{"id":682,"date":"2021-05-10T18:59:01","date_gmt":"2021-05-10T18:59:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=682"},"modified":"2021-11-17T02:44:59","modified_gmt":"2021-11-17T02:44:59","slug":"summary-of-direction-fields-and-numerical-methods","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-direction-fields-and-numerical-methods\/","title":{"raw":"Summary of Direction Fields and Numerical Methods","rendered":"Summary of Direction Fields and Numerical Methods"},"content":{"raw":"<section id=\"fs-id1170571042944\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571042951\" data-bullet-style=\"bullet\">\r\n \t<li>A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.<\/li>\r\n \t<li>Euler\u2019s Method is a numerical technique that can be used to approximate solutions to a differential equation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170571042967\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170571042973\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Euler\u2019s Method<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\begin{array}{c}{x}_{n}={x}_{0}+nh\\hfill \\\\ {y}_{n}={y}_{n - 1}+hf\\left({x}_{n - 1},{y}_{n - 1}\\right),\\text{where}h\\text{is the step size}\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1170571097524\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170571116853\">\r\n \t<dt>asymptotically semi-stable solution<\/dt>\r\n \t<dd id=\"fs-id1170571116858\">[latex]y=k[\/latex] if it is neither asymptotically stable nor asymptotically unstable<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571116871\">\r\n \t<dt>asymptotically stable solution<\/dt>\r\n \t<dd id=\"fs-id1170571116885\">[latex]y=k[\/latex] if there exists [latex]\\epsilon &gt;0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571459118\">\r\n \t<dt>asymptotically unstable solution<\/dt>\r\n \t<dd id=\"fs-id1170571459132\">[latex]y=k[\/latex] if there exists [latex]\\epsilon &gt;0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] never approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571051486\">\r\n \t<dt>direction field (slope field)<\/dt>\r\n \t<dd id=\"fs-id1170571051491\">a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571051498\">\r\n \t<dt>equilibrium solution<\/dt>\r\n \t<dd id=\"fs-id1170571042367\">any solution to the differential equation of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571042387\">\r\n \t<dt>Euler\u2019s Method<\/dt>\r\n \t<dd id=\"fs-id1170571042393\">a numerical technique used to approximate solutions to an initial-value problem<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571042397\">\r\n \t<dt>solution curve<\/dt>\r\n \t<dd id=\"fs-id1170571042402\">a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571042408\">\r\n \t<dt>step size<\/dt>\r\n \t<dd id=\"fs-id1170571042413\">the increment [latex]h[\/latex] that is added to the [latex]x[\/latex] value at each step in Euler\u2019s Method<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1170571042944\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571042951\" data-bullet-style=\"bullet\">\n<li>A direction field is a mathematical object used to graphically represent solutions to a first-order differential equation.<\/li>\n<li>Euler\u2019s Method is a numerical technique that can be used to approximate solutions to a differential equation.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1170571042967\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170571042973\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Euler\u2019s Method<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\begin{array}{c}{x}_{n}={x}_{0}+nh\\hfill \\\\ {y}_{n}={y}_{n - 1}+hf\\left({x}_{n - 1},{y}_{n - 1}\\right),\\text{where}h\\text{is the step size}\\hfill \\end{array}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1170571097524\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170571116853\">\n<dt>asymptotically semi-stable solution<\/dt>\n<dd id=\"fs-id1170571116858\">[latex]y=k[\/latex] if it is neither asymptotically stable nor asymptotically unstable<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571116871\">\n<dt>asymptotically stable solution<\/dt>\n<dd id=\"fs-id1170571116885\">[latex]y=k[\/latex] if there exists [latex]\\epsilon >0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571459118\">\n<dt>asymptotically unstable solution<\/dt>\n<dd id=\"fs-id1170571459132\">[latex]y=k[\/latex] if there exists [latex]\\epsilon >0[\/latex] such that for any value [latex]c\\in \\left(k-\\epsilon ,k+\\epsilon \\right)[\/latex] the solution to the initial-value problem [latex]{y}^{\\prime }=f\\left(x,y\\right),y\\left({x}_{0}\\right)=c[\/latex] never approaches [latex]k[\/latex] as [latex]x[\/latex] approaches infinity<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571051486\">\n<dt>direction field (slope field)<\/dt>\n<dd id=\"fs-id1170571051491\">a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571051498\">\n<dt>equilibrium solution<\/dt>\n<dd id=\"fs-id1170571042367\">any solution to the differential equation of the form [latex]y=c[\/latex], where [latex]c[\/latex] is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571042387\">\n<dt>Euler\u2019s Method<\/dt>\n<dd id=\"fs-id1170571042393\">a numerical technique used to approximate solutions to an initial-value problem<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571042397\">\n<dt>solution curve<\/dt>\n<dd id=\"fs-id1170571042402\">a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571042408\">\n<dt>step size<\/dt>\n<dd id=\"fs-id1170571042413\">the increment [latex]h[\/latex] that is added to the [latex]x[\/latex] value at each step in Euler\u2019s Method<\/dd>\n<\/dl>\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-682\">\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 2. <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\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/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-2\/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":416434,"menu_order":9,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/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-682","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/682","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/682\/revisions"}],"predecessor-version":[{"id":1038,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/682\/revisions\/1038"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/682\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=682"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=682"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=682"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=682"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}