3 ms·
Great explanatory writing, thanks. > In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a si
by nonrandomstring 3y ago
Great explanatory writing, thanks.
> In mathematics, we say that the length diverges. If you were
measuring a smooth curve, the length would converge on a single
value -- incorporating more detail would change the length, but only
by tinier and tinier amounts.
Can you say something bout the exponential curve in this context?
IIRC whatever level we zoom to we see the same curve, neither
converging nor diverging. Is there something special we should take
note of here?
- luca3v 3y agoThe exponential curve does look like a line when you zoom in very close to a point. For example, f(x) = e^x looks linear with slope e^x near x. You can see that, for small epsilon, e^(x+epsilon) - e^x is approximately epsilone^x, with an error term of the order of epsilon^2 e^x
- AdamH12113 3y agoI think you might be remembering something else -- an exponential curve is proportional to its own rate of change. Exponential curves are smooth in the way I described -- if you zoom in far enough, they look straight. The part about being proportional to its own rate of change does make exponential curves very important in calculus (and thus in science in general). Sine waves also have this property (although in a more complicated way). This is why you see exponential decay and sinusoidal oscillation so much in physics.