Chapter 2

# The Two Big Ideas of Calculus: Differentiation and Integration — plus Infinite Series

In This Chapter

Delving into the derivative: It’s a rate and a slope

Investigating the integral — addition for experts

Infinite series: Achilles versus the tortoise — place your bets

This book covers the two main topics in calculus — differentiation and integration — as well as a third topic, infinite series. All three topics touch the earth and the heavens because all are built upon the rules of ordinary algebra and geometry and all involve the idea of infinity.

## Defining Differentiation

*Differentiation* is the process of finding the *derivative* of a curve. And the word “derivative” is just the fancy calculus term for the curve’s slope or steepness. And because the slope of a curve is equivalent to a simple rate (like *miles per hour* or *profit per item*), the derivative is a rate as well as a slope.

### The derivative is a slope

In algebra, you learned about the slope of a line — it’s equal to the ratio of the *rise* to the *run*. In other words, . See Figure 2-1. Let me guess: A sudden rush of algebra ...