{"id":184,"date":"2019-08-27T17:50:32","date_gmt":"2019-08-27T17:50:32","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=184"},"modified":"2020-06-18T16:18:14","modified_gmt":"2020-06-18T16:18:14","slug":"mean-value-theorem","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/mean-value-theorem\/","title":{"rendered":"Mean Value Theorem"},"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>The Mean Value Theorem &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"277\" height=\"203\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/mean_figure1.gif?resize=277%2C203&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-369\"\/><\/figure><\/div>\n\n\n\n<p>We begin with a common-sense geometrical fact: <\/p>\n\n\n\n<p><em>somewhere between two zeros of a non-constant continuous\nfunction $f$,the function must change direction<\/em><\/p>\n\n\n\n<p> For a <em>differentiable<\/em>  function, the derivative is $0$ at the point where $f$ changes  direction.  Thus, we expect there to be a point $c$ where the tangent is  horizontal.  These ideas are precisely stated by Rolle&#8217;s Theorem:  <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Rolle&#8217;s Theorem<\/h4>\n\n\n\n<p>\n\n\t\tLet $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$.  If\n\t\t$f(a)=f(b)=0$, then there is at least one point $c$ in $(a,b)$ for\n\t\twhich $f'(c)=0$.\n<\/p>\n\n\n\n<p>\nNotice that both conditions on $f$ are necessary.  Without either one,\nthe statement is false!  \n<\/p>\n\n\n\n<p>For a <em>discontinuous<\/em> function, the conclusion of Rolle&#8217;s Theorem may not hold: <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"235\" height=\"135\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/mean_figure2.gif?resize=235%2C135&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-370\"\/><\/figure><\/div>\n\n\n\n<p>For a <em>continuous, non-differentiable<\/em> function, again this might not be the case:  <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"233\" height=\"135\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/mean_figure3.gif?resize=233%2C135&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-367\"\/><\/figure><\/div>\n\n\n\n<p>\nThough the theorem seems logical, we cannot\nbe sure that it is always true without a proof.\n<\/p>\n\n\n<style>.kt-accordion-id_7812d4-b4 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_7812d4-b4 .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_7812d4-b4 > .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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4 > .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_7812d4-b4 .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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_7812d4-b4 > .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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4: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_7812d4-b4 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_7812d4-b4 .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_7812d4-b4 kt-accordion-has-2-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_be60b5-98\"><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\">Proof of Rolle&#8217;s 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\n\n\n<p>\n<!------------------------>\n\nNote that either $f(x)$ is always $0$ on $[a,b]$ or $f$ varies on\n$[a,b]$.\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> If $f(x)$ is always $0$, then $f'(x)=0$ for all $x$ in $(a,b)$ and we are done. <br> <\/li><li> If $f(x)$ varies on $(a,b)$, then there must be points where $f(x) &gt; 0$ or points where $f(x) &lt; 0$.   <p> Assume first that there are points where $f(x) &gt; 0$.  By the <i>Value Theorem<\/i> $f$ has a maximum at some point $c$ in $(a,b)$.  Then $f(c) &gt; 0$, so $c$ is  not an endpoint.  At this maximum, $f'(c)=0$.  Now assume that there are points where $f(x) &lt; 0$.  Then, again by the <i>Value Theorem<\/i>, $f$ has a minimum at some point $c$ in $(a,b)$.  Again, $c$ is not an endpoint since $f(c) &lt; 0$ while $f(a)=f(b)=0$.  At this minimum,  $f'(c)=0$. <\/p><\/li><\/ul>\n\n\n\n<p>\n\nThis completes the proof.\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\nThe Mean Value Theorem is a generalization of Rolle&#8217;s Theorem:\n\n<\/p>\n\n\n\n<p>\nWe now let $f(a)$ and $f(b)$ have values other than $0$ and look at\nthe secant line through $(a,f(a))$ and $(b,f(b))$.  We expect that\nsomewhere between $a$ and $b$ there is a point $c$ where the tangent\nis parallel to this secant.\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignleft\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"232\" height=\"150\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/mean_figure4.gif?resize=232%2C150\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-368\"\/><\/figure><\/div>\n\n\n\n<p>\n\t\t\tIn Rolle&#8217;s Theorem, the secant was horizontal so we looked for a horizontal \n\t\t\ttangent.\n<\/p>\n\n\n\n<p> <\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>That is, the slopes of these two lines are equal.  This is formalized in the Mean Value Theorem. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Mean Value Theorem<\/h4>\n\n\n\n<p>\n\n\t\tLet $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$.  Then\n\t\tthere is at least one point $c$ in $(a,b)$ for which\n\t\t\\[f'(c)=\\frac{f(b)-f(a)}{b-a}.\\] \n<\/p>\n\n\n\n<p>\nHere, $f'(c)$ is the slope of the tangent at $c$, while $\\displaystyle\n\\frac{f(b)-f(a)}{b-a}$ is the slope of the secant through $a$ and\n$b$.  Intuitively, we see that if we translate the secant line in the\nfigure upwards, it will eventually just touch the curve at the single\npoint $c$ and will be tangent at $c$.  However, basing conclusions on\na single example can be disastrous, so we need a proof.\n\n<\/p>\n\n\n<style>.kt-accordion-id_3b846f-a5 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_3b846f-a5 .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_3b846f-a5 > .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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5 > .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_3b846f-a5 .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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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 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.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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5: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_3b846f-a5 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_3b846f-a5 .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_3b846f-a5 kt-accordion-has-2-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_c45f4e-c0\"><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\">Proof of 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\n\n\n<p>\n<!------------------------>\n\nThe equation of the secant through $(a,f(a))$ and $(b,f(b))$ is\n\\[y-f(a)=\\frac{f(b)-f(a)}{b-a}(x-a)\\]\nwhich we can rewrite as\n\\[y=\\frac{f(b)-f(a)}{b-a}(x-a)+f(a).\\]\nLet\n\\[g(x)=f(x)-\\left[\\frac{f(b)-f(a)}{b-a}(x-a)+f(a)\\right].\\]\nNote that $g(a)=g(b)=0$.  Also, $g$ is continuous on $[a,b]$ and\ndifferentiable on $(a,b)$ since $f$ is.  So by Rolle&#8217;s Theorem there\nexists $c$ in $(a,b)$ such that $g'(c)=0$.\n\n<\/p>\n\n\n\n<p>\nBut $\\displaystyle g'(x)=f'(x)-\\frac{f(b)-f(a)}{b-a}$, so\n\\[g'(c)=f'(c)-\\frac{f(b)-f(a)}{b-a}=0.\\]\nTherefore, \n\\[f'(c)=\\frac{f(b)-f(a)}{b-a}\\]\nand the proof is complete.\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Consequences of the Mean Value Theorem<\/h4>\n\n\n\n<p>\n\nThe Mean Value Theorem is behind many of the important results in\ncalculus.  The following statements, in which we assume $f$ is\ndifferentiable on an open interval $I$, are consequences of the Mean\nValue Theorem:\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> $f'(x)=0$ everywhere on $I$ if and only if $f$ is constant on\n\t\t$I$.\n\t<br><br>\n\t<\/li><li> If $f'(x)=g'(x)$ for all $x$ on $I$, then $f$ and $g$ differ at\n\t\tmost by a constant on $I$.\n\t<br><br>\n\t<\/li><li> If $f'(x)&gt;0$ for all $x$ on $I$, then $f$ is <i>increasing<\/i>\n\t\ton $I$.\n\t\t<br>\n\t\tIf $f'(x) &lt; 0$ for all $x$ on $I$, then $f$ is <i>decreasing<\/i>\n\t\ton $I$.\n<\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<b>Mean Value Theorem<\/b>\n\n\n\n<p>\nLet $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$. Then there is at \nleast one point $c$ in $(a,b)$ for which\n$$f'(c) =  \\frac{f(b)-f(a)}{b-a}.$$\n\n\n<!------------------------>\n\n\n<br>\n\n<\/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\/QZ1310\/\">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>The Mean Value Theorem &#8211; HMC Calculus Tutorial We begin with a common-sense geometrical fact: somewhere between two zeros of a non-constant continuous function $f$,the function must change direction For a differentiable function, the derivative is $0$ at the point where $f$ changes direction. Thus, we expect there to be a point $c$ where the&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-184","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/184","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=184"}],"version-history":[{"count":9,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/184\/revisions"}],"predecessor-version":[{"id":1216,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/184\/revisions\/1216"}],"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=184"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}