{"id":8242,"date":"2023-09-29T14:21:10","date_gmt":"2023-09-29T14:21:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8242"},"modified":"2024-09-12T05:07:30","modified_gmt":"2024-09-12T05:07:30","slug":"modeling-complex-scenarios-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/modeling-complex-scenarios-background-youll-need-2\/","title":{"raw":"Modeling Complex Scenarios: Background You\u2019ll Need 2","rendered":"Modeling Complex Scenarios: Background You\u2019ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Determine the rate of change<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rate of Change<\/h2>\r\n<p id=\"fs-id1165135194500\" class=\"has-noteref\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below lists the average cost, in dollars, of a gallon of gasoline for the years 2015\u20132022.[footnote]https:\/\/www.statista.com\/statistics\/204740\/retail-price-of-gasoline-in-the-united-states-since-1990\/[\/footnote] The cost of gasoline can be considered as a function of year.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\r\n<td data-align=\"center\">2015<\/td>\r\n<td data-align=\"center\">2016<\/td>\r\n<td data-align=\"center\">2017<\/td>\r\n<td data-align=\"center\">2018<\/td>\r\n<td data-align=\"center\">2019<\/td>\r\n<td data-align=\"center\">2020<\/td>\r\n<td data-align=\"center\">2021<\/td>\r\n<td data-align=\"center\">2022<\/td>\r\n<\/tr>\r\n<tr>\r\n<td width=\"20%\" data-align=\"center\">[latex]C(y)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2.43[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2.14[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2.42[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2.72[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2.60[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2.17[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]3.01[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]3.95[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2015 and 2022, we could compute that the cost per gallon had increased from [latex]$2.43[\/latex] to [latex]$3.95[\/latex], an increase of [latex]$1.52[\/latex]. While this is interesting, it might be more useful to look at how much the price changed\u00a0<em data-effect=\"italics\">per year<\/em>.<\/p>\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\u00a0<strong>rate of change<\/strong>\u00a0because 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\u00a0<strong>average rate of change<\/strong>\u00a0over 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<center>[latex] \\begin{array}{rcl} \\text{Average rate of change} &amp; = &amp; \\frac{\\text{Change in output}}{\\text{Change in input}} \\\\ &amp; = &amp; \\frac{\\Delta y}{\\Delta x} \\\\ &amp; = &amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\ &amp; = &amp; \\frac{f(x_2) - f(x_1)}{x_2 - x_1} \\\\ \\end{array} [\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p id=\"fs-id1165135471272\">The Greek letter\u00a0[latex]\\Delta[\/latex]\u00a0(delta) signifies the change in a quantity; we read the ratio as \u201cdelta-[latex]y[\/latex]\u00a0over delta-[latex]x[\/latex]\u201d or \u201cthe change in\u00a0[latex]y[\/latex]\u00a0divided by the change in\u00a0[latex]x[\/latex].\u201d Occasionally we write\u00a0[latex]\\Delta f[\/latex]\u00a0instead of\u00a0[latex]\\Delta y[\/latex]\u00a0which 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 [latex]$1.52[\/latex] from 2015 to 2022. Over [latex]7[\/latex] years, the average rate of change was<\/p>\r\n<center>[latex]\\frac{\\Delta y}{\\Delta x} = \\frac{$1.52}{7 \\text{ years}} \\approx 0.217 \\text{ dollars per year}[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about [latex]21.7\u00a2[\/latex] each year.<\/p>\r\n<p id=\"fs-id1165135397217\">Other examples of rates of change include:<\/p>\r\n<ul id=\"fs-id1165137424067\">\r\n\t<li>A population of rats increasing by [latex]40[\/latex] rats per week<\/li>\r\n\t<li>A car traveling [latex]68[\/latex] miles per hour (distance traveled changes by [latex]68[\/latex] miles each hour as time passes)<\/li>\r\n\t<li>A car driving [latex]27[\/latex] miles per gallon (distance traveled changes by [latex]27[\/latex] miles for each gallon)<\/li>\r\n\t<li>The current through an electrical circuit increasing by [latex]0.125[\/latex] amperes for every volt of increased voltage<\/li>\r\n\t<li>The amount of money in a college account decreasing by [latex]$4,000[\/latex] per quarter<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>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 \u201coutput units per input units.\u201d<\/p>\r\n<p>&nbsp;<\/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<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x} = \\frac{f(x_2) - f(x_1)}{x_2 - x_1} [\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><b>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].<\/b><\/p>\r\n<ol id=\"fs-id1165137442714\" type=\"1\">\r\n\t<li>Calculate the difference\u00a0[latex]y_2 - y_1 = \\Delta y[\/latex]<\/li>\r\n\t<li>Calculate the difference\u00a0[latex]x_2 - x_1 = \\Delta x[\/latex]<\/li>\r\n\t<li>Find the ratio\u00a0[latex]\\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Using the data in the table above, find the average rate of change of the price of gasoline between 2018 and 2020.<\/p>\r\n\r\n[reveal-answer q=\"946624\"]Show Answer[\/reveal-answer] [hidden-answer a=\"946624\"] In 2018, the price of gasoline was [latex]$2.72[\/latex]. In 2020, the cost was [latex]$2.17[\/latex]. The average rate of change is\r\n\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} \\frac{\\Delta y}{\\Delta x} &amp; = &amp; \\frac{y_2 - y_1}{x_2 - x_1} \\\\ &amp; = &amp; \\frac{\\$2.17 - $2.72}{2020 - 2018} \\\\ &amp; = &amp; \\frac{-\\$0.55}{2 \\text{ years}} \\\\ &amp; = &amp; -\\$0.275 \\text{ per year} \\\\ \\end{array} [\/latex]<\/p>\r\n<strong>Analysis<\/strong> Note that a decrease is expressed by a negative change or \u201cnegative increase.\u201d A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases. [\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13914[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine the rate of change<\/li>\n<\/ul>\n<\/section>\n<h2>Rate of Change<\/h2>\n<p id=\"fs-id1165135194500\" class=\"has-noteref\">Gasoline costs have experienced some wild fluctuations over the last several decades. The table below lists the average cost, in dollars, of a gallon of gasoline for the years 2015\u20132022.<a class=\"footnote\" title=\"https:\/\/www.statista.com\/statistics\/204740\/retail-price-of-gasoline-in-the-united-states-since-1990\/\" id=\"return-footnote-8242-1\" href=\"#footnote-8242-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> The cost of gasoline can be considered as a function of year.<\/p>\n<table>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\n<td data-align=\"center\">2015<\/td>\n<td data-align=\"center\">2016<\/td>\n<td data-align=\"center\">2017<\/td>\n<td data-align=\"center\">2018<\/td>\n<td data-align=\"center\">2019<\/td>\n<td data-align=\"center\">2020<\/td>\n<td data-align=\"center\">2021<\/td>\n<td data-align=\"center\">2022<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\" style=\"width: 20%;\">[latex]C(y)[\/latex]<\/td>\n<td data-align=\"center\">[latex]2.43[\/latex]<\/td>\n<td data-align=\"center\">[latex]2.14[\/latex]<\/td>\n<td data-align=\"center\">[latex]2.42[\/latex]<\/td>\n<td data-align=\"center\">[latex]2.72[\/latex]<\/td>\n<td data-align=\"center\">[latex]2.60[\/latex]<\/td>\n<td data-align=\"center\">[latex]2.17[\/latex]<\/td>\n<td data-align=\"center\">[latex]3.01[\/latex]<\/td>\n<td data-align=\"center\">[latex]3.95[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165133097252\">If we were interested only in how the gasoline prices changed between 2015 and 2022, we could compute that the cost per gallon had increased from [latex]$2.43[\/latex] to [latex]$3.95[\/latex], an increase of [latex]$1.52[\/latex]. While this is interesting, it might be more useful to look at how much the price changed\u00a0<em data-effect=\"italics\">per year<\/em>.<\/p>\n<h3>Finding the Average Rate of Change of a Function<\/h3>\n<p id=\"fs-id1165137834011\">The price change per year is a\u00a0<strong>rate of change<\/strong>\u00a0because 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\u00a0<strong>average rate of change<\/strong>\u00a0over 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 style=\"text-align: center;\">[latex]\\begin{array}{rcl} \\text{Average rate of change} & = & \\frac{\\text{Change in output}}{\\text{Change in input}} \\\\ & = & \\frac{\\Delta y}{\\Delta x} \\\\ & = & \\frac{y_2 - y_1}{x_2 - x_1} \\\\ & = & \\frac{f(x_2) - f(x_1)}{x_2 - x_1} \\\\ \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165135471272\">The Greek letter\u00a0[latex]\\Delta[\/latex]\u00a0(delta) signifies the change in a quantity; we read the ratio as \u201cdelta-[latex]y[\/latex]\u00a0over delta-[latex]x[\/latex]\u201d or \u201cthe change in\u00a0[latex]y[\/latex]\u00a0divided by the change in\u00a0[latex]x[\/latex].\u201d Occasionally we write\u00a0[latex]\\Delta f[\/latex]\u00a0instead of\u00a0[latex]\\Delta y[\/latex]\u00a0which 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 [latex]$1.52[\/latex] from 2015 to 2022. Over [latex]7[\/latex] years, the average rate of change was<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x} = \\frac{$1.52}{7 \\text{ years}} \\approx 0.217 \\text{ dollars per year}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137418924\">On average, the price of gas increased by about [latex]21.7\u00a2[\/latex] 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 [latex]40[\/latex] rats per week<\/li>\n<li>A car traveling [latex]68[\/latex] miles per hour (distance traveled changes by [latex]68[\/latex] miles each hour as time passes)<\/li>\n<li>A car driving [latex]27[\/latex] miles per gallon (distance traveled changes by [latex]27[\/latex] miles for each gallon)<\/li>\n<li>The current through an electrical circuit increasing by [latex]0.125[\/latex] amperes for every volt of increased voltage<\/li>\n<li>The amount of money in a college account decreasing by [latex]$4,000[\/latex] per quarter<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\">\n<h3>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 \u201coutput units per input units.\u201d<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137544638\">The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\Delta y}{\\Delta x} = \\frac{f(x_2) - f(x_1)}{x_2 - x_1}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><b>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].<\/b><\/p>\n<ol id=\"fs-id1165137442714\" type=\"1\">\n<li>Calculate the difference\u00a0[latex]y_2 - y_1 = \\Delta y[\/latex]<\/li>\n<li>Calculate the difference\u00a0[latex]x_2 - x_1 = \\Delta x[\/latex]<\/li>\n<li>Find the ratio\u00a0[latex]\\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Using the data in the table above, find the average rate of change of the price of gasoline between 2018 and 2020.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q946624\">Show Answer<\/button> <\/p>\n<div id=\"q946624\" class=\"hidden-answer\" style=\"display: none\"> In 2018, the price of gasoline was [latex]$2.72[\/latex]. In 2020, the cost was [latex]$2.17[\/latex]. The average rate of change is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} \\frac{\\Delta y}{\\Delta x} & = & \\frac{y_2 - y_1}{x_2 - x_1} \\\\ & = & \\frac{\\$2.17 - $2.72}{2020 - 2018} \\\\ & = & \\frac{-\\$0.55}{2 \\text{ years}} \\\\ & = & -\\$0.275 \\text{ per year} \\\\ \\end{array}[\/latex]<\/p>\n<p><strong>Analysis<\/strong> Note that a decrease is expressed by a negative change or \u201cnegative increase.\u201d A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases. <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13914\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13914&theme=lumen&iframe_resize_id=ohm13914&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-8242-1\">https:\/\/www.statista.com\/statistics\/204740\/retail-price-of-gasoline-in-the-united-states-since-1990\/ <a href=\"#return-footnote-8242-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":8093,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8242"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":27,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8242\/revisions"}],"predecessor-version":[{"id":14795,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8242\/revisions\/14795"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/8093"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8242\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8242"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8242"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8242"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8242"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}