The 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
Re:Coastline Paradox
by AdamH12113 on Sunday December 31, 2023 @08:24PM
(#38828984)
I 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.
> 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?