THE MEAN VALUE THEOREM

Why is it the second most important theorem in calculus?

Some Familiar (and important)

Principles

Two closely related facts: suppose we have some fixed constant C and 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 a differentiable 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?

WE SET UP THE (RELEVANT) DIFFERENCE QUOTIENTS

AND TAKE LIMITS!

Let’s try one!

Familiar (and more useful)

Principles

Two closely related facts: suppose we have some fixed constant C and 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 a differentiable 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?

PROBLEM: WE CAN’T “UN-TAKE” THE

LIMITS!

Proving these requires more “finesse.”