11/14/2008, 05:34 PM
Thank you for your comments. I generated my approximations before seeing "http://math.eretrandre.org/tetrationforum/archive/index.php/thread-33.html" or anything else on this forum. Given that these are the very first posts I have ever made to a math forum, and given that its 30 years since I took calculus, I think I did ok! 
The critical section is interesting because it is where the point of inflection in the tetration curve is. The critical section is almost linear, and the inflection points position is the most visible feature of this section of curve. That's what motivated the boundary condition for f'(x) = f'(x-1); also it makes the equations simpler. Secondly, the inflection point seems like a universal section of the curve, that applies to any tetration base larger than e^(1/e). The approximation was verified for base 2, e, 3, and base 10.
Centering the approximation around +/- 0.5 was to center the taylor series radius of convergence on the unit length being approximated. A finite taylor series doesn't have a radius of convergence, so it may not matter. Centering the series on a linear section may help with numerical approximations for a higher order derivatives.
The inflection point still seems like an important feature of the tetration curve. Jay's graphs (in the link above), show the inflection points, which is the minimum of the odd derivatives, gradually moving from the ~= -0.5 range to the +0.35 range for higher even derivatives. It would be interesting to see a table of the "x" coordinate of the inflection points, along with the slope of the tetration curve at the point of inflection for various even derivatives.
The critical section is interesting because it is where the point of inflection in the tetration curve is. The critical section is almost linear, and the inflection points position is the most visible feature of this section of curve. That's what motivated the boundary condition for f'(x) = f'(x-1); also it makes the equations simpler. Secondly, the inflection point seems like a universal section of the curve, that applies to any tetration base larger than e^(1/e). The approximation was verified for base 2, e, 3, and base 10.
Centering the approximation around +/- 0.5 was to center the taylor series radius of convergence on the unit length being approximated. A finite taylor series doesn't have a radius of convergence, so it may not matter. Centering the series on a linear section may help with numerical approximations for a higher order derivatives.
The inflection point still seems like an important feature of the tetration curve. Jay's graphs (in the link above), show the inflection points, which is the minimum of the odd derivatives, gradually moving from the ~= -0.5 range to the +0.35 range for higher even derivatives. It would be interesting to see a table of the "x" coordinate of the inflection points, along with the slope of the tetration curve at the point of inflection for various even derivatives.
