### 6.5: The Average Value of a Function

In 6.5, we learn to take our abundant knowledge of the Mean Value Theorem and adjust it in order to apply it to our new best friend: INTEGRALS!

So, for starters, let's just try to state the mean value theorem in its simplest, most "kindergarten-esque," terms:

Assuming that the graph of 'f' is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there is a number 'c' in (a, b) such that

Now, let's be really intelligent and try to apply it, at the most basic level, to integrals and their importance as the area under a curve. Let's take a look at this graph:

As you can see, I've taken the area under the curve and made an approximating rectangle in order to find it. Now, if we are told that the area under the curve is 18, we could find out what X is...6! That X value, 6, is called the average value of the function (i.e. f(c)).

Now that we know that, I can introduce the mean value theorem for integrals.

if f(x) is continuous on [a, b] then there exists a 'c' value such that

so, now we know how to find that average value of the function numerically, using the mean value of integrals:

Ok, so now let's try a practice problem to see if we really understand the concept

on the interval, [-2, 5]

Step 1: find the area under the curve by evaluating the integral of the function

Step 2: plug that integral value that we just found back into the equation for finding the average value of the equationso f(c)=4.333 or (13/3)

THIS IS THE AVERAGE VALUE OF THE EQUATION

Step 3: plug the f(c) back into the equation for y in order to find the 'c' value that gives the f(c)QUADRATIC FORMULA

YAY!!! We have finally found all the information that we need for 6.5. Let's give ourselves a pat on the back. Now everyone, let's take a second to relax our minds and read a few calculus jokes:

Q: How does a mathematician induce good behavior in her children?

A: "I've told you n times, I've told you n+1 times..."

Q: What is the first derivative of a cow?

A: Prime Rib!

Peace, Love and Calculus Soul Everyone!!

p.s. here's a fun link

Q: How does a mathematician induce good behavior in her children?

A: "I've told you n times, I've told you n+1 times..."

Q: What is the first derivative of a cow?

A: Prime Rib!

HAHAHA! LAUGH IT UP!!

dx d CABIN

--- = log x so, ------------ = log CABIN

x CABIN

/

| d CABIN

| ------------ = log CABIN + C (which is a house boat)

| CABIN

/

Peace, Love and Calculus Soul Everyone!!

p.s. here's a fun link

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