Introduction

12.1 Introduction

Calculus is the branch of mathematics that studies how the value of a function changes as the points in its domain change. This chapter is a first introduction to calculus, and it is built up in careful stages rather than all at once.

We begin with an intuitive idea of the derivative of a function, without giving it a formal definition — this is done through a physical motivating example (the problem of finding the velocity of a falling body at a single instant). Only after that intuition is in place do we give a working, informal definition of a limit and study the algebra of limits: how limits behave under addition, subtraction, multiplication and division of functions. We then return to the derivative, define it properly using the idea of a limit, and study the algebra of derivatives in the same way. The chapter closes by using these rules to obtain the derivatives of some standard functions, such as powers of $x$ and the trigonometric functions $\sin x$ and $\cos x$.