For those of you who have studied calculus, how many of you have asked yourself this question? I first learned calculus at Brooklyn Technical High School in my AB Calculus class taught by an excellent teacher named Mr. Raftery.
But while I knew the algorithmic steps behind taking limits, finding derivatives, calculating integrals, I did not quite understand what was the motivation and purpose for calculus. It was not until I went well into my math and physics courses in college that I realized what it is about.
Calculus was invented at roughly the same time independently by two people, the mathematician Gottfried Leibniz and the great mathematician, physicist Sir Isaac Newton. Everyone knows the famous story of the apple falling on Newton’s head that lead him to realize that the force bringing the apple to the ground was the same force as that which governed the motion of the planets. In order to describe this phenomenon mathematically, Newton invented a completely new branch of mathematics (calculus) that would change the way people saw the world.
Calculus takes algebra to a new level. With algebra, we could calculate things that were “easy to work with” such as lines, squares, triangles, etc. But as we all know, the real world is much more complicated than this. The real world consists of curves, changes, and weird geometries. Calculus empowers us to calculate and identify the rich mathematical properties of these irregular forms. It does this by considering changes in the structure or form that are infinitesimally small, meaning smaller than anything you can think of greater than 0. By looking at things in infinitesimally small changes and combining their contributions, you can learn about complex systems without examining the irregularities head on.
To further illustrate the idea behind calculus, imagine I have a nice straight line and a very nasty confusing curve and I want to find the lengths of both. For the straight line, this is trivial. I take a ruler and measure it. To measure the curve is not as simple. But what I can do is zoom in on a very very tiny segment of the curve, so tiny that it almost a perfectly flat line. This I can measure. I do this for every other segment on the curve until I measure the whole thing. Then I add all of their contributions and I get the length of this complicated shape. The tinier I take these segments, the more accurate the measured length will be. This is the idea behind calculus. You look at small changes and see how they contribute as a whole. Calculus is about finding the behavior of continuously changing systems. This was an ingenious way of looking at the world, and it eventually changed it.
The main application of calculus was initially to physics. Back then, if I dropped a ball from the top of my house and I wanted to know its speed, I would take the distance it fell and divide it by the time it took to fall and I would get the speed. But this is only the average speed. Calculus enables us to find the exact speed at any point of the ball’s fall, and the ball’s speed is constantly changing since it is accelerated by gravity. It turns out that studying changes in a system reveal interesting properties that enable us to make accurate predictions about how it will behave in the future. Calculus has a broad range of applications to any type of changing system including chemistry, population growth, probability, finance, and statistics.
When I realized its purpose, perhaps a little later than I should have, I developed an incredible appreciation for the field because it was such a creative insight and it revolutionized the world. Its invention has enabled us to warm our homes, power our lives, move our cars, peer into the universe, probe the atom, remain connected in a world of disconnect. It has allowed me to deliver this message to you.
Watch an excellent explanation of the motivation behind calculus starting at 13:20.
VIDEO: Newton’s Dark Secrets
