Introduction

Calculus has been called the science of change, since its tools were developed to deal with constantly changing values such as the position and velocity of planets and projectiles. Previously, there was no way to express this kind of change in a variable.

The first important topic in calculus is the derivative. This is the rate of change of a function at a given point. Straight lines follow a simple pattern known as the slope. This is the change in the y value (the rise) over a given range of x values (the run):

Figure 10.1: Slope of a line

In Figure 10.1, the y value in the line increases by 2 units for every 1-unit increase in the x value, so we divide 2 by 1 to get a slope of 2.

However, the slope of a curve isn't constant over the whole curve like it is in a line. So, as you can see in Figure 10.2, the rate of change of this function at point A is different from the rate of change at point B:

Figure 10.2: Finding...