{"id":311,"date":"2025-10-21T15:15:56","date_gmt":"2025-10-21T13:15:56","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=311"},"modified":"2025-10-15T19:13:01","modified_gmt":"2025-10-15T17:13:01","slug":"mathbio-sistemas-autovalores-y-funciones-reales-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=311","title":{"rendered":"MathBio: Sistemas, Autovalores y Funciones reales con maxima"},"content":{"rendered":"<h2>Sistemas de ecuaciones<\/h2>\n<p>En pasados d\u00edas vimos c\u00f3mo resolv\u00edamos sistemas de ecuaciones y, en particular, el problema de m\u00ednimos cuadrados mediante matrices. Hoy bordaremos estos problemas utilizando maxima.<\/p>\n<ul>\n<li><strong>linsolve<\/strong>(\\([eq_1, &#8230;, eq_m], [x_1, &#8230;, x_n]\\)): Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations. <\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver el sistema de ecuaciones \\[\\begin{matrix}2x+y-z=1 \\\\x-3y+2z=1 \\\\ -x+2y-4z=2\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1a() {\n  var htmlShow1a = document.getElementById(\"html-show1a\");\n  if (htmlShow1a.style.display === \"none\") {\n    htmlShow1a.style.display = \"block\";\n  } else {\n    htmlShow1a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1a\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_sist01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>El sistema puede tener infinitas soluciones, en cuyo caso estas se dan en forma param\u00e9trica:<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver el sistema de ecuaciones \\[\\begin{matrix}3x-y-z+t=2 \\\\ x+y-2z-5t=1 \\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2a() {\n  var htmlShow2a = document.getElementById(\"html-show2a\");\n  if (htmlShow2a.style.display === \"none\") {\n    htmlShow2a.style.display = \"block\";\n  } else {\n    htmlShow2a.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_sist02.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>En muchos casos los sistemas no tiene soluci\u00f3n; si embargo, podremos optar por encontrar una soluci\u00f3n por m\u00ednimos cuadrados.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> El pasado d\u00eda vimos una tabla en base a la ley que nos dice que <strong>el volumen de un gas a presi\u00f3n constante crece de manera lineal con respecto a la temperatura<\/strong>. En los experimentos, a Jacques Charles la experimentaci\u00f3n le proporcion\u00f3 los siguientes datos:<br \/>\n\\[\\begin{array}{l|cccc}<br \/>\nT &#038; -40 &#038; -20 &#038; 0 &#038;  \\\\ \\hline<br \/>\nV &#038; 19.1482 &#038; 20.7908 &#038;  22.4334  \\\\ \\hline<br \/>\nT &#038; 20 &#038; 40&#038; 60&#038; 80 \\\\ \\hline<br \/>\nV &#038;  24.0760 &#038;  25.7186 &#038; 27.3612 &#038; 29.0038 \\\\ \\hline<br \/>\n\\end{array}\\] donde un mol de hidr\u00f3geno se mantiene a una presi\u00f3n constante de una atm\u00f3sfera; siendo  el volumen \\(V\\) aproximado y medido en litros y la temperatura \\(T\\) grados Celsius. Determinar una recta que relacione el volumen \\(V\\) y la temperatura \\(T\\) mediante una relaci\u00f3n lineal.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a1d() {\n  var htmlShow3a1d = document.getElementById(\"html-show3a1d\");\n  if (htmlShow3a1d.style.display === \"none\") {\n    htmlShow3a1d.style.display = \"block\";\n  } else {\n    htmlShow3a1d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3a1d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a1d\" style=\"display: none;\">\nPara resolver el problema utilizaremos el m\u00e9todo de m\u00ednimos cuadrados.<br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">T<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">40<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">40<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">60<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">80<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">V<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">19<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">1482<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">20<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">7908<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">22<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">4334<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">24<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">0760<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">25<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">7186<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">27<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">3612<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">29<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_number\">0038<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Buscamos la ecuaci\u00f3n V=mT+n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">T<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}-40 &amp; 1\\\\-20 &amp; 1\\\\0 &amp; 1\\\\20 &amp; 1\\\\40 &amp; 1\\\\60 &amp; 1\\\\80 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X_apr<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">invert<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">V<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0.08213\\\\22.4334\\end{bmatrix}\\]<\/p>\n<p>Ahora ya tenemos la relaci\u00f3n lineal: \\[V=0.08213T+22.4334\\]\n<\/p><\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Encontrar la par\u00e1bola que mejor se ajuste a los puntos [1,5.5],[-1,15.5],[3,11.2][-2,26.4]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a1() {\n  var htmlShow3a1 = document.getElementById(\"html-show3a1\");\n  if (htmlShow3a1.style.display === \"none\") {\n    htmlShow3a1.style.display = \"block\";\n  } else {\n    htmlShow3a1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3a1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a1\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGminimos01.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h2>Autovalores y autovectores<\/h2>\n<p>Recordemos que los vectores propios, eigenvectores o autovectores de una matriz son los vectores no nulos que, cuando son multiplicados por la matriz, dan lugar a un m\u00faltiplo escalar de s\u00ed mismos (con lo que no cambian su direcci\u00f3n). Este escalar \\({\\displaystyle \\lambda }\\) recibe el nombre valor propio, autovalor o valor caracter\u00edstico.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1ntos autovalores distintos tiene la matriz dada? \\[\\begin{bmatrix}1&#038;0&#038;0&#038;1\\\\0&#038;1&#038;2&#038;-1\\\\ 0&#038;1&#038;-1&#038;0\\\\ 0&#038;2&#038;0&#038;-1\\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4a() {\n  var htmlShow4a = document.getElementById(\"html-show4a\");\n  if (htmlShow4a.style.display === \"none\") {\n    htmlShow4a.style.display = \"block\";\n  } else {\n    htmlShow4a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }{{\\left( -x-1\\right) }^{2}} {{\\left( 1-x\\right) }^{2}}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\left[ x=-1{,}x=1\\right] \\]<\/p>\n<\/div>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es la suma de las normas de los autovectores de la matriz anterior?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5a() {\n  var htmlShow5a = document.getElementById(\"html-show5a\");\n  if (htmlShow5a.style.display === \"none\") {\n    htmlShow5a.style.display = \"block\";\n  } else {\n    htmlShow5a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5a\" style=\"display: none;\">\n<!-- Section cell --><\/p>\n<div class=\"section\">\n<p>1 Valores propios<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">\u03bb<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }{{\\left( -\\lambda -1\\right) }^{2}} {{\\left( 1-\\lambda \\right) }^{2}}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">\u03bb<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ \\lambda =-1\\operatorname{,}\\lambda =1\\right] \\]<\/p>\n<p><!-- Section cell --><\/p>\n<div class=\"section\">\n<p>2 Autovalor -1<\/p>\n<\/div>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Veamos los autovectores para el autovalor -1.<\/p>\n<p>Primero hacemos el sistema:<\/p><\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}2 x+t\\\\2 z+2 y-t\\\\y\\\\2 y\\end{bmatrix}=\\begin{bmatrix}0\\\\0\\\\0\\\\0\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">De las ecuaciones anteriores deducimos que y=0. Por tanto, nos quedan, 2x+t=0 y 2z-t=0.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ x=-\\frac{{\\mathrm{\\% r1}}}{2}\\operatorname{,}y={\\mathrm{\\% r2}}\\operatorname{,}z=\\frac{{\\mathrm{\\% r1}}}{2}\\operatorname{,}t={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Observar que como y=0, el par\u00e1metro %r2 no es tal, ya que siempre vale cero. Luego tendremos un s\u00f3lo par\u00e1metrom, que nos proporciona un vector proporcional a [-1,0,1,2]<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%r1<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%r2<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\left[ -1\\operatorname{,}0\\operatorname{,}1\\operatorname{,}2\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Verifiquemos:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1\\\\0\\\\-1\\\\-2\\end{bmatrix}\\]<\/p>\n<p><!-- Section cell --><\/p>\n<div class=\"section\">\n<p>3 Autovecctor 1<\/p>\n<\/div>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Repitamos el proceso con el autovalor 1<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}t\\\\2 z-t\\\\y-2 z\\\\2 y-2 t\\end{bmatrix}=\\begin{bmatrix}0\\\\0\\\\0\\\\0\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora planteamos el sistema: t=0, 2z-t=0, y-2z=0, 2y-2t=0.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\]\\[solve: dependent equations eliminated: (4)\\]<\/p>\n<p>\\[\\operatorname{ }\\left[ x={\\mathrm{\\% r3}}\\operatorname{,}y=0\\operatorname{,}z=0\\operatorname{,}t=0\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">La soluci\u00f3n, pues, ser\u00e1 [1,0,0,0]. Veamoslo<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1\\\\0\\\\0\\\\0\\end{bmatrix}\\]<\/p>\n<p>Como vemos, solo nos queda calcular la norma de [-1,0,1,2] y [1,0,0,0].\n<\/p><\/div>\n<hr \/>\n<h2>Funciones<\/h2>\n<p>Una funci\u00f3n ordinaria es aquella que ha sido construida mediante cualquiera de los m\u00e9todos <strong>define<\/strong> o <strong>:=<\/strong> y que es invocada utilizando par\u00e9ntesis.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determina la suma de los valores de la funci\u00f3n \\(f(x)=x\\,e^{1-x^2}\\) para \\(x_i=\\frac{i}{4}\\) para \\(i\\in\\{1,\\ldots,5\\}\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1s() {\n  var htmlShow1s = document.getElementById(\"html-show1s\");\n  if (htmlShow1s.style.display === \"none\") {\n    htmlShow1s.style.display = \"block\";\n  } else {\n    htmlShow1s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1s\" style=\"display: none;\">\n<!-- Code cell --> <\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"> <span class=\"prompt\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">%e<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_number\">1<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">val<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">makelist<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">i<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span>),<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">sum<\/span>(<span class=\"code_variable\">val<\/span>[<span class=\"code_variable\">i<\/span>],<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>),<span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(val)<\/mtext> <\/mtd> <mtd><mo>[<\/mo><mfrac><msup> <mi>%e<\/mi> <mfrac><mn>15<\/mn><mn>16<\/mn> <\/mfrac><\/msup><mn>4<\/mn><\/mfrac><mo>,<\/mo><mfrac><msup> <mi>%e<\/mi> <mfrac><mn>3<\/mn><mn>4<\/mn> <\/mfrac><\/msup><mn>2<\/mn><\/mfrac><mo>,<\/mo><mfrac><mrow> <mn>3<\/mn> <mo>\u2062<\/mo> <msup><mi>%e<\/mi><mfrac><mn>7<\/mn><mn>16<\/mn><\/mfrac> <\/msup><\/mrow><mn>4<\/mn><\/mfrac><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo><mfrac><mrow> <mn>5<\/mn> <mo>\u2062<\/mo> <msup><mi>%e<\/mi><mrow><mi>\u2212<\/mi><mfrac> <mn>9<\/mn> <mn>16<\/mn><\/mfrac><\/mrow> <\/msup><\/mrow><mn>4<\/mn><\/mfrac><mo>]<\/mo> <\/mtd><\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o3) <\/mtext> <\/mtd> <mtd><mn>4.570748627711323<\/mn> <\/mtd><\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<ul>\n<li><strong>define<\/strong>(\\(f(x_1,\\ldots, x_n)\\), <em>expr<\/em>): Define una funci\u00f3n de nombre \\(f\\) con argumentos \\(x_1,\\ldots, x_n\\) y cuerpo <em>expr<\/em>. <strong>define<\/strong> eval\u00faa siempre su segundo argumento, a menos que se indique lo contrario con el operador de comilla simple. <\/li>\n<\/ul>\n<p>Esta funci\u00f3n la utilizaremos m\u00e1s adelante.<\/p>\n<h2>Gr\u00e1fica de una funci\u00f3n<\/h2>\n<ul>\n<li><strong>wxplot2d<\/strong>(expresi\u00f3n, [variable,m\u00ednimo,m\u00e1ximo],opciones): La funci\u00f3n <strong>wxplot2d<\/strong> representa uno o m\u00e1s gr\u00e1ficos en dos dimensiones. Las expresiones o nombres de funciones que se utilicen para definir curvas deben depender todas ellas de una \u00fanica variable var, siendo obligatorio utilizar x_range para nombrar la variable y darle sus valores m\u00ednimo y m\u00e1ximo usando la siguiente sintaxis: [variable, min, max]. <\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Dibuja la funci\u00f3n \\(f(x)=x\\,e^{1-x^2}\\) para \\(x\\in[-3,3]\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1as() {\n  var htmlShow1as = document.getElementById(\"html-show1as\");\n  if (htmlShow1as.style.display === \"none\") {\n    htmlShow1as.style.display = \"block\";\n  } else {\n    htmlShow1as.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1as()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1as\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">%e<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_number\">1<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">wxplot2d<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),[<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">3<\/span>])<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%t2) <\/mtext><\/mtd><mtd\/><\/mlabeledtr><\/mtable><\/math><img decoding=\"async\" src=\"https:\/\/uploads.jesussoto.es\/doc\/Ejer_concavo01_1.png\" width=\"598\" style=\"max-width:90%;\" loading=\"lazy\" alt=\" (Gr\u00e1ficos) \"\/><br \/><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o2) <\/mtext><\/mtd><mtd\/><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<h2>L\u00edmites<\/h2>\n<p>Si queremos calcular un limite utilizaremos:<\/p>\n<ul>\n<li><strong>limit<\/strong>(<em>expr<\/em>)<\/li>\n<li><strong>limit<\/strong>(<em>expr, x, val<\/em>)<\/li>\n<li><strong>limit<\/strong>(<em>expr, x, val, dir<\/em>): Calcula el l\u00edmite de expr cuando la variable real x se aproxima al valor val desde la direcci\u00f3n dir. El argumento dir puede ser el valor plus para un l\u00edmite por la derecha, minus para un l\u00edmite por la izquierda o simplemente se omite para indicar un l\u00edmite en ambos sentidos.<\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Calcular \\[\\lim_{x\\to 0}\\frac{3^x-3^{-x}}{3^x+3^{-x}}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1bs() {\n  var htmlShow1bs = document.getElementById(\"html-show1bs\");\n  if (htmlShow1bs.style.display === \"none\") {\n    htmlShow1bs.style.display = \"block\";\n  } else {\n    htmlShow1bs.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1bs()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1bs\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span>(<span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span>))<span class=\"code_operator\">\/<\/span>(<span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span>))<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">0<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o2) <\/mtext><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Calcular \\[\\lim_{x\\to \\pm \\infty}\\frac{3^x-3^{-x}}{3^x+3^{-x}}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1cs() {\n  var htmlShow1cs = document.getElementById(\"html-show1cs\");\n  if (htmlShow1cs.style.display === \"none\") {\n    htmlShow1cs.style.display = \"block\";\n  } else {\n    htmlShow1cs.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1cs()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1cs\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span>(<span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span>))<span class=\"code_operator\">\/<\/span>(<span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span>))<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">inf<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">inf<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o2) <\/mtext><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o3) <\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Estudiar la continuidad de la funci\u00f3n \\(f(x)=\\frac{x-1}{\\sqrt{x^2-3x+2}}\\) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1ds() {\n  var htmlShow1ds = document.getElementById(\"html-show1ds\");\n  if (htmlShow1ds.style.display === \"none\") {\n    htmlShow1ds.style.display = \"block\";\n  } else {\n    htmlShow1ds.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1ds()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1ds\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <\/td>\n<td><span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">1<\/span>)<span class=\"code_operator\">\/<\/span>(<span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span>))<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">raiz<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">2<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(raiz)<\/mtext><\/mtd><mtd><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mn>1<\/mn><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mn>2<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">raiz<\/span>[<span class=\"code_number\">1<\/span>]),<span class=\"code_variable\">minus<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">raiz<\/span>[<span class=\"code_number\">1<\/span>]),<span class=\"code_variable\">plus<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o3) <\/mtext><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o4) <\/mtext><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">raiz<\/span>[<span class=\"code_number\">2<\/span>]),<span class=\"code_variable\">minus<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">raiz<\/span>[<span class=\"code_number\">2<\/span>]),<span class=\"code_variable\">plus<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"left\">\n          <mtd>\n            <mtext>(%o5) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <munder>\n              <mo>lim<\/mo>\n              <mrow>\n                <mi>x<\/mi>\n                <mo>\u2192<\/mo>\n                <mn>2<\/mn>\n                <mo>\u2212<\/mo>\n              <\/mrow>\n            <\/munder>\n          <\/mtd>\n          <mtd>\n    <mfrac>\n      <mrow>\n        <mi>x<\/mi>\n        <mi>\u2212<\/mi>\n        <mn>1<\/mn>\n      <\/mrow>\n      <msqrt>\n        <mrow>\n          <msup>\n            <mi>x<\/mi>\n            <mn>2<\/mn>\n          <\/msup>\n          <mi>\u2212<\/mi>\n          <mn>3<\/mn>\n          <mo>\u2062<\/mo>\n          <mi>x<\/mi>\n          <mo>+<\/mo>\n          <mn>2<\/mn>\n        <\/mrow>\n      <\/msqrt>\n    <\/mfrac>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable><\/p>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o6) <\/mtext><\/mtd><mtd><mi>\u221e<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/p>\n<p>Observar que la funci\u00f3n no est\u00e1 definida para \\(x\\in (1,2]\\), de ah\u00ed que el l\u00edmite cuando \\(x\\to 2^-\\) no se puede calcular.<\/p>\n<p>Veamos si consideramos valores muy cercanos a \\(x=2\\):<br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">1<\/span><span class=\"code_number\">.<\/span><span class=\"code_number\">999<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">2<\/span><span class=\"code_number\">.<\/span><span class=\"code_number\">001<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o7) <\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>31.60696125855515<\/mn><mo>\u2062<\/mo><mi>%i<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o8) <\/mtext><\/mtd><mtd><mn>31.63858403911929<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Estudiar la continuidad de la funci\u00f3n \\(f(x)=\\frac{x^2-4x-21}{x-7}\\) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1es() {\n  var htmlShow1es = document.getElementById(\"html-show1es\");\n  if (htmlShow1es.style.display === \"none\") {\n    htmlShow1es.style.display = \"block\";\n  } else {\n    htmlShow1es.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1es()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1es\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">21<\/span>)<span class=\"code_operator\">\/<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">7<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">7<\/span>,<span class=\"code_variable\">minus<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">limit<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">7<\/span>,<span class=\"code_variable\">plus<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o2) <\/mtext><\/mtd><mtd><mn>10<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o3) <\/mtext><\/mtd><mtd><mn>10<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">21<\/span>,<span class=\"code_variable\">x<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o4) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>3<\/mn><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mn>7<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">rat<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">ev<\/span>(<span class=\"code_variable\">%<\/span>,<span class=\"code_variable\">x<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">7<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o5)\/R\/ <\/mtext><\/mtd><mtd><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o6)\/R\/ <\/mtext><\/mtd><mtd><mn>10<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Calcular los l\u00edmites de la funci\u00f3n  \\(f(x)=\\frac{x^2+2x-1}{(x-1)e^x}\\), en 0, \\(\\pm \\infty \\) y los laterales en 1. <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6s() {\n  var htmlShow6s = document.getElementById(\"html-show6s\");\n  if (htmlShow6s.style.display === \"none\") {\n    htmlShow6s.style.display = \"block\";\n  } else {\n    htmlShow6s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv6s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6s\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_limit01.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h2>Propiedades de las funciones continuas<\/h2>\n<p>En muchos casos tendremos que buscar un cero de una funci\u00f3n, aunque veremos m\u00e9todos para encontrar las soluciones de una ecuaci\u00f3n, vamos a trabajar con un resultado que nos ayudar\u00e1 a practicar m\u00e1s con maxima.<\/p>\n<blockquote>\n<p><strong>Teorema<\/strong>Sea \\(\\displaystyle f:[a,b]\\to \\mathbb{R}\\) una funci\u00f3n real continua en  \\([a,b]\\) con  \\(f(a)&lt;0&lt;f(b)\\) \\(c\\in (a,b)\\) tal que \\(f(c)=0\\).\n<\/p>\n<\/blockquote>\n<p>Este resultado nos proporciona el conocido m\u00e9todo de bisecci\u00f3n: dado un intervalo, es suficiente con verificar la diferencia de signo de la funci\u00f3n entre los extremos y el punto medio, para determinar en qu\u00e9 subintervalo se encuentra la soluci\u00f3n.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Encontrar el cero de \\(f(x)=x^3+x^2-x-2\\) con el m\u00e9todo de bisecci\u00f3n.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2s() {\n  var htmlShow2s = document.getElementById(\"html-show2s\");\n  if (htmlShow2s.style.display === \"none\") {\n    htmlShow2s.style.display = \"block\";\n  } else {\n    htmlShow2s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2s\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_biseccion02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong>Sea el sistema \\[\\begin{array}{l}2x-y-Kz=0 \\\\ x-y-2z=1 \\\\ -x+2z=K\\end{array}\\] El valor de \\(K\\) que hace el sistema incompatibles divide a <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>42<\/li>\n<li>54<\/li>\n<li>60<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>C.)<\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistema01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Sistemas de ecuaciones En pasados d\u00edas vimos c\u00f3mo resolv\u00edamos sistemas de ecuaciones y, en particular, el problema de m\u00ednimos cuadrados mediante matrices. Hoy bordaremos estos problemas utilizando maxima. linsolve(\\([eq_1, &#8230;, eq_m], [x_1,&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[5],"class_list":["post-311","post","type-post","status-publish","format-standard","hentry","category-mathbio","tag-practicas-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/311","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=311"}],"version-history":[{"count":4,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/311\/revisions"}],"predecessor-version":[{"id":314,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/311\/revisions\/314"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=311"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=311"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}