I need to build a quick curve that models the shape of what I believe is called hyperbolic growth. It should look something like this production function or this curve.
However, these curves don’t apear to my non-mathmatical eye as the same thing as wiki’s hyperbolic function.
Can anyone help?
I’m sure someone with more math skill than myself will be along soon, but, to me, it resembles a transformation done to the function f(x)=ln(x), or the natural log curve.
I thought about a natural log function, but the log function in unbounded. I think my curve should go to some kind of limit, if that’s the right word.
The curves in the OP look logarithmic to me. But if you need something bounded, try the arctangent or hyperbolic tangent functions.
“Hyperbolic growth” probably means a function y=-a/(x-x[sub]0[/sub]); as x approaches x[sub]0[/sub] from below, y approaches infinity. This function is a hyperbola, but one rotated so that its asymptotes are vertical and horizontal, not two vertically-mirrored diagonals as in the Wikipedia plot.
It might also refer to a model based on one of the hyperbolic trig functions (like tanh), though the models seem to approach the asymptotes more slowly than the hyperbolic functions do.
You could always write the autonomous first-order equation:
y’= a / (50-y)
(based loosely on one of the curves linked in the OP)
As y->50, the derivative of y (i.e. y’) tends to zero and the curve flattens.
