1. Rolle’s Theorem
Section 3.2

2. Michel Rolle ( )

3. Rolle’s Theorem
For a continuous function on the closed interval [a,b] that is differentiable on (a,b), if f(a) = f(b), then there is at least one number c in (a,b) such that f ‘(c)=0.

4. Visual Interpretation
f(2) = f(6)
Function is continuous
and differentiable on (2,6).
Is there an x-value in
(2,6) where there is a
horizontal tangent line?

5. Does Rolle’s Theorem Apply?
NO!!!!!
The function is NOT differentiable at x = -3.

6. Does Rolle’s Theorem Apply?
YES!!!
f(0)=f(π), and the function is continuous and differentiable on (0,π).

7. Determine whether Rolle’s Theorem applies on [a,b]
Determine whether Rolle’s Theorem applies on [a,b]. If so, find all values c in (a,b) such that f ‘(c) = 0.
f(x) is a polynomial…continuous everywhere.
Rolle’s Theorem applies!

8. Assignment
p.172: 1,2,3-15 odd

9. Mean Value Theorem
Section 3.2

10. Mean Value Theorem
If f is continuous on the closed interval [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that
We’ll use this to prove the Fundamental Theorem of Calculus! (coming this spring…)

11. Visual Interpretation of the Mean Value Theorem
If f is continuous on [a,b] and differentiable on (a,b), then…

12. Visual Interpretation of the Mean Value Theorem
If f is continuous on [a,b] and differentiable on (a,b), then…
…there is some c in (a,b) such that the tangent line of the graph at c is parallel to the secant line through a and b.

13. Consider on the interval [1,4].
Does the Mean Value Theorem apply? If so, find all values for c guaranteed by the theorem.
MVT applies

14. Assignment
p.172: 29-34