{"id":11286,"date":"2015-07-14T19:32:22","date_gmt":"2015-07-14T19:32:22","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&#038;p=11286"},"modified":"2021-11-08T02:02:11","modified_gmt":"2021-11-08T02:02:11","slug":"exponential-functions-with-base-e","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/exponential-functions-with-base-e\/","title":{"raw":"Exponential functions with base e","rendered":"Exponential functions with base e"},"content":{"raw":"<section id=\"fs-id1165137724961\" data-depth=\"1\">\r\n<div id=\"fs-id1165135511324\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: The Number <em data-effect=\"italics\">e<\/em><\/h3>\r\n<p id=\"fs-id1165135511335\">The letter <em data-effect=\"italics\">e<\/em> represents the irrational number<\/p>\r\n\r\n<div id=\"eip-id1165135378658\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\left(1+\\frac{1}{n}\\right)}^{n},\\text{as}n\\text{increases without bound}[\/latex]<\/div>\r\n<p id=\"fs-id1165135369344\">The letter <em data-effect=\"italics\">e <\/em>is used as a base for many real-world exponential models. To work with base <em data-effect=\"italics\">e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Using a Calculator to Find Powers of <i>e<\/i><\/h3>\r\nCalculate [latex]{e}^{3.14}[\/latex]. Round to five decimal places.\r\n\r\n[reveal-answer q=\"493647\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"493647\"]On a calculator, press the button labeled [latex]\\left[{e}^{x}\\right][\/latex]. The window shows [e^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\\approx 23.10387[\/latex]. Caution: Many scientific calculators have an \"Exp\" button, which is used to enter numbers in scientific notation. It is not used to find powers of e.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135348437\">Use a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.<\/p>\r\n\r\n[reveal-answer q=\"79904\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"79904\"]\u22480.60653[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137827923\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Investigating Continuous Growth<\/h2>\r\n<p id=\"fs-id1165137827929\">So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, <em data-effect=\"italics\">e <\/em>is used as the base for exponential functions. Exponential models that use <em>e<\/em>\u00a0as the base are called <em data-effect=\"italics\">continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.<\/p>\r\n\r\n<div id=\"fs-id1165137664673\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: The Continuous Growth\/Decay Formula<\/h3>\r\n<p id=\"fs-id1165135453868\">For all real numbers <em>t<\/em>, and all positive numbers <em>a<\/em>\u00a0and <em>r<\/em>, continuous growth or decay is represented by the formula<\/p>\r\n\r\n<div id=\"fs-id1165135536370\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/div>\r\n<p id=\"eip-101\">where<\/p>\r\n\r\n<ul id=\"fs-id1165135152052\">\r\n \t<li><em>a<\/em>\u00a0is the initial value,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the continuous growth rate per unit time,<\/li>\r\n \t<li>and <em>t<\/em>\u00a0is the elapsed time.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135560686\">If <em>r\u00a0<\/em>&gt; 0, then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt; 0, then the formula represents continuous decay.<\/p>\r\n<p id=\"fs-id1165137812323\">For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\r\n\r\n<div id=\"eip-id1165134324899\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]A\\left(t\\right)=P{e}^{rt}[\/latex]<\/div>\r\n<p id=\"eip-962\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137827330\">\r\n \t<li><em>P<\/em>\u00a0is the principal or the initial invested,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the growth or interest rate per unit time,<\/li>\r\n \t<li>and <em>t<\/em>\u00a0is the period or term of the investment.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135411368\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 id=\"fs-id1165135411373\">How To: Given the initial value, rate of growth or decay, and time <em>t<\/em>, solve a continuous growth or decay function.<\/h3>\r\n<ol id=\"fs-id1165135511371\" data-number-style=\"arabic\">\r\n \t<li>Use the information in the problem to determine <em>a<\/em>, the initial value of the function.<\/li>\r\n \t<li>Use the information in the problem to determine the growth rate <em>r<\/em>.\r\n<ol id=\"fs-id1165135188096\" data-number-style=\"lower-alpha\">\r\n \t<li>If the problem refers to continuous growth, then <em>r\u00a0<\/em>&gt; 0.<\/li>\r\n \t<li>If the problem refers to continuous decay, then <em>r\u00a0<\/em>&lt; 0.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Use the information in the problem to determine the time <em>t<\/em>.<\/li>\r\n \t<li>Substitute the given information into the continuous growth formula and solve for <em>A<\/em>(<em>t<\/em>).<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Calculating Continuous Growth<\/h3>\r\nA person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?\r\n\r\n[reveal-answer q=\"754735\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"754735\"]Since the account is growing in value, this is a continuous compounding problem with growth rate r\u00a0= 0.10. The initial investment was $1,000, so P\u00a0= 1000. We use the continuous compounding formula to find the value after t\u00a0= 1 year: [latex]\\begin{cases}A\\left(t\\right)\\hfill &amp; =P{e}^{rt}\\hfill &amp; \\text{Use the continuous compounding formula}.\\hfill \\\\ \\hfill &amp; =1000{\\left(e\\right)}^{0.1} &amp; \\text{Substitute known values for }P, r,\\text{ and }t.\\hfill \\\\ \\hfill &amp; \\approx 1105.17\\hfill &amp; \\text{Use a calculator to approximate}.\\hfill \\end{cases}[\/latex] The account is worth $1,105.17 after one year.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137895305\">A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?<\/p>\r\n\r\n[reveal-answer q=\"192743\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"192743\"]$3,659,823.44[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRadon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?\r\n\r\n[reveal-answer q=\"108147\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"108147\"]Since the substance is decaying, the rate, 17.3%, is negative. So, r\u00a0=\u00a0\u20130.173. The initial amount of radon-222 was 100 mg, so a\u00a0= 100. We use the continuous decay formula to find the value after t\u00a0= 3 days: [latex]\\begin{cases}A\\left(t\\right)\\hfill &amp; =a{e}^{rt}\\hfill &amp; \\text{Use the continuous growth formula}.\\hfill \\\\ \\hfill &amp; =100{e}^{-0.173\\left(3\\right)} &amp; \\text{Substitute known values for }a, r,\\text{ and }t.\\hfill \\\\ \\hfill &amp; \\approx 59.5115\\hfill &amp; \\text{Use a calculator to approximate}.\\hfill \\end{cases}[\/latex] So 59.5115 mg of radon-222 will remain.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135348472\">Using the data in Example 9, how much radon-222 will remain after one year?<\/p>\r\n\r\n[reveal-answer q=\"467606\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"467606\"]3.77E-26 (This is calculator notation for the number written as [latex] 3.77 \\times {10}^{-26}[\/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>","rendered":"<section id=\"fs-id1165137724961\" data-depth=\"1\">\n<div id=\"fs-id1165135511324\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: The Number <em data-effect=\"italics\">e<\/em><\/h3>\n<p id=\"fs-id1165135511335\">The letter <em data-effect=\"italics\">e<\/em> represents the irrational number<\/p>\n<div id=\"eip-id1165135378658\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\left(1+\\frac{1}{n}\\right)}^{n},\\text{as}n\\text{increases without bound}[\/latex]<\/div>\n<p id=\"fs-id1165135369344\">The letter <em data-effect=\"italics\">e <\/em>is used as a base for many real-world exponential models. To work with base <em data-effect=\"italics\">e<\/em>, we use the approximation, [latex]e\\approx 2.718282[\/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707\u20131783) who first investigated and discovered many of its properties.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Using a Calculator to Find Powers of <i>e<\/i><\/h3>\n<p>Calculate [latex]{e}^{3.14}[\/latex]. Round to five decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q493647\">Show Answer<\/span><\/p>\n<div id=\"q493647\" class=\"hidden-answer\" style=\"display: none\">On a calculator, press the button labeled [latex]\\left[{e}^{x}\\right][\/latex]. The window shows [e^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\\approx 23.10387[\/latex]. Caution: Many scientific calculators have an &#8220;Exp&#8221; button, which is used to enter numbers in scientific notation. It is not used to find powers of e.<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135348437\">Use a calculator to find [latex]{e}^{-0.5}[\/latex]. Round to five decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q79904\">Show Answer<\/span><\/p>\n<div id=\"q79904\" class=\"hidden-answer\" style=\"display: none\">\u22480.60653<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137827923\" data-depth=\"1\">\n<h2 data-type=\"title\">Investigating Continuous Growth<\/h2>\n<p id=\"fs-id1165137827929\">So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, <em data-effect=\"italics\">e <\/em>is used as the base for exponential functions. Exponential models that use <em>e<\/em>\u00a0as the base are called <em data-effect=\"italics\">continuous growth or decay models<\/em>. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.<\/p>\n<div id=\"fs-id1165137664673\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: The Continuous Growth\/Decay Formula<\/h3>\n<p id=\"fs-id1165135453868\">For all real numbers <em>t<\/em>, and all positive numbers <em>a<\/em>\u00a0and <em>r<\/em>, continuous growth or decay is represented by the formula<\/p>\n<div id=\"fs-id1165135536370\" class=\"equation\" style=\"text-align: center;\" data-type=\"equation\">[latex]A\\left(t\\right)=a{e}^{rt}[\/latex]<\/div>\n<p id=\"eip-101\">where<\/p>\n<ul id=\"fs-id1165135152052\">\n<li><em>a<\/em>\u00a0is the initial value,<\/li>\n<li><em>r<\/em>\u00a0is the continuous growth rate per unit time,<\/li>\n<li>and <em>t<\/em>\u00a0is the elapsed time.<\/li>\n<\/ul>\n<p id=\"fs-id1165135560686\">If <em>r\u00a0<\/em>&gt; 0, then the formula represents continuous growth. If <em>r\u00a0<\/em>&lt; 0, then the formula represents continuous decay.<\/p>\n<p id=\"fs-id1165137812323\">For business applications, the continuous growth formula is called the continuous compounding formula and takes the form<\/p>\n<div id=\"eip-id1165134324899\" class=\"equation unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]A\\left(t\\right)=P{e}^{rt}[\/latex]<\/div>\n<p id=\"eip-962\">where<\/p>\n<ul id=\"fs-id1165137827330\">\n<li><em>P<\/em>\u00a0is the principal or the initial invested,<\/li>\n<li><em>r<\/em>\u00a0is the growth or interest rate per unit time,<\/li>\n<li>and <em>t<\/em>\u00a0is the period or term of the investment.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135411368\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165135411373\">How To: Given the initial value, rate of growth or decay, and time <em>t<\/em>, solve a continuous growth or decay function.<\/h3>\n<ol id=\"fs-id1165135511371\" data-number-style=\"arabic\">\n<li>Use the information in the problem to determine <em>a<\/em>, the initial value of the function.<\/li>\n<li>Use the information in the problem to determine the growth rate <em>r<\/em>.\n<ol id=\"fs-id1165135188096\" data-number-style=\"lower-alpha\">\n<li>If the problem refers to continuous growth, then <em>r\u00a0<\/em>&gt; 0.<\/li>\n<li>If the problem refers to continuous decay, then <em>r\u00a0<\/em>&lt; 0.<\/li>\n<\/ol>\n<\/li>\n<li>Use the information in the problem to determine the time <em>t<\/em>.<\/li>\n<li>Substitute the given information into the continuous growth formula and solve for <em>A<\/em>(<em>t<\/em>).<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Calculating Continuous Growth<\/h3>\n<p>A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q754735\">Show Answer<\/span><\/p>\n<div id=\"q754735\" class=\"hidden-answer\" style=\"display: none\">Since the account is growing in value, this is a continuous compounding problem with growth rate r\u00a0= 0.10. The initial investment was $1,000, so P\u00a0= 1000. We use the continuous compounding formula to find the value after t\u00a0= 1 year: [latex]\\begin{cases}A\\left(t\\right)\\hfill & =P{e}^{rt}\\hfill & \\text{Use the continuous compounding formula}.\\hfill \\\\ \\hfill & =1000{\\left(e\\right)}^{0.1} & \\text{Substitute known values for }P, r,\\text{ and }t.\\hfill \\\\ \\hfill & \\approx 1105.17\\hfill & \\text{Use a calculator to approximate}.\\hfill \\end{cases}[\/latex] The account is worth $1,105.17 after one year.<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137895305\">A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q192743\">Show Answer<\/span><\/p>\n<div id=\"q192743\" class=\"hidden-answer\" style=\"display: none\">$3,659,823.44<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q108147\">Show Answer<\/span><\/p>\n<div id=\"q108147\" class=\"hidden-answer\" style=\"display: none\">Since the substance is decaying, the rate, 17.3%, is negative. So, r\u00a0=\u00a0\u20130.173. The initial amount of radon-222 was 100 mg, so a\u00a0= 100. We use the continuous decay formula to find the value after t\u00a0= 3 days: [latex]\\begin{cases}A\\left(t\\right)\\hfill & =a{e}^{rt}\\hfill & \\text{Use the continuous growth formula}.\\hfill \\\\ \\hfill & =100{e}^{-0.173\\left(3\\right)} & \\text{Substitute known values for }a, r,\\text{ and }t.\\hfill \\\\ \\hfill & \\approx 59.5115\\hfill & \\text{Use a calculator to approximate}.\\hfill \\end{cases}[\/latex] So 59.5115 mg of radon-222 will remain.<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135348472\">Using the data in Example 9, how much radon-222 will remain after one year?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q467606\">Show Answer<\/span><\/p>\n<div id=\"q467606\" class=\"hidden-answer\" style=\"display: none\">3.77E-26 (This is calculator notation for the number written as [latex]3.77 \\times {10}^{-26}[\/latex] in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)<\/div>\n<\/div>\n<\/div>\n<\/section>\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-11286\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-11286","chapter","type-chapter","status-publish","hentry"],"part":11277,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11286","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11286\/revisions"}],"predecessor-version":[{"id":16138,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11286\/revisions\/16138"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/11277"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/11286\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=11286"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=11286"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=11286"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=11286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}