{"id":2901,"date":"2024-06-10T19:17:36","date_gmt":"2024-06-10T19:17:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2901"},"modified":"2024-08-05T01:53:11","modified_gmt":"2024-08-05T01:53:11","slug":"derivatives-as-rates-of-change-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-as-rates-of-change-fresh-take\/","title":{"raw":"Derivatives as Rates of Change: Fresh Take","rendered":"Derivatives as Rates of Change: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate how quantities change on average over time<\/li>\r\n\t<li>Use rates of change to figure out how an object\u2019s position, speed, and acceleration are changing over time<\/li>\r\n\t<li>Estimate future population sizes using current data and how fast the population is growing<\/li>\r\n\t<li>Use derivatives to determine the cost and revenue of producing one more unit in a business<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Amount of Change Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Amount of Change:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Change in [latex]y[\/latex]-values over an interval [latex][a, a+h][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Given by [latex]f(a+h) - f(a)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Average Rate of Change:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Ratio of amount of change to change in [latex]x[\/latex]-values<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Formula: [latex]\\frac{f(a+h) - f(a)}{h}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Amount of Change Formula:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Approximates [latex]f(a+h)[\/latex] using [latex]f(a)[\/latex] and [latex]f'(a)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Formula: [latex]f(a+h) \\approx f(a) + f'(a)h[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Accuracy depends on the size of [latex]h[\/latex] and the behavior of [latex]f'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738905166\">Given [latex]f(10)=-5[\/latex] and [latex]f^{\\prime}(10)=6[\/latex], estimate [latex]f(10.1)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739097606\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739097606\"]<\/p>\r\n<p id=\"fs-id1169739097606\">[latex]-4.4[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Given [latex]f(3) = 2[\/latex] and [latex]f'(3) = 5[\/latex], estimate [latex]f(3.2)[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"870409\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"870409\"]\u00a0<\/p>\r\n<p>Identify known values:<\/p>\r\n<p style=\"text-align: center;\">[latex]a = 3[\/latex], [latex]f(a) = 2[\/latex], [latex]f'(a) = 5[\/latex]<\/p>\r\n<p>Calculate [latex]h[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]h = 3.2 - 3 = 0.2[\/latex]<\/p>\r\n<p>Apply the Amount of Change Formula:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} f(3.2) &amp;\\approx&amp; f(3) + f'(3)(0.2) \\\\ &amp;\\approx&amp; 2 + 5(0.2) \\\\ &amp;\\approx&amp; 2 + 1 \\\\ &amp;\\approx&amp; 3 \\end{array} [\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>\r\n<h2>Rate of Change Applications<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Motion Along a Line:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Position function: [latex]s(t)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Velocity: [latex]v(t) = s'(t)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Speed: [latex]|v(t)|[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Acceleration: [latex]a(t) = v'(t) = s''(t)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Population Change:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Population function: [latex]P(t)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Population growth rate: [latex]P'(t)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Changes in Cost and Revenue:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Marginal Cost: [latex]MC(x) = C'(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Marginal Revenue: [latex]MR(x) = R'(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Marginal Profit: [latex]MP(x) = P'(x) = MR(x) - MC(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-bold\"><strong>Application Techniques<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Motion Analysis:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use position function to find velocity and acceleration<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Analyze sign of velocity for direction of motion<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Find zeros of velocity for points where object is at rest<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Population Estimation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use current population and growth rate to estimate future population<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply Amount of Change Formula: [latex]P(t+h) \\approx P(t) + P'(t)h[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Economic Analysis:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use marginal functions to estimate changes in cost, revenue, or profit<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Approximate change: [latex]f(x+1) - f(x) \\approx f'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738941480\">A particle moves along a coordinate axis in the positive direction to the right. Its position at time [latex]t[\/latex] is given by [latex]s(t)=t^3-4t+2[\/latex]. Find [latex]v(1)[\/latex] and [latex]a(1)[\/latex] and use these values to answer the following questions.<\/p>\r\n<ol id=\"fs-id1169739274655\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Is the particle moving from left to right or from right to left at time [latex]t=1[\/latex]?<\/li>\r\n\t<li>Is the particle speeding up or slowing down at time [latex]t=1[\/latex]?<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"fs-id1169738823468\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738823468\"]<\/p>\r\n<p id=\"fs-id1169738823468\">Begin by finding [latex]v(t)[\/latex] and [latex]a(t)[\/latex].<\/p>\r\n<p id=\"fs-id1169738969278\">[latex]v(t)=s^{\\prime}(t)=3t^2-4[\/latex] and [latex]a(t)=v^{\\prime}(t)=s''(t)=6t[\/latex].<\/p>\r\n<p id=\"fs-id1169739096229\">Evaluating these functions at [latex]t=1[\/latex], we obtain [latex]v(1)=-1[\/latex] and [latex]a(1)=6[\/latex].<\/p>\r\n<ol id=\"fs-id1169738879103\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Because [latex]v(1)&lt;0[\/latex], the particle is moving from right to left.<\/li>\r\n\t<li>Because [latex]v(1)&lt;0[\/latex] and [latex]a(1)&gt;0[\/latex], velocity and acceleration are acting in opposite directions. In other words, the particle is being accelerated in the direction opposite the direction in which it is traveling, causing [latex]|v(t)|[\/latex] to decrease. The particle is slowing down.<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739343676\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)=t^2-5t+1[\/latex]. Is the particle moving from right to left or from left to right at time [latex]t=3[\/latex]?<\/p>\r\n<p>[reveal-answer q=\"789345\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"789345\"]<\/p>\r\n<p id=\"fs-id1169739236686\">Find [latex]v(3)[\/latex] and look at the sign.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739299934\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739299934\"]<\/p>\r\n<p id=\"fs-id1169739299934\">left to right<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Given the position function [latex]s(t) = t^3 - 9t^2 + 24t + 4[\/latex] for [latex]t \\geq 0[\/latex]:<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">Find the velocity function.<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">When is the particle at rest?<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"661433\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"661433\"]\u00a0<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">Velocity function: [latex]v(t) = s'(t) = 3t^2 - 18t + 24[\/latex]<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">The particle is at rest when [latex]v(t) = 0[\/latex]:<\/li>\r\n<\/ol>\r\n<p style=\"text-align: center;\">[latex]3t^2 - 18t + 24 = 0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]3(t-2)(t-4) = 0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]t = 2[\/latex] or [latex]t = 4[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">A city's population triples every [latex]5[\/latex] years. The current population is[latex] 10,000[\/latex]. Estimate the population after [latex]2[\/latex] years.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"19423\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"19423\"]<\/p>\r\n<p>Estimate growth rate:<\/p>\r\n<p style=\"text-align: center;\">[latex]P'(0) \\approx \\frac{P(5) - P(0)}{5-0} = \\frac{30,000 - 10,000}{5} = 4,000[\/latex] per year<\/p>\r\n<p>Estimate population after [latex]2[\/latex] years:<\/p>\r\n<p style=\"text-align: center;\">[latex]P(2) \\approx P(0) + 2P'(0) = 10,000 + 2(4,000) = 18,000[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Estimated population after [latex]2[\/latex] years: [latex]18,000[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736656696\">The current population of a mosquito colony is known to be [latex]3,000[\/latex]; that is, [latex]P(0)=3,000[\/latex]. If [latex]P^{\\prime}(0)=100[\/latex], estimate the size of the population in [latex]3[\/latex] days, where [latex]t[\/latex] is measured in days.<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739019878\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739019878\"]<\/p>\r\n<p id=\"fs-id1169739188550\">Use [latex]P(3)\\approx P(0)+3P^{\\prime}(0)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739019879\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739019879\"]<\/p>\r\n<p id=\"fs-id1169739019878\">[latex]3,300[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Given the revenue function [latex]R(x) = -0.03x^2 + 9x[\/latex] for [latex]0 \\leq x \\leq 300[\/latex]:\u00a0<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">Find the Marginal Revenue function.<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Estimate the revenue from selling the [latex]101[\/latex]st item.<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Calculate the actual revenue change from the [latex]100[\/latex]th to the [latex]101[\/latex]st item.<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"583714\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"583714\"]<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>Marginal Revenue function: <br \/>\r\n<br \/>\r\n[latex]MR(x) = R'(x) = -0.06x + 9[\/latex]<br \/>\r\n<br \/>\r\n<\/li>\r\n\t<li>Estimated revenue from [latex]101[\/latex]st item: <br \/>\r\n<br \/>\r\n<center>[latex]MR(100) = -0.06(100) + 9 = 3[\/latex]<\/center>Estimated additional revenue: [latex]$3[\/latex]<\/li>\r\n\t<li>Actual revenue change: <br \/>\r\n<br \/>\r\n<center>[latex]R(101) - R(100) = 602.97 - 600 = 2.97[\/latex]<\/center>Actual additional revenue: [latex]$2.97[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736589106\">Suppose that the profit obtained from the sale of [latex]x[\/latex] fish-fry dinners is given by [latex]P(x)=-0.03x^2+8x-50[\/latex]. Use the marginal profit function to estimate the profit from the sale of the [latex]101[\/latex]<sup>st<\/sup> fish-fry dinner.<\/p>\r\n<p>[reveal-answer q=\"554428\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"554428\"]<\/p>\r\n<p id=\"fs-id1169739273788\">Use [latex]P^{\\prime}(100)[\/latex] to approximate [latex]P(101)-P(100)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739208505\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739208505\"]<\/p>\r\n<p id=\"fs-id1169739208505\">[latex]$2[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate how quantities change on average over time<\/li>\n<li>Use rates of change to figure out how an object\u2019s position, speed, and acceleration are changing over time<\/li>\n<li>Estimate future population sizes using current data and how fast the population is growing<\/li>\n<li>Use derivatives to determine the cost and revenue of producing one more unit in a business<\/li>\n<\/ul>\n<\/section>\n<h2>Amount of Change Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Amount of Change:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Change in [latex]y[\/latex]-values over an interval [latex][a, a+h][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Given by [latex]f(a+h) - f(a)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Average Rate of Change:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ratio of amount of change to change in [latex]x[\/latex]-values<\/li>\n<li class=\"whitespace-normal break-words\">Formula: [latex]\\frac{f(a+h) - f(a)}{h}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Amount of Change Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Approximates [latex]f(a+h)[\/latex] using [latex]f(a)[\/latex] and [latex]f'(a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Formula: [latex]f(a+h) \\approx f(a) + f'(a)h[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Accuracy depends on the size of [latex]h[\/latex] and the behavior of [latex]f'(x)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738905166\">Given [latex]f(10)=-5[\/latex] and [latex]f^{\\prime}(10)=6[\/latex], estimate [latex]f(10.1)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739097606\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739097606\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739097606\">[latex]-4.4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Given [latex]f(3) = 2[\/latex] and [latex]f'(3) = 5[\/latex], estimate [latex]f(3.2)[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q870409\">Show Answer<\/button><\/p>\n<div id=\"q870409\" class=\"hidden-answer\" style=\"display: none\">\u00a0<\/p>\n<p>Identify known values:<\/p>\n<p style=\"text-align: center;\">[latex]a = 3[\/latex], [latex]f(a) = 2[\/latex], [latex]f'(a) = 5[\/latex]<\/p>\n<p>Calculate [latex]h[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]h = 3.2 - 3 = 0.2[\/latex]<\/p>\n<p>Apply the Amount of Change Formula:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(3.2) &\\approx& f(3) + f'(3)(0.2) \\\\ &\\approx& 2 + 5(0.2) \\\\ &\\approx& 2 + 1 \\\\ &\\approx& 3 \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n<h2>Rate of Change Applications<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Motion Along a Line:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Position function: [latex]s(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Velocity: [latex]v(t) = s'(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Speed: [latex]|v(t)|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Acceleration: [latex]a(t) = v'(t) = s''(t)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Population Change:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Population function: [latex]P(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Population growth rate: [latex]P'(t)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Changes in Cost and Revenue:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Marginal Cost: [latex]MC(x) = C'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Marginal Revenue: [latex]MR(x) = R'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Marginal Profit: [latex]MP(x) = P'(x) = MR(x) - MC(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Application Techniques<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Motion Analysis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use position function to find velocity and acceleration<\/li>\n<li class=\"whitespace-normal break-words\">Analyze sign of velocity for direction of motion<\/li>\n<li class=\"whitespace-normal break-words\">Find zeros of velocity for points where object is at rest<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Population Estimation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use current population and growth rate to estimate future population<\/li>\n<li class=\"whitespace-normal break-words\">Apply Amount of Change Formula: [latex]P(t+h) \\approx P(t) + P'(t)h[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Economic Analysis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use marginal functions to estimate changes in cost, revenue, or profit<\/li>\n<li class=\"whitespace-normal break-words\">Approximate change: [latex]f(x+1) - f(x) \\approx f'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738941480\">A particle moves along a coordinate axis in the positive direction to the right. Its position at time [latex]t[\/latex] is given by [latex]s(t)=t^3-4t+2[\/latex]. Find [latex]v(1)[\/latex] and [latex]a(1)[\/latex] and use these values to answer the following questions.<\/p>\n<ol id=\"fs-id1169739274655\" style=\"list-style-type: lower-alpha;\">\n<li>Is the particle moving from left to right or from right to left at time [latex]t=1[\/latex]?<\/li>\n<li>Is the particle speeding up or slowing down at time [latex]t=1[\/latex]?<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738823468\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738823468\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738823468\">Begin by finding [latex]v(t)[\/latex] and [latex]a(t)[\/latex].<\/p>\n<p id=\"fs-id1169738969278\">[latex]v(t)=s^{\\prime}(t)=3t^2-4[\/latex] and [latex]a(t)=v^{\\prime}(t)=s''(t)=6t[\/latex].<\/p>\n<p id=\"fs-id1169739096229\">Evaluating these functions at [latex]t=1[\/latex], we obtain [latex]v(1)=-1[\/latex] and [latex]a(1)=6[\/latex].<\/p>\n<ol id=\"fs-id1169738879103\" style=\"list-style-type: lower-alpha;\">\n<li>Because [latex]v(1)<0[\/latex], the particle is moving from right to left.<\/li>\n<li>Because [latex]v(1)<0[\/latex] and [latex]a(1)>0[\/latex], velocity and acceleration are acting in opposite directions. In other words, the particle is being accelerated in the direction opposite the direction in which it is traveling, causing [latex]|v(t)|[\/latex] to decrease. The particle is slowing down.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739343676\">A particle moves along a coordinate axis. Its position at time [latex]t[\/latex] is given by [latex]s(t)=t^2-5t+1[\/latex]. Is the particle moving from right to left or from left to right at time [latex]t=3[\/latex]?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q789345\">Hint<\/button><\/p>\n<div id=\"q789345\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739236686\">Find [latex]v(3)[\/latex] and look at the sign.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739299934\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739299934\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739299934\">left to right<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Given the position function [latex]s(t) = t^3 - 9t^2 + 24t + 4[\/latex] for [latex]t \\geq 0[\/latex]:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">Find the velocity function.<\/li>\n<li class=\"whitespace-pre-wrap break-words\">When is the particle at rest?<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q661433\">Show Answer<\/button><\/p>\n<div id=\"q661433\" class=\"hidden-answer\" style=\"display: none\">\u00a0<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">Velocity function: [latex]v(t) = s'(t) = 3t^2 - 18t + 24[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">The particle is at rest when [latex]v(t) = 0[\/latex]:<\/li>\n<\/ol>\n<p style=\"text-align: center;\">[latex]3t^2 - 18t + 24 = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]3(t-2)(t-4) = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]t = 2[\/latex] or [latex]t = 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">A city&#8217;s population triples every [latex]5[\/latex] years. The current population is[latex]10,000[\/latex]. Estimate the population after [latex]2[\/latex] years.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q19423\">Show Answer<\/button><\/p>\n<div id=\"q19423\" class=\"hidden-answer\" style=\"display: none\">\n<p>Estimate growth rate:<\/p>\n<p style=\"text-align: center;\">[latex]P'(0) \\approx \\frac{P(5) - P(0)}{5-0} = \\frac{30,000 - 10,000}{5} = 4,000[\/latex] per year<\/p>\n<p>Estimate population after [latex]2[\/latex] years:<\/p>\n<p style=\"text-align: center;\">[latex]P(2) \\approx P(0) + 2P'(0) = 10,000 + 2(4,000) = 18,000[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Estimated population after [latex]2[\/latex] years: [latex]18,000[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736656696\">The current population of a mosquito colony is known to be [latex]3,000[\/latex]; that is, [latex]P(0)=3,000[\/latex]. If [latex]P^{\\prime}(0)=100[\/latex], estimate the size of the population in [latex]3[\/latex] days, where [latex]t[\/latex] is measured in days.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739019878\">Hint<\/button><\/p>\n<div id=\"qfs-id1169739019878\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739188550\">Use [latex]P(3)\\approx P(0)+3P^{\\prime}(0)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739019879\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739019879\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739019878\">[latex]3,300[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Given the revenue function [latex]R(x) = -0.03x^2 + 9x[\/latex] for [latex]0 \\leq x \\leq 300[\/latex]:\u00a0<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">Find the Marginal Revenue function.<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Estimate the revenue from selling the [latex]101[\/latex]st item.<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Calculate the actual revenue change from the [latex]100[\/latex]th to the [latex]101[\/latex]st item.<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q583714\">Show Answer<\/button><\/p>\n<div id=\"q583714\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Marginal Revenue function:\n<p>[latex]MR(x) = R'(x) = -0.06x + 9[\/latex]<\/p>\n<\/li>\n<li>Estimated revenue from [latex]101[\/latex]st item:\n<div style=\"text-align: center;\">[latex]MR(100) = -0.06(100) + 9 = 3[\/latex]<\/div>\n<p>Estimated additional revenue: [latex]$3[\/latex]<\/li>\n<li>Actual revenue change:\n<div style=\"text-align: center;\">[latex]R(101) - R(100) = 602.97 - 600 = 2.97[\/latex]<\/div>\n<p>Actual additional revenue: [latex]$2.97[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736589106\">Suppose that the profit obtained from the sale of [latex]x[\/latex] fish-fry dinners is given by [latex]P(x)=-0.03x^2+8x-50[\/latex]. Use the marginal profit function to estimate the profit from the sale of the [latex]101[\/latex]<sup>st<\/sup> fish-fry dinner.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q554428\">Hint<\/button><\/p>\n<div id=\"q554428\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739273788\">Use [latex]P^{\\prime}(100)[\/latex] to approximate [latex]P(101)-P(100)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739208505\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739208505\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739208505\">[latex]$2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":28,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"module-header":"fresh_take","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2901"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2901\/revisions"}],"predecessor-version":[{"id":3740,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2901\/revisions\/3740"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2901\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2901"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2901"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2901"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}