Generalizing the Mean Value Theorem for Integrals Suppose
Chapter 5, Problem 69(choose chapter or problem)
Generalizing the Mean Value Theorem for Integrals Suppose f and g are continuous on 3a, b4 and let h1x2 = 1x - b2 L x a f 1t2 dt + 1x - a2 L b x g1t2 dt. a. Use Rolles theorem to show that there is a number c in (a, b) such that L c a f 1t2 dt + L b c g1t2 dt = f 1c21b - c2 + g1c21c - a2, which is a generalization of the Mean Value Theorem for Integrals. b. Show that there a number c in (a, b) such that L x a f 1t2 dt = f 1c21b - c2. c. Use a sketch to interpret part (b) geometrically. d. Use the result of part (a) to give an alternate proof of the Mean Value Theorem for Integrals. (Source: The College Mathematics Journal, 33, 5, Nov 2002)
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