{"id":58,"date":"2021-02-03T20:18:05","date_gmt":"2021-02-03T20:18:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/antiderivatives\/"},"modified":"2021-04-03T02:03:00","modified_gmt":"2021-04-03T02:03:00","slug":"antiderivatives","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/antiderivatives\/","title":{"raw":"Summary of Antiderivatives","rendered":"Summary of Antiderivatives"},"content":{"raw":"<div id=\"fs-id1165043396312\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165042323569\">\r\n \t<li>If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], then every antiderivative of [latex]f[\/latex] is of the form [latex]F(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\r\n \t<li>Solving the initial-value problem\r\n<div id=\"fs-id1165043259810\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=f(x),y(x_0)=y_0[\/latex]<\/div>\r\nrequires us first to find the set of antiderivatives of [latex]f[\/latex] and then to look for the particular antiderivative that also satisfies the initial condition.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042617549\" class=\"definition\">\r\n \t<dt>antiderivative<\/dt>\r\n \t<dd id=\"fs-id1165042617555\">a function [latex]F[\/latex] such that [latex]F^{\\prime}(x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex] is an antiderivative of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042617603\" class=\"definition\">\r\n \t<dt>indefinite integral<\/dt>\r\n \t<dd id=\"fs-id1165042617608\">the most general antiderivative of [latex]f(x)[\/latex] is the indefinite integral of [latex]f[\/latex]; we use the notation [latex]\\displaystyle\\int f(x) dx[\/latex] to denote the indefinite integral of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042617659\" class=\"definition\">\r\n \t<dt>initial value problem<\/dt>\r\n \t<dd id=\"fs-id1165042617665\">a problem that requires finding a function [latex]y[\/latex] that satisfies the differential equation [latex]\\frac{dy}{dx}=f(x)[\/latex] together with the initial condition [latex]y(x_0)=y_0[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1165043396312\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165042323569\">\n<li>If [latex]F[\/latex] is an antiderivative of [latex]f[\/latex], then every antiderivative of [latex]f[\/latex] is of the form [latex]F(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\n<li>Solving the initial-value problem\n<div id=\"fs-id1165043259810\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=f(x),y(x_0)=y_0[\/latex]<\/div>\n<p>requires us first to find the set of antiderivatives of [latex]f[\/latex] and then to look for the particular antiderivative that also satisfies the initial condition.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042617549\" class=\"definition\">\n<dt>antiderivative<\/dt>\n<dd id=\"fs-id1165042617555\">a function [latex]F[\/latex] such that [latex]F^{\\prime}(x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex] is an antiderivative of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042617603\" class=\"definition\">\n<dt>indefinite integral<\/dt>\n<dd id=\"fs-id1165042617608\">the most general antiderivative of [latex]f(x)[\/latex] is the indefinite integral of [latex]f[\/latex]; we use the notation [latex]\\displaystyle\\int f(x) dx[\/latex] to denote the indefinite integral of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042617659\" class=\"definition\">\n<dt>initial value problem<\/dt>\n<dd id=\"fs-id1165042617665\">a problem that requires finding a function [latex]y[\/latex] that satisfies the differential equation [latex]\\frac{dy}{dx}=f(x)[\/latex] together with the initial condition [latex]y(x_0)=y_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-58\">\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":39,"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-58","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/58","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":11,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions"}],"predecessor-version":[{"id":2618,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions\/2618"}],"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\/58\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=58"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=58"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=58"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}