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version of the Mean Value Theorem, called Taylor’s Theorem.
We begin with some useful notation.
Assume that f (x) is n times diﬀerentiable at x = a. Let
Rn,a (x) = f (x) − Tn,a (x)
Rn,a (x) is called the n-th degree Taylor remainder function centered at at x = a.
The error in using the Taylor polynomial to approximate f (x) is given by
Error =| Rn,a (x) |
The central problem for this approximation process is:
Problem: Given a function f (x) and a point x = a, how do we estimate the size of
The following theorem provides us with the answer to this question:
Theorem. [Taylor’s Theorem]
Assume that f (x) is n + 1-times diﬀerentiable on an interval I containing x = a. Let x ∈ I .
Then there exists a point c between x and a such that
f (x) − Tn,a (x) = Rn,a (x) = f (n+1) (c)
(x − a)n+1 .
(n + 1)! We will make three important observations about this theorem. First, since T1,a (x) = La (x),
when n = 1 the absolute value of the remainder Rn,a (x) represents the error in using the...
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