Chapter 5

Chapter 5 Overview

The derivative is supremely useful because there are so many ways to think of it. It is the instantaneous rate of change, enabling us to find velocity when position is given or acceleration when velocity is given. It also describes sensitivity, how change in the input variable is reflected in the change of the output variable. And it has a geometric meaning, as the slope of the tangent to the graph of a function. In this chapter, we will explore how all of these understandings of the derivative lead to applications.

The derivative tells us a great deal about the shape of a curve. Even though we can graph a function quickly, and usually correctly using a grapher, the derivative gives us precise information about how the curve bends and exactly where it turns. We also will see how to deduce rates of change we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point. The key to recovering functions from derivatives is the Mean Value Theorem, the theorem whose corollaries provide the gateway to integral calculus, which we begin in Chapter 6.

This chapter begins where the early developers of calculus began, with the derivative as the slope of the tangent line and the insights this gives us into finding the greatest and least values of a function.