4.2 The Mean Value Theorem

1. 4.2 The Mean Value Theorem

2. Rolle’s Theorem Let f be a function that satisfies the following three conditions: • f is continuous on the closed interval [a,b] . • f is differentiable on the open interval (a,b) . • f(a) = f(b) . Then there is a number c in (a,b) such that f ′(c)=0. Examples on the board.

3. 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 interval, the actual slope equals the average slope. Mean Value Theorem for Derivatives Note: The Mean Value Theorem only applies over a closed interval.

4. An illustration of the Mean Value Theorem. Tangent parallel to chord. Slope of tangent: Slope of chord:

5. Corollaries of the Mean Value Theorem • Corollary 1: If f ′(x)=0 for all x in an interval (a,b), then f is constant on (a,b) . • Corollary 2: If f ′ (x) = g′ (x) for all x in an interval (a,b), then f- g is constant on (a,b) , that is f(x)=g(x) + c where c is a constant. (see the next slide for an illustration of Corollary 2)

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