{"id":4777,"date":"2020-01-29T10:21:56","date_gmt":"2020-01-29T15:21:56","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/matg1013\/?p=4777"},"modified":"2026-04-05T20:10:05","modified_gmt":"2026-04-06T01:10:05","slug":"s2eva2019tii_t4-integrar-con-cuadratura-de-gauss","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s2eva20\/s2eva2019tii_t4-integrar-con-cuadratura-de-gauss\/","title":{"rendered":"s2Eva2019TII_T4 Integrar con Cuadratura de Gauss"},"content":{"rendered":"\n

Ejercicio<\/strong><\/em>: 2Eva2019TII_T4 Integrar con Cuadratura de Gauss<\/a><\/p>\n\n\n f(x) = x ln(x) <\/span>\n\n\n\n

1 \u2264x\u22644<\/p>\n\n\n\n

se requiere:<\/p>\n\n\n I = \\int_1^4 x ln(x) dx <\/span>\n\n\n\n

literal a<\/strong>. Usando el m\u00e9todo de Cuadratura de Gauss con 2 t\u00e9rminos<\/p>\n\n\n x_a = \\frac{b+a}{2} + \\frac{b-a}{2}x_0 = \\frac{4+1}{2} + \\frac{4-1}{2}\\Big(\\frac{-1}{\\sqrt{3}} \\Big)<\/span>\n\n\n\n

xa<\/sub> =1.6339745962155612<\/p>\n\n\n x_b = \\frac{b+a}{2} + \\frac{b-a}{2}x_1 = \\frac{4+1}{2} + \\frac{4-1}{2}\\Big(\\frac{1}{\\sqrt{3}} \\Big)<\/span>\n\n\n\n

xb<\/sub> =3.366025403784439<\/p>\n\n\n I \\cong \\frac{b-a}{2}(f(x_a) + f(x_b))<\/span>\n\n\n I \\cong \\frac{4-1}{2}(x_a ln(x_a) + x_b ln(x_b))<\/span>\n\n\n\n

I = 7.33164251999249<\/p>\n\n\n\n

literal b<\/strong>.  De la f\u00f3rmula , despejar el valor del error<0.0001<\/p>\n\n\n\\Big|\\frac{(b-a)}{180}h^4 f^{(4)} (\\xi)\\Big| <0.0001; \\xi \\in[a,b] <\/span>\n\n\n h^4 <0.0001\\frac{180}{(4-1)}\\frac{1}{f^{(4)} (\\xi)}<\/span>\n\n\n h^4 < 0.006\\frac{1}{f^{(4)} (\\xi)}<\/span>\n\n\n h <\\Big(0.006\\frac{1}{f^{(4)} (\\xi)}\\Big)^{1\/4}<\/span>\n\n\n\n

obteniendo la 4ta derivada de la funci\u00f3n:<\/p>\n\n\n f(x) = x ln(x) <\/span>\n\n\n f'(x) = ln(x) + x\\Big(\\frac{1}{x} \\Big) = ln(x) +1<\/span>\n\n\n f''(x) = \\frac{1}{x}<\/span>\n\n\n f'''(x) = -\\frac{1}{x^2}<\/span>\n\n\n f^{(4)}(x) = 2\\frac{1}{x^3}<\/span>\n\n\n\n

se tiene que:<\/p>\n\n\n h <\\Big(0.006\\frac{1}{f^{(4)} (\\xi)}\\Big)^{1\/4}<\/span>\n\n\n h <\\Big(0.006\\frac{1}{2\\frac{1}{\\xi^3}}\\Big)^{1\/4}<\/span>\n\n\n h <\\Big(0.003\\xi^3\\Big)^{1\/4}<\/span>\n\n\n h <(0.003)^{1\/4}\\xi^{3\/4}<\/span>\n\n\n\n

en el peor de los casos, se toma el valor menor de \u03be =1<\/p>\n\n\n h <(0.003)^{1\/4}<\/span>\n\n\n h<0.2340347319320716<\/span>\n\n\n\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Ejercicio: 2Eva2019TII_T4 Integrar con Cuadratura de Gauss 1 \u2264x\u22644 se requiere: literal a. Usando el m\u00e9todo de Cuadratura de Gauss con 2 t\u00e9rminos xa =1.6339745962155612 xb =3.366025403784439 I = 7.33164251999249 literal b.  De la f\u00f3rmula , despejar el valor del error<0.0001 obteniendo la 4ta derivada de la funci\u00f3n: se tiene que: en el peor de […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-mn-ejemplo","format":"standard","meta":{"footnotes":""},"categories":[48],"tags":[58,54],"class_list":["post-4777","post","type-post","status-publish","format-standard","hentry","category-mn-s2eva20","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/4777","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=4777"}],"version-history":[{"count":3,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/4777\/revisions"}],"predecessor-version":[{"id":23851,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/4777\/revisions\/23851"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=4777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=4777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=4777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}