MATH 191, Sections 1 and 3Calculus IFall 2007Section 4.2: The Mean Value TheoremToday’s focus is probably the second (maybe third) most important theoremfrom all of calculus.(Only the Intermediate Value Theorem, which we men-tioned when we defined continuous functions, and the Fundamental Theoremof Calculus, which we’ll see briefly at the end this semester, might outshine itin importance.) We’re going to sneak up on the Mean Value Theorem, startingwith a miniature version of it:Theorem. (Rolle’s Theorem)Suppose that the functionf1. ison the closed interval [a, b],2. ison the open interval (a, b), and3.f(a) =f(b).Then there is a numbercin (a, b) such thatf(c) =.Go ahead and use the space provided for you below to sketch aProof of Rolle’s Theorem.(Hint: it might not hurt to include a graph or twoto help visualize things, as well!)
Note.Iff(t) represents a position function, an interesting corollary of Rolle’sTheorem is that the velocity of the moving object must be 0 at some point,provided the object begins and ends at the same point.