A Gentle Introduction to Calculus

Basic concepts

Imagine we’re in a car cruising at a constant 30 miles per hour (mph). Imagine we press the accelerator pedal. If we keep it pressed down, the speed of the car increases from 35, 40, 50, to 60 mph.

In this section, we’ll explore the idea of things changing, like the speed of a car, and how to work out that change mathematically. What do we mean by mathematically? We mean understanding how things are related to each other so we can work out precisely how changes in one result in changes in another. This is called calculus.

By the end of this appendix, we’ll see that working out how things change in a mathematically precise way—because that’s all calculus really is—is not that difficult for many useful scenarios.

Even if we’ve done calculus or differentiation already, it’s worth going through this section because we’ll understand how calculus was invented. The ideas and tools used by those pioneering mathematicians can be very helpful when trying to solve different problems in the future.

If you enjoy a good historic rivalry, the drama between Leibniz and Newton, who both claimed to have invented calculus first, can be entertaining.