September 21 - Rolle's Theorem

We begin Chapter 3 with Rolle's Theorem. This is the first of several theorems that we will talk about that describe the behavior of what I have previously called "nice" functions. That is, we are better defining "nice."

Rolle's Theorem is actually a silly sort of theorem. Generalized, it becomes a more powerful and important statement about functions; alas, poor Rolle didn't generalize and so didn't get the greater glory that could have been his. (Lagrange would eventually prove the Mean Value Theorem - we will cover this tomorrow.)

Another new concept that we covered today is the idea of differentiability. We have previously talked about when a function is continuous. Now you should be able to describe when a function is differentiable. Intuitively, if a function has a "sharp point" (absolute value, x^(2/3)), then it is not differentiable at that sharp point. Additionally, if a function is not continuous at some x = c, then it is also not differentiable at x = c.

Notes from today are below.