{"id":1971,"date":"2021-08-19T16:03:43","date_gmt":"2021-08-19T16:03:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/review-for-the-fundamental-theorem-of-calculus\/"},"modified":"2021-11-17T01:32:02","modified_gmt":"2021-11-17T01:32:02","slug":"review-for-the-fundamental-theorem-of-calculus","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/review-for-the-fundamental-theorem-of-calculus\/","title":{"raw":"Skills Review for The Fundamental Theorem of Calculus","rendered":"Skills Review for The Fundamental Theorem of Calculus"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate a polynomial function at a monomial or binomial value<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Fundamental Theorem of Calculus section, we will learn how to evaluate definite integrals. Here we will review evaluating functions that have variables raised to powers.\r\n<h2>Evaluate Functions Containing Variables Raised to Powers<\/h2>\r\nWhen given the function\u00a0[latex]f(x)=x^2+7[\/latex], if asked to find the value of\u00a0[latex]f(-3t)[\/latex], you would take the variable\u00a0[latex]x[\/latex]\u00a0in the function and replace it with [latex]-3t[\/latex]. So,\r\n<p style=\"text-align: center;\">[latex]f(-3t)=(-3t)^2+7=9t^2+7[\/latex]<\/p>\r\nNote: It is important that the [latex]-3[\/latex] and the\u00a0[latex]t[\/latex] both be squared.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Functions Containing Variables Raised to Powers<\/h3>\r\nGiven\u00a0[latex]f(x)=x^3+2x-3[\/latex], find\u00a0[latex]f(-2t)[\/latex].\r\n\r\n[reveal-answer q=\"133750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133750\"]\r\n\r\nEverywhere we see [latex]x[\/latex]\u00a0in the function, we replace it with\u00a0[latex]-2t[\/latex].\r\n\r\nSo, we have:\r\n\r\n[latex]f(-2t)=(-2t)^3+2(-2t)-3=-8t^3-4t-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Functions Containing Variables Raised to Powers<\/h3>\r\nGiven\u00a0[latex]f(x)=2x^3-9[\/latex], find\u00a0[latex]f(5t^2)[\/latex].\r\n\r\n[reveal-answer q=\"133751\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133751\"]\r\n\r\nEverywhere we see\u00a0[latex]x[\/latex] in the function, we replace it with\u00a0[latex]5t^2[\/latex].\r\n\r\nSo, we have:\r\n\r\n[latex]f(5t^2)=2(5t^2)^3-9=2(125t^6)-9=250t^6-9[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote: In the above example, when we had to find\u00a0[latex](t^2)^3[\/latex], raising a power to a power requires that we multiply the exponents together to give us [latex]t^6[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven\u00a0[latex]f(x)=x^2-3x+1[\/latex], find\u00a0[latex]f(-4t^4)[\/latex].\r\n\r\n[reveal-answer q=\"133752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133752\"]\r\n\r\n[latex]f(-4t^4)=16t^8+12t^4+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate a polynomial function at a monomial or binomial value<\/li>\n<\/ul>\n<\/div>\n<p>In the Fundamental Theorem of Calculus section, we will learn how to evaluate definite integrals. Here we will review evaluating functions that have variables raised to powers.<\/p>\n<h2>Evaluate Functions Containing Variables Raised to Powers<\/h2>\n<p>When given the function\u00a0[latex]f(x)=x^2+7[\/latex], if asked to find the value of\u00a0[latex]f(-3t)[\/latex], you would take the variable\u00a0[latex]x[\/latex]\u00a0in the function and replace it with [latex]-3t[\/latex]. So,<\/p>\n<p style=\"text-align: center;\">[latex]f(-3t)=(-3t)^2+7=9t^2+7[\/latex]<\/p>\n<p>Note: It is important that the [latex]-3[\/latex] and the\u00a0[latex]t[\/latex] both be squared.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Functions Containing Variables Raised to Powers<\/h3>\n<p>Given\u00a0[latex]f(x)=x^3+2x-3[\/latex], find\u00a0[latex]f(-2t)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133750\">Show Solution<\/span><\/p>\n<div id=\"q133750\" class=\"hidden-answer\" style=\"display: none\">\n<p>Everywhere we see [latex]x[\/latex]\u00a0in the function, we replace it with\u00a0[latex]-2t[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p>[latex]f(-2t)=(-2t)^3+2(-2t)-3=-8t^3-4t-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Functions Containing Variables Raised to Powers<\/h3>\n<p>Given\u00a0[latex]f(x)=2x^3-9[\/latex], find\u00a0[latex]f(5t^2)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133751\">Show Solution<\/span><\/p>\n<div id=\"q133751\" class=\"hidden-answer\" style=\"display: none\">\n<p>Everywhere we see\u00a0[latex]x[\/latex] in the function, we replace it with\u00a0[latex]5t^2[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p>[latex]f(5t^2)=2(5t^2)^3-9=2(125t^6)-9=250t^6-9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note: In the above example, when we had to find\u00a0[latex](t^2)^3[\/latex], raising a power to a power requires that we multiply the exponents together to give us [latex]t^6[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given\u00a0[latex]f(x)=x^2-3x+1[\/latex], find\u00a0[latex]f(-4t^4)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133752\">Show Solution<\/span><\/p>\n<div id=\"q133752\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(-4t^4)=16t^8+12t^4+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1971\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":349141,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/precalculus\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1971","chapter","type-chapter","status-publish","hentry"],"part":1968,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1971","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\/349141"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1971\/revisions"}],"predecessor-version":[{"id":2458,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1971\/revisions\/2458"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1968"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1971\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1971"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1971"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1971"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}