lec14 - Lecture 14 18.01 Fall 2006 Lecture 14 Mean Value...

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Lecture 14 18.01 Fall 2006 Lecture 14: Mean Value Theorem and Inequalities Mean-Value Theorem The Mean-Value Theorem (MVT) is the underpinning of calculus. It says: If f is di ff erentiable on a < x < b , and continuous on a x b , then f ( b ) - f ( a ) = f ( c ) (for some c , a < c < b ) b - a f ( b ) - f ( a ) Here, is the slope of a secant line, while f ( c ) is the slope of a tangent line. b - a secant line slope f’(c) a b c Figure 1: Illustration of the Mean Value Theorem. Geometric Proof: Take (dotted) lines parallel to the secant line, as in Fig. 1 and shift them up from below the graph until one of them first touches the graph. Alternatively, one may have to start with a dotted line above the graph and move it down until it touches. If the function isn’t di ff erentiable, this approach goes wrong. For instance, it breaks down for the function f ( x ) = | x | . The dotted line always touches the graph first at x = 0, no matter what its slope is, and f (0) is undefined (see Fig. 2).
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