{"id":1193,"date":"2021-06-30T17:02:08","date_gmt":"2021-06-30T17:02:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-exponential-growth-and-decay\/"},"modified":"2021-11-17T02:14:19","modified_gmt":"2021-11-17T02:14:19","slug":"summary-of-exponential-growth-and-decay","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-exponential-growth-and-decay\/","title":{"raw":"Summary of Exponential Growth and Decay","rendered":"Summary of Exponential Growth and Decay"},"content":{"raw":"<div id=\"fs-id1167793546919\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793829823\">\r\n \t<li>Exponential growth and exponential decay are two of the most common applications of exponential functions.<\/li>\r\n \t<li>Systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/li>\r\n \t<li>In exponential growth, the rate of growth is proportional to the quantity present. In other words, [latex]{y}^{\\prime }=ky.[\/latex]<\/li>\r\n \t<li>Systems that exhibit exponential growth have a constant doubling time, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\r\n \t<li>Systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}.[\/latex]<\/li>\r\n \t<li>Systems that exhibit exponential decay have a constant half-life, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167793423300\" class=\"definition\">\r\n \t<dt>doubling time<\/dt>\r\n \t<dd id=\"fs-id1167793423305\">if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794146802\" class=\"definition\">\r\n \t<dt>exponential decay<\/dt>\r\n \t<dd id=\"fs-id1167794146808\">systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794146834\" class=\"definition\">\r\n \t<dt>exponential growth<\/dt>\r\n \t<dd>systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167793541831\" class=\"definition\">\r\n \t<dt>half-life<\/dt>\r\n \t<dd id=\"fs-id1167793541836\">if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1167793546919\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793829823\">\n<li>Exponential growth and exponential decay are two of the most common applications of exponential functions.<\/li>\n<li>Systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/li>\n<li>In exponential growth, the rate of growth is proportional to the quantity present. In other words, [latex]{y}^{\\prime }=ky.[\/latex]<\/li>\n<li>Systems that exhibit exponential growth have a constant doubling time, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\n<li>Systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}.[\/latex]<\/li>\n<li>Systems that exhibit exponential decay have a constant half-life, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167793423300\" class=\"definition\">\n<dt>doubling time<\/dt>\n<dd id=\"fs-id1167793423305\">if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794146802\" class=\"definition\">\n<dt>exponential decay<\/dt>\n<dd id=\"fs-id1167794146808\">systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794146834\" class=\"definition\">\n<dt>exponential growth<\/dt>\n<dd>systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793541831\" class=\"definition\">\n<dt>half-life<\/dt>\n<dd id=\"fs-id1167793541836\">if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\n<\/dl>\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-1193\">\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":17533,"menu_order":33,"template":"","meta":{"_candela_citation":"{\"2\":{\"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-1193","chapter","type-chapter","status-publish","hentry"],"part":1160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1193","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\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1193\/revisions"}],"predecessor-version":[{"id":2517,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1193\/revisions\/2517"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1193\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1193"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1193"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1193"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}