{"id":3164,"date":"2024-06-13T17:08:20","date_gmt":"2024-06-13T17:08:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3164"},"modified":"2024-08-05T02:30:54","modified_gmt":"2024-08-05T02:30:54","slug":"newtons-method-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/newtons-method-fresh-take\/","title":{"raw":"Newton\u2019s Method: Fresh Take","rendered":"Newton\u2019s Method: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Explain how Newton\u2019s method uses repetition to find roots of equations<\/li>\r\n\t<li>Recognize when Newton\u2019s method does not work<\/li>\r\n\t<li>Apply methods that repeat steps to solve different types of mathematical problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Approximating with Newton\u2019s Method<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Purpose:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Efficiently approximate roots of equations in the form [latex]f(x) = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Useful when analytical solutions are difficult or impossible to find<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The Method:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Start with an initial guess [latex]x_0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Iteratively improve the approximation using the formula: [latex]x_n = x_{n-1} - \\frac{f(x_{n-1})}{f'(x_{n-1})}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Geometric Interpretation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Each iteration finds the [latex]x[\/latex]-intercept of the tangent line at the current approximation<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Convergence:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Typically converges quickly to a root when successful<\/li>\r\n\t<li class=\"whitespace-normal break-words\">May require multiple iterations for desired accuracy<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Potential Failures:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Derivative equals zero at an approximation point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Convergence to an unintended root<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Failure to converge (e.g., oscillating between values)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Use Newton\u2019s method to approximate [latex]\\sqrt{3}[\/latex] by letting [latex]f(x)=x^2-3[\/latex] and [latex]x_0=3[\/latex]. Find [latex]x_1[\/latex] and [latex]x_2[\/latex].<\/p>\r\n<p>[reveal-answer q=\"107083\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"107083\"]<\/p>\r\n<p>For [latex]f(x)=x^2-3[\/latex], the equation for\u00a0[latex]x_n[\/latex] reduces to [latex]x_n=\\frac{x_{n-1}}{2}+\\frac{3}{2x_{n-1}}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043395413\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043395413\"]<\/p>\r\n<p>[latex]x_1=2, \\, x_2=1.75[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Explain how Newton\u2019s method uses repetition to find roots of equations<\/li>\n<li>Recognize when Newton\u2019s method does not work<\/li>\n<li>Apply methods that repeat steps to solve different types of mathematical problems<\/li>\n<\/ul>\n<\/section>\n<h2>Approximating with Newton\u2019s Method<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Efficiently approximate roots of equations in the form [latex]f(x) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Useful when analytical solutions are difficult or impossible to find<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Start with an initial guess [latex]x_0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Iteratively improve the approximation using the formula: [latex]x_n = x_{n-1} - \\frac{f(x_{n-1})}{f'(x_{n-1})}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Each iteration finds the [latex]x[\/latex]-intercept of the tangent line at the current approximation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Convergence:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Typically converges quickly to a root when successful<\/li>\n<li class=\"whitespace-normal break-words\">May require multiple iterations for desired accuracy<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Potential Failures:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Derivative equals zero at an approximation point<\/li>\n<li class=\"whitespace-normal break-words\">Convergence to an unintended root<\/li>\n<li class=\"whitespace-normal break-words\">Failure to converge (e.g., oscillating between values)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use Newton\u2019s method to approximate [latex]\\sqrt{3}[\/latex] by letting [latex]f(x)=x^2-3[\/latex] and [latex]x_0=3[\/latex]. Find [latex]x_1[\/latex] and [latex]x_2[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q107083\">Hint<\/button><\/p>\n<div id=\"q107083\" class=\"hidden-answer\" style=\"display: none\">\n<p>For [latex]f(x)=x^2-3[\/latex], the equation for\u00a0[latex]x_n[\/latex] reduces to [latex]x_n=\\frac{x_{n-1}}{2}+\\frac{3}{2x_{n-1}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043395413\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043395413\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x_1=2, \\, x_2=1.75[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":27,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3164"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3164\/revisions"}],"predecessor-version":[{"id":3894,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3164\/revisions\/3894"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/292"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3164\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3164"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3164"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3164"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}