{"id":3899,"date":"2021-05-20T18:29:56","date_gmt":"2021-05-20T18:29:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-newtons-method\/"},"modified":"2021-07-03T19:09:31","modified_gmt":"2021-07-03T19:09:31","slug":"review-for-newtons-method","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-newtons-method\/","title":{"raw":"Skills Review for Newton's Method","rendered":"Skills Review for Newton&#8217;s Method"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write the terms of a sequence defined by a recursive formula<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Newton's Method section,\u00a0 a recursive formula will be used to approximate the <i>x-<\/i>intercepts\u00a0of functions. Here we will review how a recursive formula works.\r\n<h2>Evaluate a Recursive Formula<\/h2>\r\nA <strong>recursive formula<\/strong> is\u00a0a formula that defines its value at a particular input using the result of the previous input(s).\r\n\r\nA recursive formula always has two parts: the value of an initial input and an equation defining each term in terms of preceding terms. For example, suppose we know the following:\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{x}_{1}=3 \\\\ &amp;{x}_{n}=2{x}_{n - 1}-1, \\text{ for } n\\ge 2 \\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: left;\">We can find the subsequent terms of the recursive formula using the first term.<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{x}_{1}=3\\\\ &amp;{x}_{2}=2{x}_{1}-1=2\\left(3\\right)-1=5\\\\ &amp;{x}_{3}=2{x}_{2}-1=2\\left(5\\right)-1=9\\\\ &amp;{x}_{4}=2{x}_{3}-1=2\\left(9\\right)-1=17\\end{align}[\/latex]<\/div>\r\nSo, the first four terms are [latex]3,5,9,\\text{ and},17[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating a Recursive Formula<\/h3>\r\nWrite the first five terms defined by the recursive formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;{x}_{1}=9\\\\ &amp;{x}_{n}=3{x}_{n - 1}-20\\text{, for }n\\ge 2\\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"638659\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"638659\"]\r\n\r\nThe first term is given in the formula. For each subsequent term, we replace [latex]{x}_{n - 1}[\/latex] with the value of the preceding term.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;n=1&amp;&amp; {x}_{1}=9 \\\\ &amp;n=2&amp;&amp; {x}_{2}=3{x}_{1}-20=3\\left(9\\right)-20=27 - 20=7 \\\\ &amp;n=3&amp;&amp; {x}_{3}=3{x}_{2}-20=3\\left(7\\right)-20=21 - 20=1 \\\\ &amp;n=4&amp;&amp; {x}_{4}=3{x}_{3}-20=3\\left(1\\right)-20=3 - 20=-17 \\\\ &amp;n=5&amp;&amp; {x}_{5}=3{x}_{4}-20=3\\left(-17\\right)-20=-51 - 20=-71\\end{align}[\/latex]<\/p>\r\nThe first five terms are [latex]9, 7, 1, -17, \\text{ and}, -71[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]221978[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write the terms of a sequence defined by a recursive formula<\/li>\n<\/ul>\n<\/div>\n<p>In the Newton&#8217;s Method section,\u00a0 a recursive formula will be used to approximate the <i>x-<\/i>intercepts\u00a0of functions. Here we will review how a recursive formula works.<\/p>\n<h2>Evaluate a Recursive Formula<\/h2>\n<p>A <strong>recursive formula<\/strong> is\u00a0a formula that defines its value at a particular input using the result of the previous input(s).<\/p>\n<p>A recursive formula always has two parts: the value of an initial input and an equation defining each term in terms of preceding terms. For example, suppose we know the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&{x}_{1}=3 \\\\ &{x}_{n}=2{x}_{n - 1}-1, \\text{ for } n\\ge 2 \\end{align}[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: left;\">We can find the subsequent terms of the recursive formula using the first term.<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}&{x}_{1}=3\\\\ &{x}_{2}=2{x}_{1}-1=2\\left(3\\right)-1=5\\\\ &{x}_{3}=2{x}_{2}-1=2\\left(5\\right)-1=9\\\\ &{x}_{4}=2{x}_{3}-1=2\\left(9\\right)-1=17\\end{align}[\/latex]<\/div>\n<p>So, the first four terms are [latex]3,5,9,\\text{ and},17[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating a Recursive Formula<\/h3>\n<p>Write the first five terms defined by the recursive formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&{x}_{1}=9\\\\ &{x}_{n}=3{x}_{n - 1}-20\\text{, for }n\\ge 2\\end{align}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q638659\">Show Solution<\/span><\/p>\n<div id=\"q638659\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first term is given in the formula. For each subsequent term, we replace [latex]{x}_{n - 1}[\/latex] with the value of the preceding term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&n=1&& {x}_{1}=9 \\\\ &n=2&& {x}_{2}=3{x}_{1}-20=3\\left(9\\right)-20=27 - 20=7 \\\\ &n=3&& {x}_{3}=3{x}_{2}-20=3\\left(7\\right)-20=21 - 20=1 \\\\ &n=4&& {x}_{4}=3{x}_{3}-20=3\\left(1\\right)-20=3 - 20=-17 \\\\ &n=5&& {x}_{5}=3{x}_{4}-20=3\\left(-17\\right)-20=-51 - 20=-71\\end{align}[\/latex]<\/p>\n<p>The first five terms are [latex]9, 7, 1, -17, \\text{ and}, -71[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221978\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221978&theme=oea&iframe_resize_id=ohm221978\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-3899\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen 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