{"id":3887,"date":"2021-05-20T18:29:55","date_gmt":"2021-05-20T18:29:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-derivatives-as-rates-of-change\/"},"modified":"2021-07-03T18:51:45","modified_gmt":"2021-07-03T18:51:45","slug":"review-for-derivatives-as-rates-of-change","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-derivatives-as-rates-of-change\/","title":{"raw":"Skills Review for Derivatives as Rates of Change","rendered":"Skills Review for Derivatives as Rates of Change"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the average rate of change of a function over a given interval<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Derivatives as Rates of Change section, one of the topics you will explore is how to calculate the average rate of change and instantaneous rate of change. Here we will review how to calculate the average rate of change.\r\n<h2>Finding the Average Rate of Change<\/h2>\r\nGasoline costs have experienced some wild fluctuations over the last several decades. The table below[footnote]<a href=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524\" target=\"_blank\" rel=\"noopener\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524<\/a>. Accessed 3\/5\/2014.[\/footnote]\u00a0lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.\r\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.\r\n\r\nThe price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change} &amp;=\\frac{\\text{Change in output}}{\\text{Change in input}}\\\\[2mm] &amp;=\\frac{\\Delta y}{\\Delta x}\\\\[2mm] &amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\\[2mm] &amp;=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/p>\r\nIn our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/p>\r\nOn average, the price of gas increased by about 19.6\u00a2 each year.\r\n\r\nOther examples of rates of change include:\r\n<ul>\r\n \t<li>A population of rats increasing by 40 rats per week<\/li>\r\n \t<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\r\n \t<li>A car driving 27 miles per gallon of gasoline (distance traveled changes by 27 miles for each gallon)<\/li>\r\n \t<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\r\n \t<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Computing an Average Rate of Change<\/h3>\r\nUsing the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.\r\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"982401\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982401\"]\r\n\r\nIn 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&amp;=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\\[2mm]&amp;=\\dfrac{$2.41-$2.84}{2009 - 2007}\\\\[2mm]&amp;=\\dfrac{-$0.43}{2\\text{ years}}\\\\[2mm]&amp;={-$0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that a decrease is expressed by a negative change or \"negative increase.\" A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of how to find the average rate of change between two points from a table of values.\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6405053&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=iJ_0nPUUlOg&amp;video_target=tpm-plugin-zd800o1y-iJ_0nPUUlOg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExFindTheAverageRateOfChangeFromATableTemperatures_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Ex: Find the Average Rate of Change From a Table - Temperatures\" here (opens in new window)<\/a>.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the data in the table below,\u00a0find the average rate of change between 2005 and 2010.\r\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"377325\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"377325\"]\r\n<p style=\"text-align: center;\">[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/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]1731[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the average rate of change of a function over a given interval<\/li>\n<\/ul>\n<\/div>\n<p>In the Derivatives as Rates of Change section, one of the topics you will explore is how to calculate the average rate of change and instantaneous rate of change. Here we will review how to calculate the average rate of change.<\/p>\n<h2>Finding the Average Rate of Change<\/h2>\n<p>Gasoline costs have experienced some wild fluctuations over the last several decades. The table below<a class=\"footnote\" title=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.\" id=\"return-footnote-3887-1\" href=\"#footnote-3887-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.<\/p>\n<p>The price change per year is a <strong>rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change} &=\\frac{\\text{Change in output}}{\\text{Change in input}}\\\\[2mm] &=\\frac{\\Delta y}{\\Delta x}\\\\[2mm] &=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\\[2mm] &=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/latex]<\/p>\n<p>In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/p>\n<p>On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\n<p>Other examples of rates of change include:<\/p>\n<ul>\n<li>A population of rats increasing by 40 rats per week<\/li>\n<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\n<li>A car driving 27 miles per gallon of gasoline (distance traveled changes by 27 miles for each gallon)<\/li>\n<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\n<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example: Computing an Average Rate of Change<\/h3>\n<p>Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982401\">Show Solution<\/span><\/p>\n<div id=\"q982401\" class=\"hidden-answer\" style=\"display: none\">\n<p>In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\\\[2mm]&=\\dfrac{$2.41-$2.84}{2009 - 2007}\\\\[2mm]&=\\dfrac{-$0.43}{2\\text{ years}}\\\\[2mm]&={-$0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that a decrease is expressed by a negative change or &#8220;negative increase.&#8221; A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of how to find the average rate of change between two points from a table of values.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6405053&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=iJ_0nPUUlOg&amp;video_target=tpm-plugin-zd800o1y-iJ_0nPUUlOg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExFindTheAverageRateOfChangeFromATableTemperatures_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Ex: Find the Average Rate of Change From a Table &#8211; Temperatures&#8221; here (opens in new window)<\/a>.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q377325\">Show Solution<\/span><\/p>\n<div id=\"q377325\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1731\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1731&theme=oea&iframe_resize_id=ohm1731&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-3887\">\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><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-3887-1\"><a href=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524\" target=\"_blank\" rel=\"noopener\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524<\/a>. Accessed 3\/5\/2014. <a href=\"#return-footnote-3887-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":17533,"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-3887","chapter","type-chapter","status-publish","hentry"],"part":3089,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3887","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3887\/revisions"}],"predecessor-version":[{"id":4584,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3887\/revisions\/4584"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/3089"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3887\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3887"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3887"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3887"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}