Math 152 Class Notes
November 19, 2015
10.9 Applications of Taylor Polynomials
In this section, we study how to use Taylor polynomials to approximate functions.
Suppose that a function f (x) has its Taylor series
∞
X
f (n) (a)
n=0
n!
f 00 (a)
f 000 (a)
2
(x − a) = f (a) + f (a)(x − a) +
(x − a) +
(x − a)3 + · · ·
2!
3!
0
n
The n-th partial sum
f 00 (a)
f 000 (a)
f (n) (a)
(x − a)2 +
(x − a)3 + · · · +
(x − a)n
2!
3!
n!
Taylor polynomial of f at a.
Tn (x) = f (a) + f 0 (a)(x − a) +
is called the n-th
degree
Example 1. Find the rst three Taylor polynomials for ex at x = 0 .
The
remainder
of the n-th Taylor polynomial is given by
Rn (x) = f (x) − Tn (x)
To estimate the remainder R(x), we usually use the following inequality.
If |f (n+1) (x)| < M for |x − a| < R, then the remainder Rn (x)
of the Taylor series satises the inequality
Taylor's Inequality.
|Rn (x)| ≤
M
|x − a|n+1
(n + 1)!
for |x − a| < R
Example 2. Use the 3rd degree Taylor polynomial at a = 0 to approximate ex on the
interval [−1, 1] and determine the accuracy.
Example 3. (a) Approximate the function f (x) =
two at a = 8.
√
3
x by a Taylor polynomial of degree
(b) How accurate is this approximation when 7 ≤ x ≤ 10?
Example 4. Find sin 12◦ correct to six decimal places.