{"id":1991,"date":"2021-08-19T16:07:01","date_gmt":"2021-08-19T16:07:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-direction-fields-and-numerical-methods\/"},"modified":"2021-11-19T03:10:49","modified_gmt":"2021-11-19T03:10:49","slug":"skills-review-for-direction-fields-and-numerical-methods","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-direction-fields-and-numerical-methods\/","title":{"raw":"Skills Review for Direction Fields and Numerical Methods","rendered":"Skills Review for Direction Fields and Numerical Methods"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the slope of a tangent line<\/li>\r\n \t<li>Solve polynomial equations<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Direction Fields and Numerical Methods section, we will need skills including how to find the slope of a tangent line and solve polynomial equations. These skills are reviewed here.\r\n<h2>Calculate the Slope of a Tangent Line<\/h2>\r\n<p id=\"fs-id1169737931596\">Recall that a derivative can be used to find the slope of tangent line at a specific point.<\/p>\r\n\r\n<div id=\"fs-id1169739301889\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Slope of a Tangent Line<\/h3>\r\n<p id=\"fs-id1169739242299\">Find the slope of the line tangent to the graph of [latex]f(x)=x^2-4x+6[\/latex] at [latex]x=1[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736663036\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736663036\"]\r\n<p id=\"fs-id1169736587931\">Since the slope of the tangent line at 1 is [latex]f^{\\prime}(1)[\/latex], we must find [latex]f^{\\prime}(x)[\/latex]. Using the definition of a derivative, we have<\/p>\r\n\r\n<div id=\"fs-id1169739297908\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=2x-4[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739273132\">so the slope of the tangent line is [latex]f^{\\prime}(1)=-2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736654283\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736654291\">Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[\/latex] at [latex]x=2[\/latex]. Use the point-slope form.<\/p>\r\n[reveal-answer q=\"fs-id1169739353706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739353706\"]\r\n<p id=\"fs-id1169739353706\">[latex]m=12[\/latex]<\/p>\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]33696[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169737931601\" class=\"textbook exercises\">\r\n<h3>Example: Finding The Slope of a Tangent Line<\/h3>\r\n<p id=\"fs-id1169737950789\">Find the slope of the line tangent to the curve [latex]x^2+y^2=25[\/latex] at the point [latex](3,-4)[\/latex]. Note the derivative of the equation is\u00a0[latex]\\frac{dy}{dx}=-\\frac{x}{y}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737143580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737143580\"]\r\n<p id=\"fs-id1169737144361\">The slope of the tangent line is found by substituting [latex](3,-4)[\/latex] into the derivative. Consequently, the slope of the tangent line is [latex]\\frac{dy}{dx}|_{(3,-4)} =-\\frac{3}{-4}=\\frac{3}{4}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbook exercises\">\r\n<h3>Example: Finding the Slope of the Tangent Line<\/h3>\r\n<p id=\"fs-id1169737143587\">Find the slope of the line tangent to the graph of [latex]y^3+x^3-3xy=0[\/latex] at the point [latex]\\left(\\frac{3}{2},\\frac{3}{2}\\right)[\/latex]. Note the derivative of the equation is\u00a0[latex]\\frac{dy}{dx}=\\frac{3y-3x^2}{3y^2-3x}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738217007\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738217007\"]\r\n<p id=\"fs-id1169737904704\">Substitute [latex](\\frac{3}{2},\\frac{3}{2})[\/latex] into [latex]\\frac{dy}{dx}=\\frac{3y-3x^2}{3y^2-3x}[\/latex] to find the slope of the tangent line:<\/p>\r\n\r\n<div id=\"fs-id1169738045102\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}|_{(\\frac{3}{2},\\frac{3}{2})}=-1[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738186785\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169738186793\">Find the equation of the line tangent to the hyperbola [latex]x^2-y^2=16[\/latex] at the point [latex](5,3)[\/latex]. Note the derivative of the equation is\u00a0[latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737145207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737145207\"]\r\n\r\n[latex]\\frac{5}{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solve Polynomial Equations<\/h2>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Polynomial Equations<\/h3>\r\nA polynomial of degree <em>n <\/em>is an expression of the type\r\n<div style=\"text-align: center;\">[latex]{a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\\cdot \\cdot \\cdot +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\nwhere <em>n<\/em> is a positive integer and [latex]{a}_{n},\\dots ,{a}_{0}[\/latex] are real numbers and [latex]{a}_{n}\\ne 0[\/latex].\r\n\r\nSetting the polynomial equal to zero gives a <strong>polynomial equation<\/strong>. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent <em>n<\/em>.\r\n\r\n<\/div>\r\n<div class=\"textbook exercises\">\r\n<h3>Example: Solving a Polynomial Equation<\/h3>\r\n<p id=\"fs-id1169737143587\">Solve the polynomial equation [latex](x-2)^2(x^2+5x+6)=0[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738217111\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738217111\"]\r\n<p id=\"fs-id1169737904704\">Since the equation is already set equal to 0, begin by factoring the equation entirely:<\/p>\r\n[latex](x-2)^2(x+3)(x+2)=0[\/latex]\r\n\r\nNow, set each factor equal to 0.\r\n\r\n[latex](x-2)^2=0[\/latex] yields [latex]x=2[\/latex]\r\n\r\n[latex]x+3=0[\/latex] yields [latex]x=-3[\/latex]\r\n\r\n[latex]x+2=0[\/latex] yields [latex]x=-2[\/latex]\r\n\r\nThe solutions are [latex]x=-3,-2,2[\/latex]\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]34186[\/ohm_question]\r\n[reveal-answer q=\"166577\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"166577\"]\r\nFactor by grouping.\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the slope of a tangent line<\/li>\n<li>Solve polynomial equations<\/li>\n<\/ul>\n<\/div>\n<p>In the Direction Fields and Numerical Methods section, we will need skills including how to find the slope of a tangent line and solve polynomial equations. These skills are reviewed here.<\/p>\n<h2>Calculate the Slope of a Tangent Line<\/h2>\n<p id=\"fs-id1169737931596\">Recall that a derivative can be used to find the slope of tangent line at a specific point.<\/p>\n<div id=\"fs-id1169739301889\" class=\"textbook exercises\">\n<h3>Example: Finding the Slope of a Tangent Line<\/h3>\n<p id=\"fs-id1169739242299\">Find the slope of the line tangent to the graph of [latex]f(x)=x^2-4x+6[\/latex] at [latex]x=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736663036\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736663036\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736587931\">Since the slope of the tangent line at 1 is [latex]f^{\\prime}(1)[\/latex], we must find [latex]f^{\\prime}(x)[\/latex]. Using the definition of a derivative, we have<\/p>\n<div id=\"fs-id1169739297908\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=2x-4[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739273132\">so the slope of the tangent line is [latex]f^{\\prime}(1)=-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736654283\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736654291\">Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[\/latex] at [latex]x=2[\/latex]. Use the point-slope form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739353706\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739353706\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739353706\">[latex]m=12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm33696\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33696&theme=oea&iframe_resize_id=ohm33696&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1169737931601\" class=\"textbook exercises\">\n<h3>Example: Finding The Slope of a Tangent Line<\/h3>\n<p id=\"fs-id1169737950789\">Find the slope of the line tangent to the curve [latex]x^2+y^2=25[\/latex] at the point [latex](3,-4)[\/latex]. Note the derivative of the equation is\u00a0[latex]\\frac{dy}{dx}=-\\frac{x}{y}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737143580\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737143580\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737144361\">The slope of the tangent line is found by substituting [latex](3,-4)[\/latex] into the derivative. Consequently, the slope of the tangent line is [latex]\\frac{dy}{dx}|_{(3,-4)} =-\\frac{3}{-4}=\\frac{3}{4}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook exercises\">\n<h3>Example: Finding the Slope of the Tangent Line<\/h3>\n<p id=\"fs-id1169737143587\">Find the slope of the line tangent to the graph of [latex]y^3+x^3-3xy=0[\/latex] at the point [latex]\\left(\\frac{3}{2},\\frac{3}{2}\\right)[\/latex]. Note the derivative of the equation is\u00a0[latex]\\frac{dy}{dx}=\\frac{3y-3x^2}{3y^2-3x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738217007\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738217007\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737904704\">Substitute [latex](\\frac{3}{2},\\frac{3}{2})[\/latex] into [latex]\\frac{dy}{dx}=\\frac{3y-3x^2}{3y^2-3x}[\/latex] to find the slope of the tangent line:<\/p>\n<div id=\"fs-id1169738045102\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}|_{(\\frac{3}{2},\\frac{3}{2})}=-1[\/latex]<\/div>\n<div><\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738186785\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169738186793\">Find the equation of the line tangent to the hyperbola [latex]x^2-y^2=16[\/latex] at the point [latex](5,3)[\/latex]. Note the derivative of the equation is\u00a0[latex]\\frac{dy}{dx}=\\frac{x}{y}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169737145207\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169737145207\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{5}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solve Polynomial Equations<\/h2>\n<div class=\"textbox\">\n<h3>A General Note: Polynomial Equations<\/h3>\n<p>A polynomial of degree <em>n <\/em>is an expression of the type<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+\\cdot \\cdot \\cdot +{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p>where <em>n<\/em> is a positive integer and [latex]{a}_{n},\\dots ,{a}_{0}[\/latex] are real numbers and [latex]{a}_{n}\\ne 0[\/latex].<\/p>\n<p>Setting the polynomial equal to zero gives a <strong>polynomial equation<\/strong>. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent <em>n<\/em>.<\/p>\n<\/div>\n<div class=\"textbook exercises\">\n<h3>Example: Solving a Polynomial Equation<\/h3>\n<p id=\"fs-id1169737143587\">Solve the polynomial equation [latex](x-2)^2(x^2+5x+6)=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738217111\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738217111\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737904704\">Since the equation is already set equal to 0, begin by factoring the equation entirely:<\/p>\n<p>[latex](x-2)^2(x+3)(x+2)=0[\/latex]<\/p>\n<p>Now, set each factor equal to 0.<\/p>\n<p>[latex](x-2)^2=0[\/latex] yields [latex]x=2[\/latex]<\/p>\n<p>[latex]x+3=0[\/latex] yields [latex]x=-3[\/latex]<\/p>\n<p>[latex]x+2=0[\/latex] yields [latex]x=-2[\/latex]<\/p>\n<p>The solutions are [latex]x=-3,-2,2[\/latex]<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm34186\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=34186&theme=oea&iframe_resize_id=ohm34186&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q166577\">Hint<\/span><\/p>\n<div id=\"q166577\" class=\"hidden-answer\" style=\"display: none\">\nFactor by grouping.\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-1991\">\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":6,"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-1991","chapter","type-chapter","status-publish","hentry"],"part":536,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1991","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":5,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1991\/revisions"}],"predecessor-version":[{"id":2554,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1991\/revisions\/2554"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/536"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1991\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1991"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1991"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1991"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}