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Rolles Theorem and The Mean Value Theorem 4.2

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If f (x) is a differentiable function over [ a, b ], then at some point between a and b : Mean Value Theorem for Derivatives

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If f (x) is a differentiable function over [ a, b ], then at some point between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous.

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If f (x) is a differentiable function over [ a, b ], then at some point between a and b : Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous. The Mean Value Theorem only applies over a closed interval.

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If f (x) is a differentiable function over [ a, b ], then at some point between a and b : Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the closed interval, the actual slope (instantaneous rate of change) equals the average slope (average rate of change).

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Slope of chord: Slope of tangent: Tangent parallel to chord.

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Example Suppose you have a function f(x) = x^2 in the interval [1, 3]. To find c, set derivative = to 4

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There can be more than 1 value of c that satisfy MVTD

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There is no value of c that satisfies this equation. Why? f(x) is not continuous in the given interval!

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Find the value of c that satisfies the MVTD: Since f(0) = f(12), Rolles Theorem applies

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Find the value of c that satisfies the MVTD: Since f(0) = f(1), Rolles Theorem applies

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Example A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 miles on a toll road with a speed limit of 65 mph. Can the trucker be cited for speeding? Average velocity is 159/2 = 79.5 MVTD tells us at some point in the interval, the instantaneous velocity is equal to the average velocity! Thus she must have been speeding at some point!

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