1. 3.2 Rolle’s Theorem and the Mean Value Theorem Calculus!!! We

2. Rolle’s Theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If then there is at least one number c in (a, b) . c c a b

3. Ex. Find the two x-intercepts of and show that at some point between the two intercepts. x-int. are 1 and 2 Rolles Theorem is satisfied as there is a point at x = 3/2 where .

4. Let . Find all c in the interval (-2, 2) such that . Since , we can use Rolle’s Theorem. 8 Thus, in the interval (-2, 2), the derivative is zero at each of these three x-values.

5. If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b: The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope. Mean Value Theorem for Derivatives The Mean Value Theorem only applies over a closed interval.

6. Tangent parallel to chord. Slope of tangent: Slope of chord:

7. A function is increasing over an interval if the derivative is always positive. A function is decreasing over an interval if the derivative is always negative. A couple of somewhat obvious definitions:

8. Functions with the same derivative differ by a constant. These two functions have the same slope at any value of x.

9. could be or could vary by some constant . Example 6: Find the function whose derivative is and whose graph passes through . so:

10. Example 6: HW Pg. 176 1-7 odd, 11-27 odd, 33-45 odd Find the function whose derivative is and whose graph passes through . so: Notice that we had to have initial values to determine the value of C.