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 ¢¡£¡¥¤§¦¨¤§©  £
 £© Polynomial Approximations
In Chapter 3 we discussed the problem of finding the affine function which best approx-
imates a given function about some point. In particular, we found that the best affine
approximation to a function f at a point c is given by
T (x) = f 0(c)(x − c) + f(c), ()
provided that f is differentiable at c. In this section and the next, we will extend the
ideas of Sections and to the problem of finding polynomial approximations of any
given degree to a function about some specified point. We shall see that many nonlinear
functions can be approximated to any desired level of accuracy over a specified interval
if we use polynomials of sufficiently high degree. As an example, compare the graphs of
f(x) = sin(x) and
1 1 1 1
P (x) = x − x3 + x5 − x7 + x9
6 120 5040 362, 880
in Figure . They are almost indistinguishable over the interval [−π, π]. In practical
terms, this means there is little difference in working with P (x) instead of f(x) for x
in [−π, π]. Moreover, since polynomials are the simplest of functions, involving only the
arithmetic operations of addition, subtraction, and multiplication, the substitution of P
for f can be a very helpful step in simplifying a problem.
3
2
1
- -5 - 5
-1
-2
-3
Figure Graphs of f(x) = sin(x) and an approximating polynomial
1
2 Polynomial Approximations Section
To begin, we need to recall, and then generalize, some definitions and facts from
Sections and . First, recall that a function f is said to be o(h) if
f(h)
lim = 0; ()
h→0 h
a function f is said to be O(h) if there exist constants M and such that
f(h)
≤ M ()
h
whenever − < h < . In particular, we saw that f is O(h) if
f(h)
lim
h→0 h
exists. The following definition generalizes to other powers of h this method of charac