THEOREM 4.5 Mean Value Theorem Let f be continuous over the closed interval (a, b and differentiable over the open inter

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Theorem 4 5 Mean Value Theorem Let F Be Continuous Over The Closed Interval A B And Differentiable Over The Open Inter 1 (52.08 KiB) Viewed 17 times

THEOREM 4.5 Mean Value Theorem Let f be continuous over the closed interval (a, b and differentiable over the open interval (a, b). Then, there least one point c E (a, b) such that Slope of secant line is. f'(c) = In each of the following graphs it appears that f'(c) = f(b)-f(a) for at least one point. b-a K(b)-f(a) b-a f(b)- f(a)- f(b)-f(a) b-a a Slope of tangent line is f'(c₂) y = f(x) t Slope of tangent line is f'(c₂) C₁ C₂ b Figure 4.25 The Mean Value Theorem says that for a function that meets its conditions, at some point the tangent line has the same slope as the secant line between the ends. For this function, there are two values cy and c₂ such that the tangent line to fat cand og has the same slope as the secant line. Section 4.5 Introduction: Se Then Rolle's The them. We cou we will use t Learning In this • US For each of the following, verify that the function satisfies the three hypotheses of the Mean Value Theorem on the given interval. Then find all the numbers c that satisfy the conclusion of the Mean Value Theorem. 4) f(x)=x+: [1,3] 5) f(x) = In(2x): [1,e] 6) A state patrol officer saw a car start from rest at a highway on-ramp. He radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. Can the officer conclude that the driver exceeded the 60 mile per hour speed limit?