Stewart6e4_2 - MATH 191 Sections 1 and 3 Calculus I Fall 2007 Section 4.2 The Mean Value Theorem Today's focus is probably the second(maybe third most

Stewart6e4_2 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 4.2: The Mean Value Theorem Today’s focus is probably the second (maybe third) most important theorem from all of calculus. (Only the Intermediate Value Theorem, which we men- tioned when we defined continuous functions, and the Fundamental Theorem of Calculus, which we’ll see briefly at the end this semester, might outshine it in importance.) We’re going to sneak up on the Mean Value Theorem, starting with a miniature version of it: Theorem. (Rolle’s Theorem) Suppose that the function f 1. is on the closed interval [ a, b ], 2. is on the open interval ( a, b ), and 3. f ( a ) = f ( b ). Then there is a number c in ( a, b ) such that f ( c ) = . Go ahead and use the space provided for you below to sketch a Proof of Rolle’s Theorem. ( Hint : it might not hurt to include a graph or two to help visualize things, as well!)
Note. If f ( t ) represents a position function, an interesting corollary of Rolle’s Theorem is that the velocity of the moving object must be 0 at some point, provided the object begins and ends at the same point.

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