{"id":480,"date":"2025-02-13T19:45:21","date_gmt":"2025-02-13T19:45:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-cheat-sheet\/"},"modified":"2025-02-13T19:45:21","modified_gmt":"2025-02-13T19:45:21","slug":"integration-of-exponential-logarithmic-and-hyperbolic-functions-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-cheat-sheet\/","title":{"raw":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Cheat Sheet","rendered":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Cheat Sheet"},"content":{"raw":"\n<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Integration+of+Exponential%2C+Logarithmic%2C+and+Hyperbolic+Functions.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Integrals, Exponential Functions, and Logarithms<\/strong><\/p>\n<ul id=\"fs-id1167793931233\">\n\t<li>The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.<\/li>\n\t<li>The cornerstone of the development is the definition of the natural logarithm in terms of an integral.<\/li>\n\t<li>The function [latex]{e}^{x}[\/latex] is then defined as the inverse of the natural logarithm.<\/li>\n\t<li>General exponential functions are defined in terms of [latex]{e}^{x},[\/latex] and the corresponding inverse functions are general logarithms.<\/li>\n\t<li>Familiar properties of logarithms and exponents still hold in this more rigorous context.<\/li>\n<\/ul>\n<p><strong>Exponential Growth and Decay<\/strong><\/p>\n<ul id=\"fs-id1167793829823\">\n\t<li>Exponential growth and exponential decay are two of the most common applications of exponential functions.<\/li>\n\t<li>Systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/li>\n\t<li>In exponential growth, the rate of growth is proportional to the quantity present. In other words, [latex]{y}^{\\prime }=ky.[\/latex]<\/li>\n\t<li>Systems that exhibit exponential growth have a constant doubling time, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\n\t<li>Systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}.[\/latex]<\/li>\n\t<li>Systems that exhibit exponential decay have a constant half-life, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\n<\/ul>\n<p><strong>Calculus of the Hyperbolic Functions<\/strong><\/p>\n<ul id=\"fs-id1167793618899\">\n\t<li>Hyperbolic functions are defined in terms of exponential functions.<\/li>\n\t<li>Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.<\/li>\n\t<li>With appropriate range restrictions, the hyperbolic functions all have inverses.<\/li>\n\t<li>Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.<\/li>\n\t<li>The most common physical applications of hyperbolic functions are calculations involving catenaries.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167793278406\">\n\t<li><strong>Natural logarithm function<\/strong><\/li>\n\t<li>[latex]\\text{ln}x={\\displaystyle\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] Z<\/li>\n\t<li><strong>Exponential function<\/strong>[latex]y={e}^{x}[\/latex]<\/li>\n\t<li>[latex]\\text{ln}y=\\text{ln}({e}^{x})=x[\/latex] Z<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167793432037\" class=\"definition\">\n<dt>catenary<\/dt>\n<dd id=\"fs-id1167793432042\">a curve in the shape of the function [latex]y=a\\text{cosh}(x\\text{\/}a)[\/latex] is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>doubling time<\/dt>\n<dd id=\"fs-id1167793423305\">if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794146802\" class=\"definition\">\n<dt>exponential decay<\/dt>\n<dd id=\"fs-id1167794146808\">systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794146834\" class=\"definition\">\n<dt>exponential growth<\/dt>\n<dd>systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793541831\" class=\"definition\">\n<dt>half-life<\/dt>\n<dd id=\"fs-id1167793541836\">if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>The Natural Logarithm as an Integral<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Visualize the logarithm as an area under a curve<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice using logarithm properties to manipulate expressions<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember the domain restriction [latex](x &gt; 0)[\/latex] for [latex]\\ln x[\/latex]<\/li>\n<\/ul>\n<p><strong>Properties of the Exponential Function<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Visualize [latex]e[\/latex] as the [latex]x[\/latex]-coordinate where area under [latex]\\frac{1}{t}[\/latex] curve equals [latex]1[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice using exponential properties to manipulate expressions<\/li>\n\t<li class=\"whitespace-normal break-words\">Connect differentiation and integration of exponential functions<\/li>\n<\/ul>\n<p><strong>General Logarithmic and Exponential Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Practice converting between general exponential and logarithmic forms.<\/li>\n\t<li class=\"whitespace-normal break-words\">Memorize the key derivative and integral formulas for [latex]a^x[\/latex] and [latex]\\log_a x[\/latex].<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that all logarithmic functions are constant multiples of each other.<\/li>\n\t<li class=\"whitespace-normal break-words\">When differentiating or integrating expressions with general bases, convert to natural logarithms or exponentials first if needed.<\/li>\n\t<li class=\"whitespace-normal break-words\">Pay attention to the chain rule when dealing with composite functions involving exponentials or logarithms.<\/li>\n<\/ul>\n<p><strong>Exponential Growth Model<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Understand the significance of the growth constant [latex]k[\/latex] in determining the rate of growth.<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving exponential growth problems, use logarithms to isolate the variable of interest.<\/li>\n\t<li class=\"whitespace-normal break-words\">For compound interest problems, pay attention to whether the interest is compounded continuously or at discrete intervals.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the doubling time is constant in exponential growth, regardless of the initial quantity.<\/li>\n<\/ul>\n<p><strong>Exponential Decay Model<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Understand the significance of the decay constant [latex]k[\/latex] in determining the rate of decay.<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving exponential decay problems, use logarithms to isolate the variable of interest.<\/li>\n\t<li class=\"whitespace-normal break-words\">For Newton's Law of Cooling problems, pay attention to the ambient temperature and its effect on the decay rate.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that the half-life is constant in exponential decay, regardless of the initial quantity.<\/li>\n<\/ul>\n<p><strong>Derivatives and Integrals of the Hyperbolic Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Memorize the definitions of [latex]\\sinh[\/latex] and [latex]\\cosh[\/latex] in terms of exponentials.<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice deriving the derivatives of hyperbolic functions from their definitions.<\/li>\n\t<li class=\"whitespace-normal break-words\">Notice the patterns in the derivative formulas and compare them with trigonometric function derivatives.<\/li>\n\t<li class=\"whitespace-normal break-words\">When integrating, look for opportunities to use u-substitution, especially with composites of hyperbolic functions.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]\\cosh x[\/latex] is always positive, which can simplify some integrals (e.g., when dealing with absolute value).<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving problems, consider whether using hyperbolic function identities might simplify the expression.<\/li>\n<\/ul>\n<p><strong>Calculus of Inverse Hyperbolic Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Memorize the domains and ranges of inverse hyperbolic functions.<\/li>\n\t<li class=\"whitespace-normal break-words\">Notice the similarities between derivatives of inverse hyperbolic and inverse trigonometric functions.<\/li>\n\t<li class=\"whitespace-normal break-words\">When integrating, pay attention to the form of the integrand to identify which inverse hyperbolic function to use.<\/li>\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]\\tanh^{-1} x[\/latex] and [latex]\\coth^{-1} x[\/latex] have the same derivative, so context is important when integrating [latex]\\frac{1}{1-x^2}[\/latex].<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving problems, consider whether using hyperbolic function identities might simplify the expression.<\/li>\n<\/ul>\n<p><strong>Applications of Hyperbolic Functions<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Understand the physical meaning of the parameters in the catenary equation [latex]y = a \\cosh(\\frac{x}{a})[\/latex].<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice sketching catenary curves for different values of '[latex]a[\/latex]' to understand how it affects the shape.<\/li>\n\t<li class=\"whitespace-normal break-words\">When solving catenary problems, pay attention to the units of measurement given and ensure your answer uses the correct units.<\/li>\n\t<li class=\"whitespace-normal break-words\">Practice deriving the derivative of catenary functions, as this is often needed in arc length calculations.<\/li>\n<\/ul>\n","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Integration+of+Exponential%2C+Logarithmic%2C+and+Hyperbolic+Functions.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Integrals, Exponential Functions, and Logarithms<\/strong><\/p>\n<ul id=\"fs-id1167793931233\">\n<li>The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.<\/li>\n<li>The cornerstone of the development is the definition of the natural logarithm in terms of an integral.<\/li>\n<li>The function [latex]{e}^{x}[\/latex] is then defined as the inverse of the natural logarithm.<\/li>\n<li>General exponential functions are defined in terms of [latex]{e}^{x},[\/latex] and the corresponding inverse functions are general logarithms.<\/li>\n<li>Familiar properties of logarithms and exponents still hold in this more rigorous context.<\/li>\n<\/ul>\n<p><strong>Exponential Growth and Decay<\/strong><\/p>\n<ul id=\"fs-id1167793829823\">\n<li>Exponential growth and exponential decay are two of the most common applications of exponential functions.<\/li>\n<li>Systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/li>\n<li>In exponential growth, the rate of growth is proportional to the quantity present. In other words, [latex]{y}^{\\prime }=ky.[\/latex]<\/li>\n<li>Systems that exhibit exponential growth have a constant doubling time, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\n<li>Systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}.[\/latex]<\/li>\n<li>Systems that exhibit exponential decay have a constant half-life, which is given by [latex](\\text{ln}2)\\text{\/}k.[\/latex]<\/li>\n<\/ul>\n<p><strong>Calculus of the Hyperbolic Functions<\/strong><\/p>\n<ul id=\"fs-id1167793618899\">\n<li>Hyperbolic functions are defined in terms of exponential functions.<\/li>\n<li>Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.<\/li>\n<li>With appropriate range restrictions, the hyperbolic functions all have inverses.<\/li>\n<li>Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.<\/li>\n<li>The most common physical applications of hyperbolic functions are calculations involving catenaries.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167793278406\">\n<li><strong>Natural logarithm function<\/strong><\/li>\n<li>[latex]\\text{ln}x={\\displaystyle\\int }_{1}^{x}\\frac{1}{t}dt[\/latex] Z<\/li>\n<li><strong>Exponential function<\/strong>[latex]y={e}^{x}[\/latex]<\/li>\n<li>[latex]\\text{ln}y=\\text{ln}({e}^{x})=x[\/latex] Z<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167793432037\" class=\"definition\">\n<dt>catenary<\/dt>\n<dd id=\"fs-id1167793432042\">a curve in the shape of the function [latex]y=a\\text{cosh}(x\\text{\/}a)[\/latex] is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>doubling time<\/dt>\n<dd id=\"fs-id1167793423305\">if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794146802\" class=\"definition\">\n<dt>exponential decay<\/dt>\n<dd id=\"fs-id1167794146808\">systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\\text{\u2212}kt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794146834\" class=\"definition\">\n<dt>exponential growth<\/dt>\n<dd>systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793541831\" class=\"definition\">\n<dt>half-life<\/dt>\n<dd id=\"fs-id1167793541836\">if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\\frac{(\\text{ln}2)}{k}[\/latex]<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>The Natural Logarithm as an Integral<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Visualize the logarithm as an area under a curve<\/li>\n<li class=\"whitespace-normal break-words\">Practice using logarithm properties to manipulate expressions<\/li>\n<li class=\"whitespace-normal break-words\">Remember the domain restriction [latex](x > 0)[\/latex] for [latex]\\ln x[\/latex]<\/li>\n<\/ul>\n<p><strong>Properties of the Exponential Function<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Visualize [latex]e[\/latex] as the [latex]x[\/latex]-coordinate where area under [latex]\\frac{1}{t}[\/latex] curve equals [latex]1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Practice using exponential properties to manipulate expressions<\/li>\n<li class=\"whitespace-normal break-words\">Connect differentiation and integration of exponential functions<\/li>\n<\/ul>\n<p><strong>General Logarithmic and Exponential Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice converting between general exponential and logarithmic forms.<\/li>\n<li class=\"whitespace-normal break-words\">Memorize the key derivative and integral formulas for [latex]a^x[\/latex] and [latex]\\log_a x[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Remember that all logarithmic functions are constant multiples of each other.<\/li>\n<li class=\"whitespace-normal break-words\">When differentiating or integrating expressions with general bases, convert to natural logarithms or exponentials first if needed.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the chain rule when dealing with composite functions involving exponentials or logarithms.<\/li>\n<\/ul>\n<p><strong>Exponential Growth Model<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand the significance of the growth constant [latex]k[\/latex] in determining the rate of growth.<\/li>\n<li class=\"whitespace-normal break-words\">When solving exponential growth problems, use logarithms to isolate the variable of interest.<\/li>\n<li class=\"whitespace-normal break-words\">For compound interest problems, pay attention to whether the interest is compounded continuously or at discrete intervals.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the doubling time is constant in exponential growth, regardless of the initial quantity.<\/li>\n<\/ul>\n<p><strong>Exponential Decay Model<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand the significance of the decay constant [latex]k[\/latex] in determining the rate of decay.<\/li>\n<li class=\"whitespace-normal break-words\">When solving exponential decay problems, use logarithms to isolate the variable of interest.<\/li>\n<li class=\"whitespace-normal break-words\">For Newton&#8217;s Law of Cooling problems, pay attention to the ambient temperature and its effect on the decay rate.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the half-life is constant in exponential decay, regardless of the initial quantity.<\/li>\n<\/ul>\n<p><strong>Derivatives and Integrals of the Hyperbolic Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the definitions of [latex]\\sinh[\/latex] and [latex]\\cosh[\/latex] in terms of exponentials.<\/li>\n<li class=\"whitespace-normal break-words\">Practice deriving the derivatives of hyperbolic functions from their definitions.<\/li>\n<li class=\"whitespace-normal break-words\">Notice the patterns in the derivative formulas and compare them with trigonometric function derivatives.<\/li>\n<li class=\"whitespace-normal break-words\">When integrating, look for opportunities to use u-substitution, especially with composites of hyperbolic functions.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]\\cosh x[\/latex] is always positive, which can simplify some integrals (e.g., when dealing with absolute value).<\/li>\n<li class=\"whitespace-normal break-words\">When solving problems, consider whether using hyperbolic function identities might simplify the expression.<\/li>\n<\/ul>\n<p><strong>Calculus of Inverse Hyperbolic Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the domains and ranges of inverse hyperbolic functions.<\/li>\n<li class=\"whitespace-normal break-words\">Notice the similarities between derivatives of inverse hyperbolic and inverse trigonometric functions.<\/li>\n<li class=\"whitespace-normal break-words\">When integrating, pay attention to the form of the integrand to identify which inverse hyperbolic function to use.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]\\tanh^{-1} x[\/latex] and [latex]\\coth^{-1} x[\/latex] have the same derivative, so context is important when integrating [latex]\\frac{1}{1-x^2}[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">When solving problems, consider whether using hyperbolic function identities might simplify the expression.<\/li>\n<\/ul>\n<p><strong>Applications of Hyperbolic Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand the physical meaning of the parameters in the catenary equation [latex]y = a \\cosh(\\frac{x}{a})[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Practice sketching catenary curves for different values of &#8216;[latex]a[\/latex]&#8216; to understand how it affects the shape.<\/li>\n<li class=\"whitespace-normal break-words\">When solving catenary problems, pay attention to the units of measurement given and ensure your answer uses the correct units.<\/li>\n<li class=\"whitespace-normal break-words\">Practice deriving the derivative of catenary functions, as this is often needed in arc length calculations.<\/li>\n<\/ul>\n","protected":false},"author":6,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":479,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/480"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/480\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/479"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/480\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=480"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=480"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=480"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=480"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}