Average Value of a Function
 
 
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Average Value of a Function

It is easy to calculate the average value of a finite set of numbers :

But how do we compute the average temperature during a day if infinitely many temperatures readings are possible? Figure 1 shows the graph of a temperature funtion T(t), where t is mesured in hours and T in degrees Celcius, and a guess at the average temperature, T(ave)
In general, let's try to compute the average value of a funtion , . We start by dividing the interval [a,b] into n equal subintervals, each with length . Then we choose points in successive subintervals and calculate the average of the numbers :

(For example, if f represents a temperature function and n=24, this means that we take temperature readings every hour and then average them.) Since , we can write and the average value becomes

If we let n increase, we would be computing the average value of large nmber of closely spaced values. (For example, we would be averaging temperature readings taken every minute or even every second.) The limiting value is

by the definitiion of a definite integral. Therefore, we define the average value of f on the interval [a,b] as
Example 1

Find the average value of the funtion on the interval [-1,2].

Solution: With a=-1 and b=2 we have

the question arises: Is there a number c at which the value of f is exactly equal to the average value of the funtion, that is, f(c)=f(ave)? The following theorem says that this is true for continuous funtions.

The Mean Value Theorem for Integrals states: If f is continuous on [a,b], then there exists a number c in [a,b] such that

The Mean Value theorem for Integrals is a consequenceof th eMean Value Theorem for derivatives and the Fundamental Theorem of Calculus. the proof is outlined in Excersise 21 in your book.
The geometric interpretations of th eMean Value theorem for Integrals is that, for positive funtions f, there is a number c such that the rectangle with base [a,b] and height f(c) has the same area as the region under the graph of f from a to b
Example 2

Since is continuous on the interval [-1,2], the MVT says there is a numeber c in [-1,2], such that

In this particular case we can find c explicitly. From Example 1 we know that f(ave)=2, so the value of c satisfies

Therfore so

Thus, in this case there happen to be two numbers in the interval [-1,2] that work in the MVT.
Example 3

Show that the average velocity of a car over a time interval is the same as the average of its velocities during the trip.

SOLUTION: If s(t) is the displacement of the car at time t, then, by definition, the average velocity of the car over the interval is

On the other hand, the average value of the velocity function on the interval is

=Average Velocity

Links To More Examples

Here are some links to pages that cover the Average Value of a Function

http://archives.math.utk.edu/visual.calculus/5/average.1/index.html

http://www.mathwords.com/a/average_value_function.htm

lhttp://www.calculus-help.com/probs1998/problem24.html

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