# Understanding Calculus in 4 Minutes

Calculus is the bane of many people. Much of that is the approach taken when teaching it. It’s taught using mathematical theorems and proofs rather than physical examples. I’ve had a number of people assure me that calculus doesn’t have any real-world applications. They never had any examples given when they learned it and it was taught as a purely intellectual exercise. I seek to correct that here.

Calculus describes things that change, usually with time. The most intuitive example I can think of is the relationship between distance, velocity, and acceleration. Velocity is the rate of change in distance and acceleration is the rate of change of velocity. Everyone moves, so it’s easily relatable. Here is the example I use.

Imagine you are sitting at a stop sign in your car. After stopping, you hit the gas and accelerate at a constant rate of 2 m/s² (the units don’t really matter). Your distance from the stop sign, velocity, and acceleration will look like this:

The mathematical description of how these change with time is:

Distance = time squared

Velocity = 2X time

Acceleration = 2

Now to find how to get from one value to another we need to figure out the relationships between the graphs. It turns out that the equation for the area under the curve on one graph is equal to the equation of the line on another graph. Let’s look at some examples.

Let’s look at the area under the curve for acceleration because it’s easiest since it’s just a rectangle. We know the height of the rectangle is 2 and the base is whatever time we pick. Let’s pick 3 seconds because it’s an easy number to use in calculations.

A = bh = (3)(2) = 6

Now, what is velocity doing at 3 seconds? V=2X = 2(3) = 6