1. Why is it the second most important theorem in calculus?

2. 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?

3. Let’s try one!

4. 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?

5. Proving these requires more “finesse.”