1. EXTREMELY important for the AP Exam
Chapter 4 Theorems
EXTREMELY important for the AP Exam

2. Rolle’s Theorem In the first quadrant, mark “a” and “b” on the x axis.
Plot points at f(a) and f(b) such that
f(a) = f(b)
Connect the two points however you would like but NOT in a horizontal line.
What do you notice?

3. Rolle’s Theorem f(a)=f(b) a b
What do you see between a and b on each of these graphs????

4. Rolle’s Theorem f’(c) = 0
Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). If f(a) = f(b) then there is at least one number c in (a, b) at which
f’(c) = 0

5. Mean Value Theorem
This is a more generalized version of Rolle’s Theorem.
First stated by Joseph-Louis Lagrange

6. Mean Value Theorem
Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b).
There is some point c in (a, b) at which

7. Mean Value theorem a b

8. Mean Value Theorem
Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b).
There is some point c in (a, b) at which

9. Mean Value theorem a c b

10. Extreme Value Theorem If f is continuous on a closed interval [a, b]
Then f attains both an absolute maximum M
And an absolute minimum m
In [a,b]

11. Extreme Value Theorem (cont)
That is, there are numbers x1 and x2 in [a, b]
With f(x1) = m and f(x2) = M
And m ≤ f(x) ≤ M for every other x in [a, b]

12. y = f(x) M m b a

13. y = f(x) M m a b

14. y = f(x) M m a b