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

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Unformatted text preview: 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 differentiable 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 isnt differentiable, 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.Fig....
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lec14 - Lecture 14 18.01 Fall 2006 Lecture 14: Mean Value...

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