The chief triumph of differential calculus is this:
Any nonlinear function can be approximated by a linear function.
(OK…pretty much any nonlinear function.) That approximation is the differential, aka the tangent line, aka the best affine approximation. It is valid in only a small area but that’s good enough. Because small areas can be put together to make big areas. And short lines can make nonlinear* curves.
In other words, zoom in on a function enough and it looks like a simple line. Even when the zoomed-out picture is shaky, wiggly, jumpy, scrawly, volatile, or intermittently-volatile-and-not-volatile:
Moreover, calculus says how far off those linear approximations are. So you know how tiny the straight, flat puzzle pieces should be to look like a curve when put together. That kind of advice is good enough to engineer with.
It’s surprising that you can break things down like that, because nonlinear functions can get really, really intricate. The world is, like, complicated.
So it’s reassuring to know that ideas that are built up from counting & grouping rocks on the ground, and drawing lines & circles in the sand, are in principle capable of describing ocean currents, architecture, finance, computers, mechanics, earthquakes, electronics, physics.
(OK, there are other reasons to be less optimistic.)
* What’s so terrible about nonlinear functions anyway? They’re not terrible, they’re terribly interesting. It’s just nearly impossible to generally, completely and totally solve nonlinear problems.
But lines are doable. You can project lines outward. You can solve systems of linear equations with the tap of a computer. So if it’s possible to decompose nonlinear things into linear pieces, you’re money.
Two more neat things come out from calculus.
- One can get closer to the nonlinear truth even faster by using polynomials. Put another way, the simple operations of + and ×, taught in elementary school, are good enough to do pretty much anything, so long as you do + and × enough times.
- One can also get arbitrarily truthy using trig functions. You may not remember sin & cos but they are dead simple. More later on the sexy things you can do with them (Fourier decomposition).