{"id":1631,"date":"2021-03-19T20:12:08","date_gmt":"2021-03-19T20:12:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1631"},"modified":"2021-04-02T23:50:02","modified_gmt":"2021-04-02T23:50:02","slug":"summary-of-the-mean-value-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-the-mean-value-theorem\/","title":{"raw":"Summary of the Mean Value Theorem","rendered":"Summary of the Mean Value Theorem"},"content":{"raw":"<div id=\"fs-id1165042651657\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165042651664\">\r\n \t<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex] and [latex]f(a)=0=f(b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]. This is Rolle\u2019s theorem.<\/li>\r\n \t<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that\r\n<div id=\"fs-id1165042711722\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex].<\/div>\r\nThis is the Mean Value Theorem.<\/li>\r\n \t<li>If [latex]f^{\\prime}(x)=0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is constant over [latex]I[\/latex].<\/li>\r\n \t<li>If two differentiable functions [latex]f[\/latex] and [latex]g[\/latex] satisfy [latex]f^{\\prime}(x)=g^{\\prime}(x)[\/latex] over [latex]I[\/latex], then [latex]f(x)=g(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime}(x)&gt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is increasing over [latex]I[\/latex]. If [latex]f^{\\prime}(x)&lt;0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is decreasing over [latex]I[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042595459\" class=\"definition\">\r\n \t<dt>mean value theorem<\/dt>\r\n \t<dd id=\"fs-id1165042595464\">if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that\r\n<div id=\"fs-id1165043183378\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex]<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043183432\" class=\"definition\">\r\n \t<dt>rolle\u2019s theorem<\/dt>\r\n \t<dd id=\"fs-id1165043183437\">if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], and if [latex]f(a)=f(b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1165042651657\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165042651664\">\n<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex] and [latex]f(a)=0=f(b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]. This is Rolle\u2019s theorem.<\/li>\n<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that\n<div id=\"fs-id1165042711722\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex].<\/div>\n<p>This is the Mean Value Theorem.<\/li>\n<li>If [latex]f^{\\prime}(x)=0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is constant over [latex]I[\/latex].<\/li>\n<li>If two differentiable functions [latex]f[\/latex] and [latex]g[\/latex] satisfy [latex]f^{\\prime}(x)=g^{\\prime}(x)[\/latex] over [latex]I[\/latex], then [latex]f(x)=g(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(x)>0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is increasing over [latex]I[\/latex]. If [latex]f^{\\prime}(x)<0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is decreasing over [latex]I[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042595459\" class=\"definition\">\n<dt>mean value theorem<\/dt>\n<dd id=\"fs-id1165042595464\">if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that<\/p>\n<div id=\"fs-id1165043183378\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043183432\" class=\"definition\">\n<dt>rolle\u2019s theorem<\/dt>\n<dd id=\"fs-id1165043183437\">if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], and if [latex]f(a)=f(b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/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-1631\">\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 1. <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\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/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-1\/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":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/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-1631","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1631","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1631\/revisions"}],"predecessor-version":[{"id":2551,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1631\/revisions\/2551"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1631\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1631"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1631"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1631"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}