{"id":134,"date":"2025-04-09T17:15:18","date_gmt":"2025-04-09T17:15:18","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-3-rates-of-change-and-behavior-of-graphs-college-algebra-2e-openstax\/"},"modified":"2025-08-19T17:58:54","modified_gmt":"2025-08-19T17:58:54","slug":"3-3-rates-of-change-and-behavior-of-graphs","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-3-rates-of-change-and-behavior-of-graphs\/","title":{"raw":"3.3 Rates of Change and Behavior of Graphs","rendered":"3.3 Rates of Change and Behavior of Graphs"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_f37919d7-b496-4e36-8196-431ae4733a64\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Find the average rate of change of a function.<\/li>\r\n \t<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\r\n \t<li>Use a graph to locate local maxima and local minima.<\/li>\r\n \t<li>Use a graph to locate the absolute maximum and absolute minimum.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<ul id=\"list-00001\"><\/ul>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135194500\" class=\"has-noteref\">Gasoline costs have experienced some wild fluctuations over the last several decades. Table 1 lists the average cost, in dollars, of a gallon of gasoline for the years 2017\u20132024 in the U.S. The cost of gasoline can be considered as a function of year.<\/p>\r\n\r\n<div id=\"Table_01_03_01\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_01_03_01\"><caption>Table 1<\/caption><colgroup> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/> <col data-width=\"40\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><strong>[latex] y [\/latex]\u00a0<\/strong>\r\n\r\n&nbsp;<\/td>\r\n<td data-align=\"center\">2017<\/td>\r\n<td data-align=\"center\">2018<\/td>\r\n<td data-align=\"center\">2019<\/td>\r\n<td data-align=\"center\">2020<\/td>\r\n<td data-align=\"center\">2021<\/td>\r\n<td data-align=\"center\">2022<\/td>\r\n<td data-align=\"center\">2023<\/td>\r\n<td data-align=\"center\">2024<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>[latex] C(y) [\/latex]\u00a0<\/strong>\r\n\r\n&nbsp;<\/td>\r\n<td data-align=\"center\">2.41<\/td>\r\n<td data-align=\"center\">2.72<\/td>\r\n<td data-align=\"center\">2.60<\/td>\r\n<td data-align=\"center\">2.17<\/td>\r\n<td data-align=\"center\">3.01<\/td>\r\n<td data-align=\"center\">3.95<\/td>\r\n<td data-align=\"center\">3.52<\/td>\r\n<td data-align=\"center\">3.30<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n<div>\r\n<div>\r\n\r\nIf we were interested only in how the gasoline prices changed between 2017 and 2024, we could compute that the cost per gallon had increased from $2.31 to $3.3, an increase of $0.99. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137645483\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding the Average Rate of Change of a Function<\/h2>\r\n<p id=\"fs-id1165137834011\">The price change per year is a <span id=\"term-00003\" data-type=\"term\">rate of change<\/span> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in <a class=\"autogenerated-content\" href=\"3-3-rates-of-change-and-behavior-of-graphs#Table_01_03_01\">Table 1<\/a> did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong><span id=\"term-00004\" data-type=\"term\">average rate of change<\/span><\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\text{Average rate of change} &amp;=&amp; \\frac{\\text{Change in output}}{\\text{Change in input}} \\\\&amp;=&amp; \\frac{\\Delta y}{\\Delta x} \\\\&amp;=&amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\&amp;=&amp; \\frac{f(x_2) - f(x_1)}{x_2 - x_1}\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165135471272\">The Greek letter\u00a0[latex] \\Delta [\/latex] (delta) signifies the change in a quantity; we read the ratio as \u201cdelta-<em data-effect=\"italics\">y<\/em> over delta-<em data-effect=\"italics\">x<\/em>\u201d or \u201cthe change in\u00a0[latex] y [\/latex] divided by the change in [latex] x. [\/latex]\u201d Occasionally we write\u00a0[latex] \\Delta f [\/latex] instead of\u00a0[latex] \\Delta y, [\/latex] which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\r\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\r\n<p style=\"text-align: center;\">[latex] \\frac{\\Delta y}{\\Delta x}=\\frac{\\$1.37}{7 \\ \\text{years}}\\approx0.196 \\ \\text{dollars per year} [\/latex]<\/p>\r\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\r\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\r\n\r\n<ul id=\"fs-id1165137424067\">\r\n \t<li>A population of rats increasing by 40 rats per week<\/li>\r\n \t<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\r\n \t<li>A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)<\/li>\r\n \t<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\r\n \t<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Rate of Change<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are \u201coutput units per input units.\u201d\r\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\frac{\\Delta y}{\\Delta x}=\\frac{f(x_2)-f(x_1)}{x_2-x_1} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the value <\/strong><strong>of a function at different points, calculate the average rate of change of a function for the interval between two values\u00a0[latex] x_1 [\/latex] and [latex] x_2. [\/latex]<\/strong>\r\n<ol>\r\n \t<li>Calculate the difference [latex] y_2-y_1=\\Delta y. [\/latex]<\/li>\r\n \t<li>Calculate the difference [latex] x_2-x_1=\\Delta x. [\/latex]<\/li>\r\n \t<li>Find the ratio [latex] \\frac{\\Delta y}{\\Delta x}. [\/latex]<\/li>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Computing an Average Rate of Change<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing the data in Table 1, find the average rate of change of the price of gasoline between 2022 and 2024.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>In 2022, the price of gasoline was $3.95. In 2024, the cost was $3.30. The average rate of change is\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl}\\frac{\\Delta y}{\\Delta x} &amp;=&amp; \\frac{y_2-y_1}{x_2-x_1} \\\\&amp;=&amp; \\frac{\\$3.30-\\$3.95}{2024-2022} \\\\&amp;=&amp; \\frac{-\\$0.65}{2 \\ \\text{years}} \\\\&amp;=&amp; -\\$35 \\ \\text{per year} \\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNote that a decrease is expressed by a negative change or \u201cnegative increase.\u201d A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing the data in Table 1, find the average rate of change between 2017 and 2022.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Computing Average Rate of Change from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function\u00a0[latex] g(t) [\/latex] shown in Figure 1, find the average rate of change on the interval [latex] [-1, 2]. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_512\" align=\"aligncenter\" width=\"311\"]<img class=\" wp-image-512\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1-286x300.jpeg\" alt=\"\" width=\"311\" height=\"326\" \/> Figure 1[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>At [latex] t=-1, [\/latex] Figure 2 shows\u00a0[latex] g(-1)=4. [\/latex] At\u00a0[latex] t=2, [\/latex] the graph shows [latex] g(2)-1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_513\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-513\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/> Figure 2[\/caption]\r\n\r\nThe horizontal change\u00a0[latex] \\Delta t=3 [\/latex] is shown by the red arrow, and the vertical change\u00a0[latex] \\Delta g(t)=-3 [\/latex] is shown by the turquoise arrow. The average rate of change is shown by the slope of the orange line segment. The output changes by \u20133 while the input changes by 3, giving an average rate of change of\r\n<p style=\"text-align: center;\">[latex]\\frac{1-4}{2-(-1)}=\\frac{-3}{1}=-1 [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nNote that the order we choose is very important. If, for example, we use\u00a0[latex] \\frac{y_2-y_1}{x_1-x_2} [\/latex] we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as\u00a0[latex] (x_1, y_1) [\/latex] and [latex] (x_2, y_2). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Computing Average Rate of Change from a Table<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAfter picking up a friend who lives 10 miles away and leaving on a trip, Anna records her distance from home over time. The values are shown in Table 2. Find her average speed over the first 6 hours.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 30px;\" border=\"0\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 15px;\">[latex] t [\/latex](hours)<\/td>\r\n<td style=\"width: 9.6246%; height: 15px;\">0<\/td>\r\n<td style=\"width: 11.861%; height: 15px;\">1<\/td>\r\n<td style=\"width: 8.87912%; height: 15px;\">2<\/td>\r\n<td style=\"width: 10.0506%; height: 15px;\">3<\/td>\r\n<td style=\"width: 12.9261%; height: 15px;\">4<\/td>\r\n<td style=\"width: 11.4349%; height: 15px;\">5<\/td>\r\n<td style=\"width: 11.2553%; height: 15px;\">6<\/td>\r\n<td style=\"width: 11.4683%; height: 15px;\">7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 12.5%; height: 15px;\">[latex] D(t) [\/latex](miles)<\/td>\r\n<td style=\"width: 9.6246%; height: 15px;\">10<\/td>\r\n<td style=\"width: 11.861%; height: 15px;\">55<\/td>\r\n<td style=\"width: 8.87912%; height: 15px;\">90<\/td>\r\n<td style=\"width: 10.0506%; height: 15px;\">153<\/td>\r\n<td style=\"width: 12.9261%; height: 15px;\">214<\/td>\r\n<td style=\"width: 11.4349%; height: 15px;\">240<\/td>\r\n<td style=\"width: 11.2553%; height: 15px;\">292<\/td>\r\n<td style=\"width: 11.4683%; height: 15px;\">300<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\frac{292-10}{6-0} &amp;=&amp; \\frac{282}{6} \\\\&amp;=&amp; 47 \\end{array}[\/latex]<\/p>\r\nThe average speed is 47 miles per hour.\r\n<h3>Analysis<\/h3>\r\nBecause the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_03\" class=\"ui-has-child-title\" data-type=\"example\"><header><\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165135536188\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137651998\" data-type=\"commentary\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Computing Average Rate of Change for a Function Expressed as a Formula<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCompute the average rate of change of\u00a0[latex] f(x)=x^2-\\frac{1}{x} [\/latex] on the interval [latex] [2, 4]. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can start by computing the function values at each endpoint of the interval.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl rcl}f(2) &amp;=&amp; 2^2 - \\frac{1}{2} &amp;&amp; f(4) &amp;=&amp; 4^2 - \\frac{1}{4} \\\\&amp;=&amp; 4 - \\frac{1}{2} &amp;&amp; &amp;=&amp; 16 - \\frac{1}{4} \\\\&amp;=&amp; \\frac{7}{2} &amp;&amp; &amp;=&amp; \\frac{63}{4}\\end{array} [\/latex]<\/p>\r\nNow we compute the average rate of change.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\text{Average rate of change} &amp;=&amp; \\frac{f(4)-f(2)}{4-2} \\\\ &amp;=&amp; \\frac{\\frac{63}{4}-\\frac{7}{2}}{4-2} \\\\ &amp;=&amp; \\frac{\\frac{49}{4}}{2} \\\\ &amp;=&amp; \\frac{49}{8}\\end{array} [\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the average rate of change of\u00a0[latex] f(x)=x-2\\sqrt{x} [\/latex] on the interval [latex] [1, 9]. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Finding an Average Rate of Change as an Expression<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the average rate of change of\u00a0[latex] g(t)=t^2+3t+1 [\/latex] on the interval\u00a0[latex] [0, a]. [\/latex] The answer will be an expression involving\u00a0[latex] a [\/latex] in simplest form.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We use the average rate of change formula.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll}\\text{Average rate of change} &amp;=&amp; \\frac{g(a)-g(0)}{a-0} &amp; \\text{Evaluate.} \\\\ &amp;=&amp; \\frac{(a^2+3a+1)-(0^2+3()+1)}{a-0} &amp; \\text{Simplify.} \\\\ &amp;=&amp; \\frac{a^2+3a+1-1}{a} &amp; \\text{Simplify and factor.} \\\\ &amp;=&amp; \\frac{a(a+3)}{a} &amp; \\text{Divide by the common factor} \\ a. \\\\ &amp;=&amp; a+3\\end{array} [\/latex]<\/p>\r\nThis result tells us the average rate of change in terms of\u00a0[latex] a [\/latex] between\u00a0[latex] t=0 [\/latex] and any other point\u00a0[latex] t=a\/ [\/latex] For example, on the interval\u00a0[latex] [0, 5], [\/latex] the average rate of change would be [latex] 5+3=8. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the average rate of change of\u00a0[latex] f(x)=x^2+2x-8 [\/latex] on the interval\u00a0[latex] [5, a] [\/latex] in simplest forms in terms of [latex] a. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135440486\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h2>\r\n<p id=\"fs-id1165137784644\">As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 3 shows examples of increasing and decreasing intervals on a function.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_515\" align=\"aligncenter\" width=\"322\"]<img class=\" wp-image-515\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3-300x259.jpeg\" alt=\"\" width=\"322\" height=\"278\" \/> Figure 3. The function [latex] f(x)=x^3-12x [\/latex] in increasing on [latex] (-\\infty, -2)\\cup(2, \\infty) [\/latex] and is decreasing on [latex] (-2, 2). [\/latex][\/caption]\r\n<p id=\"fs-id1165134272749\">While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is the location of a <strong><span id=\"term-00007\" data-type=\"term\">local maximum<\/span><\/strong>. The function value at that point is the local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is the location of a <strong><span id=\"term-00008\" data-type=\"term\">local minimum<\/span><\/strong>. The function value at that point is the local minimum. The plural form is \u201clocal minima.\u201d Together, local maxima and minima are called <strong><span id=\"term-00009\" data-type=\"term\">local extrema<\/span><\/strong>, or local extreme values, of the function. (The singular form is \u201cextremum.\u201d) Often, the term <em data-effect=\"italics\">local<\/em> is replaced by the term <em data-effect=\"italics\">relative<\/em>. In this text, we will use the term <em data-effect=\"italics\">local<\/em>.<\/p>\r\n<p id=\"fs-id1165134547216\">Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em data-effect=\"italics\">local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\r\n<p id=\"fs-id1165135333162\">For the function whose graph is shown in Figure 4, the local maximum is 16, and it occurs at\u00a0[latex] x=-2. [\/latex] The local minimum is\u00a0[latex] -16 [\/latex] and it occurs at [latex] x=2. [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/section><section data-depth=\"1\">\r\n\r\n[caption id=\"attachment_516\" align=\"aligncenter\" width=\"369\"]<img class=\" wp-image-516\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-300x152.jpeg\" alt=\"\" width=\"369\" height=\"187\" \/> Figure 4[\/caption]\r\n<p id=\"fs-id1165133316450\">To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5 illustrates these ideas for a local maximum.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_517\" align=\"aligncenter\" width=\"325\"]<img class=\" wp-image-517\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5-300x182.jpeg\" alt=\"\" width=\"325\" height=\"197\" \/> Figure 5. Definition of a local maximum[\/caption]\r\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Local Minima and Local Maxima<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] f [\/latex] is an <strong>increasing function<\/strong> on an open interval if\u00a0[latex] f(b)&gt;f(a) [\/latex] for any two input values\u00a0[latex] a [\/latex] and\u00a0[latex] b [\/latex] in the given interval where [latex] b&gt;a. [\/latex]\r\n\r\nA function\u00a0[latex] f [\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex] f(b)&lt; f(a) [\/latex] for any two input values [latex] a [\/latex] and [latex] b [\/latex] in the given interval where [latex] b&gt;a. [\/latex]\r\n\r\nA function [latex] f [\/latex] has a local maximum at [latex] x=b [\/latex] if there exists an interval [latex] (a, c) [\/latex] with [latex] a&lt; b [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-note-body\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Finding Increasing and Decreasing Intervals on a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function\u00a0[latex] p(t) [\/latex] in Figure 6, identify the intervals on which the function appears to be increasing.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_518\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-518\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6-300x182.jpeg\" alt=\"\" width=\"300\" height=\"182\" \/> Figure 6[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from\u00a0[latex] t=1 [\/latex] to\u00a0[latex] t=3 [\/latex] and from\u00a0[latex] t=4 [\/latex] on.\r\nIn <span id=\"term-00012\" class=\"no-emphasis\" data-type=\"term\">interval notation<\/span>, we would say the function appears to be increasing on the interval (1,3) and the interval [latex] (4, \\infty). [\/latex]\r\n<h3>Analysis<\/h3>\r\nNotice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex] t=1, t=3, [\/latex] and [latex] t=4. [\/latex] These points are the local extrema (two minima and a maximum).\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Finding Local Extreme from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph the function\u00a0[latex] f(x)=\\frac{2}{x}+\\frac{x}{3}. [\/latex] Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Using technology, we find that the graph of the function looks like that in Figure 7. It appears there is a low point, or local minimum, between\u00a0[latex] x=2 [\/latex] and\u00a0[latex] x=3, [\/latex] and a mirror-image high point, or local maximum, somewhere between\u00a0[latex] x=-3 [\/latex] and [latex] x=-2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_519\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-519\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7-300x227.jpeg\" alt=\"\" width=\"300\" height=\"227\" \/> Figure 7[\/caption]\r\n<h3>Analysis<\/h3>\r\nMost graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 8 provides screen images from two different technologies, showing the estimate for the local maximum and minimum.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_520\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-520\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8-300x119.jpeg\" alt=\"\" width=\"300\" height=\"119\" \/> Figure 8[\/caption]\r\n\r\nBased on these estimates, the function is increasing on the interval\u00a0[latex] (-\\infty, -2.449) [\/latex] and\u00a0[latex] (2.449, \\infty). [\/latex] Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at\u00a0[latex] \\pm\\sqrt{6}, [\/latex] but determining this requires calculus.)\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph the function\u00a0[latex] f(x)=x^3-6x^2-15x+20 [\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Finding Local Maxima and Minima from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the function\u00a0[latex] f [\/latex] whose graph is shown in Figure 9, find all local maxima and minima.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_521\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-521\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9-300x227.jpeg\" alt=\"\" width=\"300\" height=\"227\" \/> Figure 9[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Observe the graph of\u00a0[latex] f. [\/latex] The graph attains a local maximum at\u00a0[latex] x=1 [\/latex] because it is the highest point in an open interval around\u00a0[latex] x=1. [\/latex] The local maximum is the [latex] y [\/latex]-coordinate at\u00a0[latex] x=1, [\/latex] which is\u00a0[latex] 2. [\/latex]\r\nThe graph attains a local minimum at\u00a0[latex] x=-1 [\/latex] because it is the lowest point in an open interval around\u00a0[latex] x=-1. [\/latex] The local minimum is the [latex] y [\/latex]-coordinate at\u00a0[latex] x=-1, [\/latex] which is [latex] -2. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134544960\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h2>\r\n<p id=\"fs-id1165135704895\">We will now return to our toolkit functions and discuss their graphical behavior in Figure 10, Figure 11, and Figure 12.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_524\" align=\"aligncenter\" width=\"741\"]<img class=\" wp-image-524\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-300x162.jpeg\" alt=\"\" width=\"741\" height=\"400\" \/> Figure 10[\/caption]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_525\" align=\"aligncenter\" width=\"757\"]<img class=\" wp-image-525\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-300x162.jpeg\" alt=\"\" width=\"757\" height=\"409\" \/> Figure 11[\/caption]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_526\" align=\"aligncenter\" width=\"771\"]<img class=\" wp-image-526\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-300x162.jpeg\" alt=\"\" width=\"771\" height=\"416\" \/> Figure 12[\/caption]\r\n\r\n<\/section><section id=\"fs-id1165134381626\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h2>\r\n<p id=\"fs-id1165134381632\">There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex] y [\/latex]-coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.<\/p>\r\n<p id=\"fs-id1165131833490\">To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 13.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_527\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-527\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13-300x199.jpeg\" alt=\"\" width=\"300\" height=\"199\" \/> Figure 13[\/caption]\r\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function\u00a0[latex] f(x)=x^3 [\/latex] is one such function.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Absolute Maxima and Minima<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <strong>absolute maximum<\/strong> of\u00a0[latex] f [\/latex] at\u00a0[latex] x=c [\/latex] is\u00a0[latex] f(c) [\/latex] where\u00a0[latex] f(c)\\ge f(x) [\/latex] for all\u00a0[latex] x [\/latex] in the domain of [latex] f. [\/latex]\r\n\r\nThe <strong>absolute minimum<\/strong> of\u00a0[latex] f [\/latex] at\u00a0[latex] x=d [\/latex] is\u00a0[latex] f(d) [\/latex] where\u00a0[latex] f(d)\\le f(x) [\/latex] for all\u00a0[latex] x [\/latex] in the domain of [latex] f. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Finding Absolute Maxima and Minima from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor the function\u00a0[latex] f [\/latex] shown in Figure 14, find all absolute maxima and minima.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_528\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-528\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14-300x248.jpeg\" alt=\"\" width=\"300\" height=\"248\" \/> Figure 14[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Observe the graph of\u00a0[latex] f. [\/latex] The graph attains an absolute maximum in two locations,\u00a0[latex] x=-2 [\/latex] and\u00a0[latex] x=2 [\/latex] because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the [latex] y [\/latex]-coordinate at\u00a0[latex] x=-2 [\/latex] and\u00a0[latex] x=2, [\/latex] which is [latex] 16. [\/latex]\r\n\r\nThe graph attains an absolute minimum at\u00a0[latex] x=3, [\/latex] because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the [latex] y [\/latex]-coordinate at\u00a0[latex] x=3, [\/latex] which is [latex] -10. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess this online resource for additional instruction and practice with rates of change.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=F-7Poa3i1ZU\">Average Rate of Change<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">3.3 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135457748\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135457752\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165135457758\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135457760\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135457758-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can the average rate of change of a function be constant?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135536268\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135536270\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a function\u00a0[latex] f [\/latex] is increasing on\u00a0[latex] (a, b) [\/latex] and decreasing on\u00a0[latex] (b, c), [\/latex] then what can be said about the local extremum of\u00a0[latex] f [\/latex] on [latex] (a, c)? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137704665\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137911347\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137704665-solution\">3<\/a><span class=\"os-divider\">. <\/span>How are the absolute maximum and minimum similar to and different from the local extrema?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137911363\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137911365\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>How does the graph of the absolute value function compare to the graph of the quadratic function, [latex] y=x^2, [\/latex] in terms of increasing and decreasing intervals?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134259215\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165134259221\">For the following exercises, find the average rate of change of each function on the interval specified for real numbers\u00a0 or\u00a0 in simplest form.<\/p>\r\n\r\n<div id=\"fs-id1165134486779\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134486781\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134486779-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=4x^2-7 \\ \\text{on} \\ [1, b] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134158932\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134158934\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=2x^2-9 \\ \\text{on} \\ [4, b] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135403475\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135403477\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135403475-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex] p(x)=3x+4 \\ \\text{on} \\ [2, 2+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135237067\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135237069\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] k(x)=4x-2 \\ \\text{on} \\ [3, 3+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137579498\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137579500\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137579498-solution\">9<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex] f(x)=2x^2+1 \\ \\text{on} \\ [x, x+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134495260\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135409395\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=3x^2-2 \\ \\text{on} \\ [x, x+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134374799\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134374801\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134374799-solution\">11<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex] a(t)=\\frac{1}{t+4} \\ \\text{on} \\ [9, 9+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137693536\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137693538\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex] b(x)=\\frac{1}{x+3} \\ \\text{on} \\ [1, 1+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134104075\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134104077\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134104075-solution\">13<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex] j(x)=3x^3 \\ \\text{on} \\ [1, 1+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137851687\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137851690\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex] r(t)=4t^3 \\ \\text{on} \\ [2, 2+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137432064\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137432066\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137432064-solution\">15<\/a><span class=\"os-divider\">. <\/span>Find [latex] \\frac{f(x+h)-f(x)}{h} \\ \\ \\text{given} \\ \\ f(x)=2x^2-3x \\ \\ \\text{on} \\ \\ [x, x+h] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135409367\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135409372\">For the following exercises, consider the graph of\u00a0[latex] f [\/latex] shown in <a class=\"autogenerated-content\" href=\"3-3-rates-of-change-and-behavior-of-graphs#Figure_01_03_201\">Figure 15<\/a>.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_529\" align=\"aligncenter\" width=\"294\"]<img class=\"size-medium wp-image-529\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15-294x300.jpeg\" alt=\"\" width=\"294\" height=\"300\" \/> Figure 15[\/caption]\r\n\r\n<div id=\"fs-id1165135701406\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135701408\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span>Estimate the average rate of change from\u00a0[latex] x=1 [\/latex] to [latex] x=4. [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137769703\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137769705\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137769703-solution\">17<\/a><span class=\"os-divider\">. <\/span>Estimate the average rate of change from\u00a0[latex] x=2 [\/latex] to [latex] x=5. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165133213910\">For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.<\/p>\r\n\r\n<div id=\"fs-id1165133213915\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133213917\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165133213923\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\">\r\n<img class=\"alignnone size-full wp-image-530\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-17.jpeg\" alt=\"\" width=\"249\" height=\"251\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137897938\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137897940\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137897938-solution\">19<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137897946\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a cubic function.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-531\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-19-297x300.jpeg\" alt=\"\" width=\"297\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135600844\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135600846\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135422552\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a cubic function.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-532\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135422567\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135422569\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135422567-solution\">21<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134080979\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a reciprocal function.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-533\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-300x175.jpeg\" alt=\"\" width=\"300\" height=\"175\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134373457\">For the following exercises, consider the graph shown in Figure 16.<\/p>\r\n&nbsp;\r\n<div id=\"Figure_01_03_206\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n\r\n[caption id=\"attachment_534\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-534\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16-300x199.jpeg\" alt=\"\" width=\"300\" height=\"199\" \/> Figure 16[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134373481\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134373483\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span>Estimate the intervals where the function is increasing or decreasing.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134149087\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134149090\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134149087-solution\">23<\/a><span class=\"os-divider\">. <\/span>Estimate the point(s) at which the graph of\u00a0[latex] f [\/latex] has a local maximum or a local minimum.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135404176\">For the following exercises, consider the graph in Figure 17.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_535\" align=\"alignnone\" width=\"270\"]<img class=\"size-medium wp-image-535\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17-270x300.jpeg\" alt=\"\" width=\"270\" height=\"300\" \/> Figure 17[\/caption]\r\n\r\n<div id=\"fs-id1165135428496\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135428498\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span>If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135428507\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135428509\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135428507-solution\">25<\/a><span class=\"os-divider\">. <\/span>If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134043819\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<div id=\"fs-id1165134043824\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043826\" class=\"material-set-2\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>Table 3 gives the annual revenue (in billions of dollars) from 2015 to 2023. What was the average rate of change of annual revenue (a) between 2020 and 2021, and (b) between 2019 and 2022?\r\n<div class=\"os-problem-container\">\r\n<div>\r\n<div>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Year<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Revenue (billions of dollars)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2015<\/td>\r\n<td style=\"width: 50%;\">486<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2016<\/td>\r\n<td style=\"width: 50%;\">482<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2017<\/td>\r\n<td style=\"width: 50%;\">486<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2018<\/td>\r\n<td style=\"width: 50%;\">500<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2019<\/td>\r\n<td style=\"width: 50%;\">514<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2020<\/td>\r\n<td style=\"width: 50%;\">524<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2021<\/td>\r\n<td style=\"width: 50%;\">560<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2022<\/td>\r\n<td style=\"width: 50%;\">573<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">2023<\/td>\r\n<td style=\"width: 50%;\">611<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135547288\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135547289\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135547288-solution\">27<\/a><span class=\"os-divider\">. <\/span>Table 4 gives the population of Aurora, Colorado (in thousands) from 2019 to 2024. What was the average rate of change of population (a) between 2021 and 2023, and (b) between 2019 and 2024?\r\n<div class=\"os-problem-container\">\r\n<div>\r\n<div>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 105px;\" border=\"0\"><caption>Table 4<\/caption>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Year<\/strong><\/td>\r\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Population (thousands)<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">2019<\/td>\r\n<td style=\"width: 50%; height: 15px;\">374<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">2020<\/td>\r\n<td style=\"width: 50%; height: 15px;\">381<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">2021<\/td>\r\n<td style=\"width: 50%; height: 15px;\">386<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">2022<\/td>\r\n<td style=\"width: 50%; height: 15px;\">398<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">2023<\/td>\r\n<td style=\"width: 50%; height: 15px;\">400<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">2024<\/td>\r\n<td style=\"width: 50%; height: 15px;\">404<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134547382\">For the following exercises, find the average rate of change of each function on the interval specified.<\/p>\r\n\r\n<div id=\"fs-id1165134547387\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134547389\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex] f(x)=x^2\\ \\text{on} \\ [1, 5] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137851207\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137851209\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137851207-solution\">29<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex] h(x)=5-2x^2 \\ \\text{on} \\ [-2, 4] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134384576\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134384578\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\u00a0[latex] q(x)=x^3\\ \\text{on} \\ [-4, 2] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135453186\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135453188\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135453186-solution\">31<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex] g(x)=3x^3-1 \\ \\text{on} \\ [-3, 3] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199306\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199308\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex] y=\\frac{1}{x} \\ \\text{on} \\ [1, 3] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135317474\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135317476\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135317474-solution\">33<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex] p(t)=\\frac{(t^2-4)(t+1)}{t^2+3}\\ \\text{on} \\ [-3, 1] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135208859\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135208861\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex] k(t)=6t^2+\\frac{4}{t^3} \\ \\text{on} \\ [-1, 3] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135347321\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135347326\">For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.<\/p>\r\n\r\n<div id=\"fs-id1165134043700\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134043703\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134043700-solution\">35<\/a><span class=\"os-divider\">. [latex] f(x)=x^4-4x^3+5 [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137836516\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137836518\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. [latex] h(x)=x^5+5x^4+10x^3+10x^2-1 [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135257278\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135257281\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135257278-solution\">37<\/a><span class=\"os-divider\">. [latex] g(t)=t\\sqrt{t+3} [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137892185\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137892188\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. [latex] k(t)=3t^\\frac{2}{3}=t [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135602253\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135602256\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135602253-solution\">39<\/a><span class=\"os-divider\">. [latex] m(x)=x^4+2x^3-12x^2-10x+4 [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137400141\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137400143\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. [latex] n(x)=x^4-8x^3+18x^2-6x+2 [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135536251\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extension<\/h3>\r\n<div id=\"fs-id1165135536257\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135536258\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135536257-solution\">41<\/a><span class=\"os-divider\">. <\/span>The graph of the function\u00a0[latex] f [\/latex] is shown in Figure 18.\r\n<div class=\"os-problem-container\">\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_536\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-536\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18-300x241.jpeg\" alt=\"\" width=\"300\" height=\"241\" \/> Figure 18[\/caption]\r\n<p id=\"fs-id1165134298989\">Based on the calculator screen shot, the point\u00a0[latex] (1.333, 5.185) [\/latex] is which of the following?<\/p>\r\n(a) a relative (local) maximum of the function\r\n\r\n(b) the vertex of the function\r\n\r\n(c) the absolute maximum of the function\r\n\r\n(d) a zero of the function\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134378686\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134378688\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>Let\u00a0[latex] f(x)=\\frac{1}{x}. [\/latex] Find a number\u00a0[latex] c [\/latex] such that the average rate of change of the function\u00a0[latex] f [\/latex] on the interval\u00a0[latex] (1, c) [\/latex] is [latex] -\\frac{1}{4}. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134299101\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134299103\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134299101-solution\">43<\/a><span class=\"os-divider\">. <\/span>Let\u00a0[latex] f(x)=\\frac{1}{x}. [\/latex] Find the number\u00a0[latex] b [\/latex] such that the average rate of change of\u00a0[latex] f [\/latex] on the interval\u00a0[latex] (2, b) [\/latex] is [latex] -\\frac{1}{10}. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134151836\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165134151841\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134151843\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><span class=\"TextRun SCXW171915131 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW171915131 BCX2\">At the start of a road trip in Colorado.<\/span><\/span>, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135387234\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135387236\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135387234-solution\">45<\/a><span class=\"os-divider\">. <\/span>A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135387253\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135387255\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>Near the surface of the moon, the distance that an object falls is a function of time. It is given by\u00a0[latex] d(t)=2.6667t^2, [\/latex] where\u00a0[latex] t [\/latex] is in seconds and\u00a0[latex] d(t) [\/latex] is in feet. If an object is dropped from a certain height, find the average velocity of the object from\u00a0[latex] t=1 [\/latex] to [latex] t=2. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135571880\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135571881\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135571880-solution\">47<\/a><span class=\"os-divider\">. <\/span>The graph in Figure 19 illustrates the decay of a radioactive substance over\u00a0[latex] t [\/latex] days.\r\n<div class=\"os-problem-container\">\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_537\" align=\"alignnone\" width=\"300\"]<img class=\"size-medium wp-image-537\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19-300x249.jpeg\" alt=\"\" width=\"300\" height=\"249\" \/> Figure 19[\/caption]\r\n<p id=\"fs-id1165135517172\">Use the graph to estimate the average decay rate from\u00a0[latex] t=5 [\/latex] to<\/p>\r\n[latex] t=15. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_f37919d7-b496-4e36-8196-431ae4733a64\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Find the average rate of change of a function.<\/li>\n<li>Use a graph to determine where a function is increasing, decreasing, or constant.<\/li>\n<li>Use a graph to locate local maxima and local minima.<\/li>\n<li>Use a graph to locate the absolute maximum and absolute minimum.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section>\n<ul id=\"list-00001\"><\/ul>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135194500\" class=\"has-noteref\">Gasoline costs have experienced some wild fluctuations over the last several decades. Table 1 lists the average cost, in dollars, of a gallon of gasoline for the years 2017\u20132024 in the U.S. The cost of gasoline can be considered as a function of year.<\/p>\n<div id=\"Table_01_03_01\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_01_03_01\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/>\n<col data-width=\"40\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\"><strong>[latex]y[\/latex]\u00a0<\/strong><\/p>\n<p>&nbsp;<\/td>\n<td data-align=\"center\">2017<\/td>\n<td data-align=\"center\">2018<\/td>\n<td data-align=\"center\">2019<\/td>\n<td data-align=\"center\">2020<\/td>\n<td data-align=\"center\">2021<\/td>\n<td data-align=\"center\">2022<\/td>\n<td data-align=\"center\">2023<\/td>\n<td data-align=\"center\">2024<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>[latex]C(y)[\/latex]\u00a0<\/strong><\/p>\n<p>&nbsp;<\/td>\n<td data-align=\"center\">2.41<\/td>\n<td data-align=\"center\">2.72<\/td>\n<td data-align=\"center\">2.60<\/td>\n<td data-align=\"center\">2.17<\/td>\n<td data-align=\"center\">3.01<\/td>\n<td data-align=\"center\">3.95<\/td>\n<td data-align=\"center\">3.52<\/td>\n<td data-align=\"center\">3.30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<div>\n<div>\n<p>If we were interested only in how the gasoline prices changed between 2017 and 2024, we could compute that the cost per gallon had increased from $2.31 to $3.3, an increase of $0.99. While this is interesting, it might be more useful to look at how much the price changed <em>per year<\/em>. In this section, we will investigate changes such as these.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137645483\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding the Average Rate of Change of a Function<\/h2>\n<p id=\"fs-id1165137834011\">The price change per year is a <span id=\"term-00003\" data-type=\"term\">rate of change<\/span> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in <a class=\"autogenerated-content\" href=\"3-3-rates-of-change-and-behavior-of-graphs#Table_01_03_01\">Table 1<\/a> did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong><span id=\"term-00004\" data-type=\"term\">average rate of change<\/span><\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\text{Average rate of change} &=& \\frac{\\text{Change in output}}{\\text{Change in input}} \\\\&=& \\frac{\\Delta y}{\\Delta x} \\\\&=& \\frac{y_2 - y_1}{x_2 - x_1} \\\\&=& \\frac{f(x_2) - f(x_1)}{x_2 - x_1}\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165135471272\">The Greek letter\u00a0[latex]\\Delta[\/latex] (delta) signifies the change in a quantity; we read the ratio as \u201cdelta-<em data-effect=\"italics\">y<\/em> over delta-<em data-effect=\"italics\">x<\/em>\u201d or \u201cthe change in\u00a0[latex]y[\/latex] divided by the change in [latex]x.[\/latex]\u201d Occasionally we write\u00a0[latex]\\Delta f[\/latex] instead of\u00a0[latex]\\Delta y,[\/latex] which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{\\$1.37}{7 \\ \\text{years}}\\approx0.196 \\ \\text{dollars per year}[\/latex]<\/p>\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\n<ul id=\"fs-id1165137424067\">\n<li>A population of rats increasing by 40 rats per week<\/li>\n<li>A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)<\/li>\n<li>A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)<\/li>\n<li>The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage<\/li>\n<li>The amount of money in a college account decreasing by $4,000 per quarter<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Rate of Change<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are \u201coutput units per input units.\u201d<\/p>\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f(x_2)-f(x_1)}{x_2-x_1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the value <\/strong><strong>of a function at different points, calculate the average rate of change of a function for the interval between two values\u00a0[latex]x_1[\/latex] and [latex]x_2.[\/latex]<\/strong><\/p>\n<ol>\n<li>Calculate the difference [latex]y_2-y_1=\\Delta y.[\/latex]<\/li>\n<li>Calculate the difference [latex]x_2-x_1=\\Delta x.[\/latex]<\/li>\n<li>Find the ratio [latex]\\frac{\\Delta y}{\\Delta x}.[\/latex]<\/li>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Computing an Average Rate of Change<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using the data in Table 1, find the average rate of change of the price of gasoline between 2022 and 2024.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>In 2022, the price of gasoline was $3.95. In 2024, the cost was $3.30. The average rate of change is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}\\frac{\\Delta y}{\\Delta x} &=& \\frac{y_2-y_1}{x_2-x_1} \\\\&=& \\frac{\\$3.30-\\$3.95}{2024-2022} \\\\&=& \\frac{-\\$0.65}{2 \\ \\text{years}} \\\\&=& -\\$35 \\ \\text{per year} \\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Note that a decrease is expressed by a negative change or \u201cnegative increase.\u201d A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using the data in Table 1, find the average rate of change between 2017 and 2022.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Computing Average Rate of Change from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function\u00a0[latex]g(t)[\/latex] shown in Figure 1, find the average rate of change on the interval [latex][-1, 2].[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_512\" aria-describedby=\"caption-attachment-512\" style=\"width: 311px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-512\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1-286x300.jpeg\" alt=\"\" width=\"311\" height=\"326\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1-286x300.jpeg 286w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1-65x68.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1-225x236.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1-350x367.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-1.jpeg 358w\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" \/><figcaption id=\"caption-attachment-512\" class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>At [latex]t=-1,[\/latex] Figure 2 shows\u00a0[latex]g(-1)=4.[\/latex] At\u00a0[latex]t=2,[\/latex] the graph shows [latex]g(2)-1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_513\" aria-describedby=\"caption-attachment-513\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-513\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-225x225.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2-350x350.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-2.jpeg 374w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-513\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<p>The horizontal change\u00a0[latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change\u00a0[latex]\\Delta g(t)=-3[\/latex] is shown by the turquoise arrow. The average rate of change is shown by the slope of the orange line segment. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1-4}{2-(-1)}=\\frac{-3}{1}=-1[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Note that the order we choose is very important. If, for example, we use\u00a0[latex]\\frac{y_2-y_1}{x_1-x_2}[\/latex] we will not get the correct answer. Decide which point will be 1 and which point will be 2, and keep the coordinates fixed as\u00a0[latex](x_1, y_1)[\/latex] and [latex](x_2, y_2).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Computing Average Rate of Change from a Table<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>After picking up a friend who lives 10 miles away and leaving on a trip, Anna records her distance from home over time. The values are shown in Table 2. Find her average speed over the first 6 hours.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 30px;\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 15px;\">[latex]t[\/latex](hours)<\/td>\n<td style=\"width: 9.6246%; height: 15px;\">0<\/td>\n<td style=\"width: 11.861%; height: 15px;\">1<\/td>\n<td style=\"width: 8.87912%; height: 15px;\">2<\/td>\n<td style=\"width: 10.0506%; height: 15px;\">3<\/td>\n<td style=\"width: 12.9261%; height: 15px;\">4<\/td>\n<td style=\"width: 11.4349%; height: 15px;\">5<\/td>\n<td style=\"width: 11.2553%; height: 15px;\">6<\/td>\n<td style=\"width: 11.4683%; height: 15px;\">7<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 12.5%; height: 15px;\">[latex]D(t)[\/latex](miles)<\/td>\n<td style=\"width: 9.6246%; height: 15px;\">10<\/td>\n<td style=\"width: 11.861%; height: 15px;\">55<\/td>\n<td style=\"width: 8.87912%; height: 15px;\">90<\/td>\n<td style=\"width: 10.0506%; height: 15px;\">153<\/td>\n<td style=\"width: 12.9261%; height: 15px;\">214<\/td>\n<td style=\"width: 11.4349%; height: 15px;\">240<\/td>\n<td style=\"width: 11.2553%; height: 15px;\">292<\/td>\n<td style=\"width: 11.4683%; height: 15px;\">300<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\frac{292-10}{6-0} &=& \\frac{282}{6} \\\\&=& 47 \\end{array}[\/latex]<\/p>\n<p>The average speed is 47 miles per hour.<\/p>\n<h3>Analysis<\/h3>\n<p>Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_03\" class=\"ui-has-child-title\" data-type=\"example\">\n<header><\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1165135536188\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137651998\" data-type=\"commentary\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Computing Average Rate of Change for a Function Expressed as a Formula<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Compute the average rate of change of\u00a0[latex]f(x)=x^2-\\frac{1}{x}[\/latex] on the interval [latex][2, 4].[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can start by computing the function values at each endpoint of the interval.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl rcl}f(2) &=& 2^2 - \\frac{1}{2} && f(4) &=& 4^2 - \\frac{1}{4} \\\\&=& 4 - \\frac{1}{2} && &=& 16 - \\frac{1}{4} \\\\&=& \\frac{7}{2} && &=& \\frac{63}{4}\\end{array}[\/latex]<\/p>\n<p>Now we compute the average rate of change.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{rcl}\\text{Average rate of change} &=& \\frac{f(4)-f(2)}{4-2} \\\\ &=& \\frac{\\frac{63}{4}-\\frac{7}{2}}{4-2} \\\\ &=& \\frac{\\frac{49}{4}}{2} \\\\ &=& \\frac{49}{8}\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the average rate of change of\u00a0[latex]f(x)=x-2\\sqrt{x}[\/latex] on the interval [latex][1, 9].[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Finding an Average Rate of Change as an Expression<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the average rate of change of\u00a0[latex]g(t)=t^2+3t+1[\/latex] on the interval\u00a0[latex][0, a].[\/latex] The answer will be an expression involving\u00a0[latex]a[\/latex] in simplest form.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We use the average rate of change formula.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll}\\text{Average rate of change} &=& \\frac{g(a)-g(0)}{a-0} & \\text{Evaluate.} \\\\ &=& \\frac{(a^2+3a+1)-(0^2+3()+1)}{a-0} & \\text{Simplify.} \\\\ &=& \\frac{a^2+3a+1-1}{a} & \\text{Simplify and factor.} \\\\ &=& \\frac{a(a+3)}{a} & \\text{Divide by the common factor} \\ a. \\\\ &=& a+3\\end{array}[\/latex]<\/p>\n<p>This result tells us the average rate of change in terms of\u00a0[latex]a[\/latex] between\u00a0[latex]t=0[\/latex] and any other point\u00a0[latex]t=a\/[\/latex] For example, on the interval\u00a0[latex][0, 5],[\/latex] the average rate of change would be [latex]5+3=8.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the average rate of change of\u00a0[latex]f(x)=x^2+2x-8[\/latex] on the interval\u00a0[latex][5, a][\/latex] in simplest forms in terms of [latex]a.[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135440486\" data-depth=\"1\">\n<h2 data-type=\"title\">Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h2>\n<p id=\"fs-id1165137784644\">As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 3 shows examples of increasing and decreasing intervals on a function.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_515\" aria-describedby=\"caption-attachment-515\" style=\"width: 322px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-515\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3-300x259.jpeg\" alt=\"\" width=\"322\" height=\"278\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3-300x259.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3-65x56.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3-225x194.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3-350x302.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-3.jpeg 512w\" sizes=\"auto, (max-width: 322px) 100vw, 322px\" \/><figcaption id=\"caption-attachment-515\" class=\"wp-caption-text\">Figure 3. The function [latex] f(x)=x^3-12x [\/latex] in increasing on [latex] (-\\infty, -2)\\cup(2, \\infty) [\/latex] and is decreasing on [latex] (-2, 2). [\/latex]<\/figcaption><\/figure>\n<p id=\"fs-id1165134272749\">While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is the location of a <strong><span id=\"term-00007\" data-type=\"term\">local maximum<\/span><\/strong>. The function value at that point is the local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is the location of a <strong><span id=\"term-00008\" data-type=\"term\">local minimum<\/span><\/strong>. The function value at that point is the local minimum. The plural form is \u201clocal minima.\u201d Together, local maxima and minima are called <strong><span id=\"term-00009\" data-type=\"term\">local extrema<\/span><\/strong>, or local extreme values, of the function. (The singular form is \u201cextremum.\u201d) Often, the term <em data-effect=\"italics\">local<\/em> is replaced by the term <em data-effect=\"italics\">relative<\/em>. In this text, we will use the term <em data-effect=\"italics\">local<\/em>.<\/p>\n<p id=\"fs-id1165134547216\">Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em data-effect=\"italics\">local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\n<p id=\"fs-id1165135333162\">For the function whose graph is shown in Figure 4, the local maximum is 16, and it occurs at\u00a0[latex]x=-2.[\/latex] The local minimum is\u00a0[latex]-16[\/latex] and it occurs at [latex]x=2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<section data-depth=\"1\">\n<figure id=\"attachment_516\" aria-describedby=\"caption-attachment-516\" style=\"width: 369px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-516\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-300x152.jpeg\" alt=\"\" width=\"369\" height=\"187\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-300x152.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-768x389.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-65x33.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-225x114.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4-350x177.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-4.jpeg 800w\" sizes=\"auto, (max-width: 369px) 100vw, 369px\" \/><figcaption id=\"caption-attachment-516\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p id=\"fs-id1165133316450\">To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5 illustrates these ideas for a local maximum.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_517\" aria-describedby=\"caption-attachment-517\" style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-517\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5-300x182.jpeg\" alt=\"\" width=\"325\" height=\"197\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5-300x182.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5-225x136.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5-350x212.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-5.jpeg 487w\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" \/><figcaption id=\"caption-attachment-517\" class=\"wp-caption-text\">Figure 5. Definition of a local maximum<\/figcaption><\/figure>\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Local Minima and Local Maxima<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if\u00a0[latex]f(b)>f(a)[\/latex] for any two input values\u00a0[latex]a[\/latex] and\u00a0[latex]b[\/latex] in the given interval where [latex]b>a.[\/latex]<\/p>\n<p>A function\u00a0[latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f(b)< f(a)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a.[\/latex]<\/p>\n<p>A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex](a, c)[\/latex] with [latex]a< b[\/latex]\n\n<\/div>\n<\/div>\n<div class=\"os-note-body\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Finding Increasing and Decreasing Intervals on a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function\u00a0[latex]p(t)[\/latex] in Figure 6, identify the intervals on which the function appears to be increasing.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_518\" aria-describedby=\"caption-attachment-518\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-518\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6-300x182.jpeg\" alt=\"\" width=\"300\" height=\"182\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6-300x182.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6-65x39.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6-225x136.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6-350x212.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-6.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-518\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from\u00a0[latex]t=1[\/latex] to\u00a0[latex]t=3[\/latex] and from\u00a0[latex]t=4[\/latex] on.<br \/>\nIn <span id=\"term-00012\" class=\"no-emphasis\" data-type=\"term\">interval notation<\/span>, we would say the function appears to be increasing on the interval (1,3) and the interval [latex](4, \\infty).[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1, t=3,[\/latex] and [latex]t=4.[\/latex] These points are the local extrema (two minima and a maximum).<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Finding Local Extreme from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph the function\u00a0[latex]f(x)=\\frac{2}{x}+\\frac{x}{3}.[\/latex] Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Using technology, we find that the graph of the function looks like that in Figure 7. It appears there is a low point, or local minimum, between\u00a0[latex]x=2[\/latex] and\u00a0[latex]x=3,[\/latex] and a mirror-image high point, or local maximum, somewhere between\u00a0[latex]x=-3[\/latex] and [latex]x=-2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_519\" aria-describedby=\"caption-attachment-519\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-519\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7-300x227.jpeg\" alt=\"\" width=\"300\" height=\"227\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7-300x227.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7-65x49.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7-225x170.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7-350x264.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-7.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-519\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>Most graphing calculators and graphing utilities can estimate the location of maxima and minima. Figure 8 provides screen images from two different technologies, showing the estimate for the local maximum and minimum.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_520\" aria-describedby=\"caption-attachment-520\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-520\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8-300x119.jpeg\" alt=\"\" width=\"300\" height=\"119\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8-300x119.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8-65x26.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8-225x89.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8-350x138.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-8.jpeg 731w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-520\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<p>Based on these estimates, the function is increasing on the interval\u00a0[latex](-\\infty, -2.449)[\/latex] and\u00a0[latex](2.449, \\infty).[\/latex] Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact location of the extrema is at\u00a0[latex]\\pm\\sqrt{6},[\/latex] but determining this requires calculus.)<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph the function\u00a0[latex]f(x)=x^3-6x^2-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Finding Local Maxima and Minima from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the function\u00a0[latex]f[\/latex] whose graph is shown in Figure 9, find all local maxima and minima.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_521\" aria-describedby=\"caption-attachment-521\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-521\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9-300x227.jpeg\" alt=\"\" width=\"300\" height=\"227\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9-300x227.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9-65x49.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9-225x170.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9-350x264.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-9.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-521\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Observe the graph of\u00a0[latex]f.[\/latex] The graph attains a local maximum at\u00a0[latex]x=1[\/latex] because it is the highest point in an open interval around\u00a0[latex]x=1.[\/latex] The local maximum is the [latex]y[\/latex]-coordinate at\u00a0[latex]x=1,[\/latex] which is\u00a0[latex]2.[\/latex]<br \/>\nThe graph attains a local minimum at\u00a0[latex]x=-1[\/latex] because it is the lowest point in an open interval around\u00a0[latex]x=-1.[\/latex] The local minimum is the [latex]y[\/latex]-coordinate at\u00a0[latex]x=-1,[\/latex] which is [latex]-2.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134544960\" data-depth=\"1\">\n<h2 data-type=\"title\">Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h2>\n<p id=\"fs-id1165135704895\">We will now return to our toolkit functions and discuss their graphical behavior in Figure 10, Figure 11, and Figure 12.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_524\" aria-describedby=\"caption-attachment-524\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-524\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-300x162.jpeg\" alt=\"\" width=\"741\" height=\"400\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-300x162.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-768x414.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-65x35.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-225x121.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10-350x188.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-10.jpeg 975w\" sizes=\"auto, (max-width: 741px) 100vw, 741px\" \/><figcaption id=\"caption-attachment-524\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_525\" aria-describedby=\"caption-attachment-525\" style=\"width: 757px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-525\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-300x162.jpeg\" alt=\"\" width=\"757\" height=\"409\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-300x162.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-768x414.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-65x35.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-225x121.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11-350x188.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-11.jpeg 975w\" sizes=\"auto, (max-width: 757px) 100vw, 757px\" \/><figcaption id=\"caption-attachment-525\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_526\" aria-describedby=\"caption-attachment-526\" style=\"width: 771px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-526\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-300x162.jpeg\" alt=\"\" width=\"771\" height=\"416\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-300x162.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-768x414.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-65x35.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-225x121.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12-350x188.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-12.jpeg 975w\" sizes=\"auto, (max-width: 771px) 100vw, 771px\" \/><figcaption id=\"caption-attachment-526\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<\/section>\n<section id=\"fs-id1165134381626\" data-depth=\"1\">\n<h2 data-type=\"title\">Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h2>\n<p id=\"fs-id1165134381632\">There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y[\/latex]-coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.<\/p>\n<p id=\"fs-id1165131833490\">To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 13.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_527\" aria-describedby=\"caption-attachment-527\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-527\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13-300x199.jpeg\" alt=\"\" width=\"300\" height=\"199\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13-300x199.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13-65x43.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13-225x149.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13-350x232.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-13.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-527\" class=\"wp-caption-text\">Figure 13<\/figcaption><\/figure>\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function\u00a0[latex]f(x)=x^3[\/latex] is one such function.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Absolute Maxima and Minima<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <strong>absolute maximum<\/strong> of\u00a0[latex]f[\/latex] at\u00a0[latex]x=c[\/latex] is\u00a0[latex]f(c)[\/latex] where\u00a0[latex]f(c)\\ge f(x)[\/latex] for all\u00a0[latex]x[\/latex] in the domain of [latex]f.[\/latex]<\/p>\n<p>The <strong>absolute minimum<\/strong> of\u00a0[latex]f[\/latex] at\u00a0[latex]x=d[\/latex] is\u00a0[latex]f(d)[\/latex] where\u00a0[latex]f(d)\\le f(x)[\/latex] for all\u00a0[latex]x[\/latex] in the domain of [latex]f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Finding Absolute Maxima and Minima from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For the function\u00a0[latex]f[\/latex] shown in Figure 14, find all absolute maxima and minima.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_528\" aria-describedby=\"caption-attachment-528\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-528\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14-300x248.jpeg\" alt=\"\" width=\"300\" height=\"248\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14-300x248.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14-65x54.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14-225x186.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14-350x290.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-14.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-528\" class=\"wp-caption-text\">Figure 14<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Observe the graph of\u00a0[latex]f.[\/latex] The graph attains an absolute maximum in two locations,\u00a0[latex]x=-2[\/latex] and\u00a0[latex]x=2[\/latex] because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the [latex]y[\/latex]-coordinate at\u00a0[latex]x=-2[\/latex] and\u00a0[latex]x=2,[\/latex] which is [latex]16.[\/latex]<\/p>\n<p>The graph attains an absolute minimum at\u00a0[latex]x=3,[\/latex] because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the [latex]y[\/latex]-coordinate at\u00a0[latex]x=3,[\/latex] which is [latex]-10.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access this online resource for additional instruction and practice with rates of change.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=F-7Poa3i1ZU\">Average Rate of Change<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">3.3 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135457748\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135457752\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165135457758\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135457760\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135457758-solution\">1<\/a><span class=\"os-divider\">. <\/span>Can the average rate of change of a function be constant?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135536268\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135536270\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>If a function\u00a0[latex]f[\/latex] is increasing on\u00a0[latex](a, b)[\/latex] and decreasing on\u00a0[latex](b, c),[\/latex] then what can be said about the local extremum of\u00a0[latex]f[\/latex] on [latex](a, c)?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137704665\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137911347\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137704665-solution\">3<\/a><span class=\"os-divider\">. <\/span>How are the absolute maximum and minimum similar to and different from the local extrema?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137911363\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137911365\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>How does the graph of the absolute value function compare to the graph of the quadratic function, [latex]y=x^2,[\/latex] in terms of increasing and decreasing intervals?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134259215\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165134259221\">For the following exercises, find the average rate of change of each function on the interval specified for real numbers\u00a0 or\u00a0 in simplest form.<\/p>\n<div id=\"fs-id1165134486779\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134486781\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134486779-solution\">5<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=4x^2-7 \\ \\text{on} \\ [1, b][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134158932\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134158934\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=2x^2-9 \\ \\text{on} \\ [4, b][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135403475\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135403477\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135403475-solution\">7<\/a><span class=\"os-divider\">.<\/span> [latex]p(x)=3x+4 \\ \\text{on} \\ [2, 2+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135237067\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135237069\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]k(x)=4x-2 \\ \\text{on} \\ [3, 3+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137579498\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137579500\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137579498-solution\">9<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex]f(x)=2x^2+1 \\ \\text{on} \\ [x, x+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134495260\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135409395\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=3x^2-2 \\ \\text{on} \\ [x, x+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134374799\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134374801\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134374799-solution\">11<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex]a(t)=\\frac{1}{t+4} \\ \\text{on} \\ [9, 9+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137693536\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137693538\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex]b(x)=\\frac{1}{x+3} \\ \\text{on} \\ [1, 1+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134104075\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134104077\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134104075-solution\">13<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex]j(x)=3x^3 \\ \\text{on} \\ [1, 1+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137851687\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137851690\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex]r(t)=4t^3 \\ \\text{on} \\ [2, 2+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137432064\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137432066\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137432064-solution\">15<\/a><span class=\"os-divider\">. <\/span>Find [latex]\\frac{f(x+h)-f(x)}{h} \\ \\ \\text{given} \\ \\ f(x)=2x^2-3x \\ \\ \\text{on} \\ \\ [x, x+h][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135409367\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135409372\">For the following exercises, consider the graph of\u00a0[latex]f[\/latex] shown in <a class=\"autogenerated-content\" href=\"3-3-rates-of-change-and-behavior-of-graphs#Figure_01_03_201\">Figure 15<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_529\" aria-describedby=\"caption-attachment-529\" style=\"width: 294px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-529\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15-294x300.jpeg\" alt=\"\" width=\"294\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15-294x300.jpeg 294w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15-225x229.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15-350x357.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-15.jpeg 357w\" sizes=\"auto, (max-width: 294px) 100vw, 294px\" \/><figcaption id=\"caption-attachment-529\" class=\"wp-caption-text\">Figure 15<\/figcaption><\/figure>\n<div id=\"fs-id1165135701406\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135701408\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span>Estimate the average rate of change from\u00a0[latex]x=1[\/latex] to [latex]x=4.[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137769703\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137769705\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137769703-solution\">17<\/a><span class=\"os-divider\">. <\/span>Estimate the average rate of change from\u00a0[latex]x=2[\/latex] to [latex]x=5.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165133213910\">For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.<\/p>\n<div id=\"fs-id1165133213915\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133213917\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165133213923\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of an absolute function.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-530\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-17.jpeg\" alt=\"\" width=\"249\" height=\"251\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-17.jpeg 249w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-17-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-17-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-17-225x227.jpeg 225w\" sizes=\"auto, (max-width: 249px) 100vw, 249px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137897938\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137897940\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137897938-solution\">19<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137897946\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a cubic function.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-531\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-19-297x300.jpeg\" alt=\"\" width=\"297\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-19-297x300.jpeg 297w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-19-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-19-225x227.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-19.jpeg 342w\" sizes=\"auto, (max-width: 297px) 100vw, 297px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135600844\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135600846\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135422552\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a cubic function.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-532\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-20.jpeg 342w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135422567\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135422569\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135422567-solution\">21<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134080979\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a reciprocal function.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-533\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-300x175.jpeg\" alt=\"\" width=\"300\" height=\"175\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-300x175.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-768x447.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-65x38.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-225x131.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21-350x204.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-21.jpeg 976w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134373457\">For the following exercises, consider the graph shown in Figure 16.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"Figure_01_03_206\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_534\" aria-describedby=\"caption-attachment-534\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-534\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16-300x199.jpeg\" alt=\"\" width=\"300\" height=\"199\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16-300x199.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16-65x43.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16-225x149.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16-350x232.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-16.jpeg 435w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-534\" class=\"wp-caption-text\">Figure 16<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134373481\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134373483\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span>Estimate the intervals where the function is increasing or decreasing.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134149087\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134149090\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134149087-solution\">23<\/a><span class=\"os-divider\">. <\/span>Estimate the point(s) at which the graph of\u00a0[latex]f[\/latex] has a local maximum or a local minimum.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135404176\">For the following exercises, consider the graph in Figure 17.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_535\" aria-describedby=\"caption-attachment-535\" style=\"width: 270px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-535\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17-270x300.jpeg\" alt=\"\" width=\"270\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17-270x300.jpeg 270w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17-65x72.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17-225x250.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17-350x389.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-17.jpeg 356w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><figcaption id=\"caption-attachment-535\" class=\"wp-caption-text\">Figure 17<\/figcaption><\/figure>\n<div id=\"fs-id1165135428496\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135428498\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span>If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135428507\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135428509\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135428507-solution\">25<\/a><span class=\"os-divider\">. <\/span>If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134043819\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<div id=\"fs-id1165134043824\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043826\" class=\"material-set-2\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>Table 3 gives the annual revenue (in billions of dollars) from 2015 to 2023. What was the average rate of change of annual revenue (a) between 2020 and 2021, and (b) between 2019 and 2022?<\/p>\n<div class=\"os-problem-container\">\n<div>\n<div>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Year<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\"><strong>Revenue (billions of dollars)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2015<\/td>\n<td style=\"width: 50%;\">486<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2016<\/td>\n<td style=\"width: 50%;\">482<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2017<\/td>\n<td style=\"width: 50%;\">486<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2018<\/td>\n<td style=\"width: 50%;\">500<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2019<\/td>\n<td style=\"width: 50%;\">514<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2020<\/td>\n<td style=\"width: 50%;\">524<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2021<\/td>\n<td style=\"width: 50%;\">560<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2022<\/td>\n<td style=\"width: 50%;\">573<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">2023<\/td>\n<td style=\"width: 50%;\">611<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135547288\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135547289\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135547288-solution\">27<\/a><span class=\"os-divider\">. <\/span>Table 4 gives the population of Aurora, Colorado (in thousands) from 2019 to 2024. What was the average rate of change of population (a) between 2021 and 2023, and (b) between 2019 and 2024?<\/p>\n<div class=\"os-problem-container\">\n<div>\n<div>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 105px;\">\n<caption>Table 4<\/caption>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Year<\/strong><\/td>\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Population (thousands)<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">2019<\/td>\n<td style=\"width: 50%; height: 15px;\">374<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">2020<\/td>\n<td style=\"width: 50%; height: 15px;\">381<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">2021<\/td>\n<td style=\"width: 50%; height: 15px;\">386<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">2022<\/td>\n<td style=\"width: 50%; height: 15px;\">398<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">2023<\/td>\n<td style=\"width: 50%; height: 15px;\">400<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">2024<\/td>\n<td style=\"width: 50%; height: 15px;\">404<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134547382\">For the following exercises, find the average rate of change of each function on the interval specified.<\/p>\n<div id=\"fs-id1165134547387\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134547389\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex]f(x)=x^2\\ \\text{on} \\ [1, 5][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137851207\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137851209\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137851207-solution\">29<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex]h(x)=5-2x^2 \\ \\text{on} \\ [-2, 4][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134384576\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134384578\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\u00a0[latex]q(x)=x^3\\ \\text{on} \\ [-4, 2][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135453186\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135453188\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135453186-solution\">31<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex]g(x)=3x^3-1 \\ \\text{on} \\ [-3, 3][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199306\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199308\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex]y=\\frac{1}{x} \\ \\text{on} \\ [1, 3][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135317474\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135317476\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135317474-solution\">33<\/a><span class=\"os-divider\">.\u00a0<\/span> [latex]p(t)=\\frac{(t^2-4)(t+1)}{t^2+3}\\ \\text{on} \\ [-3, 1][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135208859\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135208861\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">.\u00a0<\/span> [latex]k(t)=6t^2+\\frac{4}{t^3} \\ \\text{on} \\ [-1, 3][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135347321\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135347326\">For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.<\/p>\n<div id=\"fs-id1165134043700\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134043703\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134043700-solution\">35<\/a><span class=\"os-divider\">. [latex]f(x)=x^4-4x^3+5[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137836516\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137836518\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. [latex]h(x)=x^5+5x^4+10x^3+10x^2-1[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135257278\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135257281\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135257278-solution\">37<\/a><span class=\"os-divider\">. [latex]g(t)=t\\sqrt{t+3}[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137892185\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137892188\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. [latex]k(t)=3t^\\frac{2}{3}=t[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135602253\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135602256\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135602253-solution\">39<\/a><span class=\"os-divider\">. [latex]m(x)=x^4+2x^3-12x^2-10x+4[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137400141\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137400143\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. [latex]n(x)=x^4-8x^3+18x^2-6x+2[\/latex]<\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135536251\" data-depth=\"2\">\n<h3 data-type=\"title\">Extension<\/h3>\n<div id=\"fs-id1165135536257\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135536258\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135536257-solution\">41<\/a><span class=\"os-divider\">. <\/span>The graph of the function\u00a0[latex]f[\/latex] is shown in Figure 18.<\/p>\n<div class=\"os-problem-container\">\n<p>&nbsp;<\/p>\n<figure id=\"attachment_536\" aria-describedby=\"caption-attachment-536\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-536\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18-300x241.jpeg\" alt=\"\" width=\"300\" height=\"241\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18-300x241.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18-65x52.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18-225x181.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18-350x281.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-18.jpeg 371w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-536\" class=\"wp-caption-text\">Figure 18<\/figcaption><\/figure>\n<p id=\"fs-id1165134298989\">Based on the calculator screen shot, the point\u00a0[latex](1.333, 5.185)[\/latex] is which of the following?<\/p>\n<p>(a) a relative (local) maximum of the function<\/p>\n<p>(b) the vertex of the function<\/p>\n<p>(c) the absolute maximum of the function<\/p>\n<p>(d) a zero of the function<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134378686\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134378688\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>Let\u00a0[latex]f(x)=\\frac{1}{x}.[\/latex] Find a number\u00a0[latex]c[\/latex] such that the average rate of change of the function\u00a0[latex]f[\/latex] on the interval\u00a0[latex](1, c)[\/latex] is [latex]-\\frac{1}{4}.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134299101\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134299103\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134299101-solution\">43<\/a><span class=\"os-divider\">. <\/span>Let\u00a0[latex]f(x)=\\frac{1}{x}.[\/latex] Find the number\u00a0[latex]b[\/latex] such that the average rate of change of\u00a0[latex]f[\/latex] on the interval\u00a0[latex](2, b)[\/latex] is [latex]-\\frac{1}{10}.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134151836\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165134151841\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134151843\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><span class=\"TextRun SCXW171915131 BCX2\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW171915131 BCX2\">At the start of a road trip in Colorado.<\/span><\/span>, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135387234\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135387236\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135387234-solution\">45<\/a><span class=\"os-divider\">. <\/span>A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135387253\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135387255\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>Near the surface of the moon, the distance that an object falls is a function of time. It is given by\u00a0[latex]d(t)=2.6667t^2,[\/latex] where\u00a0[latex]t[\/latex] is in seconds and\u00a0[latex]d(t)[\/latex] is in feet. If an object is dropped from a certain height, find the average velocity of the object from\u00a0[latex]t=1[\/latex] to [latex]t=2.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135571880\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135571881\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135571880-solution\">47<\/a><span class=\"os-divider\">. <\/span>The graph in Figure 19 illustrates the decay of a radioactive substance over\u00a0[latex]t[\/latex] days.<\/p>\n<div class=\"os-problem-container\">\n<p>&nbsp;<\/p>\n<figure id=\"attachment_537\" aria-describedby=\"caption-attachment-537\" style=\"width: 300px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-537\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19-300x249.jpeg\" alt=\"\" width=\"300\" height=\"249\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19-300x249.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19-65x54.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19-225x187.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19-350x290.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-fig-19.jpeg 393w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-537\" class=\"wp-caption-text\">Figure 19<\/figcaption><\/figure>\n<p id=\"fs-id1165135517172\">Use the graph to estimate the average decay rate from\u00a0[latex]t=5[\/latex] to<\/p>\n<p>[latex]t=15.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-134","chapter","type-chapter","status-publish","hentry"],"part":105,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/134","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":24,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/134\/revisions"}],"predecessor-version":[{"id":1495,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/134\/revisions\/1495"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/134\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=134"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=134"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=134"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}