{"id":3248,"date":"2024-06-13T19:21:11","date_gmt":"2024-06-13T19:21:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3248"},"modified":"2024-08-05T02:16:47","modified_gmt":"2024-08-05T02:16:47","slug":"the-mean-value-theorem-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-mean-value-theorem-fresh-take\/","title":{"raw":"The Mean Value Theorem: Fresh Take","rendered":"The Mean Value Theorem: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use Rolle\u2019s theorem and the Mean Value Theorem to show how functions behave between two points<\/li>\r\n\t<li>Discuss three key implications of the Mean Value Theorem for understanding function behavior<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rolle\u2019s Theorem<\/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\">Theorem Statement: For a function [latex]f[\/latex] that is:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Continuous on [latex][a,b][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(a) = f(b)[\/latex] There exists at least one [latex]c \\in (a,b)[\/latex] where [latex]f'(c) = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Geometric Interpretation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If a function starts and ends at the same [latex]y[\/latex]-value, it must have at least one horizontal tangent line between those points<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Conditions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Continuity over the closed interval<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Differentiability over the open interval<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Equal function values at endpoints<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Importance:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Foundation for the Mean Value Theorem<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Useful in proofs and theoretical analysis<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Limitations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Does not apply if the function is not differentiable at even one point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Does not guarantee uniqueness of [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Show that [latex]f(x) = x^3 - 3x + 2[\/latex] satisfies Rolle's Theorem on [latex][-1,2][\/latex], and find all values of [latex]c[\/latex] that satisfy the conclusion.<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"227703\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"227703\"]<\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Verify conditions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is a polynomial, so it's continuous and differentiable everywhere<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(-1) = -1 - 3(-1) + 2 = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(2) = 8 - 6 + 2 = 4[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(-1) \\neq f(2)[\/latex], so Rolle's Theorem doesn't apply directly<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Find roots:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x) = 0[\/latex] when [latex]x = -1[\/latex] and [latex]x = 1[\/latex] (verify: [latex]f(1) = 1 - 3 + 2 = 0[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Rolle's Theorem applies on [latex][-1,1][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Find critical points:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(x) = 3x^2 - 3[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(x) = 0[\/latex] when [latex]x = \\pm 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Conclusion:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]c = 0[\/latex] satisfies Rolle's Theorem on [latex][-1,1][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]c = 1[\/latex] is also a critical point in [latex][-1,2][\/latex], but not guaranteed by Rolle's Theorem<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>The Mean Value Theorem and Its Meaning<\/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\">Theorem Statement: For a function [latex]f[\/latex] that is:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Continuous on [latex][a,b][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">There exists at least one [latex]c \\in (a,b)[\/latex] where: [latex]f'(c) = \\frac{f(b) - f(a)}{b - a}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Geometric Interpretation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">At some point, the instantaneous rate of change (slope of tangent line) equals the average rate of change (slope of secant line)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relation to Rolle's Theorem:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Mean Value Theorem is a generalization of Rolle's Theorem<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Can be proved using Rolle's Theorem<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Velocity and displacement problems<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Proving mathematical inequalities<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Establishing properties of functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Points:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]c[\/latex] is not necessarily unique<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Theorem guarantees existence, not the exact value of [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Suppose a ball is dropped from a height of [latex]200[\/latex] ft. Its position at time [latex]t[\/latex] is [latex]s(t)=-16t^2+200[\/latex]. Find the time [latex]t[\/latex] when the instantaneous velocity of the ball equals its average velocity.<\/p>\r\n<p>[reveal-answer q=\"9003136\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"9003136\"]<\/p>\r\n<p>First, determine how long it takes for the ball to hit the ground. Then, find the average velocity of the ball from the time it is dropped until it hits the ground.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042631892\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042631892\"]<\/p>\r\n<p>[latex]\\dfrac{5}{2\\sqrt{2}}[\/latex] sec<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>A car travels [latex]240[\/latex] miles in [latex]4[\/latex] hours. Prove that at some point during the trip, the car's instantaneous speed was exactly [latex]60[\/latex] mph.<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"15564\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"15564\"]<\/p>\r\n<p>Let [latex]s(t)[\/latex] be the distance traveled (in miles) after [latex]t[\/latex] hours.<\/p>\r\n<p>[latex]a = 0[\/latex], [latex]b = 4[\/latex]<br \/>\r\n[latex]s(0) = 0[\/latex], [latex]s(4) = 240[\/latex]<\/p>\r\n<p>By the Mean Value Theorem:<\/p>\r\n<p>There exists [latex]c \\in (0,4)[\/latex] such that:<br \/>\r\n[latex]s'(c) = \\frac{s(4) - s(0)}{4 - 0} = \\frac{240 - 0}{4} = 60[\/latex]<\/p>\r\n<p>[latex]s'(t)[\/latex] represents the instantaneous speed<br \/>\r\nAt time [latex]c[\/latex], the instantaneous speed [latex]s'(c) = 60[\/latex] mph<\/p>\r\n<p>The car's speed was exactly [latex]60[\/latex] mph at some point during the trip.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Corollaries of the Mean Value Theorem<\/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\">Corollary 1: Functions with a Derivative of Zero\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f'(x) = 0[\/latex] for all [latex]x[\/latex] in an interval [latex]I[\/latex], then [latex]f(x)[\/latex] is constant on [latex]I[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Corollary 2: Constant Difference Theorem\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f'(x) = g'(x)[\/latex] for all [latex]x[\/latex] in an interval [latex]I[\/latex], then [latex]f(x) = g(x) + C[\/latex] for some constant [latex]C[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Corollary 3: Increasing and Decreasing Functions\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f'(x) &gt; 0[\/latex] on <a href=\"a,b\">latex<\/a>[\/latex], then [latex]f[\/latex] is increasing on [latex][a,b][\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f'(x) &lt; 0[\/latex] on <a href=\"a,b\">latex<\/a>[\/latex], then [latex]f[\/latex] is decreasing on [latex][a,b][\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Implications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Corollary 1 proves the converse of \"the derivative of a constant function is zero\"<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Corollary 2 shows that antiderivatives differ by at most a constant<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Corollary 3 connects the sign of the derivative to the behavior of the function<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use Rolle\u2019s theorem and the Mean Value Theorem to show how functions behave between two points<\/li>\n<li>Discuss three key implications of the Mean Value Theorem for understanding function behavior<\/li>\n<\/ul>\n<\/section>\n<h2>Rolle\u2019s Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Theorem Statement: For a function [latex]f[\/latex] that is:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Continuous on [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(a) = f(b)[\/latex] There exists at least one [latex]c \\in (a,b)[\/latex] where [latex]f'(c) = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If a function starts and ends at the same [latex]y[\/latex]-value, it must have at least one horizontal tangent line between those points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Conditions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Continuity over the closed interval<\/li>\n<li class=\"whitespace-normal break-words\">Differentiability over the open interval<\/li>\n<li class=\"whitespace-normal break-words\">Equal function values at endpoints<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Importance:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Foundation for the Mean Value Theorem<\/li>\n<li class=\"whitespace-normal break-words\">Useful in proofs and theoretical analysis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Does not apply if the function is not differentiable at even one point<\/li>\n<li class=\"whitespace-normal break-words\">Does not guarantee uniqueness of [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Show that [latex]f(x) = x^3 - 3x + 2[\/latex] satisfies Rolle&#8217;s Theorem on [latex][-1,2][\/latex], and find all values of [latex]c[\/latex] that satisfy the conclusion.<\/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=\"q227703\">Show Answer<\/button><\/p>\n<div id=\"q227703\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Verify conditions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is a polynomial, so it&#8217;s continuous and differentiable everywhere<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(-1) = -1 - 3(-1) + 2 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(2) = 8 - 6 + 2 = 4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(-1) \\neq f(2)[\/latex], so Rolle&#8217;s Theorem doesn&#8217;t apply directly<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Find roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x) = 0[\/latex] when [latex]x = -1[\/latex] and [latex]x = 1[\/latex] (verify: [latex]f(1) = 1 - 3 + 2 = 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Rolle&#8217;s Theorem applies on [latex][-1,1][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Find critical points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(x) = 3x^2 - 3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f'(x) = 0[\/latex] when [latex]x = \\pm 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conclusion:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]c = 0[\/latex] satisfies Rolle&#8217;s Theorem on [latex][-1,1][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]c = 1[\/latex] is also a critical point in [latex][-1,2][\/latex], but not guaranteed by Rolle&#8217;s Theorem<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<h2>The Mean Value Theorem and Its Meaning<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Theorem Statement: For a function [latex]f[\/latex] that is:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Continuous on [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">There exists at least one [latex]c \\in (a,b)[\/latex] where: [latex]f'(c) = \\frac{f(b) - f(a)}{b - a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">At some point, the instantaneous rate of change (slope of tangent line) equals the average rate of change (slope of secant line)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relation to Rolle&#8217;s Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Mean Value Theorem is a generalization of Rolle&#8217;s Theorem<\/li>\n<li class=\"whitespace-normal break-words\">Can be proved using Rolle&#8217;s Theorem<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Velocity and displacement problems<\/li>\n<li class=\"whitespace-normal break-words\">Proving mathematical inequalities<\/li>\n<li class=\"whitespace-normal break-words\">Establishing properties of functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]c[\/latex] is not necessarily unique<\/li>\n<li class=\"whitespace-normal break-words\">Theorem guarantees existence, not the exact value of [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Suppose a ball is dropped from a height of [latex]200[\/latex] ft. Its position at time [latex]t[\/latex] is [latex]s(t)=-16t^2+200[\/latex]. Find the time [latex]t[\/latex] when the instantaneous velocity of the ball equals its average velocity.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q9003136\">Hint<\/button><\/p>\n<div id=\"q9003136\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, determine how long it takes for the ball to hit the ground. Then, find the average velocity of the ball from the time it is dropped until it hits the ground.<\/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-id1165042631892\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042631892\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{5}{2\\sqrt{2}}[\/latex] sec<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>A car travels [latex]240[\/latex] miles in [latex]4[\/latex] hours. Prove that at some point during the trip, the car&#8217;s instantaneous speed was exactly [latex]60[\/latex] mph.<\/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=\"q15564\">Show Answer<\/button><\/p>\n<div id=\"q15564\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]s(t)[\/latex] be the distance traveled (in miles) after [latex]t[\/latex] hours.<\/p>\n<p>[latex]a = 0[\/latex], [latex]b = 4[\/latex]<br \/>\n[latex]s(0) = 0[\/latex], [latex]s(4) = 240[\/latex]<\/p>\n<p>By the Mean Value Theorem:<\/p>\n<p>There exists [latex]c \\in (0,4)[\/latex] such that:<br \/>\n[latex]s'(c) = \\frac{s(4) - s(0)}{4 - 0} = \\frac{240 - 0}{4} = 60[\/latex]<\/p>\n<p>[latex]s'(t)[\/latex] represents the instantaneous speed<br \/>\nAt time [latex]c[\/latex], the instantaneous speed [latex]s'(c) = 60[\/latex] mph<\/p>\n<p>The car&#8217;s speed was exactly [latex]60[\/latex] mph at some point during the trip.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Corollaries of the Mean Value Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Corollary 1: Functions with a Derivative of Zero\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f'(x) = 0[\/latex] for all [latex]x[\/latex] in an interval [latex]I[\/latex], then [latex]f(x)[\/latex] is constant on [latex]I[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Corollary 2: Constant Difference Theorem\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f'(x) = g'(x)[\/latex] for all [latex]x[\/latex] in an interval [latex]I[\/latex], then [latex]f(x) = g(x) + C[\/latex] for some constant [latex]C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Corollary 3: Increasing and Decreasing Functions\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f'(x) > 0[\/latex] on <a href=\"a,b\">latex<\/a>[\/latex], then [latex]f[\/latex] is increasing on [latex][a,b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]f'(x) < 0[\/latex] on <a href=\"a,b\">latex<\/a>[\/latex], then [latex]f[\/latex] is decreasing on [latex][a,b][\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Implications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Corollary 1 proves the converse of &#8220;the derivative of a constant function is zero&#8221;<\/li>\n<li class=\"whitespace-normal break-words\">Corollary 2 shows that antiderivatives differ by at most a constant<\/li>\n<li class=\"whitespace-normal break-words\">Corollary 3 connects the sign of the derivative to the behavior of the function<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":15,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"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\/3248"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3248\/revisions"}],"predecessor-version":[{"id":3834,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3248\/revisions\/3834"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3248\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3248"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3248"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3248"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}