What is calculus?

Although it is a huge subject, it is really about two fundamental ideas: differentiation and integration.

Differentiation is a sophisticated form of division, and tries to get at 0/0. Integration is a sophisticated form of multiplication and addition, and tries to get at 0*infinity.

I will discuss differentiation first. The fundamental idea in differentiation is that of determining a rate of change. If we are traveling at a constant speed, it is easy to determine. But what if we are accelerating or decelerating? What is our speed at a given moment? To get a speed, you divide the amount of distance traveled by the amount of time it took. If we go 40 miles in 1 hour, our average speed is 40/1 = 40 miles per hour. But, at the beginning of the trip, we had to accelerate to 40 mph. How fast were we going at exactly 1 second? You could try taking the distance traveled in time between 0.5 seconds and 1.5 seconds, and dividing by 1 second, but that would get you only the average speed travelled in that second, not the speed at exactly 1 second. You could take the time between .75 and 1.25 seconds, or .9 and 1.1 seconds or .9999 and 1.0001 seconds, and each would get you closer to what you wanted, but not exactly to what you wanted. The problem is that you want the speed at an *instant*, and that’s 0 seconds. In 0 seconds, you travel 0 distance, and that’s 0/0 which is nonsense. Differential calculus provides a way to answer such questions, at least if you know your rate of acceleration.

If you know your acceleration in terms of a formula, then you can differentiate that formula to get your speed at any given instant.

Integration, on the other hand, was invented to find areas of shapes that were curved in certain ways. For many shapes, there are formulas in geometry that can be used – e.g, for circles, squares, rectangles and so on. But if the shape is formed by some other functions, there are no formulas in geometry. One way to find the area of any shape is to divide it into small trapezoids, find the area of each trapezoid, and add them up. But this will not be exact, until we get to an infinite number of trapezoids, but then each has an area of 0, and infinity * 0 = nonsense. Integral calculus answers this question.