Millersville University Name Answer Key
Department of Mathematics
MATH 211, Homework 10
April 7, 2006
Page 694, Exercise 34
(a) Use a Taylor polynomial of degree 4 to approximate ln(0.9).
Suppose f0(x) = 1
x, then we can express this function as a Taylor series centered at c= 1.
f0(x) = 1
x
=1
1 + xโ1
=1
1โ(1 โx)
=
โ
X
k=0
(1 โx)k
=
โ
X
k=0
(โ1)k(xโ1)k
which converges absolutely if |xโ1|<1. Since d
dx (ln x) = f0(x), then
f(x) = ln x=
โ
X
k=0
(โ1)k
k+ 1 (xโ1)k+1 .
The Taylor polynomial of degree 4 will be denoted P4(x) with
P4(x) = (xโ1) โ1
2(xโ1)2+1
3(xโ1)3โ1
4(xโ1)4.
Thus ln(0.9) โP4(0.9) โ โ0.105358.
(b) Estimate the error in the approximation.
The maximum error in the approximation is described by the maximum magnitude of the Taylor
remainder when n= 4.
R4(0.9) = f(5)(z)
5! (0.9โ1)5
=24
z5ยท1
120(โ0.1)5
=1
5z5(โ0.1)5
where 0.9<z<1. The magnitude of the error is maximized when z= 0.9 and thus the error is no
more than
|R4(0.9)| โค ๎
๎
๎
๎
1
5(0.9)5(โ0.1)5๎
๎
๎
๎
โ3.38702 ร10โ6.
(c) Estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of 10โ10.
