I have oft gushed in these pages about the wonderfulness of calculus. Even if you hate the subject, perhaps because you are currently getting smacked around in a calculus course (I know the feeling) you must nonetheless accept a simple truth: Calculus is what separates us from the animals. If there are aliens, and there had better be, the "notable accomplishments" entry on Earthlings in the alien Wikipedia will have but two entries: (1) calculus, and (2) those glass globes you shake and it snows inside.
Still, it is possible to get to the end of a four semester calculus sequence and miss the point. It's like watching Dawn of the Dead and only years later realizing Romero was deconstructing the Freudian model of the subconscious, with zombie id, biker ego, and our scrappy band of heroes the superego. (Not really sure what the helicopter represented. The violence inherit in the system, I think.)
So we ask: Why calculus? Why suffer its outrageous slings and arrows? Wherefore art thou, calculus?
Why? Because Nature is a bitch. Not how you'll hear it explained in polite company, but that does not make it untrue.
Sit, my children. For it is time you learn the way of things.
If you look at almost any fundamental law of nature, it does not have the form:
thing of interest = a bunch of other stuff
It has the form:
how thing of interest is changing
= a bunch of other stuff
For example, take Newton's law: F = ma. If you know force (F) and mass (m), the law allows you to determine, say, the movement of a pendulum, or the orbit of a planet, or the path of a cannonball. But you don't just get to write that down. Oh, no. Nature doesn't make things easy for you. The "a" in Newton's law is acceleration. It's how velocity is changing with time (which is the derivative of velocity or, equivalently, the second derivative of position). Newton's law is a differential equation, that is, an equation that includes one or more derivatives. To get the answer you want, it's not enough to know the force(s) and the mass(es): you also have to solve the differential equation.
It's the same everywhere you look. Population growth? Differential equation. Chemical reaction? Differential equation. Diffusion? Differential equation. Heat transfer? Differential equation. Schrodinger's equation is a differential equation. Maxwell's equations are a set of four differential equations.
We don't just get to write down answers. If we want the thing, Nature demands we work backwards from a description of how the thing is changing. This is where calculus shines. Its intensive porpoise. Its whole raisin d'tree. Given a thing, the derivative tells us how the thing is changing. It was only a matter of time before people started asking whether it was possible to go in the other direction (those people were Newton and Leibniz; that time was the late 17th century). Thus was born integration and, soon after, differential equations. It was the breakthrough that made the modern world possible.
Footnote: The primary contribution of Newton and Leibniz was establishing the connection between anti-differentiation and integration, now wrapped up in the tidy Fundamental Theorem of Calculus. However, the basic gist of calculus goes all the way back to the ancient Greeks. You may have heard of Zeno's paradox, which is just a non-STEM way of describing infinite series. Good thing nobody back then could show the geometric converged or Zeno would have been out of a job.
The crown jewel of all our toilings, calculus is. Along with those glass globes you shake and it snows inside, calculus defines what it means to be human. The stage upon which we all trod. She speaks yet she says nothing; her eye discourses. Fair Juliet, our harsh mistress.
Image Credit: The Galatian Suicide by Jastrow via Wikipedia and released into the public domain.
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