Can you explain the difference between a function being "exponential in the beginning" and "having first derivative bounded away from zero at the beginning" ?
exponential in the beginning == this part of the function can be approximated by a function ae^xb where a>0 and b>1
vs.
first derivative bounded away from zero in the beginning == any function that increases, including linear functions with constant first derivative and polynomials with linear first derivative
Or do you think all increasing functions are the same..?
Consider for example the functions f(x) = x + 1 and g(x) = e^{ln(2) * x}. Then f(0) = g(0), f(1) = g(1), and f(x) > g(x) whenever 0 < x < 1.
It is easy to show that for any function whose derivative is continuous and positive at 0, there is an exponential function (properly translated such that they agree at 0) that has similar properties.
You should be specific about what properties you're talking about. What you're saying is that any function increasing function grows faster than some exponential function on a finite interval.
Still, you can observe f(x) on x ∈ [0, 1] and see that it is growing linearly.
And you can observe g(x) on ∈ [0, 1] and see that it is growing exponentially.
I do not see the value in discussing the rate of exponential growth of f. Where as for g, there is a parameter with value ln(2).
If data looks like f, don't try to fit an exponential function to it (whether it's a least-squares fit, or any other objective f > g.
