FnInt…What does it actually do?

Glad You Asked!

We first need to accept some alternative notation for f(x) and f’(x).

If you take the derivative of f(x) what do you get?

f’(x), right?

Instead of using the letter f and f primes, we are going to use the capital and the lower-case letter f.

Why Would We Do That? If you take the derivative of f(x)

what do you get?

f’(x), right? What about the second derivative of f(x)?

f’’(x) of course!

f(x) F(x)

f’(x) f(x)

f’’(x) f’(x)

f’’’(x) f’’(x)

d/dx

d/dx

So…Do You Get It? If you take the derivative of F(x)

what do you get?

f(x), right?

What about the second derivative of F(x)?

f’(x) of course!f(x) F(x)

f’(x) f(x)

f’’(x) f’(x)

f’’’(x) f’’(x)

d/dx

d/dx

We just start with F(x) and take derivatives from there!

Here’s What FnInt Does…

Let’s look at an example using functions that we are familiar with…v(t) and s(t).

s represents the capital F and v represents the lower-case f.

b

a

f x dx F b F a

b

a

v t dt s b s a

What Now?

Soon, we will be using this information to compute the value of integrals w/o using FnInt.

Today, we will be using our new capital and lower-case f notation.

The Mean Value Theorem… Revisited

That Sounds Familiar (sort of)

The Mean (Average) Value for Derivatives that you should be thinking of is as follows:

This theorem held as long as the function was continuous on [a, b] and differentiable on (a, b).

Remember???

f b f af c

b a

Don’t Exactly Remember???

Let’s look at a quick animation to refresh our memories…

Here we go!

f b f af c

b a

The slope of the secant line between two points

The slope of the tangent line at x = c

So What? Derivatives are sooo out of style, Ms. Young!

Well guess what…the MVT for derivatives has a brother…

That’s right…the MVT for integrals!

Don’t worry…I’m going to show you why in just a minute.

But first, since you know some things about integrals now, let’s change the way we write the MVT for derivatives to make it easier.

Instead of using lower-case letters to represent the functions, let’s use capital letters!

So, rewrite

as

All we’re changing is the way we represent our function, f.

Just using big f’s instead of little ones. No biggie.

f b f af c

b a

F b F af c

b a

Again…Why’d We Do That?!

Instead of F(b) – F(a) we can write

because we just learned that:

F b F af c

b a

Rewrite the left side of the equation

b

a

f x dx

b

a

f x dx F b F a

The left side is still quite unattractive.

It can be rewritten as

So the MVT for Integrals states if f is continuous on [a, b], then at some point c in [a, b]:

b

a

f x dx

f cb a

Rewrite the left side of the equation, again

1 b

a

f x dxb a

1 b

a

f x dx f cb a

Say What?

The MVT for Integrals states that a continuous function on a closed interval always assumes its average value at least once in the interval.

1 b

a

f x dx f cb a

What if I Don’t Get It? If you can compute the

average value of a function using the formula…you’ll be able to do the work. You just won’t understand what exactly you’re doing.

1Average value of a function:

b

a

f x dxb a

Let’s Look at an Example

Use FnInt to calculate the Average Value of the function over on the interval [0,3].

The average value of f(x) is indicated by the red line drawn at y=-9/5.

At what x-value(s) does the f(x) equal the average value?

x=1.066 & x=2.620.

The average value of the function is a horizontal line that is positioned so that the area above the line is equal to the area below the line.

The yellow area and the blue area are equal.

the average value y = -9/5

the x-values where the function attains the average value

Another Example

the average value y = 66.4The area above and below the average value are equal!