Calculus Essentials For Dummies

Chapter 6

Differentiation and the Shape of Curves

IN THIS CHAPTER

Locating extrema

Using the first and second derivative tests

Interpreting concavity and points of inflection

If you’ve read Chapters 4 and 5, you’re probably an expert at finding derivatives. Which is a good thing, because in this chapter you use derivatives to understand the shape of functions — where they rise, fall, max out and bottom out, how they curve, and so on.

A Calculus Road Trip

Imagine that you’re driving along the function in Figure 6-1 from left to right. Along your drive, there are several points of interest between a and l. All of them, except for the start and finish points, relate to the steepness of the road — in other words, its slope or derivative. I’m going to throw lots of new terms and definitions at you all at once here, but you shouldn’t have too much trouble because these ideas mostly involve commonsense notions like driving up or down an incline, or going over the crest of a hill.

• The function in Figure 6-1 has a derivative of zero at stationary points (level points) b, d, g, i, and k. At j, the derivative is undefined (a sharp turning point like j is a corner). These points where the derivative ...