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.
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 ...