{"id":3444,"date":"2024-06-24T16:44:58","date_gmt":"2024-06-24T16:44:58","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3444"},"modified":"2024-08-05T02:30:40","modified_gmt":"2024-08-05T02:30:40","slug":"newtons-method-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/newtons-method-apply-it\/","title":{"raw":"Newton\u2019s Method: Apply It","rendered":"Newton\u2019s Method: Apply It"},"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<h3>Iterative Processes and Chaos<\/h3>\r\n<p id=\"fs-id1165042708293\">Iterative processes can yield some very interesting behavior. In this section, we have seen several examples of iterative processes that converge to a fixed point. We also saw in the example where Newton's method fails that the iterative process bounced back and forth between two values. We call this kind of behavior a 2-<em>cycle<\/em>. Iterative processes can converge to cycles with various periodicities, such as 2-cycles, 4-cycles (where the iterative process repeats a sequence of four values), 8-cycles, and so on.<\/p>\r\n<p id=\"fs-id1165043134163\">Some iterative processes yield what mathematicians call <em>chaos<\/em>. In this case, the iterative process jumps from value to value in a seemingly random fashion and never converges or settles into a cycle. Although a complete exploration of <span class=\"no-emphasis\">chaos<\/span> is beyond the scope of this text, in this project we look at one of the key properties of a chaotic iterative process: sensitive dependence on initial conditions. This property refers to the concept that small changes in initial conditions can generate drastically different behavior in the iterative process.<\/p>\r\n<p id=\"fs-id1165043110064\">Probably the best-known example of chaos is the <span class=\"no-emphasis\">Mandelbrot set<\/span> (see Figure 7), named after Benoit Mandelbrot (1924\u20132010), who investigated its properties and helped popularize the field of chaos theory. The Mandelbrot set is usually generated by computer and shows fascinating details on enlargement, including self-replication of the set. Several colorized versions of the set have been shown in museums and can be found online and in popular books on the subject.<\/p>\r\n<div id=\"CNX_Calc_Figure_04_09_007\" class=\"wp-caption aligncenter\">\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211348\/CNX_Calc_Figure_04_09_007.jpg\" alt=\"A very complicated and organic looking fractal.\" width=\"731\" height=\"547\" \/> Figure 7. The Mandelbrot set is a well-known example of a set of points generated by the iterative chaotic behavior of a relatively simple function.[\/caption]\r\n<\/div>\r\n<p id=\"fs-id1165042955768\">In this project we use the logistic map<\/p>\r\n<div id=\"fs-id1165042707686\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=rx(1-x)[\/latex], where [latex]x\\in [0,1][\/latex] and [latex]r&gt;0[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<p id=\"fs-id1165043094009\">as the function in our iterative process. The logistic map is a deceptively simple function; but, depending on the value of [latex]r[\/latex], the resulting iterative process displays some very interesting behavior. It can lead to fixed points, cycles, and even chaos.<\/p>\r\n<p id=\"fs-id1165042713994\">To visualize the long-term behavior of the iterative process associated with the logistic map, we will use a tool called a <em>cobweb diagram<\/em>. As we did with the iterative process we examined earlier in this section, we first draw a vertical line from the point [latex](x_0,0)[\/latex] to the point [latex](x_0,f(x_0))=(x_0,x_1)[\/latex]. We then draw a horizontal line from that point to the point [latex](x_1,x_1)[\/latex], then draw a vertical line to [latex](x_1,f(x_1))=(x_1,x_2)[\/latex], and continue the process until the long-term behavior of the system becomes apparent. Figure 8 shows the long-term behavior of the logistic map when [latex]r=3.55[\/latex] and [latex]x_0=0.2[\/latex]. (The first 100 iterations are not plotted.) The long-term behavior of this iterative process is an 8-cycle.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"413\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211353\/CNX_Calc_Figure_04_09_008.jpg\" alt=\"In the first quadrant, f(x) = 3.55x(1 \u2013 x) is graphed as is y = x. From some point on the x axis, a line is drawn up to the line y = x, at which point it turns to be horizontal and continues until it touches the outside edge of f(x), at which point it turns again to be vertical until it each the line y = x. This process continues a number of times and creates an interesting series of boxes.\" width=\"413\" height=\"421\" \/> Figure 8. A cobweb diagram for [latex]f(x)=3.55x(1-x)[\/latex] is presented here. The sequence of values results in an 8-cycle.[\/caption]\r\n\r\n<ol id=\"fs-id1165043251229\">\r\n\t<li>Let [latex]r=0.5[\/latex] and choose [latex]x_0=0.2[\/latex]. Either by hand or by using a computer, calculate the first 10 values in the sequence. Does the sequence appear to converge? If so, to what value? Does it result in a cycle? If so, what kind of cycle (for example, 2-cycle, 4-cycle)?<br \/>\r\n<br \/>\r\n[reveal-answer q=\"439980\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"439980\"]<br \/>\r\n<br \/>\r\nFor [latex]r = 0.5[\/latex] and [latex]x\u2080 = 0.2[\/latex]:<br \/>\r\n<br \/>\r\n[latex]\\begin{array}{rcl}<br \/>\r\nx_1 &amp; = &amp; 0.5 \\times 0.2 \\times (1 - 0.2) = 0.08 \\\\<br \/>\r\nx_2 &amp; = &amp; 0.5 \\times 0.08 \\times (1 - 0.08) = 0.0368 \\\\<br \/>\r\nx_3 &amp; = &amp; 0.5 \\times 0.0368 \\times (1 - 0.0368) \\approx 0.0177 \\\\<br \/>\r\nx_4 &amp; \\approx &amp; 0.0087 \\\\<br \/>\r\nx_5 &amp; \\approx &amp; 0.0043 \\\\<br \/>\r\nx_6 &amp; \\approx &amp; 0.0021 \\\\<br \/>\r\nx_7 &amp; \\approx &amp; 0.0011 \\\\<br \/>\r\nx_8 &amp; \\approx &amp; 0.0005 \\\\<br \/>\r\nx_9 &amp; \\approx &amp; 0.0003 \\\\<br \/>\r\nx_{10} &amp; \\approx &amp; 0.0001 \\\\<br \/>\r\n\\end{array}[\/latex]<br \/>\r\n<br \/>\r\nThe sequence appears to converge to [latex]0[\/latex].<br \/>\r\n<br \/>\r\nIt does not result in a cycle. Instead, it exhibits monotonic convergence to a fixed point ([latex]0[\/latex] in this case).<br \/>\r\n<br \/>\r\n[\/hidden-answer]<\/li>\r\n\t<li>What happens when [latex]r=2[\/latex]?<br \/>\r\n<br \/>\r\n[reveal-answer q=\"31822\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"31822\"]<br \/>\r\n<br \/>\r\nFor [latex]r = 2[\/latex]:<br \/>\r\nThe sequence converges to [latex]0.5[\/latex].\u00a0<br \/>\r\n<br \/>\r\n[\/hidden-answer]<\/li>\r\n\t<li>For [latex]r=3.2[\/latex] and [latex]r=3.5[\/latex], calculate the first [latex]100[\/latex] sequence values.\u00a0 What is the long-term behavior in each of these cases?<br \/>\r\n<br \/>\r\n[reveal-answer q=\"613527\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"613527\"]<br \/>\r\n<br \/>\r\nFor [latex]r = 3.2[\/latex]:<br \/>\r\nThe sequence converges to a single value approximately [latex]0.7994[\/latex].<br \/>\r\n<br \/>\r\nFor [latex]r = 3.5[\/latex]:<br \/>\r\nThe sequence settles into a 4-cycle, alternating between approximately [latex]0.8270, 0.5009, 0.8746[\/latex], and [latex]0.3827[\/latex].<br \/>\r\n[\/hidden-answer]<br \/>\r\n<br \/>\r\n<\/li>\r\n\t<li>Now let [latex]r=4[\/latex]. Calculate the first [latex]100[\/latex] sequence values. What is the long-term behavior in this case?<br \/>\r\n<br \/>\r\n<br \/>\r\n[reveal-answer q=\"934037\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"934037\"]<br \/>\r\n<br \/>\r\nFor [latex]r = 4[\/latex] and [latex]x\u2080 = 0.2[\/latex]:<br \/>\r\nThe sequence exhibits chaotic behavior. It doesn't settle into any discernible pattern and appears to jump around unpredictably within the interval [latex][0,1][\/latex].<br \/>\r\n<br \/>\r\n[\/hidden-answer]<br \/>\r\n<br \/>\r\n<\/li>\r\n\t<li>Repeat the process for [latex]r=4[\/latex], but let [latex]x_0=0.201[\/latex]. How does this behavior compare with the behavior for [latex]x_0=0.2[\/latex]?<br \/>\r\n<br \/>\r\n<br \/>\r\n[reveal-answer q=\"58377\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"58377\"]<br \/>\r\n<br \/>\r\nFor [latex]r = 4[\/latex] and [latex]x\u2080 = 0.201[\/latex]:<br \/>\r\n<br \/>\r\nThe sequence also exhibits chaotic behavior, but the specific sequence of values is very different from the one with [latex]x\u2080 = 0.2[\/latex]. This demonstrates sensitive dependence on initial conditions, a key characteristic of chaos.<br \/>\r\n<br \/>\r\nAfter a few iterations, the sequences for [latex]x\u2080 = 0.2[\/latex] and [latex]x\u2080 = 0.201[\/latex] diverge significantly, despite starting very close to each other. This illustrates how small changes in initial conditions can lead to drastically different outcomes in chaotic systems.<br \/>\r\n[\/hidden-answer]<\/li>\r\n<\/ol>","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<h3>Iterative Processes and Chaos<\/h3>\n<p id=\"fs-id1165042708293\">Iterative processes can yield some very interesting behavior. In this section, we have seen several examples of iterative processes that converge to a fixed point. We also saw in the example where Newton&#8217;s method fails that the iterative process bounced back and forth between two values. We call this kind of behavior a 2-<em>cycle<\/em>. Iterative processes can converge to cycles with various periodicities, such as 2-cycles, 4-cycles (where the iterative process repeats a sequence of four values), 8-cycles, and so on.<\/p>\n<p id=\"fs-id1165043134163\">Some iterative processes yield what mathematicians call <em>chaos<\/em>. In this case, the iterative process jumps from value to value in a seemingly random fashion and never converges or settles into a cycle. Although a complete exploration of <span class=\"no-emphasis\">chaos<\/span> is beyond the scope of this text, in this project we look at one of the key properties of a chaotic iterative process: sensitive dependence on initial conditions. This property refers to the concept that small changes in initial conditions can generate drastically different behavior in the iterative process.<\/p>\n<p id=\"fs-id1165043110064\">Probably the best-known example of chaos is the <span class=\"no-emphasis\">Mandelbrot set<\/span> (see Figure 7), named after Benoit Mandelbrot (1924\u20132010), who investigated its properties and helped popularize the field of chaos theory. The Mandelbrot set is usually generated by computer and shows fascinating details on enlargement, including self-replication of the set. Several colorized versions of the set have been shown in museums and can be found online and in popular books on the subject.<\/p>\n<div id=\"CNX_Calc_Figure_04_09_007\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211348\/CNX_Calc_Figure_04_09_007.jpg\" alt=\"A very complicated and organic looking fractal.\" width=\"731\" height=\"547\" \/><figcaption class=\"wp-caption-text\">Figure 7. The Mandelbrot set is a well-known example of a set of points generated by the iterative chaotic behavior of a relatively simple function.<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165042955768\">In this project we use the logistic map<\/p>\n<div id=\"fs-id1165042707686\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(x)=rx(1-x)[\/latex], where [latex]x\\in [0,1][\/latex] and [latex]r>0[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043094009\">as the function in our iterative process. The logistic map is a deceptively simple function; but, depending on the value of [latex]r[\/latex], the resulting iterative process displays some very interesting behavior. It can lead to fixed points, cycles, and even chaos.<\/p>\n<p id=\"fs-id1165042713994\">To visualize the long-term behavior of the iterative process associated with the logistic map, we will use a tool called a <em>cobweb diagram<\/em>. As we did with the iterative process we examined earlier in this section, we first draw a vertical line from the point [latex](x_0,0)[\/latex] to the point [latex](x_0,f(x_0))=(x_0,x_1)[\/latex]. We then draw a horizontal line from that point to the point [latex](x_1,x_1)[\/latex], then draw a vertical line to [latex](x_1,f(x_1))=(x_1,x_2)[\/latex], and continue the process until the long-term behavior of the system becomes apparent. Figure 8 shows the long-term behavior of the logistic map when [latex]r=3.55[\/latex] and [latex]x_0=0.2[\/latex]. (The first 100 iterations are not plotted.) The long-term behavior of this iterative process is an 8-cycle.<\/p>\n<figure style=\"width: 413px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211353\/CNX_Calc_Figure_04_09_008.jpg\" alt=\"In the first quadrant, f(x) = 3.55x(1 \u2013 x) is graphed as is y = x. From some point on the x axis, a line is drawn up to the line y = x, at which point it turns to be horizontal and continues until it touches the outside edge of f(x), at which point it turns again to be vertical until it each the line y = x. This process continues a number of times and creates an interesting series of boxes.\" width=\"413\" height=\"421\" \/><figcaption class=\"wp-caption-text\">Figure 8. A cobweb diagram for [latex]f(x)=3.55x(1-x)[\/latex] is presented here. The sequence of values results in an 8-cycle.<\/figcaption><\/figure>\n<ol id=\"fs-id1165043251229\">\n<li>Let [latex]r=0.5[\/latex] and choose [latex]x_0=0.2[\/latex]. Either by hand or by using a computer, calculate the first 10 values in the sequence. Does the sequence appear to converge? If so, to what value? Does it result in a cycle? If so, what kind of cycle (for example, 2-cycle, 4-cycle)?\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q439980\">Show Answer<\/button><\/p>\n<div id=\"q439980\" class=\"hidden-answer\" style=\"display: none\">\n<p>For [latex]r = 0.5[\/latex] and [latex]x\u2080 = 0.2[\/latex]:<\/p>\n<p>[latex]\\begin{array}{rcl}<br \/>  x_1 & = & 0.5 \\times 0.2 \\times (1 - 0.2) = 0.08 \\\\<br \/>  x_2 & = & 0.5 \\times 0.08 \\times (1 - 0.08) = 0.0368 \\\\<br \/>  x_3 & = & 0.5 \\times 0.0368 \\times (1 - 0.0368) \\approx 0.0177 \\\\<br \/>  x_4 & \\approx & 0.0087 \\\\<br \/>  x_5 & \\approx & 0.0043 \\\\<br \/>  x_6 & \\approx & 0.0021 \\\\<br \/>  x_7 & \\approx & 0.0011 \\\\<br \/>  x_8 & \\approx & 0.0005 \\\\<br \/>  x_9 & \\approx & 0.0003 \\\\<br \/>  x_{10} & \\approx & 0.0001 \\\\<br \/>  \\end{array}[\/latex]<\/p>\n<p>The sequence appears to converge to [latex]0[\/latex].<\/p>\n<p>It does not result in a cycle. Instead, it exhibits monotonic convergence to a fixed point ([latex]0[\/latex] in this case).<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>What happens when [latex]r=2[\/latex]?\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q31822\">Show Answer<\/button><\/p>\n<div id=\"q31822\" class=\"hidden-answer\" style=\"display: none\">\n<p>For [latex]r = 2[\/latex]:<br \/>\nThe sequence converges to [latex]0.5[\/latex].\u00a0<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>For [latex]r=3.2[\/latex] and [latex]r=3.5[\/latex], calculate the first [latex]100[\/latex] sequence values.\u00a0 What is the long-term behavior in each of these cases?\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q613527\">Show Answer<\/button><\/p>\n<div id=\"q613527\" class=\"hidden-answer\" style=\"display: none\">\n<p>For [latex]r = 3.2[\/latex]:<br \/>\nThe sequence converges to a single value approximately [latex]0.7994[\/latex].<\/p>\n<p>For [latex]r = 3.5[\/latex]:<br \/>\nThe sequence settles into a 4-cycle, alternating between approximately [latex]0.8270, 0.5009, 0.8746[\/latex], and [latex]0.3827[\/latex].\n<\/div>\n<\/div>\n<\/li>\n<li>Now let [latex]r=4[\/latex]. Calculate the first [latex]100[\/latex] sequence values. What is the long-term behavior in this case?\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q934037\">Show Answer<\/button><\/p>\n<div id=\"q934037\" class=\"hidden-answer\" style=\"display: none\">\n<p>For [latex]r = 4[\/latex] and [latex]x\u2080 = 0.2[\/latex]:<br \/>\nThe sequence exhibits chaotic behavior. It doesn&#8217;t settle into any discernible pattern and appears to jump around unpredictably within the interval [latex][0,1][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>Repeat the process for [latex]r=4[\/latex], but let [latex]x_0=0.201[\/latex]. How does this behavior compare with the behavior for [latex]x_0=0.2[\/latex]?\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q58377\">Show Answer<\/button><\/p>\n<div id=\"q58377\" class=\"hidden-answer\" style=\"display: none\">\n<p>For [latex]r = 4[\/latex] and [latex]x\u2080 = 0.201[\/latex]:<\/p>\n<p>The sequence also exhibits chaotic behavior, but the specific sequence of values is very different from the one with [latex]x\u2080 = 0.2[\/latex]. This demonstrates sensitive dependence on initial conditions, a key characteristic of chaos.<\/p>\n<p>After a few iterations, the sequences for [latex]x\u2080 = 0.2[\/latex] and [latex]x\u2080 = 0.201[\/latex] diverge significantly, despite starting very close to each other. This illustrates how small changes in initial conditions can lead to drastically different outcomes in chaotic systems.\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":26,"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":"apply_it","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\/3444"}],"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\/3444\/revisions"}],"predecessor-version":[{"id":3991,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3444\/revisions\/3991"}],"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\/3444\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3444"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3444"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3444"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}