What the Heck is Calculus?

Calculus is one of those words that feels like an omen to those who do not use it regularly. "Calculus? No, I only had to take college algebra!" "Why would I need to know calculus?! I'm in marketing." Calculus seems to be the boogeyman of college classes, claiming innocent STEM sophomores only trying to cross the stage. I will not lie to you, I was not too fond of it. In fact, I felt like I would be fine never looking at any math again after completing Calculus II. The irony is that I use it constantly, but not in the way I was taught. I slaved away for hours trying to remember the chain rule and integration by substitution only to never work out a problem by hand since then. However, I deal with the formulas and utilization of calculus concepts daily. That is my main gripe with math coursework - the focus should be on the concepts as much as solving them. I'm going to talk broadly about calculus and why we need it.

The short answer is it's everywhere. Yes, even for you. The short definition of Calculus is the study of how things change. Now, if you're thinking: "But, everything changes," you are correct! And we are hardwired to notice changes because, usually, that's when the interesting stuff happens. When you drive onto the highway, do you ever zone out for a while? Maybe you're listening to the music on the radio and looking at the road but your mind tends to wander. This passive mode is because you're going at the same speed and direction for a long time; your brain is on autopilot. What would happen if the speed was suddenly reduced from 65 to 55 mph and there was a police car sitting right across from the speed limit sign? Suddenly, you're vigilantly watching the speedometer and begin to slow down. We pay close attention to when and how things change and calculus gives us a way to calculate this!

But why do we need calculus? Here comes a shocking statement: we can't measure something instantaneously. You may not believe me, but it's true. Let's return to the car example, but imagine your speedometer stopped working. If you wanted to know how fast you're going, you could calculate it. If you drove 1 mile in the past 60 seconds, you are driving at 1 mile/minute x 60 minutes/ hour = 60 miles per hour. Unfortunately, this calculation is only an average speed; it is how fast you drove over 60 seconds. What if you wanted to know how fast you were driving in a singular instant? If no time passes, you don't have any speed! Speed is the distance over time traveled: 0/0=0. This sounds frustrating, right? Scientists had to think of some workaround for this. Guess what they came up with? That's right, calculus: the ultimate cheat code.

There are two kinds of calculus: differentiation and integration. They are inverses of each other, resembling the inverse operations of multiplication and division. The driving force behind both of these is breaking up the independent variable (in most cases time) into infinitesimally small chunks. If we calculate the change in distance over a time that's extremely close to (but not exactly) zero, we have a close approximation to our instantaneous speed. This is how we find a derivative! In two-dimensional graphs, we can represent distance over time as a function. The rate of change that we get at a specific point in time is called a tangent line. If a line is tangent to a function, it just means it only touches the graph at one point (this is the speed we are looking for). Derivatives are typically expressed as dx/dt which stands for "delta x over delta t". Delta is math speak for "change". If you replace x with "distance" to fit our example, it reads as "change in distance over the change in time". Look at that, you can read math!

Alright, now imagine the opposite scenario. We know how fast we were going during our whole trip but we didn't know how far we had gone. We can figure this out using integration! Again we take small chunks of time and find the speed at each time. Say we were going 5 mph during the first second, 10 mph during the second second, and 15 mph during the third second. We can add up these speeds and find out how far we went: 5+10+15 = 30 miles/hour ÷ 3600 seconds/hour x 3 seconds = 0.025 miles. If we take chunks smaller than a second, say a microsecond, we get a closer estimation of how far we went in a single instant. We can also use this to get an average distance over a large amount of time. Say you were taking a 12-hour drive and wanted to know how far you went during hour 5, minute 43, second 15, and hour 5, minute 55, second 36. We can do this by integrating (summing up the speeds over those tiny chunks of time) over the difference in time for only the 12 minutes 21 and seconds we care about. This is the power of calculus! We usually represent these chunks of time (dt) as tiny rectangles underneath the curve of the function of speed over time. We add up the area of these rectangles (speed x dt = area) to give us an approximation of the "area under the curve". In our example, the area under the curve is the average distance we want!

When is this actually useful? Anytime we have rates of change over time! How much did my Bitcoin gain in value for this month? How fast is the ozone layer depleting this year compared to 30 years ago? How quickly is a tumor growing in a patient? All these questions are answerable through calculus. What's even better, once we understand how an object changes over time, we can begin to make predictions about how the object might change in the future. This makes it easier for us to create prediction models, aka AI. A truly humbling experience is finding out that the foundations of calculus were discovered as early as ancient Egypt (around 1800s b.c.). Modern calculus was developed by Gottfried Leibniz and Isaac Newton (the apple guy!) in the late 17th century. People have been doing calculus to solve problems...forever. Cool, right?