Unformatted text preview: d
version of the Mean Value Theorem, called Taylor’s Theorem.
We begin with some useful notation.
Deﬁnition.
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 nth 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
Rn,a (x)?
The following theorem provides us with the answer to this question:
Theorem. [Taylor’s Theorem]
Assume that f (x) is n + 1times 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...
View
Full Document
 Spring '09
 Math, Derivative, Power Series, Taylor Series, Sequences And Series, Taylor's theorem, B. Forrest

Click to edit the document details