6.5 Taylor's Theorem 329- (a) Prove that if f' exists and is bounded on an interval I (possibly
infinite) then f satisfies a Lipschitz^11 condition of order 1 on I.
(b) Use this result to prove that if a> 0, then for all n EN, the function
f(x) = J_ is uniformly continuous^12 on [a, +oo).
xn - Prove that if f satisfies a Lipschitz^11 condition of order a > 1 on an
interval I, then f is constant on I.
6.5 Taylor's Theorem
Taylor's theorem provides a way to approximate a function f that is ( n + 1 )-
times differentiable in a neighborhood of a point a by a polynomial of degree
~ n in powers of (x-a), whose coefficients can be determined by the derivatives
f', J", · · · , f(n) at a. This polynomial will be called the nth Taylor polyno-
mial for f about a, and will be denoted Tn ( x). Polynomial approximations
are significant, since polynomials are the simplest kind of function to compute.
They involve only three fundamental arithmetic operations: addition, subtrac-
tion, and multiplication. Taylor polynomials have many applications.
Taylor polynomials can be calculated without Taylor's theorem, using Def-
inition 6.5.l below. However, mere calculation of Taylor polynomials cannot
justify their use in approximating functions. We need a theorem that tells us
just how accurate we can expect a particular polynomial approximation to be.
That is what Taylor's theorem does for us.
In this section, we will use the familiar functions ex, ln x, and the trigono-
metric functions. While their formal definitions are not given until Chapters
7 and 8, we need them here as examples. Thus, we shall assume that these
functions are defined and differentiable everywhere in their domains, and that
their derivatives obey the rules set forth in elementary calculus.
TAYLOR POLYNOMIALSDefinition 6.5.1 Suppose f and its first n derivatives f', J", · · · , f(n) exist at
a. We define the nth Taylor polynomial for f about a by the formula
f"(a) J(n)(a)
Tn(x) = f(a) + f'(a)(x - a)+ - 1 -(x - a)^2 + · · · + -- 1 -(x - a)n.
n.
See Exercise 6.1.17 for a definition of Lipschitz condition of order a.
See Exercise 5.4.10.