Everyday applications of calculus

There are numerous ways one can think about change. It might be the number of coins in your pocket. It could be that thing politicians promise but often find very difficult to deliver. Instead, let’s talk about the sort of change which goes on in nature itself and how we humans begin to appreciate what happens around us. It starts with the mathematical subject of calculus. Don’t worry, you won’t have to solve an equation to understand calculus or this article.

“Calculus” is one of those words which often strikes fear and terror in the hearts of laypersons everywhere. I know it does because it once did to me too.

Even though I knew I loved the sciences and endeavored to pursue them in college, I also knew they required a rigorous dose of hardcore mathematics. At the core of those mathematics was calculus with all its weird symbols, obscure definitions and the whole host of other scary words which go along with it. Examples are derivatives, integrals and the term “instantaneous rate of change.”

Mark those last words, dear reader.

I confess I struggled with my fellow students in the introductory courses. What did all this stuff mean? How were my classmates and I ever going to do it with the same fluidity and easy grace as my teachers did? Homework and studycertainly helped us to pass the tests, but what lasting understanding could we hope to obtain from slogging our way through endless formulas and rules?

The answer lies in physics, aka the real world. Newton’s Laws of Motion, particularly Newton’s Second Law, relate the force on an object to the acceleration produced in the object. In other words, if you push an object at rest (let’s say your friend chilling out next to you) hard enough, it will begin to move with some acceleration.

This action is by itself nothing special; the movement of something is part of the everyday human experience. However, that is what makes the action so important.
The universe constantly moves; objects constantly go from rest to motion and from motion to rest. If your boyfriend or girlfriend never moved, then the universe would indeed be a very boring place to live. Try to imagine activities like sex, bowling, golfing or running without motion and you will find yourself thinking of a nightmarish Twilight Zone.

So, given that things do change, how do we make sense of these shifts? How do we bring the universe down to our level so we can understand these natural physical processes of change?

The answer: calculus.

As I learned, calculus and the real world are related because calculus is a tool used to describe the rates of change of nature. Let’s go back to acceleration. Acceleration is defined as the rate of change of velocity. It is a measure of how something going fast gets faster. Cars accelerate, planes accelerate, people accelerate.

All these different situations are governed by the same physical laws and these physical laws are described by using calculus with its special language to handle instantaneous rates of change, how something gets faster at any given moment in time.

Calculus doesn’t just describe the change in physical motion. It can also cover how fast wages change year by year, or how global temperatures change, or how space itself is expanding in the billions of years since the Big Bang.

When the connection between calculus and the real world became so readily apparent to me, calculus lost a great deal of its horrifying impenetrability. I focused less on the seemingly obscure difficulties and more on why the subject itself was so important. In my case, it was a classic instance of missing the forest for the sake of the individual trees.

Don’t let calculus intimidate you with its definitions and symbols. Those become far less challenging as long as you first understand what I missed for so long. Calculus is about how things change. Change is synonymous with the real world because the real world changes. Given that most of us live in the real world, calculus is simply a way to measure that which we already know intuitively. As the saying goes, “The more things change, the more they stay the same.”

About Myles Cupp