{"id":36,"date":"2021-06-04T18:08:17","date_gmt":"2021-06-04T18:08:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/rates-of-change-and-behavior-of-graphs\/"},"modified":"2021-06-14T23:55:01","modified_gmt":"2021-06-14T23:55:01","slug":"rates-of-change-and-behavior-of-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/chapter\/rates-of-change-and-behavior-of-graphs\/","title":{"raw":"6.3 Average Rate of Change of a Function","rendered":"6.3 Average Rate of Change of a Function"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the average rate of change of a function.<\/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. The table below[footnote]http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.[\/footnote]\u00a0lists 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 summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>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<h2>Finding the Average Rate of Change of a Function<\/h2>\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\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\r\n\r\n<div id=\"fs-id1165135452482\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{\\text{Change in output}}{\\text{Change in input}} \\\\[1mm] &amp;=\\frac{\\Delta y}{\\Delta x} \\\\[1mm] &amp;= \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &amp;= \\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/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 \"delta-<em>y<\/em> over delta-<em>x<\/em>\" or \"the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].\" 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\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=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/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<div class=\"textbox\">\r\n<h3 class=\"title\">A General Note: Rate of Change<\/h3>\r\n<p id=\"fs-id1165137780744\">A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are \"output units per input units.\"<\/p>\r\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\r\n\r\n<div id=\"fs-id1165134060431\" class=\"equation\" 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]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137762240\">How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/h3>\r\n<ol id=\"fs-id1165137442714\">\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=\"example\">\r\n<div id=\"fs-id1165135485962\" class=\"exercise\">\r\n<div id=\"fs-id1165137464225\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Computing an Average Rate of Change<\/h3>\r\n<p id=\"fs-id1165137603118\">Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"562005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"562005\"]\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<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&amp;=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &amp;=\\frac{2.41-2.84}{2009 - 2007} \\\\[1mm] &amp;=\\frac{-0.43}{2\\text{ years}} \\\\[1mm] &amp;={-0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137784092\">Note that a decrease is expressed by a negative change or \"negative increase.\" A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video provides another example of how to find the average rate of change between two points from a table of values.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=iJ_0nPUUlOg&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135160759\">Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\r\n\r\n<table summary=\"Two rows and nine columns. The first row is labeled, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\r\n<td>2005<\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n<td>2.31<\/td>\r\n<td>2.62<\/td>\r\n<td>2.84<\/td>\r\n<td>3.30<\/td>\r\n<td>2.41<\/td>\r\n<td>2.84<\/td>\r\n<td>3.58<\/td>\r\n<td>3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"175600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"175600\"]\r\n\r\n[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_02\" class=\"example\">\r\n<div id=\"fs-id1165137851963\" class=\"exercise\">\r\n<div id=\"fs-id1165137437853\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Computing Average Rate of Change from a Graph<\/h3>\r\nGiven the function [latex]g\\left(t\\right)[\/latex] shown in Figure 1, find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"806109\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"806109\"]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0022.jpg\" 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\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\nAt [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].<span id=\"fs-id1165137387448\">\r\n<\/span>\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<p style=\"text-align: center;\">[latex]\\frac{1 - 4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135538482\">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].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_03\" class=\"example\">\r\n<div id=\"fs-id1165135536188\" class=\"exercise\">\r\n<div id=\"fs-id1165137835656\" class=\"problem textbox shaded\">\r\n<h3>Example 3: 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 the table below.\u00a0Find 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, \"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><strong><em>t<\/em> (hours)<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<td>7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\r\n<td>10<\/td>\r\n<td>55<\/td>\r\n<td>90<\/td>\r\n<td>153<\/td>\r\n<td>214<\/td>\r\n<td>240<\/td>\r\n<td>282<\/td>\r\n<td>300<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"566859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"566859\"]\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<p style=\"text-align: center;\">[latex]\\frac{292 - 10}{6 - 0} =\\frac{282}{6} =47[\/latex]<\/p>\r\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137731074\">Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_01_03_04\" class=\"example\">\r\n<div id=\"fs-id1165135353057\" class=\"exercise\">\r\n<div id=\"fs-id1165135383644\" class=\"problem textbox shaded\">\r\n<h3>Example 4: 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{4].}[\/latex]<\/p>\r\n[reveal-answer q=\"222718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222718\"]\r\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(2\\right)&amp;={2}^{2}-\\frac{1}{2} &amp;&amp;&amp; f\\left(4\\right)&amp;={4}^{2}-\\frac{1}{4} \\\\ &amp;=4-\\frac{1}{2} &amp;&amp;&amp;&amp; =16-{1}{4} \\\\ &amp;=\\frac{7}{2} &amp;&amp;&amp;&amp; =\\frac{63}{4} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[1mm] &amp;=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[1mm] &amp;=\\frac{\\frac{49}{4}}{2} \\\\[1mm] &amp;=\\frac{49}{8} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video provides another example of finding the average rate of change of a function given a formula and an interval.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=g93QEKJXeu4&amp;feature=youtu.be\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\r\n[reveal-answer q=\"191250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"191250\"]\r\n\r\n[latex]\\frac{1}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]165703[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"Example_01_03_05\" class=\"example\">\r\n<div id=\"fs-id1165137772170\" class=\"exercise\">\r\n<div id=\"fs-id1165137772173\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Finding the Average Rate of Change of a Force<\/h3>\r\n<p id=\"fs-id1165135443718\">The <strong>electrostatic force<\/strong> [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[reveal-answer q=\"271117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"271117\"]\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<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change }&amp;=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2} \\\\[1mm] &amp;=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2} &amp;&amp; \\text{Simplify}. \\\\[1mm] &amp;=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4} \\\\[1mm] &amp;=\\frac{-\\frac{16}{36}}{4} &amp;&amp;\\text{Combine numerator terms}. \\\\[1mm] &amp;=-\\frac{1}{9}&amp;&amp;\\text{Simplify}\\end{align}[\/latex]<\/p>\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=\"example\">\r\n<div id=\"fs-id1165135174952\" class=\"exercise\">\r\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\r\n<h3>Example 6: 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,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\r\n[reveal-answer q=\"138559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138559\"]\r\n<p id=\"fs-id1165137418913\">We use the average rate of change formula.<\/p>\r\n<p style=\"text-align: center;\">\u200b[latex]\\begin{align}\\text{Average rate of change}&amp;=\\frac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&amp;&amp;\\text{Evaluate} \\\\[1mm] &amp;\u200b=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&amp;&amp;\\text{Simplify}\u200b \\\\[1mm] &amp;=\\frac{{a}^{2}+3a+1 - 1}{a}&amp;&amp;\\text{Simplify and factor}\u200b \\\\[1mm] &amp;=\\frac{a\\left(a+3\\right)}{a}&amp;&amp;\\text{Divide by the common factor }a\u200b \\\\[1mm] &amp;=a+3 \\end{align}[\/latex]<\/p>\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 class=\"bcc-box bcc-success\"><\/div>\r\n<section id=\"fs-id1165134544960\">\r\n<div id=\"fs-id1165135708033\" class=\"note precalculus media\"><section id=\"fs-id1165135541564\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165135358784\" summary=\"..\"><colgroup> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>Average rate of change<\/td>\r\n<td>[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{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<\/section><section id=\"fs-id1165135481945\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\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.<\/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<\/ul>\r\n<div style=\"line-height: 1.5;\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165133052921\" class=\"definition\">\r\n \t<dt><strong>average rate of change<\/strong><\/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-id1165135412050\" class=\"definition\">\r\n \t<dt><strong>rate of change<\/strong><\/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>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the average rate of change of a function.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165135194500\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below<a class=\"footnote\" title=\"http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014.\" id=\"return-footnote-36-1\" href=\"#footnote-36-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0lists 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 summary=\"Two rows and nine columns. The first row is labeled,\">\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><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>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<h2>Finding the Average Rate of Change of a Function<\/h2>\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\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the <strong>average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\n<div id=\"fs-id1165135452482\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\frac{\\text{Change in output}}{\\text{Change in input}} \\\\[1mm] &=\\frac{\\Delta y}{\\Delta x} \\\\[1mm] &= \\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &= \\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}\\end{align}[\/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 &#8220;delta-<em>y<\/em> over delta-<em>x<\/em>&#8221; or &#8220;the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].&#8221; 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 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=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/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<div class=\"textbox\">\n<h3 class=\"title\">A General Note: Rate of Change<\/h3>\n<p id=\"fs-id1165137780744\">A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are &#8220;output units per input units.&#8221;<\/p>\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\n<div id=\"fs-id1165134060431\" class=\"equation\" 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]<\/div>\n<\/div>\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137762240\">How To: Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex].<\/h3>\n<ol id=\"fs-id1165137442714\">\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=\"example\">\n<div id=\"fs-id1165135485962\" class=\"exercise\">\n<div id=\"fs-id1165137464225\" class=\"problem textbox shaded\">\n<h3>Example 1: Computing an Average Rate of Change<\/h3>\n<p id=\"fs-id1165137603118\">Using the data in the table below, find the average rate of change of the price of gasoline between 2007 and 2009.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\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><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q562005\">Show Solution<\/span><\/p>\n<div id=\"q562005\" 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<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{\\Delta y}{\\Delta x}&=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\\\[1mm] &=\\frac{2.41-2.84}{2009 - 2007} \\\\[1mm] &=\\frac{-0.43}{2\\text{ years}} \\\\[1mm] &={-0.22}\\text{ per year}\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137784092\">Note that a decrease is expressed by a negative change or &#8220;negative increase.&#8221; A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of how to find the average rate of change between two points from a table of values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Find the Average Rate of Change From a Table - Temperatures\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iJ_0nPUUlOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135160759\">Using the data in the table below,\u00a0find the average rate of change between 2005 and 2010.<\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>[latex]y[\/latex]<\/strong><\/td>\n<td>2005<\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\n<td>2.31<\/td>\n<td>2.62<\/td>\n<td>2.84<\/td>\n<td>3.30<\/td>\n<td>2.41<\/td>\n<td>2.84<\/td>\n<td>3.58<\/td>\n<td>3.68<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q175600\">Show Solution<\/span><\/p>\n<div id=\"q175600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{$2.84-$2.31}{5\\text{ years}}=\\dfrac{$0.53}{5\\text{ years}}=$0.106[\/latex] per year.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_02\" class=\"example\">\n<div id=\"fs-id1165137851963\" class=\"exercise\">\n<div id=\"fs-id1165137437853\" class=\"problem textbox shaded\">\n<h3>Example 2: Computing Average Rate of Change from a Graph<\/h3>\n<p>Given the function [latex]g\\left(t\\right)[\/latex] shown in Figure 1, find the average rate of change on the interval [latex]\\left[-1,2\\right][\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0012.jpg\" alt=\"Graph of a parabola.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q806109\">Show Solution<\/span><\/p>\n<div id=\"q806109\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010553\/CNX_Precalc_Figure_01_03_0022.jpg\" 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 class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p>At [latex]t=-1[\/latex], the graph\u00a0shows [latex]g\\left(-1\\right)=4[\/latex]. At [latex]t=2[\/latex], the graph shows [latex]g\\left(2\\right)=1[\/latex].<span id=\"fs-id1165137387448\"><br \/>\n<\/span><\/p>\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<p style=\"text-align: center;\">[latex]\\frac{1 - 4}{2-\\left(-1\\right)}=\\frac{-3}{3}=-1[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165135538482\">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].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_03\" class=\"example\">\n<div id=\"fs-id1165135536188\" class=\"exercise\">\n<div id=\"fs-id1165137835656\" class=\"problem textbox shaded\">\n<h3>Example 3: 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 the table below.\u00a0Find 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,\">\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><strong><em>t<\/em> (hours)<\/strong><\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td><strong><em>D<\/em>(<em>t<\/em>) (miles)<\/strong><\/td>\n<td>10<\/td>\n<td>55<\/td>\n<td>90<\/td>\n<td>153<\/td>\n<td>214<\/td>\n<td>240<\/td>\n<td>282<\/td>\n<td>300<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q566859\">Show Solution<\/span><\/p>\n<div id=\"q566859\" 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<p style=\"text-align: center;\">[latex]\\frac{292 - 10}{6 - 0} =\\frac{282}{6} =47[\/latex]<\/p>\n<p id=\"fs-id1165135400200\">The average speed is 47 miles per hour.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137731074\">Because the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average speed is 63 miles per hour.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_03_04\" class=\"example\">\n<div id=\"fs-id1165135353057\" class=\"exercise\">\n<div id=\"fs-id1165135383644\" class=\"problem textbox shaded\">\n<h3>Example 4: 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{4].}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222718\">Show Solution<\/span><\/p>\n<div id=\"q222718\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137595441\">We can start by computing the function values at each <strong>endpoint<\/strong> of the interval.<\/p>\n<p 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} \\\\ &=4-\\frac{1}{2} &&&& =16-{1}{4} \\\\ &=\\frac{7}{2} &&&& =\\frac{63}{4} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137427523\">Now we compute the average rate of change.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change}&=\\frac{f\\left(4\\right)-f\\left(2\\right)}{4 - 2} \\\\[1mm] &=\\frac{\\frac{63}{4}-\\frac{7}{2}}{4 - 2} \\\\[1mm] &=\\frac{\\frac{49}{4}}{2} \\\\[1mm] &=\\frac{49}{8} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video provides another example of finding the average rate of change of a function given a formula and an interval.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Find the Average Rate of Change Given a Function Rule\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/g93QEKJXeu4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137832324\">Find the average rate of change of [latex]f\\left(x\\right)=x - 2\\sqrt{x}[\/latex] on the interval [latex]\\left[1,9\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q191250\">Show Solution<\/span><\/p>\n<div id=\"q191250\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm165703\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=165703&theme=oea&iframe_resize_id=ohm165703\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"Example_01_03_05\" class=\"example\">\n<div id=\"fs-id1165137772170\" class=\"exercise\">\n<div id=\"fs-id1165137772173\" class=\"problem textbox shaded\">\n<h3>Example 5: Finding the Average Rate of Change of a Force<\/h3>\n<p id=\"fs-id1165135443718\">The <strong>electrostatic force<\/strong> [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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q271117\">Show Solution<\/span><\/p>\n<div id=\"q271117\" 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<p style=\"text-align: center;\">[latex]\\begin{align}\\text{Average rate of change }&=\\frac{F\\left(6\\right)-F\\left(2\\right)}{6 - 2} \\\\[1mm] &=\\frac{\\frac{2}{{6}^{2}}-\\frac{2}{{2}^{2}}}{6 - 2} && \\text{Simplify}. \\\\[1mm] &=\\frac{\\frac{2}{36}-\\frac{2}{4}}{4} \\\\[1mm] &=\\frac{-\\frac{16}{36}}{4} &&\\text{Combine numerator terms}. \\\\[1mm] &=-\\frac{1}{9}&&\\text{Simplify}\\end{align}[\/latex]<\/p>\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=\"example\">\n<div id=\"fs-id1165135174952\" class=\"exercise\">\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\n<h3>Example 6: 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,a\\right][\/latex]. The answer will be an expression involving [latex]a[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q138559\">Show Solution<\/span><\/p>\n<div id=\"q138559\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137418913\">We use the average rate of change formula.<\/p>\n<p style=\"text-align: center;\">\u200b[latex]\\begin{align}\\text{Average rate of change}&=\\frac{g\\left(a\\right)-g\\left(0\\right)}{a - 0}&&\\text{Evaluate} \\\\[1mm] &\u200b=\\frac{\\left({a}^{2}+3a+1\\right)-\\left({0}^{2}+3\\left(0\\right)+1\\right)}{a - 0}&&\\text{Simplify}\u200b \\\\[1mm] &=\\frac{{a}^{2}+3a+1 - 1}{a}&&\\text{Simplify and factor}\u200b \\\\[1mm] &=\\frac{a\\left(a+3\\right)}{a}&&\\text{Divide by the common factor }a\u200b \\\\[1mm] &=a+3 \\end{align}[\/latex]<\/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 class=\"bcc-box bcc-success\"><\/div>\n<section id=\"fs-id1165134544960\">\n<div id=\"fs-id1165135708033\" class=\"note precalculus media\">\n<section id=\"fs-id1165135541564\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<table id=\"eip-id1165135358784\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>Average rate of change<\/td>\n<td>[latex]\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135481945\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\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.<\/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<\/ul>\n<div style=\"line-height: 1.5;\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133052921\" class=\"definition\">\n<dt><strong>average rate of change<\/strong><\/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-id1165135412050\" class=\"definition\">\n<dt><strong>rate of change<\/strong><\/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<\/section>\n<\/div>\n<\/section>\n<\/div>\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-36\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <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-36-1\">http:\/\/www.eia.gov\/totalenergy\/data\/annual\/showtext.cfm?t=ptb0524. Accessed 3\/5\/2014. <a href=\"#return-footnote-36-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":23485,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"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-36","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/36","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/36\/revisions"}],"predecessor-version":[{"id":683,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/36\/revisions\/683"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapters\/36\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/media?parent=36"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=36"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/contributor?post=36"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-precalculus\/wp-json\/wp\/v2\/license?post=36"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}