{"id":166,"date":"2021-07-30T17:23:49","date_gmt":"2021-07-30T17:23:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=166"},"modified":"2022-10-29T01:54:26","modified_gmt":"2022-10-29T01:54:26","slug":"summary-of-tangent-planes-and-linear-approximations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-tangent-planes-and-linear-approximations\/","title":{"raw":"Summary of Tangent Planes and Linear Approximations","rendered":"Summary of Tangent Planes and Linear Approximations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.<\/li>\r\n \t<li>Tangent planes can be used to approximate values of functions near known values.<\/li>\r\n \t<li>A function is differentiable at a point if it is \u201dsmooth\u201d at that point (i.e., no corners or discontinuities exist at that point).<\/li>\r\n \t<li>The total differential can be used to approximate the change in a function [latex]z=f(x_{0},y_{0})[\/latex]\u00a0at the point\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0for given values of [latex]\\Delta{x}[\/latex] and\u00a0[latex]\\Delta{y}[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Tangent plane\r\n<\/strong>[latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Linear approximation<\/strong>\r\n[latex]L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Total differential<\/strong>\r\n[latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})(y-y_{0})dy[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Differentiability (two variables)<\/strong>\r\n[latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y)\\to (x_{0},y_{0})}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Differentiability (three variables)<\/strong>\r\n[latex]f(x,y,z)=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})+E(x,y,z)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y,z)\\to (x_{0},y_{0},z_{0})}{\\lim}\\frac{E(x,y,z)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>differentiable<\/dt>\r\n \t<dd>a function [latex]f(x,y,z)[\/latex]\u00a0is differentiable at\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0if [latex]f(x,y)[\/latex]\u00a0can be expressed in the form [latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E(x,y)[\/latex]\u00a0satisfies [latex]\\underset{(x,y)\\to{(x_{0},y_{0})}}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>linear approximation<\/dt>\r\n \t<dd>given a function [latex]f(x,y)[\/latex]\u00a0and a tangent plane to the function at a point [latex](x_{0},y_{0})[\/latex] we can approximate [latex]f(x,y)[\/latex]\u00a0for points near [latex](x_{0},y_{0})[\/latex] using the tangent plane formula<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>tangent plane<\/dt>\r\n \t<dd>given a function [latex]f(x,y)[\/latex]\u00a0that is differentiable at a point [latex](x_{0},y_{0})[\/latex] the equation of the tangent plane to the surface [latex]z=f(x,y)[\/latex]\u00a0is given by [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>total differential<\/dt>\r\n \t<dd>the total differential of the function [latex]f(x,y)[\/latex]\u00a0at [latex](x_{0},y_{0})[\/latex]\u00a0is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.<\/li>\n<li>Tangent planes can be used to approximate values of functions near known values.<\/li>\n<li>A function is differentiable at a point if it is \u201dsmooth\u201d at that point (i.e., no corners or discontinuities exist at that point).<\/li>\n<li>The total differential can be used to approximate the change in a function [latex]z=f(x_{0},y_{0})[\/latex]\u00a0at the point\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0for given values of [latex]\\Delta{x}[\/latex] and\u00a0[latex]\\Delta{y}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Tangent plane<br \/>\n<\/strong>[latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Linear approximation<\/strong><br \/>\n[latex]L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Total differential<\/strong><br \/>\n[latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})(y-y_{0})dy[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Differentiability (two variables)<\/strong><br \/>\n[latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y)\\to (x_{0},y_{0})}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Differentiability (three variables)<\/strong><br \/>\n[latex]f(x,y,z)=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})+E(x,y,z)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y,z)\\to (x_{0},y_{0},z_{0})}{\\lim}\\frac{E(x,y,z)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>differentiable<\/dt>\n<dd>a function [latex]f(x,y,z)[\/latex]\u00a0is differentiable at\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0if [latex]f(x,y)[\/latex]\u00a0can be expressed in the form [latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E(x,y)[\/latex]\u00a0satisfies [latex]\\underset{(x,y)\\to{(x_{0},y_{0})}}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>linear approximation<\/dt>\n<dd>given a function [latex]f(x,y)[\/latex]\u00a0and a tangent plane to the function at a point [latex](x_{0},y_{0})[\/latex] we can approximate [latex]f(x,y)[\/latex]\u00a0for points near [latex](x_{0},y_{0})[\/latex] using the tangent plane formula<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>tangent plane<\/dt>\n<dd>given a function [latex]f(x,y)[\/latex]\u00a0that is differentiable at a point [latex](x_{0},y_{0})[\/latex] the equation of the tangent plane to the surface [latex]z=f(x,y)[\/latex]\u00a0is given by [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>total differential<\/dt>\n<dd>the total differential of the function [latex]f(x,y)[\/latex]\u00a0at [latex](x_{0},y_{0})[\/latex]\u00a0is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-166\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-166","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":18,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/166\/revisions"}],"predecessor-version":[{"id":3766,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/166\/revisions\/3766"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/166\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=166"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=166"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=166"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}