1. 3.2a Rolle’s Theorem and the “Mean Old” Value Theorem Two theorems to help us understand graphs

2. Recall that the extreme value theorem guaranteed a maximum and minimum on a closed interval with a continuous function. Take out a piece of paper. Sketch in a coordinate plane and label points (1,3) and (5,3). Using a pencil or pen, draw a differentiable function that starts at (1,3) and ends at (5,3). Is there at least one point in the interval where the derivative would be zero? Would it be possible to draw the graph so that there isn’t a point where the derivative is zero?

3. b) If the differentiable requirement is dropped, there would still be a critical number in (a, b) but it would not yield a horizontal tangent there.

4. Ex 1 . 173 Find the two x-intercepts of and show that f ‘(x) = 0 at some point between the x-intercepts. Solution: To find x-intercepts, set f(x) = 0 and solve for x. What value of x makes the derivative zero? This confirms Rolle’s thm, since endpoints of interval are same height, and there is a point in interval where f ‘(c)=0

5. Ex 2 p. 173 Illustrating Rolle’s Theorem Let Find all values of c in interval [-3, 3] such that f ‘(c) = 0 Work with a partner to find c values. You should have found

6. Assignment 3.2a p. 176/1,2,5-29 EOO