{"id":5897,"date":"2025-08-13T16:38:56","date_gmt":"2025-08-13T11:08:56","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=5897"},"modified":"2025-08-13T16:38:56","modified_gmt":"2025-08-13T11:08:56","slug":"applications-of-derivatives","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ee\/calculus-and-differeential-equations\/applications-of-derivatives","title":{"rendered":"Applications of Derivatives"},"content":{"rendered":"<h2>There are two areas where derivatives are used<\/h2>\n<p><strong> 1.<\/strong> Increasing and Decreasing Functions<br \/>\n<strong>2.<\/strong> Maxima and Minima<\/p>\n<ul>\n<li>Relative maxima and minima<\/li>\n<li>Absolute maxima and minima<\/li>\n<\/ul>\n<p><strong>3.<\/strong> Taylor\u2019s and Maclaurin\u2019s Series Expansion of Functions<br \/>\n<strong>4.<\/strong> Slope determination of line<\/p>\n<h3>Taylor\u2019s and Maclaurin\u2019s Series Expansion of Functions<\/h3>\n<p><strong>Taylor\u2019s Series<\/strong><\/p>\n<p>If (i) f(x) and its first (n \u2013 1) derivatives be continuous in [a, a + h], and (ii) fn(x) exists for every value of x in (a, a + h), then there is at least one number \u03b8 (0 &lt; \u03b8 &lt; 1), such that<\/p>\n<p>f (a + h) = f(a) + hf\u2032(a) + h<sup>2<\/sup> \/ 2! f&#8221; (a)+&#8230;..+ h<sup>n<\/sup> \/ n! f<sup>n<\/sup> (a + \u03b8h)<\/p>\n<p>which is called Taylor\u2019s theorem with Lagrange\u2019s form of remainder, the remainder Rn being h<sup>n<\/sup> \/ n! f<sup>n<\/sup> (a + \u03b8h)<\/p>\n<p><strong>Maclaurin\u2019s Series<\/strong><\/p>\n<p>If f(x) can be expanded as an infinite series, then<\/p>\n<p>f (x) = f(0) + xf\u2032(0) + x<sup>2<\/sup> \/ 2! f&#8221;(0) + x<sup>3<\/sup> \/ 3! f&#8221;(0) + &#8230;.\u221e<\/p>\n<p>If f (x) possesses derivatives of all orders and the remainder Rn in (3) on page 154 tends to zero as n \u2192 \u221e, then the Maclaurin\u2019s theorem becomes the Maclaurin\u2019s series (1).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There are two areas where derivatives are used 1. Increasing and Decreasing Functions 2. Maxima and Minima Relative maxima and<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1733,5],"tags":[1745,1747,1746],"class_list":["post-5897","post","type-post","status-publish","format-standard","hentry","category-calculus-and-differeential-equations","category-ee","tag-maclaurins-series","tag-maxima-and-minima","tag-taylors-series"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5897","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=5897"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5897\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=5897"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=5897"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=5897"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}