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4.2 The Mean Value Theorem

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Rolle’s Theorem Let f be a function that satisfies the following three conditions: f is continuous on the closed interval [a,b] . f is differentiable on the open interval (a,b) . f(a) = f(b) . Then there is a number c in (a,b) such that f ′(c)=0. Examples on the board.

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If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b: Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the interval, the actual slope equals the average slope. Note: The Mean Value Theorem only applies over a closed interval.

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**An illustration of the Mean Value Theorem.**

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

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**Corollaries of the Mean Value Theorem**

Corollary 1: If f ′(x)=0 for all x in an interval (a,b), then f is constant on (a,b) . Corollary 2: If f ′ (x) = g′ (x) for all x in an interval (a,b), then f - g is constant on (a,b) , that is f(x)= g(x) + c where c is a constant. (see the next slide for an illustration of Corollary 2)

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**These two functions have the same slope at any value of x.**

Functions with the same derivative differ by a constant.

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