The Mean Value Theorem

1. The Mean Value Theorem Why is it the second most important theorem in calculus?

2. Some Familiar (and important) Principles Two closely related facts: suppose we have some fixed constant Cand differentiable functions f and g. • If f (x) C , then f ’(x)  0. • If f (x)  g(x) +C, then f ’(x) g ’(x) . Suppose we have adifferentiable function f. • If f is increasing on (a,b), then f ’ 0 on (a,b). • If f is decreasing on (a,b), then f ’ 0 on (a,b). How do we prove these things?

3. We set up the (relevant) Difference Quotients and Take Limits! Let’s try one!

4. Familiar (and more useful) Principles Two closely related facts: suppose we have some fixed constant Cand differentiable functions f and g. • If f ’(x)  0 , then f(x) C. • If f ’(x) g ’(x), then f(x)  g(x) +C. Suppose we have adifferentiable function f. • If f ’ 0 on (a,b), then f is increasing on (a,b). • If f ’ 0 on (a,b), then f is decreasing on (a,b). How do we prove these things?

5. Problem: we can’t “Un-Take” the Limits! Proving these requires more “finesse.”