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 findings 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).

Tags: best affine approximation, calculus, differential, education, Fourier decomposition, long reads, math, mathematics, maths, science, Taylor's theorem

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