FnInt …

1. FnInt… What does it actually do?

2. 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.

3. 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! d/dx d/dx

4. 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! We just start with F(x) and take derivatives from there! d/dx d/dx

5. 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.

6. 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.

7. The Mean Value Theorem… Revisited

8. 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???

9. Don’t Exactly Remember??? • Let’s look at a quick animation to refresh our memories… • Here we go! The slope of the secant line between two points The slope of the tangent line at x = c

10. 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.

11. 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.

12. Again…Why’d We Do That?! Rewrite the left side of the equation • Instead of F(b) – F(a) we can write because we just learned that:

13. Rewrite the left side of the equation, again • 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]:

14. 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.

15. 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.

16. Let’s Look at an Example • Use FnInt to calculate the Average Value of the function over on the interval [0,3].

17. 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.

18. the x-values where the function attains the average value the average value y = -9/5 • 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.

19. Another Example the average value y = 66.4 The area above and below the average value are equal!