Warm Up

1. Warm Up • Sketch a rectangular coordinate plane on a piece of paper. • Label the points (1, 3) and (5, 3). • Draw the graph of a differentiable function that starts at (1, 3) and ends at (5, 3). • WHAT DO YOU NOTICE ABOUT YOUR GRAPH?

2. 4.2Mean value theorem

3. Goal • Apply the Mean value theorem and determine when a function is increasing or decreasing.

4. Rolle’s theorem • Iff is differentiable on (a, b) AND f(a)=f(b) • Then there is at least one number c in (a, b) such that f’(c) = 0. In other words, if the stipulations above hold true, there is at least one point where the derivative equals 0 (there exists at least one critical point in (a, b) ALSO f has at least an absolute max or min on (a, b)).

5. Rolle’s theorem • Example: Show that f(x) = x4 – 2x2satisfies the stipulations to Rolle’s Theorem on [-2, 2] and find where Rolle’s Theorem holds true. • f is differentiable on (-2, 2) • f(-2) = f(2) = 8

6. Mean Value theorem • Consider the graph below: Slope of tangent at c? f’(c) Slope of secant from a to b? a c b

7. Mean value theorem • If • f is differentiable at every point on its interior (a, b) • Then there is at least one point c in (a, b) at which In other words, there is at least one point where the instantaneous slope equals the average slope.

8. Mean value theorem • Example: a.) Show that the function f(x) = x2 + 2x – 1 on [0, 1] satisfies the Mean Value Theorem. b.) Find each value of c that satisfies the MVT. a.) f is continuous and differentiable on [0, 1] b.) So,

9. You try • Example: Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) that satisfies the Mean Value Theorem. So,

10. Homework • 5.2: 1-3 • 2 new note cards! • Mean value theorem and Rolle’s Thm.