{"id":44,"date":"2019-01-29T21:28:09","date_gmt":"2019-01-29T21:28:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=44"},"modified":"2025-03-07T22:37:37","modified_gmt":"2025-03-07T22:37:37","slug":"rates-of-change-and-behavior-of-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/rates-of-change-and-behavior-of-graphs\/","title":{"raw":"1.3 Rates of Change and Behavior of Graphs","rendered":"1.3 Rates of Change and Behavior of Graphs"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\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<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. <a class=\"autogenerated-content\" href=\"#Table_01_03_01\">Table 1<\/a>[footnote]<a href=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524<\/a>. Accessed 3\/5\/2014.[\/footnote] lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\r\n\r\n<table id=\"Table_01_03_01\" style=\"height: 28px;\" summary=\"Two rows and nine columns. The first row is labeled, \u201cy\u201d, and the second row is labeled, \u201cC(y)\u201d. Reading the rows as ordered pairs, we have: (2005, 2.31), (2006, 2.62), (2007, 2.84), (2008, 3.30), (2009, 2.41), (2010, 2.84), (2011, 3.58), and (2012, 3.68).\"><caption>Table 1<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2005<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2006<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2007<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2008<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2009<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2010<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2011<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2.31<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2.62<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2.84<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">3.30<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2.41<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2.84<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">3.58<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. 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>\r\n\r\n<div id=\"fs-id1165137645483\" class=\"bc-section section\">\r\n<h3>Finding the Average Rate of Change of a Function<\/h3>\r\n<p id=\"fs-id1165137834011\">The price change per year is a <strong>rate of change<\/strong> 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=\"#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 average rate of change 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\nIf we consider the two points [latex]\\left({x}_{1}, {y}_{1}\\right)[\/latex] and\u00a0[latex]\\left({x}_{2}, {y}_{2}\\right)[\/latex] on the graph of a function [latex]f,[\/latex] we can talk about the average rate of change on the interval of input values [latex]\\left[{x}_{1}, {x}_{2}\\right].[\/latex]\r\n<div id=\"fs-id1165135452482\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\text{Average rate of change}&amp;=\\frac{\\text{Change in output}}{\\text{Change in input}}&amp;\\text{ } \\\\ \\text{ }&amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\hfill&amp;\\text{using coordinates of the point,} \\\\ \\text{ }&amp;=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\hfill&amp;\\text{using function notation,}\\\\ \\text{ }&amp;=\\frac{\\Delta y}{\\Delta x}&amp;\\text{using delta notation.}\\hfill \\\\\\end{align*}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135471272\">The Greek letter [latex]\\Delta[\/latex] (delta) signifies the change in a quantity; we read the ratio as \u201cdelta-<em>y<\/em> over delta-<em>x<\/em>\u201d or \u201cthe change in [latex]y[\/latex] divided by the change in [latex]x.[\/latex]\u201d Occasionally we write [latex]\\Delta f[\/latex] instead of [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\nYou may also see the interval given as [latex]\\left[a,b\\right].[\/latex]\u00a0 This means that our points would be [latex]\\left(a, f\\left(a\\right)\\right)[\/latex] and [latex]\\left(b, f\\left(b\\right)\\right).[\/latex] This would lead to the alternate form for the average rate of change shown below.\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}\\text{ Average rate of change }=\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}\\end{align*}[\/latex]<\/p>\r\nYou should be comfortable working with any of these presentations of the material.\r\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\r\n\r\n<div id=\"fs-id1165137526960\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{\\text{\\$}1.37}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\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[latex]\\\\[\/latex]\r\n<div id=\"fs-id1165137642836\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165137780744\">A <strong>rate of change<\/strong> 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>\r\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function output values [latex]f\\left({x}_{1}\\right)[\/latex] and [latex]f\\left({x}_{2}\\right)[\/latex] divided by the change in the input values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135530407\" class=\"precalculus howto examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137762240\"><strong>Given the value of a function at different points, calculate the average rate of change of a function for the interval between two input values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}.[\/latex] <\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137442714\" type=\"1\">\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<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_03_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135485962\">\r\n<div id=\"fs-id1165137464225\">\r\n<h3>Example 1:\u00a0 Computing an Average Rate of Change<\/h3>\r\n<p id=\"fs-id1165137603118\">Using the data in <a class=\"autogenerated-content\" href=\"#Table_01_03_01\">Table 1<\/a>, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137593409\">[reveal-answer q=\"fs-id1165137593409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137593409\"]\r\n<p id=\"fs-id1165135209401\">In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\r\n\r\n<div id=\"fs-id1165137812332\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{\\Delta y}{\\Delta x}&amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\hfill \\\\\\text{ }&amp;=\\frac{2.41-2.84}{2009-2007}\\\\\\text{ }&amp;=\\frac{-0.43}{2}\\hfill \\\\ \\text{ }&amp;=-0.22\\hfill \\end{align*}[\/latex]<\/div>\r\n<div>On average, the price of gasoline goes down $0.22 or 22 cents per year between 2007 and 2009.<\/div>\r\n<div><\/div>\r\n<h3>Analysis<\/h3>\r\n<div>Note that a decrease is expressed by a negative change or \"negative increase.\"\u00a0 A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137593409\">\r\n<div><\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133267246\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_01_03_01\">\r\n<div id=\"fs-id1165135160758\">\r\n<p id=\"fs-id1165135160759\">Using the data in <a class=\"autogenerated-content\" href=\"#Table_01_03_01\">Table 1<\/a>, find the average rate of change between 2005 and 2010.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137401356\">[reveal-answer q=\"fs-id1165137401356\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137401356\"]\r\n<p id=\"fs-id1165137453887\">[latex]\\frac{$2.84-$2.31}{5\\text{ years}}=\\frac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137851963\">\r\n<div id=\"fs-id1165137437853\">\r\n<h3>Example 2:\u00a0 Computing Average Rate of Change from a Graph<\/h3>\r\n<p id=\"fs-id1165137415140\">Given the function [latex]g\\left(t\\right)[\/latex] shown in <a class=\"autogenerated-content\" href=\"#Figure_01_03_001\">Figure 1<\/a>, find the average rate of change on the interval [latex]\\left[-1,2\\right].[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_3109\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3109 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14161725\/c4474b079af00abf4118ea94530062e2ea966ecd.jpeg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/> Figure 1[\/caption]\r\n\r\n<div id=\"Figure_01_03_001\" class=\"small\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137501924\">[reveal-answer q=\"fs-id1165137501924\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137501924\"]\r\n<p id=\"fs-id1165137598939\">At [latex]t=-1,[\/latex] <a class=\"autogenerated-content\" href=\"#Figure_01_03_002\">Figure 2<\/a> shows [latex]g\\left(-1\\right)=4.[\/latex] At [latex]t=2,[\/latex] the graph shows [latex]g\\left(2\\right)=1.[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_3110\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3110\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14161856\/887f5c84cd5b446004895f538bc0a8c2b5f2ade6.jpeg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/> Figure 2[\/caption]\r\n\r\n<div id=\"Figure_01_03_002\" class=\"small\"><\/div>\r\n<p id=\"fs-id1165137591169\">The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\r\n\r\n<div id=\"fs-id1165137414241\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\frac{1-4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/div>\r\n<h3>Analysis<\/h3>\r\n<div class=\"unnumbered\">Note that the order we choose is very important. If, for example, we use [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 [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right).[\/latex]<\/div>\r\n<div class=\"unnumbered\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135536188\">\r\n<div id=\"fs-id1165137835656\">\r\n<h3>Example 3:\u00a0 Computing Average Rate of Change from a Table<\/h3>\r\n<p id=\"fs-id1165135515898\">After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in <a class=\"autogenerated-content\" href=\"#Table_01_03_02\">Table 2<\/a>. Find her average speed over the first 6 hours.<\/p>\r\n\r\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled, \u201ct (hours)\u201d, and the second row is labeled, \u201cD(t) (miles)\u201d. Reading the rows as ordered pairs, we have: (0, 10), (1, 55), (2, 90), (3, 153), (4, 214), (5, 240), (6, 292), and (7, 300).\"><caption>Table 2<\/caption><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong><em>t<\/em> (hours)<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">55<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">90<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">153<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">214<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">240<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">292<\/td>\r\n<td class=\"border\" style=\"text-align: center;\">300<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137934383\">[reveal-answer q=\"fs-id1165137934383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137934383\"]\r\n<p id=\"fs-id1165137891478\">Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of<\/p>\r\n\r\n<div id=\"fs-id1165137789030\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{292-10}{6-0}&amp;=\\frac{282}{6}\\hfill \\\\\\text{ }&amp;=47\\hfill \\end{align*}[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nBecause the speed is not constant, the <em>average speed<\/em> depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135353057\">\r\n<div id=\"fs-id1165135383644\">\r\n<h3>Example 4:\u00a0 Computing Average Rate of Change for a Function Expressed as a Formula<\/h3>\r\n<p id=\"fs-id1165131958324\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{ }\\text{4].}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137595439\">[reveal-answer q=\"fs-id1165137595439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137595439\"]\r\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <span class=\"no-emphasis\">endpoint<\/span> of the interval.<\/p>\r\n\r\n<div id=\"fs-id1165137701107\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(2\\right)&amp;={2}^{2}-\\frac{1}{2}&amp; f\\left(4\\right)&amp;={4}^{2}-\\frac{1}{4}\\\\\\text{ }&amp;=4-\\frac{1}{2} &amp; \\text{ }&amp;=16-\\frac{1}{4}\\hfill \\\\ \\text{ }&amp;=\\frac{7}{2}\\hfill &amp; \\text{ }&amp;=\\frac{63}{4}\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change from [latex]x=2[\/latex] to [latex]x=4[\/latex] using the formula [latex]\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137401823\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}&amp;\\frac{f\\left(4\\right)-f\\left(2\\right)}{4-2}\\hfill\\text{ }&amp;&amp;\\text{Plug 2 and 4 into the formula.} \\\\ \\text{ }&amp;=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4-2}\\hfill&amp;&amp;\\text{Substitute values for }f\\left(4\\right)\\text{ and }f\\left(2\\right). \\\\ \\text{ }&amp;=\\frac{\\frac{49}{4}}{2}\\hfill &amp;&amp;\\text{Simplify.}\\\\ \\text{ }&amp;=\\frac{49}{8}\\hfill \\end{align*}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137436764\" class=\"precalculus tryit\">\r\n<h3>Try it #2<\/h3>\r\n<div id=\"ti_01_03_02\">\r\n<div id=\"fs-id1165137832323\">\r\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x-2\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] on the interval [latex]\\left[1,\\text{ }9\\right].[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137455570\">[reveal-answer q=\"fs-id1165137455570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137455570\"]\r\n<p id=\"fs-id1165137455571\">[latex]\\frac{1}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137772170\">\r\n<div id=\"fs-id1165137772173\">\r\n<h3>Example 5:\u00a0 Finding the Average Rate of Change of a Force<\/h3>\r\n<p id=\"fs-id1165135443718\">The <span class=\"no-emphasis\">electrostatic force\u00a0<\/span>[latex]F,[\/latex] measured in newtons, between two charged particles can be related to the distance between the particles [latex]d,[\/latex] in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}.[\/latex] Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137696464\">[reveal-answer q=\"fs-id1165137696464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137696464\"]\r\n<p id=\"fs-id1165137770364\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right].[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135613444\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\text{Average rate of change }&amp;=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6-2}\\hfill &amp;&amp; \\hfill \\\\&amp;=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6-2}\\hfill &amp;&amp; \\text{Simplify}.\\hfill \\\\ &amp;=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4}\\hfill &amp;&amp; \\hfill \\\\ &amp;=\\frac{-\\frac{16}{36}}{4}\\hfill &amp;&amp; \\text{Combine numerator terms}.\\hfill \\\\ &amp;=-\\frac{1}{9}\\hfill &amp;&amp; \\text{Simplify.}\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135543242\">The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135174952\">\r\n<div id=\"fs-id1165135174954\">\r\n<h3>Example 6:\u00a0 Finding an Average Rate of Change as an Expression<\/h3>\r\n<p id=\"fs-id1165135155397\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,\\text{ }a\\right].[\/latex] The answer will be an expression involving [latex]a.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137418911\">[reveal-answer q=\"fs-id1165137418911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137418911\"]\r\n<p id=\"fs-id1165137418913\">We use the average rate of change formula with input values [latex]a[\/latex] and 0.<\/p>\r\n\r\n<div id=\"fs-id1165134198703\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}&amp;\\frac{g\\left(a\\right)-g\\left(0\\right)}{a-0}\\hfill &amp;&amp; \\text{Evaluate}.\\hfill \\\\ \\text{ }&amp;=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a-0}\\hfill &amp;&amp; \\text{Simplify}.\\hfill \\\\ \\text{ }&amp;=\\frac{{a}^{2}+3a+1-1}{a}\\hfill &amp;&amp; \\text{Simplify and factor}.\\hfill \\\\ \\text{ }&amp;=\\frac{a\\left(a+3\\right)}{a} \\hfill &amp;&amp;\\text{Cancel }a.\\\\\\text{ }&amp;=a+3\\hfill \\end{align*}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165133316469\">This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a.[\/latex] For example, on the interval [latex]\\left[0,5\\right],[\/latex] the average rate of change would be [latex]5+3=8.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137453128\" class=\"precalculus tryit\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"ti_01_03_03\">\r\n<div id=\"fs-id1165134149845\">\r\n<p id=\"fs-id1165134149846\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x-8[\/latex] on the interval [latex]\\left[5,\\text{ }a\\right].[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133065696\">[reveal-answer q=\"fs-id1165133065696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133065696\"]\r\n<p id=\"fs-id1165133065698\">[latex]a+7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135440486\" class=\"bc-section section\">\r\n<h3>Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h3>\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.<\/p>\r\nThe average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. <a class=\"autogenerated-content\" href=\"#Figure_01_03_004\">Figure 3<\/a> shows examples of increasing and decreasing intervals on a function.\r\n\r\n[caption id=\"attachment_3111\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3111 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162020\/56ee4dd1a9e462cff7a40c566285b1405e1744ba.jpeg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/> <strong>Figure 3<\/strong> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}\\text{ }-\\text{2}\\right){{\\cup }^{\\text{\u200b}}}^{\\text{\u200b}}\\left(2,\\text{ }\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}\\text{ }2\\right).[\/latex][\/caption]\r\n<div id=\"Figure_01_03_004\" class=\"small\"><\/div>\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 called a<strong> local maximum<\/strong>. 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 called a <strong>local minimum<\/strong>. The plural form is \u201clocal minima.\u201d Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is \u201cextremum.\u201d) Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>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>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 <a class=\"autogenerated-content\" href=\"#Figure_01_03_015\">Figure 4<\/a>, the local maximum occurs when [latex]x=-2[\/latex]\u00a0 The maximum value is the output value of 16.\u00a0 The local minimum\u00a0occurs when [latex]x=2[\/latex] .\u00a0 The minimum value is the output value of [latex]-16[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_3112\" align=\"aligncenter\" width=\"731\"]<img class=\"wp-image-3112 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162141\/745142cc108b26ac588ea701460c9fec96f6974c-1.jpeg\" alt=\"The graph shows a curve going up until (-2, 16) and then goes down until (2, -16) and then goes up after that.\" width=\"731\" height=\"467\" \/> <strong>Figure 4<\/strong>[\/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. <a class=\"autogenerated-content\" href=\"#Figure_01_03_005\">Figure 5<\/a> illustrates these ideas for a local maximum.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3113\" align=\"aligncenter\" width=\"487\"]<img class=\"wp-image-3113 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162256\/151daece91ddcd0ad086eb4a6a63f0dedfd44ac1.jpeg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/> <strong>Figure 5<\/strong> Definition of a local maximum[\/caption]\r\n\r\n<div id=\"Figure_01_03_005\" class=\"small\"><\/div>\r\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\r\n\r\n<div id=\"fs-id1165134169419\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165134169426\">A function [latex]f[\/latex] is an <strong>increasing function on an open interval<\/strong> if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for every two input values [latex]a[\/latex] and [latex]b[\/latex] in the interval where [latex]b&gt;a.[\/latex]<\/p>\r\n<p id=\"fs-id1165137668624\">A function [latex]f[\/latex] is a <strong>decreasing function on an open interva<\/strong>l if [latex]f\\left( b \\right)\\lt f\\left(a\\right)[\/latex] for every two input values [latex]a[\/latex] and [latex]b[\/latex] in the interval where [latex]b&gt;a.[\/latex]<\/p>\r\n<p id=\"fs-id1165135389881\">A function [latex]f[\/latex] has a <strong>local maximum<\/strong> at a point [latex]b[\/latex] in an open interval [latex]\\left(a,c\\right)[\/latex] if [latex]f\\left(b\\right)[\/latex] is greater than or equal to [latex]f\\left(x\\right)[\/latex] for every point [latex]x[\/latex] ([latex]x[\/latex] does not equal [latex]b[\/latex]) in the interval. Likewise, [latex]f[\/latex] has a <strong>local minimum<\/strong> at a point [latex]b[\/latex] in [latex]\\left(a,c\\right)[\/latex] if [latex]f\\left(b\\right)[\/latex] is less than or equal to [latex]f\\left(x\\right)[\/latex] for every [latex]x[\/latex] ([latex]x[\/latex] does not equal [latex]b[\/latex]) in the interval.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134266716\">\r\n<div id=\"fs-id1165134266718\">\r\n<h3>Example 7:\u00a0 Finding Increasing and Decreasing Intervals on a Graph<\/h3>\r\n<p id=\"fs-id1165134266723\">Given the function [latex]p\\left(t\\right)[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_03_006\">Figure 6<\/a>, identify the intervals on which the function appears to be increasing and decreasing.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3114\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3114\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162527\/dab6d2577d252f65a3a3cd76c792f97abc482af3-1.jpeg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"295\" \/> <strong>Figure 6<\/strong>[\/caption]\r\n\r\n<div id=\"Figure_01_03_006\" class=\"small\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133067194\">[reveal-answer q=\"fs-id1165133067194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133067194\"]\r\n<p id=\"fs-id1165133067197\">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 [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on. The function appears to be decreasing until [latex]t=1[\/latex] and then again from [latex]t=3[\/latex] to [latex]t=4.[\/latex]<\/p>\r\n<p id=\"fs-id1165135369127\">In <span class=\"no-emphasis\">interval notation<\/span>, we would say the function appears to be increasing [latex]\\left(1,\\text{ }3\\right)\\cup\\left(4,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-\\infty,\\text{ }1\\right)\\cup\\left(3,\\text{ }4\\right).[\/latex]<\/p>\r\n\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[\/latex], [latex]t=3[\/latex],\u00a0 and [latex]t=4[\/latex]. These points are the local extrema (two minima and a maximum).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_08\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135261521\">\r\n<div id=\"fs-id1165135261523\">\r\n<h3>Example 8:\u00a0 Finding Local Extrema from a Graph<\/h3>\r\n<p id=\"fs-id1165135261528\">Use technology to graph the function [latex]f\\left(x\\right)=\\frac{2}{x}+\\frac{x}{3}.[\/latex] Then use features of your graphing utility to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134255027\">[reveal-answer q=\"fs-id1165134255027\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134255027\"]Using technology, we find that the graph of the function looks like that in <a class=\"autogenerated-content\" href=\"#Figure_01_03_007\">Figure 7<\/a>. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3,[\/latex] and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2.[\/latex]\r\n\r\n[caption id=\"attachment_3115\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3115\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162819\/241fa93a9ce8124d48204ada8fe6c8edfa4b69e0.jpeg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/> Figure 7[\/caption]\r\n\r\n<div id=\"Figure_01_03_007\" class=\"small\">\r\n<h3>Analysis<\/h3>\r\nMost graphing calculators and graphing utilities can estimate the location of maxima and minima. <a class=\"autogenerated-content\" href=\"#Figure_01_03_008\">Figure 8<\/a> provides screen images from two different technologies, showing the estimate for the local maximum and minimum.\r\n\r\n[caption id=\"attachment_3116\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3116\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162900\/b2c81159521af2d8d476a1a0fb2d335bee24c34e-1.jpeg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/> Figure 8[\/caption]\r\n<p id=\"fs-id1165134075625\">Based on these estimates, the function is increasing on the interval [latex](-\\infty \\text{,}-\\text{2}\\text{.449)}[\/latex] and [latex]\\left(2.449\\text{,}\\infty \\right).[\/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 [latex]\u00b1\\sqrt{6},[\/latex] but determining this requires calculus.)<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135160233\" class=\"precalculus tryit\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"ti_01_03_04\">\r\n<div id=\"fs-id1165135640966\">\r\n<p id=\"fs-id1165135640967\">Use technology to graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] and to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135471082\">[reveal-answer q=\"fs-id1165135471082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135471082\"]\r\n<p id=\"fs-id1165135471083\">Using technology, we find the local maximum of 28 appears to occur at [latex]\\left(-1,28\\right),[\/latex] and the local minimum of -80 appears to occur at [latex]\\left(5,-80\\right).[\/latex] The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right).[\/latex]<\/p>\r\n<span id=\"fs-id1165134043615\"><img class=\"aligncenter wp-image-3174 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/15150135\/11c7b4fdf95bfa15af5b8e130c14018e252f69b8.jpeg\" alt=\"\" width=\"487\" height=\"328\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_09\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135367558\">\r\n<div id=\"fs-id1165137896103\">\r\n<h3>Example 9:\u00a0 Finding Local Maxima and Minima from a Graph<\/h3>\r\n<p id=\"fs-id1165135209636\">For the function [latex]f[\/latex] whose graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_03_011\">Figure 9<\/a>, find all local maxima and minima.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3117\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3117\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163053\/2addb41469229c8466f45cbccda65866650c2c47.jpeg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"368\" \/> Figure 9[\/caption]\r\n\r\n<div id=\"Figure_01_03_011\" class=\"small\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135527083\">[reveal-answer q=\"fs-id1165135527083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135527083\"]\r\n<p id=\"fs-id1165135527085\">Observe the graph of [latex]f.[\/latex] The graph attains a local maximum at [latex]x=1[\/latex] because the highest point in an open interval occurs when [latex]x=1.[\/latex] The local maximum is the [latex]y[\/latex]-coordinate at [latex]x=1,[\/latex] which is [latex]2.[\/latex]<\/p>\r\n<p id=\"fs-id1165134485672\">The graph attains a local minimum at [latex]x=-1[\/latex] because the lowest point in an open interval occurs when [latex]x=-1.[\/latex] The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1,[\/latex] which is\u00a0 [latex]-2.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134544960\" class=\"bc-section section\">\r\n<h3>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h3>\r\n<p id=\"fs-id1165135704895\">We will now return to our toolkit functions and discuss their graphical behavior in <a class=\"autogenerated-content\" href=\"#Figure_01_03_012\">Figure 10<\/a>, <a class=\"autogenerated-content\" href=\"#Figure_01_03_016\">Figure 11<\/a>, and <a class=\"autogenerated-content\" href=\"#Figure_01_03_017\">Figure 12<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3118\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3118\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163211\/f22b41248dd22d00a63dcb02f358ed1391fb7eca.jpeg\" alt=\"Table showing the increasing and decreasing intervals of the toolkit functions.\" width=\"975\" height=\"525\" \/> Figure 10[\/caption]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_3119\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3119\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163250\/a67cdc93a5dc0fbc3016a4949587f64a835c309a.jpeg\" alt=\"Table showing the increasing and decreasing intervals of the toolkit functions.\" width=\"975\" height=\"525\" \/> Figure 11[\/caption]\r\n\r\n[caption id=\"attachment_3120\" align=\"aligncenter\" width=\"975\"]<img class=\"size-full wp-image-3120\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163352\/cb19bea012bc887dec6e4cc83ebcfd51da042cd0.jpeg\" alt=\"Table showing the increasing and decreasing intervals of the toolkit functions.\" width=\"975\" height=\"525\" \/> Figure 12[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134381626\" class=\"bc-section section\">\r\n<h3>Use A Graph to Locate the Absolute Maximum and Absolute Minimum (Optional)<\/h3>\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\\text{-}[\/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 <a class=\"autogenerated-content\" href=\"#Figure_01_03_014\">Figure 13<\/a>.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3121\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3121\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163447\/bb08f4a7682898031911c3e11d4103b49d309391.jpeg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/> Figure 13[\/caption]\r\n\r\n<div id=\"Figure_01_03_014\" class=\"small\"><\/div>\r\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\r\n\r\n<div id=\"fs-id1165135251290\">\r\n<div class=\"textbox definitions\">\r\n<h3>Definition<\/h3>\r\n<p id=\"fs-id1165132939786\">The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f.[\/latex]<\/p>\r\n<p id=\"fs-id1165137932685\">The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134047533\">\r\n<div id=\"fs-id1165134047535\">\r\n<h3>Example 10:\u00a0 Finding Absolute Maxima and Minima from a Graph<\/h3>\r\n<p id=\"fs-id1165133394704\">For the function [latex]f[\/latex] shown in <a class=\"autogenerated-content\" href=\"#Figure_01_03_013\">Figure 14<\/a>, find all absolute maxima and minima.<\/p>\r\n\r\n\r\n[caption id=\"attachment_3122\" align=\"aligncenter\" width=\"487\"]<img class=\"size-full wp-image-3122\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163530\/913701fd366ea33bac8906953907480a87bfd2cf.jpeg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/> Figure 14[\/caption]\r\n\r\n<div id=\"Figure_01_03_013\" class=\"small\"><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135532368\">[reveal-answer q=\"fs-id1165135532368\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135532368\"]\r\n<p id=\"fs-id1165135532371\">Observe the graph of [latex]f.[\/latex] The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [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 <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2,[\/latex] which is [latex]16.[\/latex]<\/p>\r\nThe graph attains an absolute minimum at [latex]x=3,[\/latex] because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3,[\/latex] which is [latex]-10.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135708033\" class=\"precalculus media\">\r\n<p id=\"fs-id1165135708038\">Access this online resource for additional instruction and practice with rates of change.<\/p>\r\n\r\n<ul id=\"fs-id1165135708041\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/aroc\">Average Rate of Change<\/a><\/li>\r\n<\/ul>\r\nhttps:\/\/www.youtube.com\/watch?v=F-7Poa3i1ZU\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135541564\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165135358784\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Average rate of change<\/td>\r\n<td class=\"border\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165135481945\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165135481952\">\r\n \t<li>A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data over an interval.<\/li>\r\n \t<li>Identifying points that mark the interval on a graph can be used to find the average rate of change.<\/li>\r\n \t<li>Comparing pairs of input and output values in a table can also be used to find the average rate of change.<\/li>\r\n \t<li>An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.<\/li>\r\n \t<li>The average rate of change can sometimes be determined as an expression.<\/li>\r\n \t<li>A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.<\/li>\r\n \t<li>A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.<\/li>\r\n \t<li>A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.<\/li>\r\n \t<li>Minima and maxima are also called extrema.<\/li>\r\n \t<li>We can find local extrema from a graph.<\/li>\r\n \t<li>The highest and lowest points on a graph indicate the absolute maxima and minima.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165133052903\">\r\n \t<dt>absolute maximum<\/dt>\r\n \t<dd id=\"fs-id1165133052908\">the greatest value of a function over an interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133052911\">\r\n \t<dt>absolute minimum<\/dt>\r\n \t<dd id=\"fs-id1165133052916\">the lowest value of a function over an interval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133052921\">\r\n \t<dt>average rate of change<\/dt>\r\n \t<dd id=\"fs-id1165133052926\">the difference in the output values of a function found for two values of the input divided by the difference between the inputs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135264639\">\r\n \t<dt>decreasing function<\/dt>\r\n \t<dd id=\"fs-id1165135264645\">a function is decreasing in some open interval if [latex]f\\left(b\\right) \\lt f\\left(a\\right)[\/latex] for any two input values\u00a0 [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135639824\">\r\n \t<dt>increasing function<\/dt>\r\n \t<dd id=\"fs-id1165135639829\">a function is increasing in some open interval if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135536408\">\r\n \t<dt>local extrema<\/dt>\r\n \t<dd id=\"fs-id1165135536413\">collectively, all of a function's local maxima and minima<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135536416\">\r\n \t<dt>local maximum<\/dt>\r\n \t<dd id=\"fs-id1165135412035\">a value of the input where a function changes from increasing to decreasing as the input value increases.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135412040\">\r\n \t<dt>local minimum<\/dt>\r\n \t<dd id=\"fs-id1165135412046\">a value of the input where a function changes from decreasing to increasing as the input value increases.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135412050\">\r\n \t<dt>rate of change<\/dt>\r\n \t<dd id=\"fs-id1165135412054\">the change of an output quantity relative to the change of the input quantity<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\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<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. <a class=\"autogenerated-content\" href=\"#Table_01_03_01\">Table 1<\/a><a class=\"footnote\" title=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.\" id=\"return-footnote-44-1\" href=\"#footnote-44-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> lists the average cost, in dollars, of a gallon of gasoline for the years 2005\u20132012. The cost of gasoline can be considered as a function of year.<\/p>\n<table id=\"Table_01_03_01\" style=\"height: 28px;\" summary=\"Two rows and nine columns. The first row is labeled, \u201cy\u201d, and the second row is labeled, \u201cC(y)\u201d. Reading the rows as ordered pairs, we have: (2005, 2.31), (2006, 2.62), (2007, 2.84), (2008, 3.30), (2009, 2.41), (2010, 2.84), (2011, 3.58), and (2012, 3.68).\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]y[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">2005<\/td>\n<td class=\"border\" style=\"text-align: center;\">2006<\/td>\n<td class=\"border\" style=\"text-align: center;\">2007<\/td>\n<td class=\"border\" style=\"text-align: center;\">2008<\/td>\n<td class=\"border\" style=\"text-align: center;\">2009<\/td>\n<td class=\"border\" style=\"text-align: center;\">2010<\/td>\n<td class=\"border\" style=\"text-align: center;\">2011<\/td>\n<td class=\"border\" style=\"text-align: center;\">2012<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center;\"><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">2.31<\/td>\n<td class=\"border\" style=\"text-align: center;\">2.62<\/td>\n<td class=\"border\" style=\"text-align: center;\">2.84<\/td>\n<td class=\"border\" style=\"text-align: center;\">3.30<\/td>\n<td class=\"border\" style=\"text-align: center;\">2.41<\/td>\n<td class=\"border\" style=\"text-align: center;\">2.84<\/td>\n<td class=\"border\" style=\"text-align: center;\">3.58<\/td>\n<td class=\"border\" style=\"text-align: center;\">3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. 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 id=\"fs-id1165137645483\" class=\"bc-section section\">\n<h3>Finding the Average Rate of Change of a Function<\/h3>\n<p id=\"fs-id1165137834011\">The price change per year is a <strong>rate of change<\/strong> 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=\"#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 average rate of change 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>If we consider the two points [latex]\\left({x}_{1}, {y}_{1}\\right)[\/latex] and\u00a0[latex]\\left({x}_{2}, {y}_{2}\\right)[\/latex] on the graph of a function [latex]f,[\/latex] we can talk about the average rate of change on the interval of input values [latex]\\left[{x}_{1}, {x}_{2}\\right].[\/latex]<\/p>\n<div id=\"fs-id1165135452482\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\text{Average rate of change}&=\\frac{\\text{Change in output}}{\\text{Change in input}}&\\text{ } \\\\ \\text{ }&=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\hfill&\\text{using coordinates of the point,} \\\\ \\text{ }&=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\hfill&\\text{using function notation,}\\\\ \\text{ }&=\\frac{\\Delta y}{\\Delta x}&\\text{using delta notation.}\\hfill \\\\\\end{align*}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135471272\">The Greek letter [latex]\\Delta[\/latex] (delta) signifies the change in a quantity; we read the ratio as \u201cdelta-<em>y<\/em> over delta-<em>x<\/em>\u201d or \u201cthe change in [latex]y[\/latex] divided by the change in [latex]x.[\/latex]\u201d Occasionally we write [latex]\\Delta f[\/latex] instead of [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>You may also see the interval given as [latex]\\left[a,b\\right].[\/latex]\u00a0 This means that our points would be [latex]\\left(a, f\\left(a\\right)\\right)[\/latex] and [latex]\\left(b, f\\left(b\\right)\\right).[\/latex] This would lead to the alternate form for the average rate of change shown below.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align*}\\text{ Average rate of change }=\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}\\end{align*}[\/latex]<\/p>\n<p>You should be comfortable working with any of these presentations of the material.<\/p>\n<p id=\"fs-id1165137539940\">In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\n<div id=\"fs-id1165137526960\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{\\text{\\$}1.37}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\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<p>[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1165137642836\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165137780744\">A <strong>rate of change<\/strong> 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 output values [latex]f\\left({x}_{1}\\right)[\/latex] and [latex]f\\left({x}_{2}\\right)[\/latex] divided by the change in the input values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530407\" class=\"precalculus howto examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137762240\"><strong>Given the value of a function at different points, calculate the average rate of change of a function for the interval between two input values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}.[\/latex] <\/strong><\/p>\n<ol id=\"fs-id1165137442714\" type=\"1\">\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<\/ol>\n<\/div>\n<div id=\"Example_01_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165135485962\">\n<div id=\"fs-id1165137464225\">\n<h3>Example 1:\u00a0 Computing an Average Rate of Change<\/h3>\n<p id=\"fs-id1165137603118\">Using the data in <a class=\"autogenerated-content\" href=\"#Table_01_03_01\">Table 1<\/a>, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\n<\/div>\n<div id=\"fs-id1165137593409\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137593409\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137593409\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135209401\">In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is<\/p>\n<div id=\"fs-id1165137812332\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{\\Delta y}{\\Delta x}&=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\hfill \\\\\\text{ }&=\\frac{2.41-2.84}{2009-2007}\\\\\\text{ }&=\\frac{-0.43}{2}\\hfill \\\\ \\text{ }&=-0.22\\hfill \\end{align*}[\/latex]<\/div>\n<div>On average, the price of gasoline goes down $0.22 or 22 cents per year between 2007 and 2009.<\/div>\n<div><\/div>\n<h3>Analysis<\/h3>\n<div>Note that a decrease is expressed by a negative change or &#8220;negative increase.&#8221;\u00a0 A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/div>\n<\/div>\n<div id=\"fs-id1165137593409\">\n<div><\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133267246\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_01_03_01\">\n<div id=\"fs-id1165135160758\">\n<p id=\"fs-id1165135160759\">Using the data in <a class=\"autogenerated-content\" href=\"#Table_01_03_01\">Table 1<\/a>, find the average rate of change between 2005 and 2010.<\/p>\n<\/div>\n<div id=\"fs-id1165137401356\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137401356\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137401356\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137453887\">[latex]\\frac{$2.84-$2.31}{5\\text{ years}}=\\frac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137851963\">\n<div id=\"fs-id1165137437853\">\n<h3>Example 2:\u00a0 Computing Average Rate of Change from a Graph<\/h3>\n<p id=\"fs-id1165137415140\">Given the function [latex]g\\left(t\\right)[\/latex] shown in <a class=\"autogenerated-content\" href=\"#Figure_01_03_001\">Figure 1<\/a>, find the average rate of change on the interval [latex]\\left[-1,2\\right].[\/latex]<\/p>\n<div id=\"attachment_3109\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3109\" class=\"wp-image-3109 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14161725\/c4474b079af00abf4118ea94530062e2ea966ecd.jpeg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/><\/p>\n<p id=\"caption-attachment-3109\" class=\"wp-caption-text\">Figure 1<\/p>\n<\/div>\n<div id=\"Figure_01_03_001\" class=\"small\"><\/div>\n<\/div>\n<div id=\"fs-id1165137501924\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137501924\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137501924\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137598939\">At [latex]t=-1,[\/latex] <a class=\"autogenerated-content\" href=\"#Figure_01_03_002\">Figure 2<\/a> shows [latex]g\\left(-1\\right)=4.[\/latex] At [latex]t=2,[\/latex] the graph shows [latex]g\\left(2\\right)=1.[\/latex]<\/p>\n<div id=\"attachment_3110\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3110\" class=\"size-full wp-image-3110\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14161856\/887f5c84cd5b446004895f538bc0a8c2b5f2ade6.jpeg\" alt=\"Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.\" width=\"487\" height=\"296\" \/><\/p>\n<p id=\"caption-attachment-3110\" class=\"wp-caption-text\">Figure 2<\/p>\n<\/div>\n<div id=\"Figure_01_03_002\" class=\"small\"><\/div>\n<p id=\"fs-id1165137591169\">The horizontal change [latex]\\Delta t=3[\/latex] is shown by the red arrow, and the vertical change [latex]\\Delta g\\left(t\\right)=-3[\/latex] is shown by the turquoise arrow. The output changes by \u20133 while the input changes by 3, giving an average rate of change of<\/p>\n<div id=\"fs-id1165137414241\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\frac{1-4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/div>\n<h3>Analysis<\/h3>\n<div class=\"unnumbered\">Note that the order we choose is very important. If, for example, we use [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 [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right).[\/latex]<\/div>\n<div class=\"unnumbered\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135536188\">\n<div id=\"fs-id1165137835656\">\n<h3>Example 3:\u00a0 Computing Average Rate of Change from a Table<\/h3>\n<p id=\"fs-id1165135515898\">After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in <a class=\"autogenerated-content\" href=\"#Table_01_03_02\">Table 2<\/a>. Find her average speed over the first 6 hours.<\/p>\n<table id=\"Table_01_03_02\" summary=\"Two rows and nine columns. The first row is labeled, \u201ct (hours)\u201d, and the second row is labeled, \u201cD(t) (miles)\u201d. Reading the rows as ordered pairs, we have: (0, 10), (1, 55), (2, 90), (3, 153), (4, 214), (5, 240), (6, 292), and (7, 300).\">\n<caption>Table 2<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\"><strong><em>t<\/em> (hours)<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">0<\/td>\n<td class=\"border\" style=\"text-align: center;\">1<\/td>\n<td class=\"border\" style=\"text-align: center;\">2<\/td>\n<td class=\"border\" style=\"text-align: center;\">3<\/td>\n<td class=\"border\" style=\"text-align: center;\">4<\/td>\n<td class=\"border\" style=\"text-align: center;\">5<\/td>\n<td class=\"border\" style=\"text-align: center;\">6<\/td>\n<td class=\"border\" style=\"text-align: center;\">7<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center;\">10<\/td>\n<td class=\"border\" style=\"text-align: center;\">55<\/td>\n<td class=\"border\" style=\"text-align: center;\">90<\/td>\n<td class=\"border\" style=\"text-align: center;\">153<\/td>\n<td class=\"border\" style=\"text-align: center;\">214<\/td>\n<td class=\"border\" style=\"text-align: center;\">240<\/td>\n<td class=\"border\" style=\"text-align: center;\">292<\/td>\n<td class=\"border\" style=\"text-align: center;\">300<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137934383\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137934383\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137934383\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137891478\">Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of<\/p>\n<div id=\"fs-id1165137789030\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\frac{292-10}{6-0}&=\\frac{282}{6}\\hfill \\\\\\text{ }&=47\\hfill \\end{align*}[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\n<h3>Analysis<\/h3>\n<p>Because the speed is not constant, the <em>average speed<\/em> depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135353057\">\n<div id=\"fs-id1165135383644\">\n<h3>Example 4:\u00a0 Computing Average Rate of Change for a Function Expressed as a Formula<\/h3>\n<p id=\"fs-id1165131958324\">Compute the average rate of change of [latex]f\\left(x\\right)={x}^{2}-\\frac{1}{x}[\/latex] on the interval [latex]\\text{[2,}\\text{ }\\text{4].}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137595439\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137595439\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137595439\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <span class=\"no-emphasis\">endpoint<\/span> of the interval.<\/p>\n<div id=\"fs-id1165137701107\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}f\\left(2\\right)&={2}^{2}-\\frac{1}{2}& f\\left(4\\right)&={4}^{2}-\\frac{1}{4}\\\\\\text{ }&=4-\\frac{1}{2} & \\text{ }&=16-\\frac{1}{4}\\hfill \\\\ \\text{ }&=\\frac{7}{2}\\hfill & \\text{ }&=\\frac{63}{4}\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change from [latex]x=2[\/latex] to [latex]x=4[\/latex] using the formula [latex]\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex].<\/p>\n<div id=\"fs-id1165137401823\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}&\\frac{f\\left(4\\right)-f\\left(2\\right)}{4-2}\\hfill\\text{ }&&\\text{Plug 2 and 4 into the formula.} \\\\ \\text{ }&=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4-2}\\hfill&&\\text{Substitute values for }f\\left(4\\right)\\text{ and }f\\left(2\\right). \\\\ \\text{ }&=\\frac{\\frac{49}{4}}{2}\\hfill &&\\text{Simplify.}\\\\ \\text{ }&=\\frac{49}{8}\\hfill \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436764\" class=\"precalculus tryit\">\n<h3>Try it #2<\/h3>\n<div id=\"ti_01_03_02\">\n<div id=\"fs-id1165137832323\">\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x-2\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] on the interval [latex]\\left[1,\\text{ }9\\right].[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137455570\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137455570\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137455570\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137455571\">[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137772170\">\n<div id=\"fs-id1165137772173\">\n<h3>Example 5:\u00a0 Finding the Average Rate of Change of a Force<\/h3>\n<p id=\"fs-id1165135443718\">The <span class=\"no-emphasis\">electrostatic force\u00a0<\/span>[latex]F,[\/latex] measured in newtons, between two charged particles can be related to the distance between the particles [latex]d,[\/latex] in centimeters, by the formula [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}.[\/latex] Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.<\/p>\n<\/div>\n<div id=\"fs-id1165137696464\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137696464\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137696464\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137770364\">We are computing the average rate of change of [latex]F\\left(d\\right)=\\frac{2}{{d}^{2}}[\/latex] on the interval [latex]\\left[2,6\\right].[\/latex]<\/p>\n<div id=\"fs-id1165135613444\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}\\text{Average rate of change }&=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6-2}\\hfill && \\hfill \\\\&=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6-2}\\hfill && \\text{Simplify}.\\hfill \\\\ &=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4}\\hfill && \\hfill \\\\ &=\\frac{-\\frac{16}{36}}{4}\\hfill && \\text{Combine numerator terms}.\\hfill \\\\ &=-\\frac{1}{9}\\hfill && \\text{Simplify.}\\hfill \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135543242\">The average rate of change is [latex]-\\frac{1}{9}[\/latex] newton per centimeter.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135174952\">\n<div id=\"fs-id1165135174954\">\n<h3>Example 6:\u00a0 Finding an Average Rate of Change as an Expression<\/h3>\n<p id=\"fs-id1165135155397\">Find the average rate of change of [latex]g\\left(t\\right)={t}^{2}+3t+1[\/latex] on the interval [latex]\\left[0,\\text{ }a\\right].[\/latex] The answer will be an expression involving [latex]a.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137418911\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137418911\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137418911\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137418913\">We use the average rate of change formula with input values [latex]a[\/latex] and 0.<\/p>\n<div id=\"fs-id1165134198703\" class=\"unnumbered\" style=\"text-align: center;\">[latex]\\begin{align*}&\\frac{g\\left(a\\right)-g\\left(0\\right)}{a-0}\\hfill && \\text{Evaluate}.\\hfill \\\\ \\text{ }&=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a-0}\\hfill && \\text{Simplify}.\\hfill \\\\ \\text{ }&=\\frac{{a}^{2}+3a+1-1}{a}\\hfill && \\text{Simplify and factor}.\\hfill \\\\ \\text{ }&=\\frac{a\\left(a+3\\right)}{a} \\hfill &&\\text{Cancel }a.\\\\\\text{ }&=a+3\\hfill \\end{align*}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165133316469\">This result tells us the average rate of change in terms of [latex]a[\/latex] between [latex]t=0[\/latex] and any other point [latex]t=a.[\/latex] For example, on the interval [latex]\\left[0,5\\right],[\/latex] the average rate of change would be [latex]5+3=8.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137453128\" class=\"precalculus tryit\">\n<h3>Try it #3<\/h3>\n<div id=\"ti_01_03_03\">\n<div id=\"fs-id1165134149845\">\n<p id=\"fs-id1165134149846\">Find the average rate of change of [latex]f\\left(x\\right)={x}^{2}+2x-8[\/latex] on the interval [latex]\\left[5,\\text{ }a\\right].[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133065696\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165133065696\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165133065696\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133065698\">[latex]a+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440486\" class=\"bc-section section\">\n<h3>Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant<\/h3>\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.<\/p>\n<p>The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. <a class=\"autogenerated-content\" href=\"#Figure_01_03_004\">Figure 3<\/a> shows examples of increasing and decreasing intervals on a function.<\/p>\n<div id=\"attachment_3111\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3111\" class=\"wp-image-3111 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162020\/56ee4dd1a9e462cff7a40c566285b1405e1744ba.jpeg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/><\/p>\n<p id=\"caption-attachment-3111\" class=\"wp-caption-text\"><strong>Figure 3<\/strong> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}\\text{ }-\\text{2}\\right){{\\cup }^{\\text{\u200b}}}^{\\text{\u200b}}\\left(2,\\text{ }\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}\\text{ }2\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"Figure_01_03_004\" class=\"small\"><\/div>\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 called a<strong> local maximum<\/strong>. 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 called a <strong>local minimum<\/strong>. The plural form is \u201clocal minima.\u201d Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is \u201cextremum.\u201d) Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>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>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 <a class=\"autogenerated-content\" href=\"#Figure_01_03_015\">Figure 4<\/a>, the local maximum occurs when [latex]x=-2[\/latex]\u00a0 The maximum value is the output value of 16.\u00a0 The local minimum\u00a0occurs when [latex]x=2[\/latex] .\u00a0 The minimum value is the output value of [latex]-16[\/latex]<\/p>\n<div id=\"attachment_3112\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3112\" class=\"wp-image-3112 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162141\/745142cc108b26ac588ea701460c9fec96f6974c-1.jpeg\" alt=\"The graph shows a curve going up until (-2, 16) and then goes down until (2, -16) and then goes up after that.\" width=\"731\" height=\"467\" \/><\/p>\n<p id=\"caption-attachment-3112\" class=\"wp-caption-text\"><strong>Figure 4<\/strong><\/p>\n<\/div>\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. <a class=\"autogenerated-content\" href=\"#Figure_01_03_005\">Figure 5<\/a> illustrates these ideas for a local maximum.<\/p>\n<div id=\"attachment_3113\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3113\" class=\"wp-image-3113 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162256\/151daece91ddcd0ad086eb4a6a63f0dedfd44ac1.jpeg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/><\/p>\n<p id=\"caption-attachment-3113\" class=\"wp-caption-text\"><strong>Figure 5<\/strong> Definition of a local maximum<\/p>\n<\/div>\n<div id=\"Figure_01_03_005\" class=\"small\"><\/div>\n<p id=\"eip-673\">These observations lead us to a formal definition of local extrema.<\/p>\n<div id=\"fs-id1165134169419\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165134169426\">A function [latex]f[\/latex] is an <strong>increasing function on an open interval<\/strong> if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for every two input values [latex]a[\/latex] and [latex]b[\/latex] in the interval where [latex]b>a.[\/latex]<\/p>\n<p id=\"fs-id1165137668624\">A function [latex]f[\/latex] is a <strong>decreasing function on an open interva<\/strong>l if [latex]f\\left( b \\right)\\lt f\\left(a\\right)[\/latex] for every two input values [latex]a[\/latex] and [latex]b[\/latex] in the interval where [latex]b>a.[\/latex]<\/p>\n<p id=\"fs-id1165135389881\">A function [latex]f[\/latex] has a <strong>local maximum<\/strong> at a point [latex]b[\/latex] in an open interval [latex]\\left(a,c\\right)[\/latex] if [latex]f\\left(b\\right)[\/latex] is greater than or equal to [latex]f\\left(x\\right)[\/latex] for every point [latex]x[\/latex] ([latex]x[\/latex] does not equal [latex]b[\/latex]) in the interval. Likewise, [latex]f[\/latex] has a <strong>local minimum<\/strong> at a point [latex]b[\/latex] in [latex]\\left(a,c\\right)[\/latex] if [latex]f\\left(b\\right)[\/latex] is less than or equal to [latex]f\\left(x\\right)[\/latex] for every [latex]x[\/latex] ([latex]x[\/latex] does not equal [latex]b[\/latex]) in the interval.<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165134266716\">\n<div id=\"fs-id1165134266718\">\n<h3>Example 7:\u00a0 Finding Increasing and Decreasing Intervals on a Graph<\/h3>\n<p id=\"fs-id1165134266723\">Given the function [latex]p\\left(t\\right)[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_01_03_006\">Figure 6<\/a>, identify the intervals on which the function appears to be increasing and decreasing.<\/p>\n<div id=\"attachment_3114\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3114\" class=\"size-full wp-image-3114\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162527\/dab6d2577d252f65a3a3cd76c792f97abc482af3-1.jpeg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"295\" \/><\/p>\n<p id=\"caption-attachment-3114\" class=\"wp-caption-text\"><strong>Figure 6<\/strong><\/p>\n<\/div>\n<div id=\"Figure_01_03_006\" class=\"small\"><\/div>\n<\/div>\n<div id=\"fs-id1165133067194\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165133067194\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165133067194\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133067197\">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 [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on. The function appears to be decreasing until [latex]t=1[\/latex] and then again from [latex]t=3[\/latex] to [latex]t=4.[\/latex]<\/p>\n<p id=\"fs-id1165135369127\">In <span class=\"no-emphasis\">interval notation<\/span>, we would say the function appears to be increasing [latex]\\left(1,\\text{ }3\\right)\\cup\\left(4,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-\\infty,\\text{ }1\\right)\\cup\\left(3,\\text{ }4\\right).[\/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[\/latex], [latex]t=3[\/latex],\u00a0 and [latex]t=4[\/latex]. These points are the local extrema (two minima and a maximum).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165135261521\">\n<div id=\"fs-id1165135261523\">\n<h3>Example 8:\u00a0 Finding Local Extrema from a Graph<\/h3>\n<p id=\"fs-id1165135261528\">Use technology to graph the function [latex]f\\left(x\\right)=\\frac{2}{x}+\\frac{x}{3}.[\/latex] Then use features of your graphing utility to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\n<\/div>\n<div id=\"fs-id1165134255027\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134255027\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134255027\" class=\"hidden-answer\" style=\"display: none\">Using technology, we find that the graph of the function looks like that in <a class=\"autogenerated-content\" href=\"#Figure_01_03_007\">Figure 7<\/a>. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3,[\/latex] and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2.[\/latex]<\/p>\n<div id=\"attachment_3115\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3115\" class=\"size-full wp-image-3115\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162819\/241fa93a9ce8124d48204ada8fe6c8edfa4b69e0.jpeg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/><\/p>\n<p id=\"caption-attachment-3115\" class=\"wp-caption-text\">Figure 7<\/p>\n<\/div>\n<div id=\"Figure_01_03_007\" class=\"small\">\n<h3>Analysis<\/h3>\n<p>Most graphing calculators and graphing utilities can estimate the location of maxima and minima. <a class=\"autogenerated-content\" href=\"#Figure_01_03_008\">Figure 8<\/a> provides screen images from two different technologies, showing the estimate for the local maximum and minimum.<\/p>\n<div id=\"attachment_3116\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3116\" class=\"size-full wp-image-3116\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14162900\/b2c81159521af2d8d476a1a0fb2d335bee24c34e-1.jpeg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/><\/p>\n<p id=\"caption-attachment-3116\" class=\"wp-caption-text\">Figure 8<\/p>\n<\/div>\n<p id=\"fs-id1165134075625\">Based on these estimates, the function is increasing on the interval [latex](-\\infty \\text{,}-\\text{2}\\text{.449)}[\/latex] and [latex]\\left(2.449\\text{,}\\infty \\right).[\/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 [latex]\u00b1\\sqrt{6},[\/latex] but determining this requires calculus.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160233\" class=\"precalculus tryit\">\n<h3>Try it #4<\/h3>\n<div id=\"ti_01_03_04\">\n<div id=\"fs-id1165135640966\">\n<p id=\"fs-id1165135640967\">Use technology to graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] and 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 id=\"fs-id1165135471082\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135471082\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135471082\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135471083\">Using technology, we find the local maximum of 28 appears to occur at [latex]\\left(-1,28\\right),[\/latex] and the local minimum of -80 appears to occur at [latex]\\left(5,-80\\right).[\/latex] The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right).[\/latex]<\/p>\n<p><span id=\"fs-id1165134043615\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3174 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/15150135\/11c7b4fdf95bfa15af5b8e130c14018e252f69b8.jpeg\" alt=\"\" width=\"487\" height=\"328\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_09\" class=\"textbox examples\">\n<div id=\"fs-id1165135367558\">\n<div id=\"fs-id1165137896103\">\n<h3>Example 9:\u00a0 Finding Local Maxima and Minima from a Graph<\/h3>\n<p id=\"fs-id1165135209636\">For the function [latex]f[\/latex] whose graph is shown in <a class=\"autogenerated-content\" href=\"#Figure_01_03_011\">Figure 9<\/a>, find all local maxima and minima.<\/p>\n<div id=\"attachment_3117\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3117\" class=\"size-full wp-image-3117\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163053\/2addb41469229c8466f45cbccda65866650c2c47.jpeg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"368\" \/><\/p>\n<p id=\"caption-attachment-3117\" class=\"wp-caption-text\">Figure 9<\/p>\n<\/div>\n<div id=\"Figure_01_03_011\" class=\"small\"><\/div>\n<\/div>\n<div id=\"fs-id1165135527083\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135527083\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135527083\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135527085\">Observe the graph of [latex]f.[\/latex] The graph attains a local maximum at [latex]x=1[\/latex] because the highest point in an open interval occurs when [latex]x=1.[\/latex] The local maximum is the [latex]y[\/latex]-coordinate at [latex]x=1,[\/latex] which is [latex]2.[\/latex]<\/p>\n<p id=\"fs-id1165134485672\">The graph attains a local minimum at [latex]x=-1[\/latex] because the lowest point in an open interval occurs when [latex]x=-1.[\/latex] The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1,[\/latex] which is\u00a0 [latex]-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134544960\" class=\"bc-section section\">\n<h3>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h3>\n<p id=\"fs-id1165135704895\">We will now return to our toolkit functions and discuss their graphical behavior in <a class=\"autogenerated-content\" href=\"#Figure_01_03_012\">Figure 10<\/a>, <a class=\"autogenerated-content\" href=\"#Figure_01_03_016\">Figure 11<\/a>, and <a class=\"autogenerated-content\" href=\"#Figure_01_03_017\">Figure 12<\/a>.<\/p>\n<div id=\"attachment_3118\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3118\" class=\"size-full wp-image-3118\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163211\/f22b41248dd22d00a63dcb02f358ed1391fb7eca.jpeg\" alt=\"Table showing the increasing and decreasing intervals of the toolkit functions.\" width=\"975\" height=\"525\" \/><\/p>\n<p id=\"caption-attachment-3118\" class=\"wp-caption-text\">Figure 10<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_3119\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3119\" class=\"size-full wp-image-3119\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163250\/a67cdc93a5dc0fbc3016a4949587f64a835c309a.jpeg\" alt=\"Table showing the increasing and decreasing intervals of the toolkit functions.\" width=\"975\" height=\"525\" \/><\/p>\n<p id=\"caption-attachment-3119\" class=\"wp-caption-text\">Figure 11<\/p>\n<\/div>\n<div id=\"attachment_3120\" style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3120\" class=\"size-full wp-image-3120\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163352\/cb19bea012bc887dec6e4cc83ebcfd51da042cd0.jpeg\" alt=\"Table showing the increasing and decreasing intervals of the toolkit functions.\" width=\"975\" height=\"525\" \/><\/p>\n<p id=\"caption-attachment-3120\" class=\"wp-caption-text\">Figure 12<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134381626\" class=\"bc-section section\">\n<h3>Use A Graph to Locate the Absolute Maximum and Absolute Minimum (Optional)<\/h3>\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\\text{-}[\/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 <a class=\"autogenerated-content\" href=\"#Figure_01_03_014\">Figure 13<\/a>.<\/p>\n<div id=\"attachment_3121\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3121\" class=\"size-full wp-image-3121\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163447\/bb08f4a7682898031911c3e11d4103b49d309391.jpeg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/><\/p>\n<p id=\"caption-attachment-3121\" class=\"wp-caption-text\">Figure 13<\/p>\n<\/div>\n<div id=\"Figure_01_03_014\" class=\"small\"><\/div>\n<p id=\"fs-id1165137692066\">Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\n<div id=\"fs-id1165135251290\">\n<div class=\"textbox definitions\">\n<h3>Definition<\/h3>\n<p id=\"fs-id1165132939786\">The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f.[\/latex]<\/p>\n<p id=\"fs-id1165137932685\">The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165134047533\">\n<div id=\"fs-id1165134047535\">\n<h3>Example 10:\u00a0 Finding Absolute Maxima and Minima from a Graph<\/h3>\n<p id=\"fs-id1165133394704\">For the function [latex]f[\/latex] shown in <a class=\"autogenerated-content\" href=\"#Figure_01_03_013\">Figure 14<\/a>, find all absolute maxima and minima.<\/p>\n<div id=\"attachment_3122\" style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3122\" class=\"size-full wp-image-3122\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/01\/14163530\/913701fd366ea33bac8906953907480a87bfd2cf.jpeg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/><\/p>\n<p id=\"caption-attachment-3122\" class=\"wp-caption-text\">Figure 14<\/p>\n<\/div>\n<div id=\"Figure_01_03_013\" class=\"small\"><\/div>\n<\/div>\n<div id=\"fs-id1165135532368\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135532368\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135532368\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135532371\">Observe the graph of [latex]f.[\/latex] The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [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 <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2,[\/latex] which is [latex]16.[\/latex]<\/p>\n<p>The graph attains an absolute minimum at [latex]x=3,[\/latex] because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3,[\/latex] which is [latex]-10.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135708033\" class=\"precalculus media\">\n<p id=\"fs-id1165135708038\">Access this online resource for additional instruction and practice with rates of change.<\/p>\n<ul id=\"fs-id1165135708041\">\n<li><a href=\"http:\/\/openstax.org\/l\/aroc\">Average Rate of Change<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Average Rate of Change\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F-7Poa3i1ZU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541564\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165135358784\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td class=\"border\">Average rate of change<\/td>\n<td class=\"border\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135481945\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135481952\">\n<li>A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data over an interval.<\/li>\n<li>Identifying points that mark the interval on a graph can be used to find the average rate of change.<\/li>\n<li>Comparing pairs of input and output values in a table can also be used to find the average rate of change.<\/li>\n<li>An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.<\/li>\n<li>The average rate of change can sometimes be determined as an expression.<\/li>\n<li>A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.<\/li>\n<li>A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.<\/li>\n<li>A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.<\/li>\n<li>Minima and maxima are also called extrema.<\/li>\n<li>We can find local extrema from a graph.<\/li>\n<li>The highest and lowest points on a graph indicate the absolute maxima and minima.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165133052903\">\n<dt>absolute maximum<\/dt>\n<dd id=\"fs-id1165133052908\">the greatest value of a function over an interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133052911\">\n<dt>absolute minimum<\/dt>\n<dd id=\"fs-id1165133052916\">the lowest value of a function over an interval<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133052921\">\n<dt>average rate of change<\/dt>\n<dd id=\"fs-id1165133052926\">the difference in the output values of a function found for two values of the input divided by the difference between the inputs<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135264639\">\n<dt>decreasing function<\/dt>\n<dd id=\"fs-id1165135264645\">a function is decreasing in some open interval if [latex]f\\left(b\\right) \\lt f\\left(a\\right)[\/latex] for any two input values\u00a0 [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135639824\">\n<dt>increasing function<\/dt>\n<dd id=\"fs-id1165135639829\">a function is increasing in some open interval if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135536408\">\n<dt>local extrema<\/dt>\n<dd id=\"fs-id1165135536413\">collectively, all of a function&#8217;s local maxima and minima<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135536416\">\n<dt>local maximum<\/dt>\n<dd id=\"fs-id1165135412035\">a value of the input where a function changes from increasing to decreasing as the input value increases.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135412040\">\n<dt>local minimum<\/dt>\n<dd id=\"fs-id1165135412046\">a value of the input where a function changes from decreasing to increasing as the input value increases.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135412050\">\n<dt>rate of change<\/dt>\n<dd id=\"fs-id1165135412054\">the change of an output quantity relative to the change of the input quantity<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-44\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Rates of Change and Behavior of Graphs. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:O5iKX5Vm@10\/Rates-of-Change-and-Behavior-of-Graphs\">https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:O5iKX5Vm@10\/Rates-of-Change-and-Behavior-of-Graphs<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 1. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-44-1\"><a href=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524<\/a>. Accessed 3\/5\/2014. <a href=\"#return-footnote-44-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":311,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Rates of Change and Behavior of Graphs\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/l3_8ZlRi@1.94:O5iKX5Vm@10\/Rates-of-Change-and-Behavior-of-Graphs\",\"project\":\"Essential Precalcus, Part 1\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-44","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/44","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":29,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions"}],"predecessor-version":[{"id":2938,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions\/2938"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/44\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=44"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=44"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=44"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}