{"id":219,"date":"2019-09-10T16:31:41","date_gmt":"2019-09-10T16:31:41","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=219"},"modified":"2020-06-18T17:40:30","modified_gmt":"2020-06-18T17:40:30","slug":"tangent-planes","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/tangent-planes\/","title":{"rendered":"Tangent Planes"},"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>Tangent Planes and Linear Approximation &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nJust as we can visualize the line tangent to a curve at a point in 2-space, in 3-space we can\npicture the <b>plane<\/b> tangent to a <b>surface<\/b> at a point.\n\n<\/p>\n\n\n\n<p> Consider the surface given by $z = f (x, y)$. Let $(x_0 , y_0 , z_0 )$ be any point on this surface. If $f (x, y)$ is <b>differentiable<\/b> at $(x_0 , y_0 )$, then the surface has a <b>tangent plane<\/b> at $(x_0 , y_0 , z_0 )$.<\/p>\n\n\n<style>.kt-accordion-id_aa09ba-e6 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_aa09ba-e6 .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_aa09ba-e6 > .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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6 > .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_aa09ba-e6 .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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_aa09ba-e6 > .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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6: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_aa09ba-e6 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_aa09ba-e6 .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_aa09ba-e6 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_e19632-5b\"><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\">Differentiability<\/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<p>\n\n A function $f(x,y)$ is <strong>differentiable<\/strong> at the point $(x_0,y_0)$ if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ exist and $\\Delta f = f(x_0+\\Delta x,y_0+\\Delta y) &#8211; f(x_0,y_0)$ can be written in the form $$ \\Delta f = f_x(x_0,y_0)\\Delta x + f_y(x_0,y_0)\\Delta y + \\varepsilon_1 \\Delta x + \\varepsilon_2 \\Delta y $$ where $\\varepsilon_1$ and $\\varepsilon_2$ are functions of $\\Delta x$ and $\\Delta y$ such that $$ \\lim_{(\\Delta x,\\Delta y) \\rightarrow (0,0)}\\varepsilon_1 = \\lim_{(\\Delta x,\\Delta y) \\rightarrow (0,0)}\\varepsilon_2 = 0.  $$  \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_192f04-19\"><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\">Tangent Plane<\/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<p>\n\nLet $(x_0, y_0, z_0)$ be any point on the surface $z = f(x,y)$. If the tangent lines at $(x_0, y_0, z_0)$ to all smooth curves on the surface passing through $(x_0, y_0, z_0)$ lie on a common plane, then we call that plane the <strong>tangent plane<\/strong> to $z = f(x,y)$ at $(x_0, y_0, z_0)$.   \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>The equation of the tangent plane at $(x_0 , y_0 , z_0 )$ is given by $$f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ) &#8211; (z &#8211; z_0 ) = 0.$$<\/p>\n\n\n\n<center>\n<font size=\"+2\">Notes<\/font>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> Recall that the equation of the plane containing a point $(x_0 , y_0 , z_0 )$ and normal to the vector ${\\bf n} = (a, b, c)$ is $$ a(x &#8211; x_0 ) + b(y &#8211; y_0 ) + c(z &#8211; z_0 ) = 0. $$ The <b>derivation<\/b> of the equation for the tangent plane just involves showing that the tangent plane is normal to the vector ${\\bf n} = (f_x (x_0 , y_0 ), f_y (x_0 , y_0 ), -1)$. <br><\/li><\/ul>\n\n\n<style>.kt-accordion-id_528d68-eb .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_528d68-eb .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_528d68-eb > .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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb > .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_528d68-eb .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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_528d68-eb > .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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb: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_528d68-eb .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_528d68-eb .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_528d68-eb 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_57fda3-0d\"><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\">Derivation of the Tangent Plane<\/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<p>\n<!------------------------>\n\nLet $(x_0,y_0,z_0)$ be any point on the surface $z=f(x,y)$ such that\n$f(x,y)$ is differentiable at $(x_0,y_0)$.  We well show that the\ntangent plane is normal to the vector ${\\bf n} =\n(f_x(x_0,y_0),f_y(x_0,y_0),-1)$.\n\n<\/p>\n\n\n\n<p>\nConsider any smooth curve $C$ on the surface that passes through\n$(x_0,y_0,z_0)$.  Parametrize the curve as\n\\begin{eqnarray*}\n\tx &amp; = &amp; x(s) \\\\\n\ty &amp; = &amp; y(s) \\\\\n\tz &amp; = &amp; z(s).\n\\end{eqnarray*}\nLet $s=s_0$ satisfy $x_0=x(s_0), \\quad y_0=y(s_0), \\quad z_0=z(s_0).$  Then\n$$\nz(s) = f(x(s),y(s))\n$$\nfor all $s$.  Using the Chain Rule to differentiate both sides of this\nequation,\n$$\n\\frac{dz}{ds}=\\frac{\\partial f}{\\partial x}\\frac{dx}{ds} +\n\\frac{\\partial f}{\\partial y}\\frac{dy}{ds}.\n$$\nThus,\n$$\n\\frac{\\partial f}{\\partial x}\\frac{dx}{ds} +\n\\frac{\\partial f}{\\partial y}\\frac{dy}{ds} &#8211; \\frac{dz}{ds}=0.\n$$\nSwitching notation and writing the equation in vector form,\n$$\n(f_x(x,y),f_y(x,y),-1) \\cdot (x'(s),y'(s),z'(s))=0.\n$$\nAt $s=s_0$,\n$$\n(f_x(x_0,y_0),f_y(x_0,y_0),-1) \\cdot (x'(s_0),y'(s_0),z'(s_0))=0.\n$$\nBut $(x'(s_0),y'(s_0),z'(s_0))$ is the tangent vector to the curve $C$\nat $(x_0,y_0,z_0)$.  Thus, we have that the tangent vector to any\nsmooth curve $C$ on the surface that passes through $(x_0,y_0,z_0)$ is\nnormal to the vector ${\\bf n} = (f_x(x_0,y_0),f_y(x_0,y_0),-1)$ and so\nis given by the equation \n$$\nf_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)-(z-z_0)=0.\n$$\n\n<\/p>\n\n\n\n<p>\n(This proof is taken from <i>Calculus<\/i>, by Howard Anton.)\n\n\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\"><li>  For surfaces $F (x, y, z) = 0$ that are not easily solved for $z$, the equation of the tangent plane at $(x_0 , y_0 , z_0 )$ is $$ F_x (x_0 , y_0 , z_0 )(x &#8211; x_0 ) + F_y (x_0 , y_0 , z_0 )(y &#8211; y_0 ) + F_z (x_0 , y_0 , z_0 )(z &#8211; z_0 ) = 0 $$ provided that $\\nabla F (x_0 , y_0 , z_0 ) \\neq 0$. Note that if we let $F (x, y, z) = f (x, y) &#8211; z$, we obtain the equation given for the tangent plane to $z = f (x, y)$ at $(x_0 , y_0 , z_0 )$.  <\/li><\/ul>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\nLet&#8217;s find the equation of the plane tangent to the surface $z = 4x^3 y^2 + 2y$ at the point\n$(1, -2, 12)$.\n\n<\/p>\n\n\n\n<p>\nSince $f (x, y) = 4x^3 y^2 + 2y$,\n                            $$ f_x (x, y) = 12x^2 y^2 \\textrm{ and } f_y (x, y) = 8x^3 y + 2.$$\n<\/p>\n\n\n\n<p>\nWith $x = 1$ and $y = -2$,\n\\begin{eqnarray*}\n\tf_x (1, -2) &amp;=&amp; 12(1)^2 (-2)^2 = 48 \\\\\n\tf_y (1, -2) &amp;=&amp; 8(3)^3(-2) + 2 = -14. \n\\end{eqnarray*}\n\n<\/p>\n\n\n\n<p>\nThus, the tangent plane has normal vector $ {\\bf n} = (48, -14, -1) $ at $(1, -2, 12)$ and the equation\nof the tangent plane is given by\n$$ 48(x &#8211; 1) &#8211; 14 (y &#8211; (-2)) &#8211; (z &#8211; 12) = 0.$$\nSimplifying, \n$$ 48x &#8211; 14y &#8211; z = 64. $$ \n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Linear Approximation<\/h4>\n\n\n\n<p>\nThe tangent plane to a surface at a point stays close to the surface near the point. In fact,\nif $f (x, y)$ is differentiable at the point $(x_0 , y_0 )$, the tangent plane to the surface $z = f (x, y)$\nat $(x_0 , y_0 )$ provides a good approximation to $f (x, y)$ near $(x_0 , y_0 )$:\n\n<\/p>\n\n\n\n<p>\n$\\qquad$ Solving $f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ) &#8211; (z &#8211; z_0 ) = 0$ for $z$,\n$$ z = z_0 + f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ). $$\n$\\qquad$ Since $z_0 = f (x_0 , y_0 )$, we have that\n$$ z = f (x_0 , y_0 ) + f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ). $$\n$\\qquad$ Near $(x_0 , y_0 )$, the surface is close to the tangent plane. Thus,\n$$ f (x, y) \\approx f (x_0 , y_0 ) + f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ). $$\nWe call this the <b>linear approximation<\/b> or <b>local linearization<\/b> of $f (x, y)$ near $(x_0 , y_0 )$.\n\n<br><br>\n<\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> The linear approximation is really just the multivariable Taylor polynomial of degree\n\t\t1 for $f (x, y)$ about $(x_0 , y_0 )$. It is only accurate near $(x_0 , y_0 )$. Better approximations\n\t\tcan be obtained by using higher-order Taylor polynomials.\n\t\t\n\t<br><br>\n\t<\/li><li> These concepts can be extended to functions of more than two variables:\n\t\t$$ f (x, y, z) \\approx f (x_0 , y_0 , z_0 )+f_x (x_0 , y_0 , z_0 )(x-x_0 )+f_y (x_0 , y_0 , z_0 )(y-y_0 )+f_z (x_0 , y_0 , z_0 )(z-z_0 ) $$\n\t\twhere $f (x, y, z)$ is differentiable at $(x_0 , y_0 , z_0 )$.\n<\/li><\/ul>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nFrom our work in the previous example, the linear approximation to $f (x, y) = 4x^3 y^2 + 2y$\nnear $x = 1,\\quad y = -2$ is\n                                    $$ f (x, y) \\approx 48x &#8211; 14y &#8211; 64. $$   \nThis is, of course, exact at $x = 1, \\quad y = -2$:\n                            $$ f (1, -2) = 12 = 48(1) &#8211; 14(-2) &#8211; 64. $$\nAt $x = 1.1$ and $y = -1.9$, according to the linear approximation,\n                        $$ f (1.1, -1.9) \\approx 48(1.1) &#8211; 14(-1.9) &#8211; 64 = 15.4, $$\nwhich is indeed very close to the exact value $f (1.1, -1.9) = 15.41964$.\n\n<\/p>\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<ul class=\"wp-block-list\"><li> <b>Tangent Plane to a Surface<\/b>\n\t\t<p>\n\t\tLet $(x_0 , y_0 , z_0 )$ be any point on the surface $z = f (x, y)$. If $f (x, y)$ is differentiable at\n\t\t$(x_0 , y_0 )$, then the surface has a tangent plane at $(x_0 , y_0 , z_0 )$ given by\n                $$ f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ) &#8211; (z &#8211; z_0 ) = 0. $$\n                     \n\t<br><br>\n\t<\/p><\/li><li> <b>Linear Approximation to a Surface<\/b>\n\t\t<p>\n\t\tIf $f (x, y)$ is differentiable at $(x_0 , y_0 )$, then near $(x_0 , y_0 )$\n                $$ f (x, y) \\approx f (x_0 , y_0 ) + f_x (x_0 , y_0 )(x &#8211; x_0 ) + f_y (x_0 , y_0 )(y &#8211; y_0 ). $$\n<\/p><\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p> [<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ3010\/\">I&#8217;m ready to take the quiz.<\/a>] [<a href=\"#top\">I need to review more.<\/a>]<br> <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tangent Planes and Linear Approximation &#8211; HMC Calculus Tutorial Just as we can visualize the line tangent to a curve at a point in 2-space, in 3-space we can picture the plane tangent to a surface at a point. Consider the surface given by $z = f (x, y)$. Let $(x_0 , y_0 , z_0&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":59,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-219","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/219","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=219"}],"version-history":[{"count":8,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/219\/revisions"}],"predecessor-version":[{"id":1242,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/219\/revisions\/1242"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/59"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=219"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}