- dt is a moment of your life
- ∫ dt is the moments as they accumulate
is the whole thing and what it meant.
Posts tagged with differential
The chief triumph of differential calculus is this:
(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.