Prototypes: More serious questions about Taylor polynomials
Beyond just writing out Taylor expansions, we could actually use them to approximate things in a more serious way. There are roughly three different sorts of serious questions that one can ask in this context. They all use similar words, so a careful reading of such questions is necessary to be sure of answering the question asked.
(The word ‘tolerance’ is a synonym for ‘error estimate’, meaning that we know that the error is no worse than such-and-such)
- Given a Taylor polynomial approximation to a function, expanded at some given point, and given a required tolerance, on how large an interval around the given point does the Taylor polynomial achieve that tolerance?
- Given a Taylor polynomial approximation to a function, expanded at some given point, and given an interval around that given point, within what tolerance does the Taylor polynomial approximate the function on that interval?
- Given a function, given a fixed point, given an interval around that fixed point, and given a required tolerance, find how many terms must be used in the Taylor expansion to approximate the function to within the required tolerance on the given interval.
As a special case of the last question, we can consider the question of approximating $f(x)$ to within a given tolerance/error in terms of $f(x_o), f'(x_o), f''(x_o)$ and higher derivatives of $f$ evaluated at a given point $x_o$.
In ‘real life’ this last question is not really so important as the third of the questions listed above, since evaluation at just one point can often be achieved more simply by some other means. Having a polynomial approximation that works all along an interval is a much more substantive thing than evaluation at a single point.
It must be noted that there are also other ways to approach the issue of best approximation by a polynomial on an interval. And beyond worry over approximating the values of the function, we might also want the values of one or more of the derivatives to be close, as well. The theory of splines is one approach to approximation which is very important in practical applications.