Mean Value Theorem
Date: 04/18/99 at 12:50:08 From: Matt Porter Subject: AP Practice Test We are in the process of practicing for the AP practice test. We starred questions that we didn't know and asked our teacher about them. Most, he said he still needed to teach us. Some he helped us with, but one we all didn't get, so I figured I would ask you. Here is the question: If f(x) = sin (x/2), then there exists a number c in the interval pi/2< x < 3pi/2 that satisfies the conclusion of the Mean Value Theorem. Which of the following would be c? (A) 2pi/3 (B) 3pi/4 (C) 5pi/6 (D) pi (E) 3pi/2 The answer we come up with is nowhere near what theirs is, which is D. We used a formula 1/(b-a) times the integral from 3pi/2 to pi/2 of sin (x/2). If you use this you don't get d. Please help!
Date: 04/18/99 at 18:02:25 From: Doctor Pat Subject: Re: AP Practice Test Hi Matt, I think you are confusing the mean value theorem with the average value of a function. The mean value theorem says that for a function that is continuous and differentiable everywhere between a and b, there is a point where the slope of the function is equal to the slope of the line through a,f(a) and b,f(b). Find the value of sin(x/2) at each end of the interval and find the slope of the line through these two points. Now take the first derivative and find the value of x that makes the slope equal to the slope of the secant. It may happen more than once (although not in this problem) but the theorem just guarentees that it will happen at least once. Hope that helps. - Doctor Pat, The Math Forum http://mathforum.org/dr.math/
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