1. 1 3.2 The Mean Value Theorem

2. 2 Rolle’s Theorem

3. 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

4. 4 The Mean Value Theorem This theorem is an extension of Rolle’s Theorem

5. 5 Example Show that f(x) satisfies the Mean Value Theorem on [a,b] f (x) = x 3 – x, Interval: a = 0, b = 2. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 2] and differentiable on (0, 2). Therefore, by the Mean Value Theorem, there is a number c in (0, 2) such that f (2) – f (0) = f (c)(2 – 0)

6. 6 Example - proof f (2) = 6, f (0) = 0, and f (x) = 3x 2 – 1, so this equation becomes: 6 = (3c 2 – 1)2 = 6c 2 – 2 which gives that is, c = But c must lie in (0, 2), so

7. 7 The Mean Value Theorem The Mean Value Theorem can be used to establish some of the basic facts of differential calculus.