{"id":3900,"date":"2021-05-20T18:29:56","date_gmt":"2021-05-20T18:29:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-antiderivatives\/"},"modified":"2021-07-03T19:10:39","modified_gmt":"2021-07-03T19:10:39","slug":"review-for-antiderivatives","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-antiderivatives\/","title":{"raw":"Skills Review for Antiderivatives","rendered":"Skills Review for Antiderivatives"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide polynomials by monomials<\/li>\r\n \t<li>Write function equations using given conditions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Antiderivatives section, we will sometimes simplify rational expressions by dividing a polynomial in the numerator by a monomial in the denominator.\r\n<h2>Divide a Polynomial by a Monomial<\/h2>\r\nWhen dividing a polynomial by a monomial, it is important that every term of the polynomial is divided by the monomial. You will then want to simplify. Here we will review this topic.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Dividing a Polynomial by a Monomial<\/h3>\r\nPerform each division. Be sure to simplify your answer.\r\n<ol>\r\n \t<li>[latex]\\dfrac{9{x}^{3}+6x}{3{x}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{x}^{9}+3{x}^{6}}{{x}^{4}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"717838\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"717838\"]\r\n\r\nUse the quotient rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]\\dfrac{9{x}^{3}+6x}{3{x}^{2}}=\\dfrac{9{x}^{3}}{3{x}^{2}}+\\dfrac{6x}{3{x}^{2}}=3{x}^{3-2}+2{x}^{1-2}=3x+{x}^{-1}[\/latex]\u00a0Just like we did for derivatives, when finding antiderivatives, we will leave [latex]{x}^{-1}[\/latex] as it is rather than writing as\u00a0[latex]\\dfrac{1}{{x}}[\/latex].<\/li>\r\n \t<li>[latex]\\dfrac{{x}^{9}+3{x}^{6}}{{x}^{4}}=\\dfrac{{x}^{9}}{{x}^{4}}+\\dfrac{3{x}^{6}}{{x}^{4}}={x}^{9-4}+3{x}^{6-4}={x}^{5}+3{x}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]221979[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Write Function Equations Using Given Conditions<\/h2>\r\nSometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Function Equation from given conditions<\/h3>\r\nGiven [latex]f(2)=-1[\/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[\/latex].\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"338564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338564\"]\r\n\r\nTo find c, we use the fact that [latex]f(2)=-1[\/latex], that is, the function's value is -1 when [latex]x=2[\/latex].\r\n\r\n[latex]\\begin{array}{l}-1=3(2)^3-4(2)^2-2+c\\hfill \\\\ -1=3(8)-4(4)-2+c\\hfill \\\\ -1=24-16-2+c\\hfill \\\\ -1=6+c\\hfill \\\\ -7=c \\end{array}[\/latex]\r\n\r\nThe function equation is\u00a0[latex]f(x)=3x^3-4x^2-x-7[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]f(1)=5[\/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[\/latex].\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"338565\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338565\"]\r\n\r\nThe function equation is [latex]f(x)=-2x^2+3x+4[\/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>Divide polynomials by monomials<\/li>\n<li>Write function equations using given conditions<\/li>\n<\/ul>\n<\/div>\n<p>In the Antiderivatives section, we will sometimes simplify rational expressions by dividing a polynomial in the numerator by a monomial in the denominator.<\/p>\n<h2>Divide a Polynomial by a Monomial<\/h2>\n<p>When dividing a polynomial by a monomial, it is important that every term of the polynomial is divided by the monomial. You will then want to simplify. Here we will review this topic.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Dividing a Polynomial by a Monomial<\/h3>\n<p>Perform each division. Be sure to simplify your answer.<\/p>\n<ol>\n<li>[latex]\\dfrac{9{x}^{3}+6x}{3{x}^{2}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{x}^{9}+3{x}^{6}}{{x}^{4}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q717838\">Show Solution<\/span><\/p>\n<div id=\"q717838\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the quotient rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\dfrac{9{x}^{3}+6x}{3{x}^{2}}=\\dfrac{9{x}^{3}}{3{x}^{2}}+\\dfrac{6x}{3{x}^{2}}=3{x}^{3-2}+2{x}^{1-2}=3x+{x}^{-1}[\/latex]\u00a0Just like we did for derivatives, when finding antiderivatives, we will leave [latex]{x}^{-1}[\/latex] as it is rather than writing as\u00a0[latex]\\dfrac{1}{{x}}[\/latex].<\/li>\n<li>[latex]\\dfrac{{x}^{9}+3{x}^{6}}{{x}^{4}}=\\dfrac{{x}^{9}}{{x}^{4}}+\\dfrac{3{x}^{6}}{{x}^{4}}={x}^{9-4}+3{x}^{6-4}={x}^{5}+3{x}^{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221979\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221979&theme=oea&iframe_resize_id=ohm221979&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Write Function Equations Using Given Conditions<\/h2>\n<p>Sometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Function Equation from given conditions<\/h3>\n<p>Given [latex]f(2)=-1[\/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338564\">Show Solution<\/span><\/p>\n<div id=\"q338564\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find c, we use the fact that [latex]f(2)=-1[\/latex], that is, the function&#8217;s value is -1 when [latex]x=2[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}-1=3(2)^3-4(2)^2-2+c\\hfill \\\\ -1=3(8)-4(4)-2+c\\hfill \\\\ -1=24-16-2+c\\hfill \\\\ -1=6+c\\hfill \\\\ -7=c \\end{array}[\/latex]<\/p>\n<p>The function equation is\u00a0[latex]f(x)=3x^3-4x^2-x-7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]f(1)=5[\/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338565\">Show Solution<\/span><\/p>\n<div id=\"q338565\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function equation is [latex]f(x)=-2x^2+3x+4[\/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-3900\">\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":17533,"menu_order":9,"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 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