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.