{"id":4397,"date":"2019-11-27T06:00:31","date_gmt":"2019-11-27T11:00:31","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/matg1013\/?p=4397"},"modified":"2026-02-26T11:16:25","modified_gmt":"2026-02-26T16:16:25","slug":"s1eva2019tii_t4-concentracion-de-quimico","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s1eva20\/s1eva2019tii_t4-concentracion-de-quimico\/","title":{"rendered":"s1Eva2019TII_T4 Concentraci\u00f3n de qu\u00edmico"},"content":{"rendered":"\n

Ejercicio<\/strong>: 1Eva2019TII_T4 Concentraci\u00f3n de qu\u00edmico<\/a><\/p>\n\n\n\n

formula a usar:<\/p>\n\n\nC = C_{ent} ( 1 - e^{-0.04t})+C_{0} e^{-0.03t}<\/span>\n\n\n\n

Se sustituyen los valores dados con:
C0<\/sub> = 4, Cent<\/sub> = 10, C = 0.93 Cent<\/sub>.<\/p>\n\n\n0.93(10) = 10 ( 1 - e^{-0.04t}) + 4 e^{-0.03t}<\/span>\n\n\n\n

igualando a cero para forma estandarizada del algoritmo,<\/p>\n\n\n 10( 1 - e^{-0.04t}) + 4 e^{-0.03t} - 9.3 = 0<\/span>\n\n\n 0.7 - 10 e^{-0.04t} + 4 e^{-0.03t} = 0<\/span>\n\n\n\n

\"1e2019tii_t4newtonraphson01_gif\"<\/figure>\n\n\n\n

Se usan las funciones f(t) y f'(t) para el m\u00e9todo de Newton-Raphson,<\/p>\n\n\n f(t) = 0.7 - 10 e^{-0.04t} + 4 e^{-0.03t}<\/span>\n\n\n f'(t) = - 10(-0.04) e^{-0.04t} + 4(-0.03) e^{-0.03t}<\/span>\n\n\n f'(t) = 0.4 e^{-0.04t} - 0.12 e^{-0.03t}<\/span>\n\n\n\n

con lo que se pueden realizar los c\u00e1lculos de forma iterativa.<\/p>\n\n\n t_{i+1} = t_i -\\frac{f(t_i)}{f'(t_i)} <\/span>\n\n\n\n

De no disponer de la gr\u00e1fica de f(t), y se desconoce el valor inicial para t0<\/sub> se usa 0. Como no se indica la tolerancia, se estima en 10-4<\/sup><\/p>\n\n\n\n

Iteraci\u00f3n 1<\/strong><\/em><\/p>\n\n\n\n

t0<\/sub> = 0<\/p>\n\n\n f(0) = 0.7 - 10 e^{-0.04(0)} + 4 e^{-0.03(0)} = 5.3<\/span>\n\n\n f'(0) = 0.4 e^{-0.04(0)} - 0.12 e^{-0.03(0)} = -0.28<\/span>\n\n\n t_{1} = 0 -\\frac{5.3}{-0.28} = 18.92 <\/span>\n\n\n error = |18.92-0| = 18.92<\/span>\n\n\n\n

Iteraci\u00f3n 2<\/strong><\/em><\/p>\n\n\n f(18.92) = 0.7 - 10 e^{-0.04(18.92)} + 4 e^{-0.03(18.92)} = -1.72308<\/span>\n\n\n f'(18.92) = 0.4 e^{-0.04(18.92)} - 0.12 e^{-0.03(18.92)} = 0.119593<\/span>\n\n\n t_{2} = 18.92 -\\frac{-1.723087}{0.119593} = 33.3365 <\/span>\n\n\n error = |33.3365 - 18.92| = 14.4079<\/span>\n\n\n\n

Iteraci\u00f3n 3<\/strong><\/em><\/p>\n\n\n f(33.3365) = 0.7 - 10 e^{-0.04(33.3365)} + 4 e^{-0.03(33.3365)} = -0.4642<\/span>\n\n\n f'(33.3365) = 0.4 e^{-0.04(33.3365)} - 0.12 e^{-0.03(33.3365)} = 0.06128<\/span>\n\n\n t_{3} = 33.3365 -\\frac{-0.46427}{-5.8013} = 40.912 <\/span>\n\n\n error = |40.912 - 33.3365| = 7.5755<\/span>\n\n\n\n

Observando que los errores disminuyen entre cada iteraci\u00f3n, se encuentra que el m\u00e9todo converge.<\/p>\n\n\n\n

y se forma la siguiente tabla:<\/p>\n\n\n\n

['xi' ,  'xnuevo', 'error']\n[ 0.      18.9286  18.9286]\n[18.9286  33.3365  14.4079]\n[33.3365  40.912    7.5755]\n[40.912   42.654     1.742]\n[42.654   42.7316   0.0776]\n[4.2732e+01 4.2732e+01 1.4632e-04]\nraiz en:  42.731721341402796<\/code><\/pre>\n\n\n\n
Observando la gr\u00e1fica de la funci\u00f3n puede observar el resultado:<\/p>\n\n\n\n
\"1e2019tii_t4newtonraphson01_gif\"<\/figure>\n\n\n\n
\n\n\n\n
Algoritmo en Python<\/h2>\n\n\n
\n# 1Eva_IIT2019_T4\n# M\u00e9todo de Newton-Raphson\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# INGRESO\nfx = lambda t: 0.7-10*np.exp(-0.04*t)+4*np.exp(-0.03*t)\ndfx = lambda t:0.40*np.exp(-0.04*t)-0.12*np.exp(-0.03*t)\n\nx0 = 0\ntolera = 0.001\n\na = 0\nb = 60\nmuestras = 21\n\n# PROCEDIMIENTO\ntabla = []\ntramo = abs(2*tolera)\nxi = x0\nwhile (tramo>=tolera):\n    xnuevo = xi - fx(xi)\/dfx(xi)\n    tramo = abs(xnuevo-xi)\n    tabla.append([xi,xnuevo,tramo])\n    xi = xnuevo\n\ntabla = np.array(tabla)\nn=len(tabla)\n\n# para la gr\u00e1fica\nxk = np.linspace(a,b,muestras)\nfk = fx(xk)\n\n# SALIDA\nprint(['xi', 'xnuevo', 'error'])\nnp.set_printoptions(precision = 4)\nfor i in range(0,n,1):\n    print(tabla[i])\nprint('raiz en: ', xi)\n\n# grafica\nplt.plot(xk,fk)\nplt.axhline(0, color='black')\nplt.xlabel('t')\nplt.ylabel('f(t)')\nplt.show()\n<\/pre><\/div>","protected":false},"excerpt":{"rendered":"
Ejercicio: 1Eva2019TII_T4 Concentraci\u00f3n de qu\u00edmico formula a usar: Se sustituyen los valores dados con:C0 = 4, Cent = 10, C = 0.93 Cent. igualando a cero para forma estandarizada del algoritmo, Se usan las funciones f(t) y f'(t) para el m\u00e9todo de Newton-Raphson, con lo que se pueden realizar los c\u00e1lculos de forma iterativa. 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":[45],"tags":[58,54],"class_list":["post-4397","post","type-post","status-publish","format-standard","hentry","category-mn-s1eva20","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/4397","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=4397"}],"version-history":[{"count":5,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/4397\/revisions"}],"predecessor-version":[{"id":21806,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/4397\/revisions\/21806"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=4397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=4397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=4397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}