{"id":2826,"date":"2025-08-13T18:45:21","date_gmt":"2025-08-13T18:45:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2826"},"modified":"2026-01-31T02:09:41","modified_gmt":"2026-01-31T02:09:41","slug":"introduction-to-calculus-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-calculus-background-youll-need-3\/","title":{"raw":"Introduction to Calculus: Background You'll Need 3","rendered":"Introduction to Calculus: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine the average rate of change of a function over a given interval<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Average Rate of Change<\/h2>\r\nThe <strong>average rate of change<\/strong> of a function describes how the output of the function changes, on average, over a specific interval of input values. It is calculated using two points on the function.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Average rate of change<\/h3>\r\nThe <b>average rate of change<\/b> of a function from [latex]x=a[\/latex] to [latex]x=b[\/latex] is\r\n[latex]\\frac{f(b)-f(a)}{b-a}[\/latex].\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Supposed [latex]f(x)=x^2[\/latex]. Find the average rate of change on the interval [latex][1,3][\/latex].[reveal-answer q=\"281588\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"281588\"]Start by evaluating the function at the interval endpoints:[latex]f(1)=1^2 =1[\/latex] and[latex]f(3)=3^2=9[\/latex]\r\n\r\nNow use the formula:\r\n\r\n[latex]\\begin{align}\\dfrac{f(3)-f(1)}{3-1}&amp;=\\dfrac{9-1}{3-1}\\\\&amp;=\\dfrac{8}{2}\\\\&amp;=4\\end{align}[\/latex]\r\n\r\nThe average rate of change is [latex]4[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]319416[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine the average rate of change of a function over a given interval<\/li>\n<\/ul>\n<\/section>\n<h2 id=\"Introduction\" class=\"no-indent\" style=\"text-align: left;\">Average Rate of Change<\/h2>\n<p>The <strong>average rate of change<\/strong> of a function describes how the output of the function changes, on average, over a specific interval of input values. It is calculated using two points on the function.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Average rate of change<\/h3>\n<p>The <b>average rate of change<\/b> of a function from [latex]x=a[\/latex] to [latex]x=b[\/latex] is<br \/>\n[latex]\\frac{f(b)-f(a)}{b-a}[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Supposed [latex]f(x)=x^2[\/latex]. Find the average rate of change on the interval [latex][1,3][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q281588\">Show Solution<\/button><\/p>\n<div id=\"q281588\" class=\"hidden-answer\" style=\"display: none\">Start by evaluating the function at the interval endpoints:[latex]f(1)=1^2 =1[\/latex] and[latex]f(3)=3^2=9[\/latex]<\/p>\n<p>Now use the formula:<\/p>\n<p>[latex]\\begin{align}\\dfrac{f(3)-f(1)}{3-1}&=\\dfrac{9-1}{3-1}\\\\&=\\dfrac{8}{2}\\\\&=4\\end{align}[\/latex]<\/p>\n<p>The average rate of change is [latex]4[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm319416\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=319416&theme=lumen&iframe_resize_id=ohm319416&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2826"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2826\/revisions"}],"predecessor-version":[{"id":5480,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2826\/revisions\/5480"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2826\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2826"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2826"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2826"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}