{"id":1642,"date":"2021-03-19T21:36:56","date_gmt":"2021-03-19T21:36:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1642"},"modified":"2021-04-05T19:29:57","modified_gmt":"2021-04-05T19:29:57","slug":"summary-of-linear-approximations-and-differentials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-linear-approximations-and-differentials\/","title":{"raw":"Summary of Linear Approximations and Differentials","rendered":"Summary of Linear Approximations and Differentials"},"content":{"raw":"<div id=\"fs-id1165043309075\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165043309846\">\r\n \t<li>A differentiable function [latex]y=f(x)[\/latex] can be approximated at [latex]a[\/latex] by the linear function\r\n<div id=\"fs-id1165042638768\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/div><\/li>\r\n \t<li>For a function [latex]y=f(x)[\/latex], if [latex]x[\/latex] changes from [latex]a[\/latex] to [latex]a+dx[\/latex], then\r\n<div id=\"fs-id1165043309024\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/div>\r\nis an approximation for the change in [latex]y[\/latex]. The actual change in [latex]y[\/latex] is\r\n<div id=\"fs-id1165042514151\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\Delta y=f(a+dx)-f(a)[\/latex]<\/div><\/li>\r\n \t<li>A measurement error [latex]dx[\/latex] can lead to an error in a calculated quantity [latex]f(x)[\/latex]. The error in the calculated quantity is known as the <em>propagated error<\/em>. The propagated error can be estimated by\r\n<div id=\"fs-id1165042582705\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy\\approx f^{\\prime}(x) \\, dx[\/latex]<\/div><\/li>\r\n \t<li style=\"text-align: left;\">To estimate the relative error of a particular quantity [latex]q[\/latex], we estimate [latex]\\dfrac{\\Delta q}{q}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165042713534\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1165042390258\">\r\n \t<li><strong>Linear approximation<\/strong>\r\n[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/li>\r\n \t<li><strong>A differential<\/strong>\r\n[latex]dy=f^{\\prime}(x) \\, dx[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165043099979\" class=\"definition\">\r\n \t<dt>differential<\/dt>\r\n \t<dd id=\"fs-id1165043315303\">the differential [latex]dx[\/latex] is an independent variable that can be assigned any nonzero real number; the differential [latex]dy[\/latex] is defined to be [latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043199989\" class=\"definition\">\r\n \t<dt>differential form<\/dt>\r\n \t<dd id=\"fs-id1165043422532\">given a differentiable function [latex]y=f^{\\prime}(x)[\/latex], the equation [latex]dy=f^{\\prime}(x) \\, dx[\/latex] is the differential form of the derivative of [latex]y[\/latex] with respect to [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043380416\" class=\"definition\">\r\n \t<dt>linear approximation<\/dt>\r\n \t<dd id=\"fs-id1165043380421\">the linear function [latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex] is the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043257667\" class=\"definition\">\r\n \t<dt>percentage error<\/dt>\r\n \t<dd id=\"fs-id1165042478964\">the relative error expressed as a percentage<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042478968\" class=\"definition\">\r\n \t<dt>propagated error<\/dt>\r\n \t<dd id=\"fs-id1165042321686\">the error that results in a calculated quantity [latex]f(x)[\/latex] resulting from a measurement error [latex]dx[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042370920\" class=\"definition\">\r\n \t<dt>relative error<\/dt>\r\n \t<dd id=\"fs-id1165042370925\">given an absolute error [latex]\\Delta q[\/latex] for a particular quantity, [latex]\\dfrac{\\Delta q}{q}[\/latex] is the relative error.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042318986\" class=\"definition\">\r\n \t<dt>tangent line approximation (linearization)<\/dt>\r\n \t<dd id=\"fs-id1165043393042\">since the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is defined using the equation of the tangent line, the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is also known as the tangent line approximation to [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1165043309075\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165043309846\">\n<li>A differentiable function [latex]y=f(x)[\/latex] can be approximated at [latex]a[\/latex] by the linear function\n<div id=\"fs-id1165042638768\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/div>\n<\/li>\n<li>For a function [latex]y=f(x)[\/latex], if [latex]x[\/latex] changes from [latex]a[\/latex] to [latex]a+dx[\/latex], then\n<div id=\"fs-id1165043309024\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/div>\n<p>is an approximation for the change in [latex]y[\/latex]. The actual change in [latex]y[\/latex] is<\/p>\n<div id=\"fs-id1165042514151\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\Delta y=f(a+dx)-f(a)[\/latex]<\/div>\n<\/li>\n<li>A measurement error [latex]dx[\/latex] can lead to an error in a calculated quantity [latex]f(x)[\/latex]. The error in the calculated quantity is known as the <em>propagated error<\/em>. The propagated error can be estimated by\n<div id=\"fs-id1165042582705\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy\\approx f^{\\prime}(x) \\, dx[\/latex]<\/div>\n<\/li>\n<li style=\"text-align: left;\">To estimate the relative error of a particular quantity [latex]q[\/latex], we estimate [latex]\\dfrac{\\Delta q}{q}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165042713534\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1165042390258\">\n<li><strong>Linear approximation<\/strong><br \/>\n[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/li>\n<li><strong>A differential<\/strong><br \/>\n[latex]dy=f^{\\prime}(x) \\, dx[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165043099979\" class=\"definition\">\n<dt>differential<\/dt>\n<dd id=\"fs-id1165043315303\">the differential [latex]dx[\/latex] is an independent variable that can be assigned any nonzero real number; the differential [latex]dy[\/latex] is defined to be [latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043199989\" class=\"definition\">\n<dt>differential form<\/dt>\n<dd id=\"fs-id1165043422532\">given a differentiable function [latex]y=f^{\\prime}(x)[\/latex], the equation [latex]dy=f^{\\prime}(x) \\, dx[\/latex] is the differential form of the derivative of [latex]y[\/latex] with respect to [latex]x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043380416\" class=\"definition\">\n<dt>linear approximation<\/dt>\n<dd id=\"fs-id1165043380421\">the linear function [latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex] is the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043257667\" class=\"definition\">\n<dt>percentage error<\/dt>\n<dd id=\"fs-id1165042478964\">the relative error expressed as a percentage<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042478968\" class=\"definition\">\n<dt>propagated error<\/dt>\n<dd id=\"fs-id1165042321686\">the error that results in a calculated quantity [latex]f(x)[\/latex] resulting from a measurement error [latex]dx[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042370920\" class=\"definition\">\n<dt>relative error<\/dt>\n<dd id=\"fs-id1165042370925\">given an absolute error [latex]\\Delta q[\/latex] for a particular quantity, [latex]\\dfrac{\\Delta q}{q}[\/latex] is the relative error.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042318986\" class=\"definition\">\n<dt>tangent line approximation (linearization)<\/dt>\n<dd id=\"fs-id1165043393042\">since the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is defined using the equation of the tangent line, the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is also known as the tangent line approximation to [latex]f[\/latex] at [latex]x=a[\/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-1642\">\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 1. <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\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/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-1\/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":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/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-1642","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1642","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1642\/revisions"}],"predecessor-version":[{"id":2662,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1642\/revisions\/2662"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1642\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1642"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1642"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1642"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1642"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}