why is it the second most important theorem in calculus?
TRANSCRIPT
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.”