{"id":4092,"date":"2022-04-14T18:15:53","date_gmt":"2022-04-14T18:15:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-motion-in-space\/"},"modified":"2022-04-19T21:09:16","modified_gmt":"2022-04-19T21:09:16","slug":"skills-review-for-motion-in-space","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-motion-in-space\/","title":{"raw":"Skills Review for Motion in Space","rendered":"Skills Review for Motion in Space"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Apply basic derivative rules<\/li>\r\n \t<li>Use the product rule for finding the derivative of a product of functions<\/li>\r\n \t<li>Use the quotient rule for finding the derivative of a quotient of functions<\/li>\r\n \t<li>Apply the chain rule together with the power and product rule<\/li>\r\n \t<li>Write function equations using given conditions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Motion in Space section, we will explore motion along plane and space curves. Here we will review properties for taking derivatives and how to write function equations using given conditions.\r\n<h2>Basic Derivative Rules<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong>\r\n<h2>The Product Rule<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong>\r\n<h2>The Quotient Rule<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong>\r\n<h2>The Chain Rule<\/h2>\r\n<strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong>\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>Apply basic derivative rules<\/li>\n<li>Use the product rule for finding the derivative of a product of functions<\/li>\n<li>Use the quotient rule for finding the derivative of a quotient of functions<\/li>\n<li>Apply the chain rule together with the power and product rule<\/li>\n<li>Write function equations using given conditions<\/li>\n<\/ul>\n<\/div>\n<p>In the Motion in Space section, we will explore motion along plane and space curves. Here we will review properties for taking derivatives and how to write function equations using given conditions.<\/p>\n<h2>Basic Derivative Rules<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong><\/p>\n<h2>The Product Rule<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong><\/p>\n<h2>The Quotient Rule<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong><\/p>\n<h2>The Chain Rule<\/h2>\n<p><strong><em>(See <a href=\"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-calculus-of-vector-valued-functions\/\" target=\"_blank\" rel=\"noopener\">Module 3, Skills Review for Calculus of Vector-Valued Functions<\/a>)<\/em><\/strong><\/p>\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","protected":false},"author":349141,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4092","chapter","type-chapter","status-publish","hentry"],"part":4118,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4092","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4092\/revisions"}],"predecessor-version":[{"id":4247,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4092\/revisions\/4247"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/4118"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4092\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=4092"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=4092"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=4092"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=4092"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}