{"id":819,"date":"2021-08-27T20:06:35","date_gmt":"2021-08-27T20:06:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=819"},"modified":"2022-10-29T00:32:44","modified_gmt":"2022-10-29T00:32:44","slug":"limits-and-continuity-of-a-vector-valued-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/limits-and-continuity-of-a-vector-valued-function\/","title":{"raw":"Limits and Continuity of a Vector-Valued Function","rendered":"Limits and Continuity of a Vector-Valued Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span class=\"os-abstract-content\">Define the limit of a vector-valued function.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nWe now take a look at the <strong><span id=\"term119\" data-type=\"term\">limit of a vector-valued function<\/span><\/strong>. This is important to understand to study the calculus of vector-valued functions.\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nA vector-valued function [latex]{\\bf{r}}[\/latex] approaches the limit [latex]\\bf{L}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex], written\r\n<div style=\"text-align: center;\">[latex]\\displaystyle{\\lim_{t\\to{a}}}\\;{\\bf{r}}\\,(t)={\\bf{L}}[\/latex],<\/div>\r\n&nbsp;\r\n\r\nprovided\r\n<div style=\"text-align: center;\">[latex]\\displaystyle{\\lim_{t\\to{a}}}\\left\\|{\\bf{r}}\\,(t)-{\\bf{L}}\\right\\|=0[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>This is a rigorous definition of the limit of a vector-valued function. In practice, we use the following theorem:<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Limit of a Vector-Valued Function Theorem<\/h3>\r\n\r\n<hr \/>\r\n\r\nLet [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] be functions of [latex]t[\/latex]. Then the limit of a vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex] is given by\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}\\,(t)=\\bigg[\\displaystyle\\lim_{t{\\to}a}f\\,(t)\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}a}g\\,(t)\\bigg]{\\bf{j}}[\/latex],<\/div>\r\n&nbsp;\r\n\r\nprovided the limits [latex]\\displaystyle\\lim_{t{\\to}a}f\\,(t)[\/latex] and [latex]\\displaystyle\\lim_{t{\\to}a}g\\,(t)[\/latex] exist. Similarly, the limit of the vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}+h\\,(t)\\,{\\bf{k}}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex] is given by\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}\\,(t)=\\bigg[\\displaystyle\\lim_{t{\\to}a}f\\,(t)\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}a}g\\,(t)\\bigg]{\\bf{j}}+\\bigg[\\displaystyle\\lim_{t{\\to}a}h\\,(t)\\bigg]\\,{\\bf{k}}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nprovided the limits [latex]\\displaystyle\\lim_{t{\\to}a}f\\,(t),\\ \\displaystyle\\lim_{t{\\to}a}g\\,(t)[\/latex], and [latex]\\displaystyle\\lim_{t{\\to}a}h\\,(t)[\/latex] exist.\r\n\r\n<\/div>\r\nIn the following example, we show how to calculate the limit of a vector-valued function.\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793900968\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating the limit of a vector-valued function<\/h3>\r\nFor each of the following vector-valued functions, calculate [latex]\\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)[\/latex] for\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol style=\"list-style-type: lower-alpha;\">a. [latex]{\\bf{r}}\\,(t)=(t^{2}-3t+4)\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}[\/latex]<\/ol>\r\n<ol style=\"list-style-type: lower-alpha;\">b. [latex]{\\bf{r}}\\,(t)=\\frac{2t-4}{t+1}\\,{\\bf{i}}+\\frac{t}{t^{2}+1}\\,{\\bf{j}}+(4t-3)\\,{\\bf{k}}[\/latex]<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1167793461764\" class=\"exercise\">[reveal-answer q=\"fs-id1167795055165\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167795055165\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Use the Limit of a Vector-Valued Function theorem and substitute the value [latex]t=3[\/latex] into the two component expression:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)&amp;=\\hfill&amp;{\\displaystyle\\lim_{t{\\to}3}\\big[(t^{2}-3t+4)\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}\\big]}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\bigg[\\displaystyle\\lim_{t{\\to}3}{(t^{2}-3t+4)}\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(4t+3)}\\bigg]\\,{\\bf{j}}} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; {4\\,{\\bf{i}}+15\\,{\\bf{j}}}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;<\/li>\r\n \t<li>Use the Limit of a Vector-Valued Function theorem and substitute the value [latex]t=3[\/latex] into the three component expression:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)&amp;=\\hfill&amp;{\\displaystyle\\lim_{t{\\to}3}\\bigg(\\frac{2t-4}{t+1}\\,{\\bf{i}}+\\frac{t}{t^{2}+1}\\,{\\bf{j}}+(4t-3){\\bf{k}}\\bigg)}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\bigg[\\displaystyle\\lim_{t{\\to}3}{(\\frac{2t-4}{t+1})}\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(\\frac{t}{t^{2}+1})}\\bigg]\\,{\\bf{j}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(4t-3)}\\bigg]\\,{\\bf{k}}} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\frac{1}{2}\\,{\\bf{i}}+\\frac{3}{10}\\,{\\bf{j}}+9\\,{\\bf{k}}}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793958097\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nCalculate [latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)[\/latex] for the function [latex]{\\bf{r}}\\,(t)=\\sqrt{t^{2}-3t-1}\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}+\\sin{\\frac{(t+1)\\,\\pi}{2}}\\,{\\bf{k}}[\/latex].\r\n<div id=\"fs-id1167793940339\" class=\"exercise\">[reveal-answer q=\"fs-id1167794933124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794933124\"]\r\n<div>[latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)=3\\,{\\bf{i}}-5\\,{\\bf{j}}-{\\bf{k}}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7949584&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=UkThcL4VPfc&amp;video_target=tpm-plugin-x7y3bb4i-UkThcL4VPfc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP3.3_transcript.html\">transcript for \u201cCP 3.3\u201d here (opens in new window).<\/a><\/center>Now that we know how to calculate the limit of a vector-valued function, we can define continuity at a point for such a function.\r\n<div id=\"fs-id1167793372221\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nLet <em data-effect=\"italics\">f, g,<\/em> and <em data-effect=\"italics\">h<\/em> be functions of <em data-effect=\"italics\">t.<\/em> Then, the vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex] is continuous at point [latex]t=a[\/latex] if the following three conditions hold:\r\n<ol>\r\n \t<li>[latex]{\\bf{r}}\\,(a)[\/latex] exists<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)[\/latex] exists<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)={\\bf{r}}\\,(a)[\/latex]<\/li>\r\n<\/ol>\r\nSimilarly, the vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}+h\\,(t)\\,{\\bf{k}}[\/latex] is continuous at point [latex]t=a[\/latex] if the following three conditions hold:\r\n<ol>\r\n \t<li>[latex]{\\bf{r}}\\,(a)[\/latex] exists<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)[\/latex] exists<\/li>\r\n \t<li>[latex]\\lim_{t{\\to}a}{\\bf{r}}\\,(t)={\\bf{r}}\\,(a)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span class=\"os-abstract-content\">Define the limit of a vector-valued function.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>We now take a look at the <strong><span id=\"term119\" data-type=\"term\">limit of a vector-valued function<\/span><\/strong>. This is important to understand to study the calculus of vector-valued functions.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>A vector-valued function [latex]{\\bf{r}}[\/latex] approaches the limit [latex]\\bf{L}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex], written<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle{\\lim_{t\\to{a}}}\\;{\\bf{r}}\\,(t)={\\bf{L}}[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p>provided<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle{\\lim_{t\\to{a}}}\\left\\|{\\bf{r}}\\,(t)-{\\bf{L}}\\right\\|=0[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div>This is a rigorous definition of the limit of a vector-valued function. In practice, we use the following theorem:<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Limit of a Vector-Valued Function Theorem<\/h3>\n<hr \/>\n<p>Let [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] be functions of [latex]t[\/latex]. Then the limit of a vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex] is given by<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}\\,(t)=\\bigg[\\displaystyle\\lim_{t{\\to}a}f\\,(t)\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}a}g\\,(t)\\bigg]{\\bf{j}}[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p>provided the limits [latex]\\displaystyle\\lim_{t{\\to}a}f\\,(t)[\/latex] and [latex]\\displaystyle\\lim_{t{\\to}a}g\\,(t)[\/latex] exist. Similarly, the limit of the vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}+h\\,(t)\\,{\\bf{k}}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex] is given by<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{t\\to a}{\\bf{r}}\\,(t)=\\bigg[\\displaystyle\\lim_{t{\\to}a}f\\,(t)\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}a}g\\,(t)\\bigg]{\\bf{j}}+\\bigg[\\displaystyle\\lim_{t{\\to}a}h\\,(t)\\bigg]\\,{\\bf{k}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>provided the limits [latex]\\displaystyle\\lim_{t{\\to}a}f\\,(t),\\ \\displaystyle\\lim_{t{\\to}a}g\\,(t)[\/latex], and [latex]\\displaystyle\\lim_{t{\\to}a}h\\,(t)[\/latex] exist.<\/p>\n<\/div>\n<p>In the following example, we show how to calculate the limit of a vector-valued function.<\/p>\n<\/div>\n<div id=\"fs-id1167793900968\" class=\"textbook exercises\">\n<h3>Example: Evaluating the limit of a vector-valued function<\/h3>\n<p>For each of the following vector-valued functions, calculate [latex]\\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)[\/latex] for<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: lower-alpha;\"> <\/ol>\n<ol style=\"list-style-type: lower-alpha;\"> <\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1167793461764\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167795055165\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167795055165\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Use the Limit of a Vector-Valued Function theorem and substitute the value [latex]t=3[\/latex] into the two component expression:\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)&=\\hfill&{\\displaystyle\\lim_{t{\\to}3}\\big[(t^{2}-3t+4)\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}\\big]}\\hfill \\\\ \\hfill & =\\hfill & {\\bigg[\\displaystyle\\lim_{t{\\to}3}{(t^{2}-3t+4)}\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(4t+3)}\\bigg]\\,{\\bf{j}}} \\hfill \\\\ \\hfill & =\\hfill & {4\\,{\\bf{i}}+15\\,{\\bf{j}}}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/li>\n<li>Use the Limit of a Vector-Valued Function theorem and substitute the value [latex]t=3[\/latex] into the three component expression:\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)&=\\hfill&{\\displaystyle\\lim_{t{\\to}3}\\bigg(\\frac{2t-4}{t+1}\\,{\\bf{i}}+\\frac{t}{t^{2}+1}\\,{\\bf{j}}+(4t-3){\\bf{k}}\\bigg)}\\hfill \\\\ \\hfill & =\\hfill & {\\bigg[\\displaystyle\\lim_{t{\\to}3}{(\\frac{2t-4}{t+1})}\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(\\frac{t}{t^{2}+1})}\\bigg]\\,{\\bf{j}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(4t-3)}\\bigg]\\,{\\bf{k}}} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{1}{2}\\,{\\bf{i}}+\\frac{3}{10}\\,{\\bf{j}}+9\\,{\\bf{k}}}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793958097\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Calculate [latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)[\/latex] for the function [latex]{\\bf{r}}\\,(t)=\\sqrt{t^{2}-3t-1}\\,{\\bf{i}}+(4t+3)\\,{\\bf{j}}+\\sin{\\frac{(t+1)\\,\\pi}{2}}\\,{\\bf{k}}[\/latex].<\/p>\n<div id=\"fs-id1167793940339\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794933124\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794933124\" class=\"hidden-answer\" style=\"display: none\">\n<div>[latex]\\displaystyle\\lim_{t{\\to}-2}{\\bf{r}}\\,(t)=3\\,{\\bf{i}}-5\\,{\\bf{j}}-{\\bf{k}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7949584&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=UkThcL4VPfc&amp;video_target=tpm-plugin-x7y3bb4i-UkThcL4VPfc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP3.3_transcript.html\">transcript for \u201cCP 3.3\u201d here (opens in new window).<\/a><\/div>\n<p>Now that we know how to calculate the limit of a vector-valued function, we can define continuity at a point for such a function.<\/p>\n<div id=\"fs-id1167793372221\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>Let <em data-effect=\"italics\">f, g,<\/em> and <em data-effect=\"italics\">h<\/em> be functions of <em data-effect=\"italics\">t.<\/em> Then, the vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}[\/latex] is continuous at point [latex]t=a[\/latex] if the following three conditions hold:<\/p>\n<ol>\n<li>[latex]{\\bf{r}}\\,(a)[\/latex] exists<\/li>\n<li>[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)[\/latex] exists<\/li>\n<li>[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)={\\bf{r}}\\,(a)[\/latex]<\/li>\n<\/ol>\n<p>Similarly, the vector-valued function [latex]{\\bf{r}}\\,(t)=f\\,(t)\\,{\\bf{i}}+g\\,(t)\\,{\\bf{j}}+h\\,(t)\\,{\\bf{k}}[\/latex] is continuous at point [latex]t=a[\/latex] if the following three conditions hold:<\/p>\n<ol>\n<li>[latex]{\\bf{r}}\\,(a)[\/latex] exists<\/li>\n<li>[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)[\/latex] exists<\/li>\n<li>[latex]\\lim_{t{\\to}a}{\\bf{r}}\\,(t)={\\bf{r}}\\,(a)[\/latex]<\/li>\n<\/ol>\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-819\">\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>CP 3.3. <strong>Authored by<\/strong>: Ryan Melton. <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>Calculus Volume 3. <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\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/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-3\/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":349141,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 3.3\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"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-819","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/819","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":55,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/819\/revisions"}],"predecessor-version":[{"id":4073,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/819\/revisions\/4073"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/819\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=819"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=819"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=819"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=819"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}