:25:38.459-05:00Escribe fórmulas LaTeX en Word<h1 class="pagetitle" style="border-bottom-color: rgb(225, 225, 225); border-bottom-style: solid; border-bottom-width: 7px; clear: both; color: #646464; font-family: 'trebuchet ms', arial, sans-serif; font-size: 22px; font-weight: bold; margin-bottom: 0.4em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 2px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 22px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 22px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Aurora</span></span></h1><div class="column1-unit-left" style="font-family: verdana, arial, sans-serif; font-size: 10px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><div style="font-size: 12px; line-height: 1.5em; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Aurora le permite utilizar LaTeX en Microsoft ® Word </span></span><span style="font-weight: normal; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">,</span></span></span><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> PowerPoint ® Visio ®, Excel ®, y muchos otros programas.</span> <span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Se asegura de que sus fórmulas se ve bien, bellamente impresa, y jugar bien con el resto del texto.</span> <span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Aurora se encarga de las pequeñas cosas como la numeración de ecuaciones y colocarlos en la página, y se queda fuera de su camino el resto del tiempo.</span></span></div><div style="font-size: 12px; line-height: 1.5em; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Esto es lo que Aurora con el siguiente aspecto:</span></span></div><div style="font-size: 12px; line-height: 1.5em; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><img alt="Screenshot of using Aurora" class="center" src="http://elevatorlady.ca/data/aurora/aurora-example.png" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; clear: both; display: block; float: none; margin-bottom: 0px; margin-left: auto; margin-right: auto; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;" title="Screenshot of using Aurora" /></div><div style="font-size: 12px; line-height: 1.5em; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Si usted es un veterano de LaTeX, Aurora le hará sentirse como en casa.</span> <span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">E incluso si usted no está, la documentación de Aurora, resaltado de sintaxis, y cuenta con vista previa tendrá que TeX más rápido que escribir en ningún momento.</span></span></div><hr class="clear-contentunit" style="background-attachment: initial; background-clip: initial; background-color: #d2d2d2; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; clear: both; color: #d2d2d2; height: 0.1em; margin-bottom: 0px; margin-left: auto; margin-right: auto; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: 440px;" /><h1 style="clear: both; color: #505050; font-family: 'trebuchet ms', arial, sans-serif; font-size: 21px; font-weight: normal; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 1em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 21px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 21px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">¿Por qué Aurora es bueno para usted</span></span></h1><ul style="font-size: 12px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://elevatorlady.ca/data/img/bg_bullet_full_1.gif); background-origin: initial; background-position: 0px 0.5em; background-repeat: no-repeat no-repeat; line-height: 1.4em; margin-bottom: 0.2em; margin-left: 2px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 12px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: #e6ecf9; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; color: black; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">N del ratón más la caza de los símbolos a entrar en </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: #e6ecf9; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; color: black; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">el futuro matemáticas gran rapidez y eficacia</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: #e6ecf9; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; color: black; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> utilizando el lenguaje estándar de composición tipográfica científica.</span></span></li>
<li style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://elevatorlady.ca/data/img/bg_bullet_full_1.gif); background-origin: initial; background-position: 0px 0.5em; background-repeat: no-repeat no-repeat; line-height: 1.4em; margin-bottom: 0.2em; margin-left: 2px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 12px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Expresar cualquier concepto científico de </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">las matemáticas</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> , </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">la informática</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> ,</span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">la química</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> , </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">la ingeniería</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> , y muchas otras áreas.</span></span></li>
<li style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://elevatorlady.ca/data/img/bg_bullet_full_1.gif); background-origin: initial; background-position: 0px 0.5em; background-repeat: no-repeat no-repeat; line-height: 1.4em; margin-bottom: 0.2em; margin-left: 2px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 12px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">La integración completa con </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Microsoft ® Word </span></span><span style="font-weight: normal; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">,</span></span></span><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> PowerPoint ® </span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">, </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Visio ®</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> y</span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Excel ®</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> .</span></span></li>
<li style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://elevatorlady.ca/data/img/bg_bullet_full_1.gif); background-origin: initial; background-position: 0px 0.5em; background-repeat: no-repeat no-repeat; line-height: 1.4em; margin-bottom: 0.2em; margin-left: 2px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 12px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Incorporado </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">convertidor de documentos LaTeX</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> .</span></span></li>
<li style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://elevatorlady.ca/data/img/bg_bullet_full_1.gif); background-origin: initial; background-position: 0px 0.5em; background-repeat: no-repeat no-repeat; line-height: 1.4em; margin-bottom: 0.2em; margin-left: 2px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 12px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Soporte avanzado para la </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">ecuación de numeración</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> .</span></span></li>
<li style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://elevatorlady.ca/data/img/bg_bullet_full_1.gif); background-origin: initial; background-position: 0px 0.5em; background-repeat: no-repeat no-repeat; line-height: 1.4em; margin-bottom: 0.2em; margin-left: 2px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 12px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Funciona con </span></span><b style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">cualquier aplicación de Windows</span></span></b><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> que tiene un "Insertar objeto ..." la función o le permite pegar imágenes.</span></span></li>
</ul><div style="font-size: 12px; line-height: 1.5em; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">Más información:</div><div style="font-size: 12px; line-height: 1.5em; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><a href="http://elevatorlady.ca/" target="_blank">http://elevatorlady.ca/</a></div></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com0tag:blogger.com,1999:blog-2619881317292576475.post-87394384443790130632011-04-26T22:52:00.098-05:002011-04-30T05:50:53.759-05:00Aplicación de las funciones cuadráticas en la vida realLas funciones cuadráticas modelan gran parte de situaciones del mundo físico. Aquí se muestra una de ellas, con la proposición y desarrollo del siguiente<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">Ejercicio Explicativo</span></b>.<br />
<a href="http://4.bp.blogspot.com/-o1c4hUlurjg/Tbkc8U9BuBI/AAAAAAAABNM/DK9k680tAIc/s1600/GoldenGateBridge-001-San-Francisco.jpg" target="_blank" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="150" src="http://4.bp.blogspot.com/-o1c4hUlurjg/Tbkc8U9BuBI/AAAAAAAABNM/DK9k680tAIc/s200/GoldenGateBridge-001-San-Francisco.jpg" width="200" /></a>El puente Golden Gate enmarca la entrada a la bahía de San Francisco. Sus torres de 746 pies de altura están separadas por una distancia de 4200 pies. El puente está suspendido de dos enormes cables que miden 3 pies de diámetro: el ancho de la calzada es de 90 pies y ésta se encuentra aproximadamente a 220 pies del nivel del agua. Los cables forman una parábola y tocan la calzada en el centro del puente. <i><b>Determinar la altura de los cables a una distancia de 1000 pies del centro del puente.</b></i><br />
<br />
<b><span class="Apple-style-span" style="color: blue;"><i><u>Solución.</u></i></span></b><br />
<b></b>Empezarnos seleccionando la ubicación de los ejes de coordenadas de modo que el eje \(x\) coincida en la calzada y el origen coincida en el centro del puente.<br />
Como resultado de esto, las torres gemelas quedarán a 746-220=526 pies arriba de la calzada y ubicadas a \(\frac{4200}{2}=2100\) pies del centro.<br />
Los cables de forma parabólica se extenderán desde las torres, abriendo hacia arriba, y tendrán su vértice en \((0,0)\) como se ilustra en la figura de abajo<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-KTZMDxe_QpY/Tbj2jpCwUsI/AAAAAAAABNI/_wGIXbmwOdw/s1600/golden+gate..png" target="_blank" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="226" src="http://4.bp.blogspot.com/-KTZMDxe_QpY/Tbj2jpCwUsI/AAAAAAAABNI/_wGIXbmwOdw/s640/golden+gate..png" width="555" /></a></div><br />
La manera en que seleccionamos la colocación de los ejes nos permite identificar la ecuación de una parábola como \[y = ax^2 \quad,\quad a>0.\] Obsérvese que los puntos \((-2100, 526)\) y \((2100, 526)\) están en la gráfica parabólica.<br />
<br />
Con base en estos datos podemos encontrar el valor de \(a\) en \(y = ax^2\): \[\begin{gathered}<br />
y = a{x^2} \\<br />
526 = a{(2100)^2} \\<br />
a = \frac{{526}}{{{{(2100)}^2}}} \\<br />
\end{gathered} \] Así, la ecuación de la parábola es \[y = \frac{{526}}{{{{(2100)}^2}}}{x^2}\] La altura del cable cuando \(x=1000\) es \[y = \frac{{526}}{{{{(2100)}^2}}}{(1000)^2} \approx 119.3\,\,{\text{pies}}\] Por tanto, el cable mide 119.3 pies de altura cuando se está a una distancia de 1000 pies del centro del puente.<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">Biliografía.</span></b><br />
1. Sullivan, Michael. <a href="https://doc-0o-28-docsviewer.googleusercontent.com/viewer/securedownload/dhefkric6mjclna010p4f98h48l7oe98/oj2ec1htuci5d5ulldgujs0a4jqi0vao/1303980300000/ZXhwbG9yZXI=/AGZ5hq9YtKxbdHDWJiMZ11V0AqOy/MEI2ZU5jZlk1REhHa01UWmtZVEF4Tm1FdFpEUTNOaTAwWm1VekxXSmpZbVV0TWprNFlUZGpZMll3TjJNMQ==?a=dl&filename=Page186_+Precalculo_Sullivan.pdf&sec=AHSqidasvhtr84vh2Kw8caBpgiTfHcvr73XYxl6_XyTJHmA9jtfsLpU3Cda6UJDSanBy_sVTYf1TT9fk-_qMUCXsthicZNuuIY961HiE8f8OR3wyus0M7gRYX3k5z4dIi_agmcYBXxOrvQQvwbvlbm_o4YKILH-VDWNz8eB0sA6E92ykR3tVOPURt1SSmeRcd6KN2THlPbhGTsKcgP3xHe4t-t1uUPFjJwrg8_ytFWxXlkz-sM_W3rtXyK9cp7OANrgj3_FPdLcQ4cueXbxrD9wLR76bucEbVNIF5vbysJlCW0Frgxr8M_qVYbQZQNpBNUrgb4i6wXlA173-AwQN6XaWTmbFdPu3sDGe4xeJmwpqImmxBzPUTQmM1bUg3kgWCmFVbBuykhz-cREla2fekAyt8fPdBwYzSZhPfrvkMt-9zfDkmOw71rX8DGDgq6oc0wJQnrJQcmnYNCgTTa1044TY4msi7Y668g&nonce=sjuahn1naoo7u&user=AGZ5hq9YtKxbdHDWJiMZ11V0AqOy&hash=aa5sh14nmq7e16eqs6mtskv7pv8lb1rr" target="_blank">Precálculo</a>. 4ta ed. Prentice Hall Hispanoamericana SA. Naulcalpan de Juarez, México. 1997. <b>pag Nº 186</b> de 842 pp ISBN: 968-880-964-0.<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com28tag:blogger.com,1999:blog-2619881317292576475.post-61985494183446927722011-04-26T20:24:00.018-05:002017-04-29T01:45:05.043-05:00Dos Demostraciones Directas en Inferencia Lógica (VIDEO)<div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;">
<img border="0" height="80" src="https://2.bp.blogspot.com/-ppOQTJ6A9so/TbdwqaAWg_I/AAAAAAAABNE/raXAnLAReDE/s200/snap605.jpg" width="100" /></div>
<br />
<div style="text-align: center;">
<b><span class="Apple-style-span" style="color: #660000;">Inferencias lógicas.</span></b> </div>
<div style="text-align: center;">
<span class="Apple-style-span" style="color: #990000;">Demostración directa o Prueba formal de validez</span></div>
<span class="Apple-style-span" style="color: #990000;"><br />
</span><br />
Dos ejercicios resueltos explicados paso a paso. La novedad del vídeo es que resuelvo los ejercicios sólo con 4 leyes de inferencia:<br />
<br />
<ol>
<li><span class="Apple-style-span" style="color: #073763;">Modus Ponens</span></li>
<li><span class="Apple-style-span" style="color: #073763;">Modus Tollens</span></li>
<li><span class="Apple-style-span" style="color: #073763;">Tollendo Ponens y</span></li>
<li><span class="Apple-style-span" style="color: #073763;">Ley de la Doble Negación</span></li>
</ol>
<br />
Resalto la importancia de la doble negación, que no se incluyó en anteriores vídeos.<br />
<br />
<div style="text-align: center;">
<object height="305" width="489"><param name="movie" value="https://www.youtube.com/v/JNskt6JkP9A&hl=en_US&feature=player_embedded&version=3"></param>
<param name="allowFullScreen" value="true"></param>
<param name="allowScriptAccess" value="always"></param>
<embed src="https://www.youtube.com/v/JNskt6JkP9A&hl=en_US&feature=player_embedded&version=3" type="application/x-shockwave-flash" allowfullscreen="true" allowscriptaccess="always" width="489" height="305"></embed></object></div>
<br />
<div style="text-align: center;">
<br /></div>
<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com2tag:blogger.com,1999:blog-2619881317292576475.post-51264099410049766202011-04-23T07:33:00.040-05:002017-05-16T05:09:34.941-05:00Aplicación de la Función exponencial \({\rm a}^x\) o \({\rm e}^x\)<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
La función exponencial se produce con mucha frecuencia en los modelos matemáticos de la naturaleza y de la sociedad. Aquí se indicará brevemente cómo surge en la descripción del crecimiento de poblaciones.<br />
<br />
<span class="Apple-style-span" style="color: #660000;"><b>Propagación de Bacterias</b></span><br />
Considere una población de bacterias en un medio nutritivo homogéneo. Suponga que haciendo un muestreo de la población a ciertos intervalos se determina que la población se duplica cada hora. Si el número de bacterias en el instante $t$ es \(p(t)\), donde \(t\) se mide en horas, y la población inicial es \(p(0)=1000\) habitantes, entonces tenemos: \[\begin{gathered} p(1) = 2p(0) = 2 \times 1000\\<br />
p(2) = 2p(1) = 2^2 \times 1000\\<br />
p(3) = 2p(2) = 2^3 \times 1000\\<br />
\end{gathered} \] Con base en este patrón, en general se cumple \[p(t) = {2^t} \times 1000 = (1000){2^t}\] Esta función de población es un múltiplo escalar de la función exponencial \(y=2^x\) (cambie \(x\) por \(t\) ), entonces posee un rápido crecimiento, tal como se puede apreciar en su respectiva gráfica:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-eKvmkDlHdHc/TbK-C4KbAzI/AAAAAAAABMU/3yNOzRpLgP8/s1600/funcion+exponencial2..png" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="400" src="https://1.bp.blogspot.com/-eKvmkDlHdHc/TbK-C4KbAzI/AAAAAAAABMU/3yNOzRpLgP8/s400/funcion+exponencial2..png" width="278" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">La función \(y=2^x\) crece incluso más rápido<br />
que la fucnión \(y=x^2\) cuando \(x>0\)</span></td></tr>
</tbody></table>
En condiciones ideales (espacio y nutrición ilimitados con ausencia de enfermedad), este crecimiento exponencial es típico de lo que realmente ocurre en la naturaleza.<br />
<br />
<span class="Apple-style-span" style="color: #660000;"><b>Población Humana Mundial</b></span><br />
La siguiente tabla muestra los datos de la población mundial en el siglo XX y en la figura siguiente se muestra el diagrama de dispersión correspondiente. \[\begin{array}{*{20}{c}}<br />
{\begin{array}{*{20}{|c}}<br />
{{\text{Año}}}&\begin{gathered}<br />
{\text{Población}}\\<br />
{\text{(millones)}}\\ <br />
\end{gathered} \\ \hline<br />
{{\text{1900}}}&{{\text{1650}}} \\ <br />
{{\text{1910}}}&{{\text{1750}}} \\ <br />
{{\text{1920}}}&{{\text{1860}}} \\ <br />
{{\text{1930}}}&{{\text{2070}}} \\ <br />
{{\text{1940 }}}&{{\text{2300}}} \\ <br />
{{\text{1950 }}}&{{\text{2560}}} \\ <br />
{{\text{1960 }}}&{{\text{3040}}} \\ <br />
{{\text{1970 }}}&{{\text{3710}}} \\ <br />
{{\text{1980 }}}&{{\text{4450}}} \\ <br />
{{\text{1990 }}}&{{\text{5280}}} \\ <br />
{{\text{2000}}}&{{\text{6080}}} <br />
\end{array}} \\ <br />
{{\text{Población humana mundial}}} <br />
\end{array}\]<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-JxLovED-zvQ/TbLawTT8eeI/AAAAAAAABMs/qf__fF6K1go/s1600/snap586.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" src="https://1.bp.blogspot.com/-JxLovED-zvQ/TbLawTT8eeI/AAAAAAAABMs/qf__fF6K1go/s1600/snap586.jpg" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Dispersión para el crecimiento de la población mundial</span></td></tr>
</tbody></table>
Si ponemos los datos de la tabla en una hoja de Excel podemos encontrar fácilmente la función exponencial que se ajusta a los datos, con la finalidad de obtener interpolaciones (valores intermedios) y extrapolaciones (valores externos) necesarias para una predicción bien aproximada de la población en los años siguientes.<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-oX-G5T0yBRY/TbLRObrV8lI/AAAAAAAABMk/my_RpgBBI_c/s1600/snap584.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="275" src="https://1.bp.blogspot.com/-oX-G5T0yBRY/TbLRObrV8lI/AAAAAAAABMk/my_RpgBBI_c/s320/snap584.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Obtenido con Excel 2007, <br />
herramienta <span class="Apple-style-span" style="font-size: xx-small;">inserción de gráficos</span> -<span class="Apple-style-span" style="font-size: xx-small;">></span> <span class="Apple-style-span" style="font-size: xx-small;">dispersión</span> -<span class="Apple-style-span" style="font-size: xx-small;">> línea de tendencia</span> </span></td></tr>
</tbody></table>
Excel usó el método de mínimos cuadrados para encontrar la función que ajusta a los datos (en excel es llamada línea de tendencia) la cual viene a ser: \(y={\text{(8}} \cdot {\text{1}}{{\text{0}}^{{\text{ - 9}}}}{\text{)}}{{\text{e}}^{0.0136x}}.\) Interpretando el resultado:<br />
<br />
La población mundial en el año \(t\) es: \[{\text{p(t) = (8}} \cdot {\text{1}}{{\text{0}}^{{\text{ - 9}}}}{\text{)}}{{\text{e}}^{0.0136t}}\;\;\text{millones de habitantes}\]<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-nGdAojTIz7o/TbLZ_R3APzI/AAAAAAAABMo/qSYPcPl2i8Y/s1600/snap585.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" src="https://4.bp.blogspot.com/-nGdAojTIz7o/TbLZ_R3APzI/AAAAAAAABMo/qSYPcPl2i8Y/s1600/snap585.jpg" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">El modelo exponencial de crecimiento de población mundial en el año \(t\)</span></td></tr>
</tbody></table>
<b><span class="Apple-style-span" style="color: #660000;">Bibliografía</span></b><br />
1. Stewart, James. 2008. <a href="http://bit.ly/iL862V" target="_blank">Calculus Early Trascendentals</a>. 6th ed. Belmont, CA. USA : Thomson Learning Inc, 2008. pág. 18 de 1336 pp. ISBN 0-495-01166-5.<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com10tag:blogger.com,1999:blog-2619881317292576475.post-17925126025797454752011-04-22T03:54:00.041-05:002011-05-03T02:13:02.909-05:00La Función Signo de un Número Real [ y = sgn(x) ]La función signo de un número real \(x\) es una función de valor real cuya regla de correspondencia viene dada por: \[\operatorname{sgn} (x) = \left\{ {\begin{array}{*{20}{c}}<br />
1&,&{{\text{si}}}&{x > 0} \\<br />
0&,&{{\text{si}}}&{x = 0} \\<br />
{ - 1}&,&{{\text{si}}}&{x < 0}<br />
\end{array}} \right.\] su gráfica es la de una función de dos escalones con un salto en \(x=0\)<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-wRIXfksT1ic/TbEHQDf1OZI/AAAAAAAABLY/yWbugF8Q7tc/s1600/funcion+signo.png" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="blank"><img border="0" height="219" src="http://2.bp.blogspot.com/-wRIXfksT1ic/TbEHQDf1OZI/AAAAAAAABLY/yWbugF8Q7tc/s320/funcion+signo.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">\(\operatorname{sgn}(x)\) se lee: <i><b>signo del número real</b></i> \(\boldsymbol{x}\)</span></td></tr>
</tbody></table><div class="separator" style="clear: both; text-align: center;"></div>También puede expresarse de la forma: \[f = \left\{ {(x,y) \in \mathbb{R} \times \mathbb{R}\;|\;y = \operatorname{sgn} (x)} \right\}\] donde su dominio y rango son respectivamente: \[\begin{gathered}<br />
{\text{Dom}}(f) = \mathbb{R} \\<br />
{\text{Ran}}(f) = \{-1,0,1\} \\<br />
\end{gathered} \] Existen situaciones en que se debe hallar el dominio y gráfico de funciones signo compuestas por funciones algebraicas o funciones elementales como veremos en el desarrollo del siguiente ejercicio:<br />
<span class="Apple-style-span" style="color: red;"><b>Trazar el gráfico de la función</b> \[\boldsymbol{f(x) = \operatorname{sgn} \left( {|{x^2}-1|-1} \right)}\] </span><span class="Apple-style-span" style="color: blue;"><i><u><b>Solución</b></u></i></span><br />
<span class="Apple-style-span" style="color: #660000;">Primera forma</span><br />
<div><input onclick="if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar Contenido';}else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar Contenido';}" type="button" value="Mostrar" /><br />
<div style="display: none;">Por definición de función signo: \[\operatorname{sgn} \left( {|{x^2}-1| -1} \right) = \left\{ {\begin{array}{*{20}{c}}<br />
1&,&{\left| {{x^2} -1}\right| -1 >0\,\,\,...\,\,\,(1)} \\<br />
0&,&{\left| {{x^2} -1}\right| -1 = \,0\,\,\,...\,\,(2)} \\<br />
{ -1}&,&{\left| {{x^2} - 1}\right| -1<0\,\,\,...\,\,(3)}<br />
\end{array}} \right.\] Analizando las tres condiciones:<br />
En la condición (1) se tiene: \[\begin{gathered}<br />
\left|{{x^2} - 1} \right| > 1 \\<br />
{x^2}-1 > 1 \quad \vee\quad {x^2} -1< -1 \\<br />
{x^2} > 2 \quad \vee\quad {x^2} < 0\\<br />
{x^2} - 2 > 0 \\<br />
(x + \sqrt 2 )(x - \sqrt 2 ) > 0\\<br />
\end{gathered} \] por el método de puntos críticos<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-_FQZQ0-EPQc/TbEWPJdd1dI/AAAAAAAABLg/HgjgLPcBatI/s1600/puntos_criticos..png" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="71" src="http://2.bp.blogspot.com/-_FQZQ0-EPQc/TbEWPJdd1dI/AAAAAAAABLg/HgjgLPcBatI/s320/puntos_criticos..png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Se elige los intervalos (+) porque la inecuación dice: \(>0\)</span></td></tr>
</tbody></table>así la condición (1) es: \[x\in\left\langle { - \infty , - \sqrt 2 } \right\rangle \cup \left\langle {\sqrt 2 ,\infty } \right\rangle\] En la condición (2) se tiene: \[\begin{gathered}<br />
\left| {{x^2} - 1} \right| = \,1 \\<br />
{x^2} - 1 = \,1\,\,\,\,\,\,\,\, \vee \,\,\,\,\,{x^2} - 1 = \, - 1 \\<br />
{x^2} = \,2\,\,\,\,\,\,\,\, \vee \,\,\,\,\,\,\,\,\,{x^2} = \,0 \\<br />
x = \pm \,\sqrt{2}\,\,\,\,\,\,\,\, \vee \,\,\,\,\,\,\,\,\,x = \,0 \\<br />
\end{gathered} \]<br />
así la condición (2) es: \[x \in \left\{ { - \sqrt 2 \;,\;0\;,\;\sqrt 2 } \right\}\]<br />
En la condición (3) se tiene: \[\begin{gathered}<br />
\left| {{x^2} - 1} \right| < 1 \\<br />
- 1 < {x^2} - 1 < 1 \\<br />
0 < {x^2} < 2 \\<br />
0 < {x^2}\,\,\,\,\,\,\,\,\, \wedge \,\,\,\,\,\,\,\,\,{x^2} < 2 \\<br />
\left[ {x < 0 \vee x > 0} \right]\quad \wedge \quad \left[ {{x^2} - 2 < 0} \right]\\ \left[ {x < 0 \vee x > 0} \right] \wedge \left[ {\left( {x - \sqrt 2 } \right)\left( {x - \sqrt 2 } \right) < 0} \right] \\<br />
\end{gathered} \] <br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-5uR7lTYmX1M/TbTrrf3k_LI/AAAAAAAABMw/0FJR5HGDiZY/s1600/puntos_criticos2..png" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="104" src="http://3.bp.blogspot.com/-5uR7lTYmX1M/TbTrrf3k_LI/AAAAAAAABMw/0FJR5HGDiZY/s320/puntos_criticos2..png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: #274e13;">Se elige el intervalo (-) porque la inecuación tiene: \(<0\)</span><br />
<span class="Apple-style-span" style="color: #274e13;">luego se intersecta con \(x<0\) y</span><span class="Apple-style-span" style="color: #274e13;"> \(x>0\)</span><span class="Apple-style-span" style="color: #274e13;"> </span></td></tr>
</tbody></table>entonces la condición (3) es: \[x \in \left\langle { - \sqrt 2 {\kern 1pt} \,{\kern 1pt} ,\,\,0{\kern 1pt} } \right\rangle \cup \left\langle {0\,\,,\sqrt 2 } \right\rangle \] Ahora de lo hallado de (1), (2) y (3) la función \(f(x)\) está dada por: \[f(x) = \left\{ {\begin{array}{*{20}{c}}<br />
1&,&{x \in \left\langle { - \infty , - \sqrt 2 } \right\rangle \cup \left\langle {\sqrt 2 ,\infty } \right\rangle } \\<br />
0&,&{x \in \left\{ { - \sqrt 2 \,\,,0\,,\sqrt 2 } \right\}} \\<br />
{ - 1}&,&{x \in \left\langle { - \sqrt 2 ,0} \right\rangle \cup \left\langle {0,\sqrt 2 } \right\rangle }<br />
\end{array}} \right.\] Finalmente con los intervalos hallados en las 3 condiciones se construye el gráfico de \(f(x)\)<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-8nNJnvGR55g/TbTxOhZqYjI/AAAAAAAABM4/rqIGf5FJ_7M/s1600/funcion+signo+compuesta+con+valor+absoluto5..png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" src="http://3.bp.blogspot.com/-8nNJnvGR55g/TbTxOhZqYjI/AAAAAAAABM4/rqIGf5FJ_7M/s1600/funcion+signo+compuesta+con+valor+absoluto5..png" /></a></div></div></div><br />
<span class="Apple-style-span" style="color: #660000;">Segunda forma</span><br />
<div><input onclick="if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar Contenido';}else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Mostrar Contenido';}" type="button" value="Mostrar" /><br />
<div style="display: none;">Este problema también puede resolverse mediante una <b><i>forma gráfica</i></b>.<br />
En efecto, con los criterios para gráficos de la función cuadrática y valor absoluto partimos graficando \(y = \left| {{x^2} - 1} \right|\) está representada por:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-v-Qwsv6k7Vg/TbExmkIKr7I/AAAAAAAABLw/TzG3AAPMT9w/s1600/funcion+signo+compuesta+con+valor+absoluto.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" height="260" src="http://3.bp.blogspot.com/-v-Qwsv6k7Vg/TbExmkIKr7I/AAAAAAAABLw/TzG3AAPMT9w/s320/funcion+signo+compuesta+con+valor+absoluto.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">al restarle <b>1</b> al valor de esta función su gráfica se desplaza una unidad hacia abajo</div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/---jc4HSb6p4/TbEzIvzMubI/AAAAAAAABL0/5J1q4XTv5qY/s1600/funcion+signo+compuesta+con+valor+absoluto2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" height="239" src="http://2.bp.blogspot.com/---jc4HSb6p4/TbEzIvzMubI/AAAAAAAABL0/5J1q4XTv5qY/s320/funcion+signo+compuesta+con+valor+absoluto2.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">proyectando el gráfico de esta función sobre el eje <b style="font-style: italic;">x</b>, vemos claramente las partes negativas (en rojo) y las partes positivas (azul) y los puntos críticos (o ceros) (en naranja), lo que nos ayuda a ver cual será el gráfico de \(f(x)\) de acuerdo a la definición de función signo. En otras palabras se ha encontrado las 3 condiciones anteriormente calculadas.</div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-NijeT5BSafg/TbE4Ovcx7lI/AAAAAAAABL4/d6qOZzWfI5w/s1600/funcion+signo+compuesta+con+valor+absoluto3..png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" height="236" src="http://4.bp.blogspot.com/-NijeT5BSafg/TbE4Ovcx7lI/AAAAAAAABL4/d6qOZzWfI5w/s320/funcion+signo+compuesta+con+valor+absoluto3..png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"></div>Entonces el gráfico de la función \(f(x)\) es:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Su6HtjB3bJ0/TbE-fX-xesI/AAAAAAAABL8/DZR1lZLAA5E/s1600/funcion+signo+compuesta+con+valor+absoluto4..png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" height="293" src="http://1.bp.blogspot.com/-Su6HtjB3bJ0/TbE-fX-xesI/AAAAAAAABL8/DZR1lZLAA5E/s400/funcion+signo+compuesta+con+valor+absoluto4..png" width="400" /></a></div></div></div><br />
<b><span class="Apple-style-span" style="color: blue;">Bibliografía:</span></b><br />
Espinoza Ramos E, <i>Matemática Básica</i>. Editorial <i>Serv. Graf. J.J</i>. Lima - Perú. 2002.<br />
Figueroa García R, <i>Matemática Basica</i>. Editorial <i>América.</i> Lima - Perú. 1992.<br />
<br />
<div style="text-align: center;"><b><a href="http://bit.ly/itXo2Q" target="_blank">Descargar este artículo en versión de archivo PDF</a></b></div><div class="separator" style="clear: both; text-align: center;"><img border="0" src="http://1.bp.blogspot.com/-MK7xj7AnvJg/TbT798TWGTI/AAAAAAAABNA/zf4-7Zs1cfQ/s1600/snap416.jpg" /></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com3tag:blogger.com,1999:blog-2619881317292576475.post-76035186165753537142011-04-20T23:37:00.083-05:002017-04-28T04:40:50.231-05:00Aplicaciones de la función Máximo Entero o Mayor Entero en la vida real<br />
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El contenido de esta entrada se ha trasladado a este sitio</div>
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<a href="http://matematicauniversitaria.com/aplicaciones-del-maximo-entero-en-la-vida-real/"><b>http://matematicauniversitaria.com/aplicaciones-del-maximo-entero-en-la-vida-real/</b></a></div>
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Todas las demás entradas relacionadas con máximo entero</div>
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<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com2tag:blogger.com,1999:blog-2619881317292576475.post-87257570041159276552011-04-19T11:13:00.044-05:002017-06-01T02:05:50.786-05:00El Mayor Entero o Máximo Entero de un número real<a href="http://1.bp.blogspot.com/-xjD_4l2HWKs/TZdYU_8pOMI/AAAAAAAABIA/lz3zCPUROaI/s1600/snap410.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="100" src="https://1.bp.blogspot.com/-xjD_4l2HWKs/TZdYU_8pOMI/AAAAAAAABIA/lz3zCPUROaI/s200/snap410.jpg" width="120" /></a>El mayor entero de un número real \(x\), denotado por \(\left[\kern-0.17em\left[x \right]\kern-0.17em\right]\), es un número entero \(n\) el cual es el máximo de todos los números enteros menores o iguales que \(x\), es decir:<br />
<br />
\[\boldsymbol{\left[\kern-0.17em\left[x \right]\kern-0.17em\right] = n\qquad \leftrightarrow \qquad n = \text{máx} \left\{ {m \in {\Bbb Z}\;|\; m \leq x} \right\}}\]<br />
<div style="text-align: center;">
<a href="http://3.bp.blogspot.com/-Hfw7KItckVA/Tbm8ZA35pII/AAAAAAAABNU/JLnxunxzyYM/s1600/maximo+entero+-+mayor+entero.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" height="76" src="https://3.bp.blogspot.com/-Hfw7KItckVA/Tbm8ZA35pII/AAAAAAAABNU/JLnxunxzyYM/s640/maximo+entero+-+mayor+entero.PNG" width="555" /></a></div>
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<b><span class="Apple-style-span" style="color: #660000;">Ejemplos</span></b></div>
<div class="separator" style="clear: both;">
<br /></div>
<b><span class="Apple-style-span" style="color: #660000;">1.</span></b> \(\left[\kern-0.17em\left[ {2.4}\right]\kern-0.17em\right] = 2\) , puesto que \[ \begin{eqnarray}<br />
2 &=&\text{máx}\left\{ {m \in \mathbb{Z}\,\,\,\, | \,\,\,m \leq 2.4} \right\} \\<br />
&=&\text{máx}\left\{ {\ldots, - 1,\,\,0,\,\,1,\,\,2} \right\}<br />
\end{eqnarray}<br />
\] <i>tal como se aprecia en el gráfico de arriba.</i><br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">2.</span></b> \(\left[\kern-0.17em\left[ {-2.4}\right]\kern-0.17em\right] = -3\) , puesto que \[ \begin{eqnarray}<br />
-3 &=& \text{máx}\left\{ {m \in \mathbb{Z}\,\,\,\, | \,\,\,m \leq -2.4} \right\} \\<br />
&=& \text{máx}\left\{ {\ldots, -6,\,\,-5,\,\,-4,\,\,-3} \right\}<br />
\end{eqnarray}<br />
\]<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">3.</span></b> \(\left[\kern-0.17em\left[ {5}\right]\kern-0.17em\right] = 5\) , puesto que \[ \begin{eqnarray}<br />
5 &=& \text{máx}\left\{ {m \in \mathbb{Z}\,\,\,\, | \,\,\,m \leq 5} \right\} \\<br />
&=& \text{máx}\left\{ {\ldots, 2,\,\,3,\,\,4,\,\,5} \right\}<br />
\end{eqnarray}<br />
\]<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">4.</span></b> \(\left[\kern-0.17em\left[ {-4}\right]\kern-0.17em\right] = -4\) , puesto que \[ \begin{eqnarray}<br />
-4 &=& \text{máx}\left\{ {m \in \mathbb{Z}\,\,\,\, | \,\,\,m \leq -4} \right\} \\<br />
&=& \text{máx}\left\{ {\ldots, -7,\,\,-6,\,\,-5,\,\,-4} \right\}<br />
\end{eqnarray}<br />
\]<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">5.</span></b> \(\left[\kern-0.17em\left[ {\pi}\right]\kern-0.17em\right] = 3\) , puesto que \(\pi \approx 3.14\) \[ \begin{eqnarray}<br />
3 &=& \text{máx}\left\{ {m \in \mathbb{Z}\,\,\,\, | \,\,\,m \leq 3.14} \right\} \\<br />
&=& \text{máx}\left\{ {\ldots, 0,\,\,1,\,\,2,\,\,3} \right\}<br />
\end{eqnarray}<br />
\]<br />
<br />
<b><span class="Apple-style-span" style="color: #660000;">6.</span></b> \(\left[\kern-0.29em\left[{1-\sqrt{2}}\right]\kern-0.29em\right] = -1\) , puesto que \( \left(1-\sqrt{2}\right)\approx -0.41\) \[ \begin{eqnarray}<br />
-1 &=& \text{máx}\left\{ {m \in \mathbb{Z}\,\,\,\, | \,\,\,m \leq -0.41} \right\} \\<br />
&=& \text{máx}\left\{ {\ldots, -4,\,\,-3,\,\,-2,\,\,-1} \right\}<br />
\end{eqnarray}<br />
\]<br />
<div style="text-align: center;">
Pulse <b><a href="http://matematicauniversitaria.com/maximo-entero-o-mayor-entero-definicion-y-propiedades/">aquí</a> </b>para ver las</div>
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<b><span class="Apple-style-span" style="color: #660000;"><a href="http://matematicauniversitaria.com/maximo-entero-o-mayor-entero-definicion-y-propiedades/">>> Propiedades y ejercicios resueltos <<</a></span></b><br />
<b>de Máximo Entero</b><br />
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<b><br /></b></div>
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<b><span class="Apple-style-span" style="color: #660000;">Bibliografía.</span></b><br />
Figueroa G, Ricardo. <i>Matemática Básica</i>. Editorial América S.R.L., Lima-Perú, 1995.<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com1tag:blogger.com,1999:blog-2619881317292576475.post-36485006975946053052011-04-17T00:30:00.002-05:002017-05-16T05:48:00.798-05:00Ejercicios Resueltos de Física - EstáticaProblemas que se resuelven usando la primera y segunda condición de equilibrio.<br />
Institución: <b>I.E. San Agustín </b><br />
Chiclayo - Perú<br />
4to Año de secundaria<br />
<br />
<div style="text-align: center;">
<b><span class="Apple-style-span" style="color: red;">ESTÁTICA – EJERCICIOS RESUELTOS</span></b></div>
<br />
<span class="Apple-style-span" style="color: red;"><b>1.<span class="Apple-tab-span" style="white-space: pre;"> </span></b></span>La barra mostrada pesa 20N y está en reposo. Calcular la longitud de la barra, si además se sabe que la reacción en el apoyo B es 5N.<br />
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<a href="http://4.bp.blogspot.com/-rexa0YINa2k/TapvZ7wsK6I/AAAAAAAABKs/R9o4g95hrhs/s1600/image002.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-rexa0YINa2k/TapvZ7wsK6I/AAAAAAAABKs/R9o4g95hrhs/s1600/image002.gif" /></a></div>
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<a href="http://1.bp.blogspot.com/-xQh6zgfFjug/TapvaK3ABsI/AAAAAAAABKw/HciFNYQz7T4/s1600/image004.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-xQh6zgfFjug/TapvaK3ABsI/AAAAAAAABKw/HciFNYQz7T4/s1600/image004.gif" /></a></div>
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<span class="Apple-style-span" style="color: red;">Solución</span><br />
Sea \(L\) la longitud de la barra, entonces su diagrama de cuerpo libre (DCL) para la barra es:<br />
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<a href="http://3.bp.blogspot.com/-PTT0Caj7G2k/TapwJxes9-I/AAAAAAAABK0/fzLDMbENj9M/s1600/snap541.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="129" src="https://3.bp.blogspot.com/-PTT0Caj7G2k/TapwJxes9-I/AAAAAAAABK0/fzLDMbENj9M/s320/snap541.jpg" width="320" /></a></div>
De la segunda condición de equilibrio “<span class="Apple-style-span" style="color: blue;"><i>La suma de momentos con respecto al punto O es nula</i></span>”, se tiene:<br />
<div style="text-align: right;">
\[\sum {{M_0}} = 0\; \ldots \;(1.1)\] </div>
reemplazando el lado izquierdo de (1.1):<br />
\[M_0^{{F_1}} + M_0^W + M_0^{{F_2}} = 0\] calculando los momentos de \({F_1}\), \(W\) y \({F_2}\) respectivamente: \[\boxed{2 \cdot {F_1} - \frac{L}{2} \cdot W + L \cdot {F_2} = 0}\; \ldots \;(1.2)\] por dato del problema: \[\begin{gathered}<br />
{F_2} = 5\,\,{\text{N}} \\ <br />
W = 20\,\,{\text{N}} \\ <br />
\end{gathered} \] entonces: \[\begin{gathered}<br />
2 \cdot {F_1} - \frac{L}{2} \cdot 20 + L \cdot 5 = 0 \\ <br />
2{F_1} - 10L + 5L = 0 \\ <br />
\end{gathered} \] \[\boxed{2{F_1} - 5L = 0}\; \ldots \;(1.3)\] Por otro lado de la primera condición de equilibrio “la suma de todas las fuerzas en y es igual a cero” entonces:<br />
\[\sum {{F_y}} = 0\]\[\begin{gathered}<br />
{F_1} + {F_2} - W = 0 \\ <br />
{F_1} + 5 - 20 = 0 \\ <br />
{F_1} - 15 = 0 \\ <br />
\end{gathered} \] \[\left. {\underline {\, <br />
{{F_1} = 15} \,}}\! \right| \] Reemplazando \({F_1}\) en (1.3): \[\begin{gathered}<br />
2(15) - 5L = 0 \\ <br />
2(15) = 5L \\ <br />
2(3) = L \\ <br />
\left. {\underline {\, <br />
{L = 6\,\,{\text{m}}} \,}}\! \right| \\ <br />
\end{gathered} \] <br />
<div style="text-align: right;">
<span class="Apple-style-span" style="color: blue;"><b>Rpta: a)</b></span></div>
<div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com12tag:blogger.com,1999:blog-2619881317292576475.post-64987595138410052002011-04-12T00:23:00.013-05:002011-04-20T20:44:20.036-05:00Descarga de documentos<div class="separator" style="clear: both; text-align: center;"><img border="0" height="79" src="http://2.bp.blogspot.com/-k2wa3K0NXq8/TaPiyUaSPcI/AAAAAAAABKc/SSTj3go90OU/s200/snap416.jpg" width="90" /></div><span class="Apple-style-span" style="color: red; font-size: large; font-weight: bold;"></span><br />
<div style="text-align: center;"><style="clear: 1em;"="" 1em;="" float:="" left;="" margin-bottom:="" margin-right:=""><span class="Apple-style-span" style="color: red; font-size: large; font-weight: bold;"><span class="Apple-style-span" style="color: black; font-weight: normal;"><span class="Apple-style-span" style="color: red; font-weight: bold;">Para descargar los documentos de este sitio</span></span><span class="Apple-style-span" style="font-weight: normal;"><span class="Apple-style-span"><b>:<span class="Apple-style-span" style="color: black;"> </span></b></span></span></span></style="clear:></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763;"><br />
</span><br />
<span class="Apple-style-span" style="color: #073763;"><b></b>1º) En la ventana del documento póngalo<b> </b></span><span class="Apple-style-span" style="color: #073763;">en modo <i><b>pantalla completa</b></i> </span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763;">haciendo clic en el boton:</span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><img border="0" src="http://3.bp.blogspot.com/-W5t0c5zdDZY/TaNh_9Q2xBI/AAAAAAAABKA/nsiD3Z9E-_0/s1600/snap474.jpg" /></div></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763;">2º)<span class="Apple-style-span" style="color: #073763;"> P</span>ulse: <b><i>Menú</i></b> luego en <i><b>Download or Share</b></i> (<i>descargar o compartir)</i> </span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763; font-size: x-small;"></span></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="color: #073763; font-size: x-small;"><a href="http://4.bp.blogspot.com/-mOY8mB4YU0Y/TaPWR_tWhfI/AAAAAAAABKE/Ye3F6g8x1i4/s1600/Snap1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="94" src="http://4.bp.blogspot.com/-mOY8mB4YU0Y/TaPWR_tWhfI/AAAAAAAABKE/Ye3F6g8x1i4/s200/Snap1.jpg" width="180" /></a></span></div><div class="separator" style="clear: both; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763;">3º) Hacer clic en en <b style="font-style: italic;">Download </b><i>(</i>o<i> descargar)</i></span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-fkG9UjX9Zlg/TaPWoyX6OiI/AAAAAAAABKI/pJBDOskARCc/s1600/snap475.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-fkG9UjX9Zlg/TaPWoyX6OiI/AAAAAAAABKI/pJBDOskARCc/s1600/snap475.jpg" /></a></div><span class="Apple-style-span" style="color: #073763;"><i><b><br />
</b></i></span></div><div class="separator" style="clear: both; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"></div><div class="separator" style="clear: both; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763;">4ª) En la nueva ventana clic en <i><b>Pick location </b>(elegir ubicación)</i></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Zwq9t8lcAFY/TaPXTnY9D7I/AAAAAAAABKM/_1ETPqngYKY/s1600/Snap3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="138" src="http://1.bp.blogspot.com/-Zwq9t8lcAFY/TaPXTnY9D7I/AAAAAAAABKM/_1ETPqngYKY/s320/Snap3.jpg" width="320" /></a></div><div class="separator" style="clear: both; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"><span class="Apple-style-span" style="color: #073763;"><b><i><br />
</i></b></span></div><div class="separator" style="clear: both; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-align: center;"></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com0tag:blogger.com,1999:blog-2619881317292576475.post-48677893106909335452011-04-11T14:56:00.032-05:002011-04-12T02:00:42.899-05:00Ejercicios Resueltos de Matemática Básica<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-CrVcEMh2rPE/TaPbwQu6TCI/AAAAAAAABKQ/xnzBUBhfU98/s1600/snap476.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="130" src="http://2.bp.blogspot.com/-CrVcEMh2rPE/TaPbwQu6TCI/AAAAAAAABKQ/xnzBUBhfU98/s200/snap476.jpg" width="110" /></a></div>Después de varias horas de desarrollo de una práctica que me encargaron, paso a compartirla en formato PDF escaneado.<br />
<br />
<span class="Apple-style-span" style="color: #660000;">Área: </span>Pre-Cálculo<br />
<span class="Apple-style-span" style="color: #660000;">Nº de Ejercicios:</span> 21<br />
<span class="Apple-style-span" style="color: #660000;">Universidad: </span>Santo Toribio de Mogrovejo (Chiclayo-Perú)<br />
<span class="Apple-style-span" style="color: #660000;"><b>Contenido:</b></span><br />
<b><span class="Apple-style-span" style="color: #073763;">Funciones Reales de Variable Real</span></b><span class="Apple-style-span" style="color: #0c343d;"> (\(f:\mathbb{R} \to \mathbb{R}\))</span><br />
<ul><li><span class="Apple-style-span" style="color: #0c343d;">Dominio, rango y gráfica de funciones: </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Lineales \(f(x) = ax + b\), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Cuadráticas \(f(x) = a{x^2} + bx + c\), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Raíz cuadrada \(f(x) = \sqrt x \), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Valor absoluto \(f(x) = |x|\), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;">Máximo entero \(f(x) = [\kern-0.15em[ x ]\kern-0.15em] \)</span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Composición de funciones</span><span class="Apple-style-span" style="color: #0c343d;"> \[(f \circ g)(x)=f\left( {g(x)} \right)\]</span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Aplicaciones a la vida real. (cálculo de funciones que modelen a un fenómeno en particular)</span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Algebra de funciones:</span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Adición (\(f + g\)), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Sustracción (\(f - g\)), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">Multiplicación (\(f \cdot g\)), </span></li>
<li><span class="Apple-style-span" style="color: #0c343d;"></span><span class="Apple-style-span" style="color: #0c343d;">División </span><span class="Apple-style-span" style="color: #0c343d;">\(\left( {\frac{f}{g}} \right)\)</span></li>
</ul><div style="text-align: center;"><b><span class="Apple-style-span" style="color: #0c343d;">Previsualización</span></b><br />
<span class="Apple-style-span" style="color: blue;"><b></b><span class="Apple-style-span" style="font-size: x-small;">Pulsa en el botón <img border="0" src="http://2.bp.blogspot.com/-hIhINdMSPjU/TaP0gWlnV0I/AAAAAAAABKg/xFF2oa1TZO0/s1600/Snap10.jpg" /> para verlo en pantalla completa</span></span><br />
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Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com3tag:blogger.com,1999:blog-2619881317292576475.post-21871476499383218432011-04-04T03:52:00.091-05:002011-04-06T22:58:45.725-05:00Fórmulas Clásicas de Geometría ElementalEnumero aquí las formulas más usuales de la geometría elemental.<br />
<div style="text-align: left;"><span class="Apple-style-span" style="color: #660000;"><br />
</span></div><div style="text-align: left;"></div><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-wpeZJBU6vRA/TZlzOhgmc7I/AAAAAAAABJk/FkbsL_-FhfU/s1600/snap421.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="135" src="http://3.bp.blogspot.com/-wpeZJBU6vRA/TZlzOhgmc7I/AAAAAAAABJk/FkbsL_-FhfU/s200/snap421.jpg" width="170" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: blue;">Triángulo Rectángulo</span></td></tr>
</tbody></table><b><span class="Apple-style-span" style="color: #660000;">1. Teorema de Pitágoras</span></b><br />
<i>"En todo triangulo rectángulo el cuadrado de la hipotenusa es igual a la suma de los cuadrados de los catetos"</i> <span class="Apple-style-span" style="color: blue;">\[\boldsymbol{c^2 = a^2 + b^2}\]</span> donde : \(c\) := hipotenusa y \(a,b\) := catetos<br />
<div style="text-align: center;"><div style="text-align: left;"><span class="Apple-style-span" style="color: blue;"><b><span class="Apple-style-span" style="color: #660000;">2. Áreas de un triángulo.</span></b></span></div><div style="text-align: left;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-8CqlBiA4VZQ/TZlwWaKNw5I/AAAAAAAABJg/jTth4niNdoo/s1600/snap420.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="94" src="http://4.bp.blogspot.com/-8CqlBiA4VZQ/TZlwWaKNw5I/AAAAAAAABJg/jTth4niNdoo/s200/snap420.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: blue;">Cualquier triángulo</span></td></tr>
</tbody></table><b><span class="Apple-style-span" style="color: #990000;">2.1 Aréa I.-</span><span class="Apple-style-span" style="color: blue;"> </span>"</b><i>En todo triángulo el área de su superficie \(A\) es igual al producto de su base \(b\) por su correspondiente altura \(h\) dividida entre 2</i>". <span class="Apple-style-span" style="color: blue;"><span class="Apple-style-span" style="color: blue;">\[\boldsymbol{A_{\scriptscriptstyle \Delta}=\frac{bh}{2}}\] </span><span class="Apple-style-span" style="color: #134f5c;"><b>Nota:</b> <i>Si el triángulo es rectángulo, su área es el producto de sus catetos sobre dos</i>.</span></span></div><div style="text-align: left;"><div style="text-align: left;"><span class="Apple-style-span" style="color: #990000;"><b>2.2 Área II.-</b> </span>"<i>En<b> </b>todo triángulo el área de su superficie \(A\) es igual a la raíz cuadrada del producto de su semiperímetro \(p\) con las diferencias del mismo con cada uno de sus lados</i>".<span class="Apple-style-span" style="color: blue;"> </span></div><div style="text-align: center;"><span class="Apple-style-span" style="color: blue;">\[\boldsymbol{A_{\scriptscriptstyle \Delta}=\sqrt{p(p-a)(p-b)(p-c)}}\] </span><span class="Apple-style-span" style="color: #073763;">(fórmula de Herón)</span></div><span class="Apple-style-span" style="color: blue;"><span class="Apple-style-span" style="color: blue;"> </span></span></div><div style="text-align: left;">siendo el semiperímetro: \(p=\frac{a+b+c}{2}\) y por el postulado de Euclides: \(\alpha+\beta+\gamma=180º.\)</div><div style="text-align: left;"><b><span class="Apple-style-span" style="color: #990000;">2.3 Área III.-</span></b><span class="Apple-style-span" style="color: blue;"> </span>"<i>En<b> </b>todo triángulo el área de su superficie \(A\) es igual al semiproducto del seno de uno de sus ángulos con el producto de las longitudes de los lados que forman dicho ángulo</i>".</div><div style="text-align: left;"><span class="Apple-style-span" style="color: blue;"><span class="Apple-style-span" style="color: blue;"></span><span class="Apple-style-span" style="color: blue;">\[\boldsymbol{A_{\scriptscriptstyle \Delta}=\frac{ab\sin\gamma}{2}\;,\quad A_{\scriptscriptstyle \Delta}=\frac{bc\sin\alpha}{2}\;,\quad A_{\scriptscriptstyle \Delta}=\frac{ac\sin\beta}{2}}\]</span></span></div><div style="text-align: left;"><span class="Apple-style-span" style="color: blue;"><b><span class="Apple-style-span" style="color: #660000;">3. Círculo y circunferencia.</span></b></span></div><div style="text-align: center;"><span class="Apple-style-span" style="clear: left; color: blue; float: left; margin-bottom: 1em; margin-right: 1em;"></span><br />
<div style="text-align: left;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img border="0" height="100" src="http://2.bp.blogspot.com/-jEr9DBTmog0/TZmEdgj_ChI/AAAAAAAABJo/uzh2o1SBzgg/s200/snap423.jpg" style="margin-left: auto; margin-right: auto;" width="100" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: blue;">Círculo de radio \(r\)</span></td></tr>
</tbody></table><i>"El <b>área del círculo</b> </i>(<i>A</i>)<i>, es directamente proporcional al cuadrado su radio".</i><span class="Apple-style-span" style="color: blue;"> </span><span class="Apple-style-span" style="color: blue;">\[\boldsymbol{A_{\scriptscriptstyle C}=\pi r^2}\] </span><i>"La<span class="Apple-style-span" style="color: blue;"> </span><b>longitud de la circunferencia</b></i> (<i>L</i>) es directamente proporcional a su radio.<span class="Apple-style-span" style="color: blue;"> <span class="Apple-style-span" style="color: blue;">\[\boldsymbol{L_{\scriptscriptstyle C}=2\pi r}\]</span> </span><i>donde \(r\) := radio y \(O\) es el centro del círculo".</i><br />
<b><span class="Apple-style-span" style="color: #660000;"><span class="Apple-style-span" style="color: black; font-weight: normal;"><b><span class="Apple-style-span" style="color: #660000;">4. Sector circular.</span></b></span></span></b><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-bhGeqDuc7Bc/TZoXi5GoOeI/AAAAAAAABJ8/nqZ3MnBeQPU/s1600/snap426.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="150" src="http://1.bp.blogspot.com/-bhGeqDuc7Bc/TZoXi5GoOeI/AAAAAAAABJ8/nqZ3MnBeQPU/s200/snap426.jpg" width="129" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="color: blue;">Sector circular de radio \(r\)</span></td></tr>
</tbody></table><span class="Apple-style-span"><i>"El área del sector circular es el semiproducto del cuadrado de su radio por el valor numérico de su ángulo medido en radianes"</i></span><i><span class="Apple-style-span" style="color: blue;">\[\boldsymbol{A_{\scriptscriptstyle SC}=\frac{\theta r^2}{2}}\]</span><br />
<b><span class="Apple-style-span" style="color: #660000;">5. Ley de Cosenos.</span></b><br />
<i>"El cuadrado de la longitud de cualquier lado de un triángulo oblicuángulo es igual a la suma de los cuadrados de los otros dos lados menos el doble producto de las longitudes de los mismos por el coseno del ángulo que éstos forman entre sí".</i><br />
</i><br />
<div class="separator" style="clear: both; text-align: center;"><i><a href="http://4.bp.blogspot.com/-oph3SYsghCE/TZnGUrW3iiI/AAAAAAAABJ4/QyHqYtz7cjI/s1600/snap425.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-oph3SYsghCE/TZnGUrW3iiI/AAAAAAAABJ4/QyHqYtz7cjI/s1600/snap425.jpg" /></a></i></div><i>\[\begin{array}{c}<br />
\boldsymbol{{c^2} = {a^2} + {b^2} - 2ab\cos \gamma} \\<br />
\boldsymbol{{b^2} = {a^2} + {c^2} - 2ac\cos \beta} \\<br />
\boldsymbol{{a^2} = {b^2} + {c^2} - 2bc\cos \alpha} <br />
\end{array}\]<br />
</i><br />
<div style="text-align: left;"><i><i><span class="Apple-style-span" style="color: blue;"></span></i></i><br />
<i><i><span class="Apple-style-span" style="color: blue;"> </span></i></i></div><div style="text-align: left;"><i><i><span class="Apple-style-span" style="color: blue;"><b><span class="Apple-style-span" style="color: lime; font-size: large;"><i>[continua....]</i></span></b></span></i></i></div></div></div></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com1tag:blogger.com,1999:blog-2619881317292576475.post-73665876264563907592011-04-01T11:38:00.010-05:002011-04-29T14:04:50.825-05:00Símbolos matemáticos en tu correo electrónico<div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><strong><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">GmailTeX</span></span></strong><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> es un plugin que añade la capacidad del procesador de texto científico \(\LaTeX\) tu cuenta de Gmail, de modo que usted puede enviar y recibir correo electrónico texto en LaTeX.</span> Aqui la<span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> imagen lo muestra mejor, así que en pocas palabras esto es lo que hace:</span></span></div><div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><div style="text-align: center;"><img height="413" src="http://alexeev.org/gmailtex.png" width="480" /></div></div><div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">GmailTeX trabaja en todos los navegadores modernos: Mozilla Firefox (versión 3.6 y posteriores), Google Chrome, Opera, Safari (versión 5), y Microsoft Internet Explorer (versión 9).</span></span></div><div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">GmailTeX utiliza </span></span><a href="http://www.mathjax.org/" style="color: #4486c7; text-decoration: none;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">MathJax</span></span></a><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> como su motor de TeX.</span></span></div><div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Para LaTeX en el chat de Gmail, puede utilizar el </span></span><a href="http://alexeev.org/gmailchattex.html" style="color: #4486c7; text-decoration: none;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">GmailChatTeX</span></span></a><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> userscript.</span> <span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Para ver de látex en las páginas web donde un diseñador de páginas web no proporcionan un servidor-junto al camino de hacerlo (por ejemplo, arXiv.org, front.math.ucdavis.edu), puede utilizar </span></span><a href="http://alexeev.org/display-latex2.html" style="color: #4486c7; text-decoration: none;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">la pantalla-latex2</span></span></a><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"> userscript.</span></span></div><div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><br />
</span></span></div><div style="font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 21px; margin-top: 0px;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-color: transparent; background-image: initial; background-origin: initial; border-bottom-width: 0px; border-color: initial; border-left-width: 0px; border-right-width: 0px; border-style: initial; border-top-width: 0px; display: inline; font-size: 12px; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: initial; outline-width: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; vertical-align: baseline;">Más información <a href="http://alexeev.org/gmailtex.html" target="_blank"><b><span class="Apple-style-span" style="color: blue;">Aqui</span></b></a></span></span></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com0tag:blogger.com,1999:blog-2619881317292576475.post-14425635380372011892011-01-07T05:02:00.013-05:002011-04-29T14:00:22.067-05:00Video Repaso de la Guía de Lógica Inferencial<a href="http://1.bp.blogspot.com/-16FLdIP5Ffo/TZfLmsBlbeI/AAAAAAAABIw/bHwB141GA4Y/s1600/play_video-logica.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-16FLdIP5Ffo/TZfLmsBlbeI/AAAAAAAABIw/bHwB141GA4Y/s1600/play_video-logica.jpg" /></a>Habiendo hecho una <a href="http://aportemath.blogspot.com/2010/12/guia-de-logica-inferencial.html" target="_blank">guía de aprendizaje</a> de Logica Inferencial, pensé que sería bueno colgar videos explicativa de la misma. Ya estoy casi por terminar, en la próxima semana estaré colgando más ejercicios resueltos sobre método directo e indirecto, en vídeo claro está.<br />
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<div style="text-align: center;"><object height="385" width="480"><param name="movie" value="http://www.youtube.com/p/701664F8A92E48F7?hl=es_ES&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/p/701664F8A92E48F7?hl=es_ES&fs=1" type="application/x-shockwave-flash" width="480" height="385" allowscriptaccess="always" allowfullscreen="true"></embed></object></div><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com0tag:blogger.com,1999:blog-2619881317292576475.post-54605756721715672672010-12-30T21:41:00.009-05:002011-04-29T13:59:40.938-05:00Guía de Lógica Inferencial<div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" height="158" src="http://3.bp.blogspot.com/-pTQpvNbfdyg/TZeOaHbzQoI/AAAAAAAABIE/p2oaQpsowDw/s200/snap411.jpg" width="200" /></div>Después de poco más de un mes he terminado de elaborar, usando la clase Beamer de Latex, una guía teórica de Inferencia lógica. Las inferencias aquí se refiere a razonamientos deductivos en general. La última novedad que le agregué es enlaces a una serie de video resúmenes con gran calidad y efectos para maximizar la claridad de exposición (ver dentro del documento: diapositivas 'video resumen'). encontraremos aquí algunos conceptos y modus operandi de las reglas de validación. Se agradece su gentil opinión al respecto.<br />
<a href="http://es.scribd.com/doc/46055466/Logica-Inferencial" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto; text-decoration: underline;" title="View Logica_Inferencial on Scribd">Logica_Inferencial</a><iframe class="scribd_iframe_embed" data-aspect-ratio="1.33115468409586" data-auto-height="true" frameborder="0" height="600" id="doc_22794" scrolling="no" src="http://www.scribd.com/embeds/46055466/content?start_page=1&view_mode=slideshow&access_key=key-1vsw1lpgb1ryyqt97h1z" width="100%"></iframe><script type="text/javascript">
(function() { var scribd = document.createElement("script"); scribd.type = "text/javascript"; scribd.async = true; scribd.src = "/javascripts/embed_code/inject.js?1300826227"; var s = document.getElementsByTagName("script")[0]; s.parentNode.insertBefore(scribd, s); })();
</script><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com6tag:blogger.com,1999:blog-2619881317292576475.post-34010156468923001882010-12-14T09:08:00.004-05:002011-04-02T19:32:21.170-05:00Tecnicas de Integración de Funciones Racionales Irracionales e Irracionales Trigonométricas<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-zFyHRjXAXBo/TZfADEc6hgI/AAAAAAAABIs/8FhqA1PvQyo/s1600/snap413.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-zFyHRjXAXBo/TZfADEc6hgI/AAAAAAAABIs/8FhqA1PvQyo/s1600/snap413.jpg" /></a></div>Varios ejercicios resueltos sobre integral indefinida.<br />
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<a href="http://www.scribd.com/doc/45263304/Matematica-II-Tecnicas-de-Integracion" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto; text-decoration: underline;" title="View Matemática II - Tecnicas de Integración on Scribd">Matemática II - Tecnicas de Integración</a> <object data="http://d1.scribdassets.com/ScribdViewer.swf" height="600" id="doc_207422223127472" name="doc_207422223127472" style="outline: none;" type="application/x-shockwave-flash" width="100%"><span class="Apple-tab-span" style="white-space:pre"> </span><param name="movie" value="http://d1.scribdassets.com/ScribdViewer.swf"><span class="Apple-tab-span" style="white-space:pre"> </span><param name="wmode" value="opaque"><span class="Apple-tab-span" style="white-space:pre"> </span><param name="bgcolor" value="#ffffff"><span class="Apple-tab-span" style="white-space:pre"> </span><param name="allowFullScreen" value="true"><span class="Apple-tab-span" style="white-space:pre"> </span><param name="allowScriptAccess" value="always"><span class="Apple-tab-span" style="white-space:pre"> </span><param name="FlashVars" value="document_id=45263304&access_key=key-2a6v99y14j7j49qgrjti&page=1&viewMode=list"><span class="Apple-tab-span" style="white-space:pre"> </span><embed id="doc_207422223127472" name="doc_207422223127472" src="http://d1.scribdassets.com/ScribdViewer.swf?document_id=45263304&access_key=key-2a6v99y14j7j49qgrjti&page=1&viewMode=list" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="600" width="100%" wmode="opaque" bgcolor="#ffffff"></embed> <span class="Apple-tab-span" style="white-space:pre"> </span></object><span class="Apple-tab-span" style="white-space: pre;"> </span><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com2tag:blogger.com,1999:blog-2619881317292576475.post-73650166489727956482010-12-12T09:41:00.013-05:002011-04-06T14:44:46.609-05:00Cómo se calcula la trasformada de Laplace de Coseno(t) al Cubo<div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" height="166" src="http://4.bp.blogspot.com/-w2SbfZzuLvE/TZfeh2qizSI/AAAAAAAABJI/D9ROu_QYcXQ/s200/snap414.jpg" width="200" /></div><a href="http://www.scribd.com/doc/45146652/Calculando-la-trasformada-de-Laplace-de-Coseno-t-al-Cubo" style="display: block; font-family: Helvetica, Arial, sans-serif; font-size: 14px; font-style: normal; font-variant: normal; line-height: normal; margin-bottom: 6px; margin-left: auto; margin-right: auto; margin-top: 12px;" title="View Calculando la trasformada de Laplace de Coseno(t) al Cubo on Scribd"><b>La trasformada de Laplace de \(cos^3(t)\)</b></a> <object data="http://d1.scribdassets.com/ScribdViewer.swf" height="600" id="doc_389366786683236" name="doc_389366786683236" style="outline: none;" type="application/x-shockwave-flash" width="100%"> <param name="movie" value="http://d1.scribdassets.com/ScribdViewer.swf"><param name="wmode" value="opaque"><param name="bgcolor" value="#ffffff"><param name="allowFullScreen" value="true"><param name="allowScriptAccess" value="always"><param name="FlashVars" value="document_id=45146652&access_key=key-1iqy3u38c18agrzs4x70&page=1&viewMode=list"><embed id="doc_389366786683236" name="doc_389366786683236" src="http://d1.scribdassets.com/ScribdViewer.swf?document_id=45146652&access_key=key-1iqy3u38c18agrzs4x70&page=1&viewMode=list" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="600" width="100%" wmode="opaque" bgcolor="#ffffff"></embed> </object><div class="blogger-post-footer">Thanks for reading.
Copyright 2011. Aportes Matemáticos, by MathSalomon.</div>salin1http://www.blogger.com/profile/10209204897471210415noreply@blogger.com0