{"id":809,"date":"2025-06-20T17:14:20","date_gmt":"2025-06-20T17:14:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=809"},"modified":"2025-07-22T14:58:47","modified_gmt":"2025-07-22T14:58:47","slug":"direction-fields-and-eulers-method-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/direction-fields-and-eulers-method-learn-it-1\/","title":{"raw":"Direction Fields and Euler's Method: Learn It 1","rendered":"Direction Fields and Euler&#8217;s Method: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Create direction fields for first-order differential equations<\/li>\r\n \t<li>Use a direction field to sketch solution curves<\/li>\r\n \t<li>Use Euler's Method to find approximate solutions step by step<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Direction Fields<\/h2>\r\n<strong>Direction fields<\/strong> (also called <strong>slope fields<\/strong>) provide a powerful visual tool for investigating first-order differential equations. We'll work specifically with equations of the form:\r\n<p style=\"text-align: center;\">[latex]y^{\\prime} = f(x,y)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Let's start with a real-world example from Newton's law of cooling:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]T^{\\prime}(t) = -0.4(T - 72)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Here [latex]T(t)[\/latex] represents temperature (in degrees Fahrenheit) of an object at time [latex]t[\/latex], and the ambient temperature is [latex]72\u00b0F[\/latex]. Figure 1 shows what the direction field looks like for this equation.<\/p>\r\n<p id=\"fs-id1170571277998\"><\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_08_02_001\">[caption id=\"\" align=\"aligncenter\" width=\"484\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233925\/CNX_Calc_Figure_08_02_009.jpg\" alt=\"A graph of a direction field for the given differential equation in quadrants one and two. The arrows are pointing directly to the right at y = 72. Below that line, the arrows have increasingly positive slope as y becomes smaller. Above that line, the arrows have increasingly negative slope as y becomes larger. The arrows point to convergence at y = 72. Two solutions are drawn: one for initial temperature less than 72, and one for initial temperatures larger than 72. The upper solution is a decreasing concave up curve, approaching y = 72 as t goes to infinity. The lower solution is an increasing concave down curve, approaching y = 72 as t goes to infinity.\" width=\"484\" height=\"509\" data-media-type=\"image\/jpeg\" \/> Figure 1. Direction field for the differential equation [latex]{T}^{\\prime }\\left(t\\right)=-0.4\\left(T - 72\\right)[\/latex]. Two solutions are plotted: one with initial temperature less than [latex]72^\\circ\\text{F}[\/latex] and the other with initial temperature greater than [latex]72^\\circ\\text{F}\\text{.}[\/latex][\/caption]<\/figure>\r\nThe idea behind a direction field is the fact that the derivative of a function evaluated at a given point is the slope of the tangent line to the graph of that function at the same point.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Here are other differential equations where direction fields prove useful:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-pre-wrap break-words\">[latex]y^{\\prime} = 3x + 2y - 4[\/latex]<\/li>\r\n \t<li class=\"whitespace-pre-wrap break-words\">[latex]y^{\\prime} = x^2 - y^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-pre-wrap break-words\">[latex]y^{\\prime} = \\frac{2x + 4}{y - 2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Creating a Direction Field<\/h3>\r\n<p class=\"whitespace-normal break-words\">Let's work through the process of creating a direction field using one of our examples: [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">The key idea is to pick any point [latex](x_0, y_0)[\/latex] in the coordinate plane and substitute these coordinates into the right-hand side of the differential equation.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">If we choose [latex]x = 1[\/latex] and [latex]y = 2[\/latex], substituting into the right-hand side gives us:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]y^{\\prime} = 3x + 2y - 4 = 3(1) + 2(2) - 4 = 3[\/latex]<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">This tells us that if a solution to the differential equation [latex]y^{\\prime} = 3x + 2y - 4[\/latex] passes through the point [latex](1, 2)[\/latex], then the slope of the solution at that point must equal [latex]3[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">To start creating the direction field, we put a short line segment at the point [latex](1, 2)[\/latex] having slope [latex]3[\/latex]. We can do this for any point in the domain of the function [latex]f(x, y) = 3x + 2y - 4[\/latex], which consists of all ordered pairs [latex](x, y)[\/latex] in [latex]\\mathbb{R}^2[\/latex].<\/p>\r\nThis means<strong> every point<\/strong> in the Cartesian plane has a slope associated with it, assuming that a solution to the differential equation passes through that point. The direction field for the differential equation [latex]y^{\\prime} = 3x + 2y - 4[\/latex] is shown in Figure 2.\r\n<p id=\"fs-id1170571291390\"><\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_08_02_002\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233929\/CNX_Calc_Figure_08_02_001.jpg\" alt=\"A graph of the direction field for the differential equation y\u2019 = 3 x + 2 y \u2013 4 in all four quadrants. In quadrants two and three, the arrows point down and slightly to the right. On a diagonal line, roughly y = -x + 2, the arrows point further and further to the right, curve, and then point up above that line.\" width=\"487\" height=\"467\" data-media-type=\"image\/jpeg\" \/> Figure 2. Direction field for the differential equation [latex]y^{\\prime} =3x+2y - 4[\/latex].[\/caption]<\/figure>\r\nWe can generate a direction field of this type for any differential equation of the form [latex]y^{\\prime} =f\\left(x,y\\right)[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>direction field<\/h3>\r\nA <strong>direction field (slope field)<\/strong> is 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.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Create direction fields for first-order differential equations<\/li>\n<li>Use a direction field to sketch solution curves<\/li>\n<li>Use Euler&#8217;s Method to find approximate solutions step by step<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Direction Fields<\/h2>\n<p><strong>Direction fields<\/strong> (also called <strong>slope fields<\/strong>) provide a powerful visual tool for investigating first-order differential equations. We&#8217;ll work specifically with equations of the form:<\/p>\n<p style=\"text-align: center;\">[latex]y^{\\prime} = f(x,y)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Let&#8217;s start with a real-world example from Newton&#8217;s law of cooling:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]T^{\\prime}(t) = -0.4(T - 72)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Here [latex]T(t)[\/latex] represents temperature (in degrees Fahrenheit) of an object at time [latex]t[\/latex], and the ambient temperature is [latex]72\u00b0F[\/latex]. Figure 1 shows what the direction field looks like for this equation.<\/p>\n<p id=\"fs-id1170571277998\">\n<figure id=\"CNX_Calc_Figure_08_02_001\">\n<figure style=\"width: 484px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233925\/CNX_Calc_Figure_08_02_009.jpg\" alt=\"A graph of a direction field for the given differential equation in quadrants one and two. The arrows are pointing directly to the right at y = 72. Below that line, the arrows have increasingly positive slope as y becomes smaller. Above that line, the arrows have increasingly negative slope as y becomes larger. The arrows point to convergence at y = 72. Two solutions are drawn: one for initial temperature less than 72, and one for initial temperatures larger than 72. The upper solution is a decreasing concave up curve, approaching y = 72 as t goes to infinity. The lower solution is an increasing concave down curve, approaching y = 72 as t goes to infinity.\" width=\"484\" height=\"509\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1. Direction field for the differential equation [latex]{T}^{\\prime }\\left(t\\right)=-0.4\\left(T - 72\\right)[\/latex]. Two solutions are plotted: one with initial temperature less than [latex]72^\\circ\\text{F}[\/latex] and the other with initial temperature greater than [latex]72^\\circ\\text{F}\\text{.}[\/latex]<\/figcaption><\/figure>\n<\/figure>\n<p>The idea behind a direction field is the fact that the derivative of a function evaluated at a given point is the slope of the tangent line to the graph of that function at the same point.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Here are other differential equations where direction fields prove useful:<\/p>\n<ul>\n<li class=\"whitespace-pre-wrap break-words\">[latex]y^{\\prime} = 3x + 2y - 4[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">[latex]y^{\\prime} = x^2 - y^2[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">[latex]y^{\\prime} = \\frac{2x + 4}{y - 2}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<h3>Creating a Direction Field<\/h3>\n<p class=\"whitespace-normal break-words\">Let&#8217;s work through the process of creating a direction field using one of our examples: [latex]y^{\\prime} = 3x + 2y - 4[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">The key idea is to pick any point [latex](x_0, y_0)[\/latex] in the coordinate plane and substitute these coordinates into the right-hand side of the differential equation.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">If we choose [latex]x = 1[\/latex] and [latex]y = 2[\/latex], substituting into the right-hand side gives us:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]y^{\\prime} = 3x + 2y - 4 = 3(1) + 2(2) - 4 = 3[\/latex]<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">This tells us that if a solution to the differential equation [latex]y^{\\prime} = 3x + 2y - 4[\/latex] passes through the point [latex](1, 2)[\/latex], then the slope of the solution at that point must equal [latex]3[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">To start creating the direction field, we put a short line segment at the point [latex](1, 2)[\/latex] having slope [latex]3[\/latex]. We can do this for any point in the domain of the function [latex]f(x, y) = 3x + 2y - 4[\/latex], which consists of all ordered pairs [latex](x, y)[\/latex] in [latex]\\mathbb{R}^2[\/latex].<\/p>\n<p>This means<strong> every point<\/strong> in the Cartesian plane has a slope associated with it, assuming that a solution to the differential equation passes through that point. The direction field for the differential equation [latex]y^{\\prime} = 3x + 2y - 4[\/latex] is shown in Figure 2.<\/p>\n<p id=\"fs-id1170571291390\">\n<figure id=\"CNX_Calc_Figure_08_02_002\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233929\/CNX_Calc_Figure_08_02_001.jpg\" alt=\"A graph of the direction field for the differential equation y\u2019 = 3 x + 2 y \u2013 4 in all four quadrants. In quadrants two and three, the arrows point down and slightly to the right. On a diagonal line, roughly y = -x + 2, the arrows point further and further to the right, curve, and then point up above that line.\" width=\"487\" height=\"467\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2. Direction field for the differential equation [latex]y^{\\prime} =3x+2y - 4[\/latex].<\/figcaption><\/figure>\n<\/figure>\n<p>We can generate a direction field of this type for any differential equation of the form [latex]y^{\\prime} =f\\left(x,y\\right)[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>direction field<\/h3>\n<p>A <strong>direction field (slope field)<\/strong> is 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.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":669,"module-header":"- Select Header -","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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/809"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/809\/revisions"}],"predecessor-version":[{"id":1395,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/809\/revisions\/1395"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/669"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/809\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=809"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=809"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=809"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=809"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}