{"id":409,"date":"2025-11-11T15:15:39","date_gmt":"2025-11-11T14:15:39","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=409"},"modified":"2025-11-11T12:08:03","modified_gmt":"2025-11-11T11:08:03","slug":"mathbio-calculo-diferencial-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=409","title":{"rendered":"MathBio: C\u00e1lculo diferencial con maxima"},"content":{"rendered":"<p>Abordemos c\u00f3mo hacer la derivada a funciones reales de una variable real:<\/p>\n<ul>\n<li><strong>diff<\/strong>(<em>expr, variable, veces<\/em>): Calcula la derivada de una Funci\u00f3n que depende de la variable el n\u00famero de veces indicado. El n\u00famero veces puede eludirse si es uno. Si aparecen otras variables en <em>expr<\/em> son consideradas como constantes.<\/li>\n<\/ul>\n<p>Cuando queremos utilizar la derivada como funci\u00f3n si es conveniente usar <strong>define<\/strong>:<\/p>\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<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar el valor de \\(f^\\prime(1)\\) donde \\[f(x)=e^{\\sin \\left(x^2\\right)}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3() {\n  var htmlShow3 = document.getElementById(\"html-show3\");\n  if (htmlShow3.style.display === \"none\") {\n    htmlShow3.style.display = \"block\";\n  } else {\n    htmlShow3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" 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\">%e<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_function\">sin<\/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\">define<\/span>(<span class=\"code_function\">df<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_function\">diff<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/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>(%o2) <\/mtext><\/mtd><mtd><mrow><mi>df<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/mrow><mo>:=<\/mo><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>\u2062<\/mo><msup><mi>%e<\/mi><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>)<\/mo><\/mrow><\/mrow><\/msup><mo>\u2062<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>)<\/mo><\/mrow><\/mrow><\/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\">(%i3)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_function\">df<\/span>(<span class=\"code_number\">1<\/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>(%o3) <\/mtext><\/mtd><mtd><mn>2.506761534986894<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar el valor de \\(f^\\prime(1)\\) donde \\[f(x)=\\sin \\left(x^2+\\frac{1}{x}\\right)\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a() {\n  var htmlShow3a = document.getElementById(\"html-show3a\");\n  if (htmlShow3a.style.display === \"none\") {\n    htmlShow3a.style.display = \"block\";\n  } else {\n    htmlShow3a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a\" 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_function\">sin<\/span>(<span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">x<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">define<\/span>(<span class=\"code_function\">df<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_function\">diff<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/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>(%o2) <\/mtext><\/mtd><mtd><mrow><mi>df<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/mrow><mo>:=<\/mo><mrow><mo>(<\/mo><mrow><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mfrac><mn>1<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mrow><mo>)<\/mo><\/mrow><mo>\u2062<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/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\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">df<\/span>(<span class=\"code_number\">1<\/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>(%o3) <\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>0.4161468365471424<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(T(x)\\) la recta tangente a \\(y=x^2\\sin(x\/2)\\) en \\(x=\\pi\/3\\), calcula \\(T(5)\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a91() {\n  var htmlShow3a91 = document.getElementById(\"html-show3a91\");\n  if (htmlShow3a91.style.display === \"none\") {\n    htmlShow3a91.style.display = \"block\";\n  } else {\n    htmlShow3a91.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3a91()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a91\" 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_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/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\">\u00b7<\/span><span class=\"code_function\">sin<\/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\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">p<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">%pi<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">define<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">df<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">diff<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }{df}(x){:=}\\frac{\\cos{\\left( \\frac{x}{2}\\right) } {{x}^{2}}}{2}+2 \\sin{\\left( \\frac{x}{2}\\right) } x\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Definimos la tangente en x=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 style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">define<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">T<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">df<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">p<\/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>\\[{ }{T}(x){:=}\\frac{\\left( 3 {{{\\pi} }^{2}}+4 {{3}^{\\frac{3}{2}}} {\\pi} \\right)x-{{{\\pi} }^{3}}-2 \\sqrt{3} {{{\\pi} }^{2}}}{4 {{3}^{\\frac{5}{2}}}}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora sustituimos T(5)<\/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\">T<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }6.564670821992515\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(N(x)\\) la recta normal a \\(y=\\frac{-{{x}^{2}}-11 }{2 {{x}^{3}}}\\) en \\(x=-2\\), calcula \\(N(5)\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a9() {\n  var htmlShow3a9 = document.getElementById(\"html-show3a9\");\n  if (htmlShow3a9.style.display === \"none\") {\n    htmlShow3a9.style.display = \"block\";\n  } else {\n    htmlShow3a9.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3a9()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a9\" 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_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">11<\/span><span class=\"code_operator\">)<\/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_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">define<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">df<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">diff<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }{df}(x){:=}-\\frac{3 \\left( -{{x}^{2}}-11\\right) }{2 {{x}^{4}}}-\\frac{1}{{{x}^{2}}}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Definimos la normal en x=-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\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">define<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">N<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ratsimp<\/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_function\">df<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_function\">f<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/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>\\[{ }{N}(x){:=}-\\frac{512 x+469}{592}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora sustituimos N(5)<\/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 style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">N<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }-5.116554054054054\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Concavidad y convexidad de una funci\u00f3n<\/h2>\n<p>Veamos c\u00f3mo utilizamos la derivada para estimar la concavidad o convexidad de una funci\u00f3n. En este caso utilizaremos tambi\u00e9n otra herramienta para gr\u00e1ficos:<\/p>\n<ul>\n<li><strong>draw2d<\/strong>(): capaz de gestionar, uno o varios gr\u00e1ficos simult\u00e1neamente, que pueden ser de tipos diferentes (f\u00f3rmulas en expl\u00edcitas, en impl\u00edcitas, en param\u00e9tricas, en polares&#8230;) cada uno con sus propios par\u00e1metros y opciones (que se precisan antes de declarar la figura); tambi\u00e9n existen opciones globales que afectan a todos los elementos (por ejemplo, el t\u00edtulo del conjunto) y que es conveniente (pero no imprescindible) colocar al principio<\/li>\n<\/ul>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es el m\u00f3dulo del vector formado por del punto (1,1) y el punto la par\u00e1bola \\(y^2=2x\\) m\u00e1s cercano a el?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv17() {\n  var htmlShow17 = document.getElementById(\"html-show17\");\n  if (htmlShow17.style.display === \"none\") {\n    htmlShow17.style.display = \"block\";\n  } else {\n    htmlShow17.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv17()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show17\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_minimos.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe><\/p>\n<p>Observemos que lo anterior responde a la pregunta \u00bfcu\u00e1l es la norma del punto considerado como vector?. <\/p>\n<p>La soluci\u00f3n correcta a la pregunta ser\u00eda:<br \/>\n\\[||[1,1]-\\left[\\frac{1}{2^{1\/3}},\\sqrt{\\frac{2}{ 2^{1\/3}}}\\right]||=\\sqrt{{{\\left( 1-{{2}^{\\frac{1}{3}}}\\right) }^{2}}+{{\\left( 1-\\frac{1}{{{2}^{\\frac{1}{3}}}}\\right) }^{2}}}=0.3318\\]<\/p>\n<p>Este ser\u00eda el resultado de sustituir \\(x=\\frac{1}{2^\\frac{1}{3}}\\) en la funci\u00f3n \\(dt(x)\\) escrita en maxima. Es decir<br \/>\n\\[dt\\left(\\frac{1}{2^\\frac{1}{3}}\\right)=\\sqrt{{{\\left( 1-{{2}^{\\frac{1}{3}}}\\right) }^{2}}+{{\\left( 1-\\frac{1}{{{2}^{\\frac{1}{3}}}}\\right) }^{2}}}=0.3318\\]\n<\/p><\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> En qu\u00e9 intervalo la funci\u00f3n \\(f(x)=x\\,e^{1-x^2}\\) es c\u00f3ncava<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1() {\n  var htmlShow1 = document.getElementById(\"html-show1\");\n  if (htmlShow1.style.display === \"none\") {\n    htmlShow1.style.display = \"block\";\n  } else {\n    htmlShow1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_concavo01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h2>M\u00e9todo de Newton<\/h2>\n<p>Como hemos observado necesitamos conocer las soluciones de las ecuaciones que plantean las funciones para encontrar m\u00e1ximos y m\u00ednimos o los cambios de concavidad y convexidad. Repasemos el m\u00e9todo de  bisecci\u00f3n para encontrar un cero, y apliquemos un m\u00e9todo m\u00e1s preciso:El M\u00e9todo de Newton.<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Encontrar los ceros de la segunda derivada de \\(f(x)=x\\,e^{1-x^2}\\) con el m\u00e9todo de bisecci\u00f3n.<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2() {\n  var htmlShow2 = document.getElementById(\"html-show2\");\n  if (htmlShow2.style.display === \"none\") {\n    htmlShow2.style.display = \"block\";\n  } else {\n    htmlShow2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_biseccion01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>M\u00e9todo de Newton<\/strong> Sea \\( f:[a,b]\\to \\mathbb {R}\\) una funci\u00f3n derivable definida en el intervalo real \\([a,b]\\) y \\(x_p\\in [a,b]\\) tal que \\(f(x_p)=0\\). Entonces para cierto \\(x_{0}\\in [a,b]\\) la sucesi\u00f3n \\[x_{{n+1}}=x_{n}-{\\frac{f(x_{n})}{f^\\prime(x_{n})}},\\]<br \/>\ncumple que \\(\\lim_{n\\to\\infty}x_n=x_p\\).\n<\/p><\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> Encontrar el cero de \\(f(x)=x^3-x^2+2x+1\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv43() {\n  var htmlShow43 = document.getElementById(\"html-show43\");\n  if (htmlShow43.style.display === \"none\") {\n    htmlShow43.style.display = \"block\";\n  } else {\n    htmlShow43.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv43()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show43\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_newton01.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(s\\) la abscisa donde las curvas \\(y=x^2-x\\) e \\(y=\\sqrt{x}\\) se cortan en el intervalo [1,2], \u00bfcu\u00e1l es la parte entera del valor de \\(e^s+e^{s+1}\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv43s() {\n  var htmlShow43s = document.getElementById(\"html-show43s\");\n  if (htmlShow43s.style.display === \"none\") {\n    htmlShow43s.style.display = \"block\";\n  } else {\n    htmlShow43s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv43s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show43s\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"Matem\u00e1tica aplicada con m\u00e1xima - M\u00e9todo de Newton Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/DuKUPqFHgJ0?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<h2>Funciones de varias variables<\/h2>\n<p>El siguiente comando nos permitir\u00e1 realizar la gr\u00e1fica de un campo escalar.<\/p>\n<ul>\n<li><strong>plot3d<\/strong>(<em>expresi\u00f3n, [variable_x m\u00ednimo, m\u00e1ximo], [variable_y m\u00ednimo, m\u00e1ximo], opciones<\/em>):<br \/>\nLa expresi\u00f3n es del tipo f(x,y) y corresponde a z=f(x,y) para coordenadas cartesianas<\/li>\n<\/ul>\n<blockquote><p><strong>Ejemplo:<\/strong> Dibujar la gr\u00e1fica de \\(f(x,y)=(x^2-y^2)\/(x^2+y^2)\\) en [-2,2]x[-2,2].<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4a31() {\n  var htmlShow4a31 = document.getElementById(\"html-show4a31\");\n  if (htmlShow4a31.style.display === \"none\") {\n    htmlShow4a31.style.display = \"block\";\n  } else {\n    htmlShow4a31.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a31()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a31\" 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_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">:<\/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\">\u2212<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/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\">+<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">wxplot3d<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">f<\/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_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_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><img decoding=\"async\" src=\"https:\/\/uploads.jesussoto.es\/campo_escalar1.png\" width=\"1198\" style=\"max-width:90%;\" loading=\"lazy\" alt=\" (Graphics) \"\/><\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(f(x,y)=\\frac{x}{(x+y)^2}\\). \u00bfCu\u00e1nto vale \\(f_x(1,1)+f_y(1,1)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4a3() {\n  var htmlShow4a3 = document.getElementById(\"html-show4a3\");\n  if (htmlShow4a3.style.display === \"none\") {\n    htmlShow4a3.style.display = \"block\";\n  } else {\n    htmlShow4a3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a3\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"Matem\u00e1tica aplicada - Ej.1 Derivadas parciales - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/pQq-EWBG9c8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(f(x,y)=\\frac{x}{(x+y)^2}\\). \u00bfCu\u00e1nto vale \\(\\frac{dy}{dx}(1,0)\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4g() {\n  var htmlShow4g = document.getElementById(\"html-show4g\");\n  if (htmlShow4g.style.display === \"none\") {\n    htmlShow4g.style.display = \"block\";\n  } else {\n    htmlShow4g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4g\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_parcial02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar la ecuaci\u00f3n de la recta tangente a la curva \\(xy-y^2-2y^3=0\\), en el punto (1,-1) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5g() {\n  var htmlShow5g = document.getElementById(\"html-show5g\");\n  if (htmlShow5g.style.display === \"none\") {\n    htmlShow5g.style.display = \"block\";\n  } else {\n    htmlShow5g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5g\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_parcial04.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar el vector gradiente de \\(f(x,y,z)=x\\,\\sin^2(y)+z\\,\\cos^2(y)\\) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6() {\n  var htmlShow6 = document.getElementById(\"html-show6\");\n  if (htmlShow6.style.display === \"none\") {\n    htmlShow6.style.display = \"block\";\n  } else {\n    htmlShow6.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv6()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_parcial03.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(f(x,y,z)=x\\,\\sin^2(y)+ze^{2x}\\). \u00bfCu\u00e1nto vale \\(\\left \\|\\nabla f\\left(1,\\tfrac{\\pi}{4},0\\right) \\right \\|\\) ? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6b() {\n  var htmlShow6b = document.getElementById(\"html-show6b\");\n  if (htmlShow6b.style.display === \"none\") {\n    htmlShow6b.style.display = \"block\";\n  } else {\n    htmlShow6b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv6b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6b\" 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_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>)<span class=\"code_operator\">:<\/span><span class=\"code_operator\">=<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_function\">sin<\/span>(<span class=\"code_variable\">y<\/span>)<span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">+<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">%e<\/span><span class=\"code_operator\">^<\/span>(<span class=\"code_number\">2<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">x<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">define<\/span>(<span class=\"code_function\">grf<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>),<span class=\"code_function\">ratsimp<\/span>([<span class=\"code_function\">diff<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>),<span class=\"code_variable\">x<\/span>),<span class=\"code_function\">diff<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>),<span class=\"code_variable\">y<\/span>),<span class=\"code_function\">diff<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>),<span class=\"code_variable\">z<\/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><mrow><mi>grf<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo>,<\/mo><mi>z<\/mi><\/mrow><mo>)<\/mo><\/mrow><\/mrow><mo>:=<\/mo><mo>[<\/mo><mn>2<\/mn><mo>\u2062<\/mo><msup><mi>%e<\/mi><mrow><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><\/mrow><\/msup><mo>\u2062<\/mo><mi>z<\/mi><mo>+<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mi>y<\/mi><mo>)<\/mo><\/mrow><\/mrow><mn>2<\/mn><\/msup><mo>,<\/mo><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>\u2062<\/mo><mrow><mi>cos<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mi>y<\/mi><mo>)<\/mo><\/mrow><\/mrow><mo>\u2062<\/mo><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mi>y<\/mi><mo>)<\/mo><\/mrow><\/mrow><mo>,<\/mo><msup><mi>%e<\/mi><mrow><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><\/mrow><\/msup><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\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">vn<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">grf<\/span>(<span class=\"code_number\">1<\/span>,<span class=\"code_variable\">%pi<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">4<\/span>,<span class=\"code_number\">0<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">vn<\/span>.<span class=\"code_variable\">vn<\/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>(vn)<\/mtext><\/mtd><mtd><mo>[<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo><msup><mi>%e<\/mi><mn>2<\/mn><\/msup><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o5) <\/mtext><\/mtd><mtd><mn>7.473<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong>  calcular la derivada direccional de \\(f(x,y,z)=x\\,\\sin(y)+yz^2\\) en el punto P(1,\\(\\frac{\\pi}{2}\\),-1) y en la direcci\u00f3n del vector u=(4,3,0).<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv7() {\n  var htmlShow7 = document.getElementById(\"html-show7\");\n  if (htmlShow7.style.display === \"none\") {\n    htmlShow7.style.display = \"block\";\n  } else {\n    htmlShow7.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv7()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show7\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_parcial05.html\" width=\"650\" height=\"350\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>&nbsp;<\/p>\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 \\(f(x,y,z)=\\cos^2(x)+ze^{-y}\\). \u00bfCu\u00e1nto vale \\(\\left \\|\\nabla f\\left(\\pi,-1,1\\right) \\right \\|\\) ? <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>9.21<\/li>\n<li>7.45<\/li>\n<li>3.84<\/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\/Ejer_grad02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Abordemos c\u00f3mo hacer la derivada a funciones reales de una variable real: diff(expr, variable, veces): Calcula la derivada de una Funci\u00f3n que depende de la variable el n\u00famero de veces indicado. El&#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-409","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\/409","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=409"}],"version-history":[{"count":4,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/409\/revisions"}],"predecessor-version":[{"id":469,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/409\/revisions\/469"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}