I'm working on approximating functions in A.I., and I noticed that everyday functions seem to, at some point, have either a cyclical, or constant, derivative.
For example, a straight line has a constant derivative, and so we can easily approximate a linear function by simply taking the difference between the range values.
Consider f(x) = 2x + 3:
I would start with a dataset that contains the range values, and then try to extrapolate a function, so I would be given (x,y) pairs -
(1,5); (2,7); (3,9); (4,11).
If we take the difference over the y values, it's 2, which is constant, and of course the slope of the line. This means that we can generate the function by simply taking the initial y value 5, and iteratively adding 2.
If you do the same for a sin function, eventually you'll find the derivative repeats.
If we consider a parabola, the function itself does not have a constant rate of change, since its slope is a linear function. But, this of course implies that its derivative will have a constant rate of change.
As a general matter, I'm wondering if there's a word that describes this class of functions that eventually have a constant, or cyclical derivative.
Note that it is necessarily the case that this class of functions can be very easily approximated, since you just bootstrap up from the first constant / cyclical derivative, until you get to the original function.
As a result, it looks like this is a pretty good method for approximating normal data that doesn't have a wildly complex underlying function.