{"id":160,"date":"2019-08-22T18:45:58","date_gmt":"2019-08-22T18:45:58","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=160"},"modified":"2020-06-17T18:53:54","modified_gmt":"2020-06-17T18:53:54","slug":"arc-length","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/arc-length\/","title":{"rendered":"Arc Length"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Arc Length &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<center><font size=\"+3\">\nArc Length\n<\/font><\/center>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"188\" height=\"119\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/figure1.gif?resize=188%2C119&#038;ssl=1\" alt=\"A graph of the differentiable function f from a to b\" class=\"wp-image-332\"\/><\/figure><\/div>\n\n\n\n<p>\n<!------------------------>\n\n\nSuppose $f$ is continuously differentiable on the interval $[a,b]$.  \n\n<\/p>\n\n\n\n<p>Suppose $f$ is continuously differentiable on the interval $[a,b]$.   <\/p>\n\n\n\n<p> <b>Let&#8217;s derive a formula for the length $L$ of the curve on the interval, called the <i>arc length<\/i> over $[a,b]$.<\/b> <\/p>\n\n\n<style>.kt-accordion-id_bcc2cc-c6 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_bcc2cc-c6 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_bcc2cc-c6 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_bcc2cc-c6 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_bcc2cc-c6 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_bcc2cc-c6 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_bcc2cc-c6 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_bcc2cc-c6:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_bcc2cc-c6 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_bcc2cc-c6 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_bcc2cc-c6 kt-accordion-has-3-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_276bde-17\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Mean Value Theorem<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>Mean Value Theorem<\/title>\n<center><font size=\"+2\">\nMean Value Theorem\n<\/font><\/center>\n\n\n\n<p>\n\n Let $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$.  Then there exists at least one point $c$ in $(a,b)$ for which  \\[ f'(c)=\\frac{f(b)-f(a)}{b-a}{\\small\\textrm{.}}\\]  \n\n<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane_b66517-df\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Riemann Sum<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>Riemann Sum<\/title>\n<center><font size=\"+2\">\nRiemann Sum\n<\/font><\/center>\n\n\n\n<p>   Let $f$ be continuous and non-negative on $[a,b]$ and let \\[a=x_0&lt; x_1&lt; \\ldots&lt; x_n=b\\] be a partition of $[a,b]$.    <\/p>\n\n\n\n<p>\n\nFor each $k$, $ 1 \\leq k \\leq n$, let $x_k^*$ be any point in the interval $[x_{k-1}, x_k]$.  Then the sum \\[\\sum_{i=1}^n f(x_i^*)\\, \\Delta x_i\\] is called a <strong>Riemann sum<\/strong> for $f$ on $[a,b]$.  \n\n<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane_e3192a-b6\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Definite Integral<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>Definite Integral<\/title>\n<center><font size=\"+2\">\nDefinite Integral\n<\/font><\/center>\n\n\n\n<p>\n\n Let $f$ be defined on $[a,b]$ and let \\[ a = x_0 &lt; x_1 &lt; ~&#8230;~ &lt; x_n = b \\] be a partition of $[a,b]$. For each $k$, $1 \\leq k \\leq n$, let $x_k^*$ be any point in $[x_{k-1}, x_k]$.  Then the <strong>definite integral<\/strong> of $f$ over $[a,b]$ is defined as \\[\\int^b_a f(x)\\, dx=\\lim_{\\max \\Delta x_i\\to 0}  \\left(\\sum_{i=1}^n f\\left(x_i^*\\right)\\Delta x_i\\right).\\]  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p>We&#8217;ll start by subdividing the interval $[a,b]$ into $n$ subintervals $[x_0, x_1], [x_1, x_2],~&#8230;~, [x_{n-1},x_n]$ <br> where $ a=x_0 &lt; x_1&lt; &#8230; &lt; x_{n-1} &lt; x_n=b $. <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"188\" height=\"119\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/figure2.gif?resize=188%2C119&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-333\"\/><\/figure><\/div>\n\n\n\n<p>\n\nIntroduce the line segments between <br> $(x_0, f(x_0)) {\\small\\textrm{ and }} (x_1,\nf(x_1)),$ <br> $(x_1, f(x_1)) {\\small\\textrm{ and }} (x_2, f(x_2)), &#8230; ,$ <br>\n$(x_{n-1}, f(x_{n-1})) {\\small\\textrm{ and }} (x_n, f(x_n))$.  \n\n<\/p>\n\n\n\n<p>\nThe resulting polygonal path approximates the curve given by $y=f(x)$,\nand its length approximates the arc length of $f(x)$ over $[a,b]$.\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"143\" height=\"60\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/figure3.gif?resize=143%2C60&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-334\"\/><\/figure><\/div>\n\n\n\n<p> Let&#8217;s find the length of the polygonal path by adding up the lengths of the individual line segments.  The $k$th line segment is the hypotenuse of a triangle with base $\\Delta x_k$ and height $f(x_k)-f(x_{k-1})$, and so has length<\/p>\n\n\n\n<p>\\[L_k=\\sqrt{\\left(\\Delta x_k \\right)^2+\\left[f(x_k)-f(x_{k-1})\\right]^2}  {\\small\\textrm{.}} \\] By the Mean Value Theorem, there exists $x_k^*\\in [x_{k-1},x_k]$ such that  \\[\\frac{f(x_k)-f(x_{k-1})}{x_k-x_{k-1}}=f'(x_k^*)\\] so  \\[f(x_k)-f(x_{k-1})=f'(x_k^*)(x_k-x_{k-1})=f'(x_k^*)\\Delta x_k.\\] Thus, \\[L_k=\\sqrt{(\\Delta x_k)^2+[f'(x_k^*)\\Delta x_k]^2}=\\sqrt{1+[f'(x_k^*)]^2}\\, \\Delta x_k.\\] Finally, the length of the entire polygonal path is \\[\\sum^n_{k=1} L_k=\\sum^n_{k=1} \\sqrt{1+[f'(x_k^*)]^2}\\, \\Delta x_k\\] which has the form of a Riemann sum.   Increasing the number of subintervals such that $\\max \\Delta x_k \\to 0$, $\\,  \\sum^n_{k=1} L_k \\to L$. That is,  \\[L=\\lim_{\\max \\Delta x_k\\to 0}\\sum^n_{k=1} \\sqrt{1+[f'(x_k^*)]^2}\\, \\Delta x_k=\\int^b_a \\sqrt{1+[f'(x)]^2}\\, dx\\]  by the definition of the definite integral as a limit of Riemann sums.  Thus, we have proved the following:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Arc Length<\/h4>\n\n\n\n<p>\n\nLet $f(x)$ be continuously differentiable on $[a,b]$.  Then the arc\nlength $L$ of $f(x)$ over $[a,b]$ is given by \n\\[L=\\int^b_a \\sqrt{1+[f'(x)]^2}\\, dx.\\]\nSimilarly, if $x=g(y)$ with $g$ continuously differentiable on\n$[c,d]$, then the arc length $L$ of $g(y)$ over $[c,d]$ is given by\n\\[L=\\int^d_c \\sqrt{1+[g'(y)]^2}\\, dy.\\]\nThese integrals often can only be computed using numerical methods.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Example<\/h4>\n\n\n\n<p>\n\nWe can compute the arc length of the graph of $f(x)=x^{3\/2}$ over\n$[0,1]$ as follows:\n\\begin{eqnarray*}\n\tL=\\int^1_0 \\sqrt{1+[f'(x)]^2}\\, dx\n\t&amp;=&amp; \\int^1_0 \\sqrt{1+[3x^{1\/2}\/2]^2}\\, dx\\\\\n\t&amp;=&amp; \\int^1_0 \\sqrt{1+9x\/4}\\, dx\\\\\n\t&amp;=&amp; \\left.\\frac{8}{27}(1+9x\/4)^{3\/2}\\right|^1_0\\\\\n\t&amp;=&amp; (1+9\/4)^{3\/2}-(1)^{3\/2}\\\\\n\t&amp;=&amp; (13\/4)^{3\/2}-1\\\\\n\t&amp;\\approx &amp; 1.44.\n\\end{eqnarray*}\n\n\n<\/p>\n\n\n\n<center>\n<applet codebase=\"..\/..\/java\" code=\"ArcLength.class\" width=\"640\" height=\"480\"><\/applet>\n\n<hr color=\"blue\">\n\n<h3>Key Concepts<\/h3>\n<\/center>\n\n\n\n<p>\n\nLet $f(x)$ be continuously differentiable on $[a,b]$. Then the arc length $L$ \nof $f(x)$ over $[a,b]$ is given by\n\\[L = \\int^b_a \\sqrt{1+[f'(x)]^2}\\, dx\\]\n\nSimilarly, if $x = g(y)$ with $g$ continuously differentiable on $[c,d]$, then \nthe arc length $L$ of $g(y)$ over $[c,d]$ is given by\n\\[L = \\int^d_c \\sqrt{1+[g'(y)]^2}\\, dy\\]\n\n\n\n<br><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1610\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Arc Length &#8211; HMC Calculus Tutorial Arc Length Suppose $f$ is continuously differentiable on the interval $[a,b]$. Suppose $f$ is continuously differentiable on the interval $[a,b]$. Let&#8217;s derive a formula for the length $L$ of the curve on the interval, called the arc length over $[a,b]$. We&#8217;ll start by subdividing the interval $[a,b]$ into $n$&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-160","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=160"}],"version-history":[{"count":9,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/160\/revisions"}],"predecessor-version":[{"id":1209,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/160\/revisions\/1209"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}