{"id":2145,"date":"2021-03-27T17:56:55","date_gmt":"2021-03-27T17:56:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2145"},"modified":"2021-04-03T01:46:16","modified_gmt":"2021-04-03T01:46:16","slug":"summary-of-newtons-method","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-newtons-method\/","title":{"raw":"Summary of Newton's Method","rendered":"Summary of Newton&#8217;s Method"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Key Concepts<\/h3>\r\n<ul>\r\n \t<li>Newton\u2019s method approximates roots of [latex]f(x)=0[\/latex] by starting with an initial approximation [latex]x_0[\/latex], then uses tangent lines to the graph of [latex]f[\/latex] to create a sequence of approximations [latex]x_1,x_2,x_3, \\cdots[\/latex].<\/li>\r\n \t<li>Typically, Newton\u2019s method is an efficient method for finding a particular root. In certain cases, Newton\u2019s method fails to work because the list of numbers [latex]x_0,x_1,x_2, \\cdots[\/latex] does not approach a finite value or it approaches a value other than the root sought.<\/li>\r\n \t<li>Any process in which a list of numbers [latex]x_0,x_1,x_2, \\cdots[\/latex] is generated by defining an initial number [latex]x_0[\/latex] and defining the subsequent numbers by the equation [latex]x_n=F(x_{n-1})[\/latex] for some function [latex]F[\/latex] is an iterative process. Newton\u2019s method is an example of an iterative process, where the function [latex]F(x)=x-\\left[\\frac{f(x)}{f^{\\prime}(x)}\\right][\/latex] for a given function [latex]f[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165043426273\" class=\"definition\">\r\n \t<dt>iterative process<\/dt>\r\n \t<dd id=\"fs-id1165043426278\">process in which a list of numbers [latex]x_0,x_1,x_2,x_3, \\cdots[\/latex] is generated by starting with a number [latex]x_0[\/latex] and defining [latex]x_n=F(x_{n-1})[\/latex] for [latex]n \\ge 1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042323633\" class=\"definition\">\r\n \t<dt>Newton\u2019s method<\/dt>\r\n \t<dd id=\"fs-id1165042323638\">method for approximating roots of [latex]f(x)=0[\/latex]; using an initial guess [latex]x_0[\/latex], each subsequent approximation is defined by the equation [latex]x_n=x_{n-1}-\\dfrac{f(x_{n-1})}{f^{\\prime}(x_{n-1})}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Key Concepts<\/h3>\n<ul>\n<li>Newton\u2019s method approximates roots of [latex]f(x)=0[\/latex] by starting with an initial approximation [latex]x_0[\/latex], then uses tangent lines to the graph of [latex]f[\/latex] to create a sequence of approximations [latex]x_1,x_2,x_3, \\cdots[\/latex].<\/li>\n<li>Typically, Newton\u2019s method is an efficient method for finding a particular root. In certain cases, Newton\u2019s method fails to work because the list of numbers [latex]x_0,x_1,x_2, \\cdots[\/latex] does not approach a finite value or it approaches a value other than the root sought.<\/li>\n<li>Any process in which a list of numbers [latex]x_0,x_1,x_2, \\cdots[\/latex] is generated by defining an initial number [latex]x_0[\/latex] and defining the subsequent numbers by the equation [latex]x_n=F(x_{n-1})[\/latex] for some function [latex]F[\/latex] is an iterative process. Newton\u2019s method is an example of an iterative process, where the function [latex]F(x)=x-\\left[\\frac{f(x)}{f^{\\prime}(x)}\\right][\/latex] for a given function [latex]f[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165043426273\" class=\"definition\">\n<dt>iterative process<\/dt>\n<dd id=\"fs-id1165043426278\">process in which a list of numbers [latex]x_0,x_1,x_2,x_3, \\cdots[\/latex] is generated by starting with a number [latex]x_0[\/latex] and defining [latex]x_n=F(x_{n-1})[\/latex] for [latex]n \\ge 1[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042323633\" class=\"definition\">\n<dt>Newton\u2019s method<\/dt>\n<dd id=\"fs-id1165042323638\">method for approximating roots of [latex]f(x)=0[\/latex]; using an initial guess [latex]x_0[\/latex], each subsequent approximation is defined by the equation [latex]x_n=x_{n-1}-\\dfrac{f(x_{n-1})}{f^{\\prime}(x_{n-1})}[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2145\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":35,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2145","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2145\/revisions"}],"predecessor-version":[{"id":2610,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2145\/revisions\/2610"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2145\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2145"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2145"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2145"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}