4-4 Average Value of a Function
A MathematicsAcademy
Production
Let’s get started:
• You already know about functions and how to take the average of some finite set.
• Today we’re going to take the average over infinitely many values (those that the function takes on over some interval)…which means CALCULUS!
• Before I show how to do this…let’s talk about WHY we might want to do this?
Consider the following picture:
• How high would the water level be if the waves all settled?
Okay! So, now that you have seen that this an interesting
question...
Let’s forget about real life, and...
Do Some Math
Suppose we have a “nice”function and we need to find its average value over the interval [a,b].
Let’s apply our knowledge of how to find the average over a finite set of values to this problem:
First, we partition the interval [a,b] into n subintervals ofequal length to get back to the finite situation:
In the above graph, we have n=8
nabx /)(
Let us set up our notation:
ix comes from the i-th interval
Now we can get an estimate for theaverage value:
nxfxfxff n
average)(...)()( 21
Let’s try to clean this up a little:
nxfxfxff n
average)(...)()( 21
)](...)()([ 21 nxfxfxfabx
Since nabx /)(
In a more condensed form, we now get:
n
i
iaverage xxfab
f1
)(1
But we want to get out of the finite,and into the infinite!
How do we do this?
Take Limits!!!
b
a
n
i
inaverage dxxfab
xfab
f )(1)(1lim1
In this way, we get the average value of f(x) over the interval [a,b]:
So, if f is a “nice” function (i.e. wecan compute its integral) then we havea precise solution to our problem.
Let’s look back at our graph:
Remember the original MVT?
Average Value Theorem (for definite integrals)
If f is continuous on then at some point c in (a, b), ,a b
1 b
af c f x dx
b a
abafbfcf
)()()(
When looking at anti-derivatives and definite integrals, we write it another way:
abaFbFcf
)()()(
ab
dxxfb
a
)(
So we just say that: Average Value of f (x)
Average value of a function
The average value of function f on the interval [a, b] is defined as
Note: For a positive function, we can think of this definition as saying area/width = average height
Example: Find the average value of f(x)=x3 on [0,2].
b
aave dxxfab
f )(1
242
21
421
021 42
0
42
0
3
xdxxfave
So we’ve solved our problem!
If I give you the equation 13)( 2 xxfand ask you to find it’s average valueover the interval [0,2], you’ll all say
b
a
average dxxfab
f )(1
dxx 1302
1 2
0
2
2
0
3 ][21 xx
5)010(21
Now we can answer our fish tank question!(That is, if the waves were described by an integrable function)
The Mean Value Theorem for Integrals
Theorem: If f is continuous on [a, b], then there exists a number c in [a, b] such that
))(()(is,that
)(1)(
abcfdxxf
dxxfab
fcf
b
a
b
aave