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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   
Associated Topics:
High School Analysis
High School Calculus

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