# Second Derivatives and Beyond

### Intro

Dr. Shmeuss and his trusty companion, Bella, have decided to start a new hobby, spelunking. Not only are they looking for adventure, they are looking for treasure. Dr. Shmeuss is a huge fan of *The Hobbit*, and he has hopes of becoming as lucky as Bilbo. They gather their ropes, don their headlamps, and head into the cave. They don't go far when they run into a drove of convicts who are, of course, also spelunking for treasure. They've walked into a con-cave. The cave starts spinning. Which way is (concave) up or (concave) down? They hightail it out of there, changing their velocity with respect to time until they reach safety.

What does a con-cave need to do with second derivatives? Nothing. However, the term **concave** actually does. Not to mention acceleration is the second derivative of distance with respect to time.

This unit deals with derivatives beyond the first one. For some functions we can take second derivatives, third derivatives, fourth derivatives, etc.

The second derivative is tied to the shape of a function. The first derivative tells whether the slope of the original function is increasing or decreasing. The second derivative describes how the original function curves.

We can also use the first and second derivatives to find maximum and minimum points of a function.

For the grand finale, we'll draw graphs of wild functions without calculators. Once we know where a function turns around and where it changes shape, we can draw a good picture of what it looks like overall. Keep the calculator close, but mostly for finding approximate values and for checking answers.