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### Mean Value Theorem

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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
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
```

```
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/
```
Associated Topics:
High School Analysis
High School Calculus

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