TV Taylor expansion is approximating a function F (normally from |R ---> |R) by a 'Taylor' polynomial T of degree N which is based on the first N derivatives at a point a of the domain of F.
The idea is to make a polynomial by looking at the function value, and make it follow the first N derivatives at a point a as an approximation of F around F(a).
This works so well that a lot of relevant theory is based on the main theorem:
2 (n) n
t (x) = f(a) + f'(a)(x-a) + f''(a) (x-a) + ... + f (a) (x-a)
n ------ -------
2! n!
It can be proven that the absolute error in the approximation is upper bounded by the next term of the expansion [no, but it's not far off].
In various theoretical mathematical proofs, the theorem has a prominent place (also in engineering because of its practical value) mainly because of its general applicability: any continuous, bounded function [not true, says CL; it's "any analytic function", although that's far from the most useful form of the theory] can be fully represented, without error by an infinite order (or as many orders as the function has) Taylor expansion in one point only.
It would be interesting to have a math lib which can differentiate on lists in Tcl, but at this point, I didn't write one, or know of one. Wolfram licences are expensive (at least they were 10 years ago). [ CL speculates that this paragraph is about symbolic calculation, which has been done in Tcl. Results are certainly available without paying Wolfram.]
Why are pages like these being added to the Tclers Wiki? -jcw