Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 4. CALCULUS

To create a good philosophy you should renounce metaphysics but be a good mathematician.

BERTRAND RUSSELL (1872–1970)

We have taken an overview of mathematics in history. In this part of the book, we turn to two major subjects of mathematics, calculus and number theory, to be studied in detail. The first of these, calculus, is the focus of this chapter. Calculus has a rich historical tradition and is widely used in scientific endeavors today.

4.1 What Is Calculus?

Calculus is the mathematics of change. It deals with rates of change of all kinds. We can use calculus to compute physical rates, such as the rate at which a rocket rises or a bomb falls, or the rate at which a radioactive substance decays. We can compute rates in biology, such as the rate at which a bacteria colony grows, the rate at which solutions diffuse across a membrane, or the rate at which a disease spreads through a population. It has applications in economics, such as marginal cost, marginal profit, and elasticity of demand. We use calculus to solve engineering problems, like how to make a roller coaster exciting but not dangerous.

Using calculus, we can maximize and minimize quantities. How can we produce a can of green beans in the most efficient way possible? Calculus is part of the answer. How can we maximize the profit of a company, or an industry, or an economy? Calculus plays a part in the models that answer these questions.

If something changes, and nearly everything interesting changes, calculus may have a role in describing it or modeling it.