{"id":813,"date":"2025-06-20T17:15:15","date_gmt":"2025-06-20T17:15:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=813"},"modified":"2025-10-09T21:13:01","modified_gmt":"2025-10-09T21:13:01","slug":"direction-fields-and-eulers-method-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/direction-fields-and-eulers-method-apply-it\/","title":{"raw":"Direction Fields and Euler's Method: Apply It","rendered":"Direction Fields and Euler&#8217;s Method: Apply It"},"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 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Coffee Temperature Analysis<\/h2>\r\n<p class=\"whitespace-normal break-words\">You're working as a barista at a local coffee shop and want to understand how quickly different beverages cool down to ensure the best customer experience. The rate at which a hot beverage cools follows Newton's law of cooling, which can be modeled by the differential equation:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]T'(t) = -k(T - A)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">where [latex]T(t)[\/latex] is the temperature of the beverage at time [latex]t[\/latex] (in minutes), [latex]k[\/latex] is the cooling constant that depends on the beverage container and environment, and [latex]A[\/latex] is the ambient room temperature.<\/p>\r\n<p class=\"whitespace-normal break-words\">For your analysis, you're studying a specialty latte that starts at [latex]160\u00b0F[\/latex] in a ceramic mug. The room temperature is [latex]72\u00b0F[\/latex], and through experimentation, you've determined that [latex]k = 0.08[\/latex]. This gives you the specific differential equation:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]T'(t) = -0.08(T - 72)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">You want to create a direction field to visualize the cooling behavior and use Euler's method to predict temperatures at specific times to help train other baristas about optimal serving times.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313465[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313466[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313467[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313468[\/ohm_question]<\/p>\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 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Coffee Temperature Analysis<\/h2>\n<p class=\"whitespace-normal break-words\">You&#8217;re working as a barista at a local coffee shop and want to understand how quickly different beverages cool down to ensure the best customer experience. The rate at which a hot beverage cools follows Newton&#8217;s law of cooling, which can be modeled by the differential equation:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]T'(t) = -k(T - A)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">where [latex]T(t)[\/latex] is the temperature of the beverage at time [latex]t[\/latex] (in minutes), [latex]k[\/latex] is the cooling constant that depends on the beverage container and environment, and [latex]A[\/latex] is the ambient room temperature.<\/p>\n<p class=\"whitespace-normal break-words\">For your analysis, you&#8217;re studying a specialty latte that starts at [latex]160\u00b0F[\/latex] in a ceramic mug. The room temperature is [latex]72\u00b0F[\/latex], and through experimentation, you&#8217;ve determined that [latex]k = 0.08[\/latex]. This gives you the specific differential equation:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]T'(t) = -0.08(T - 72)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">You want to create a direction field to visualize the cooling behavior and use Euler&#8217;s method to predict temperatures at specific times to help train other baristas about optimal serving times.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313465\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313465&theme=lumen&iframe_resize_id=ohm313465&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313466\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313466&theme=lumen&iframe_resize_id=ohm313466&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313467\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313467&theme=lumen&iframe_resize_id=ohm313467&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm313468\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313468&theme=lumen&iframe_resize_id=ohm313468&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"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\/813"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/813\/revisions"}],"predecessor-version":[{"id":2419,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/813\/revisions\/2419"}],"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\/813\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=813"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=813"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=813"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=813"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}