4-4 Average Value of a Function

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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