Math 166
Quiz 9
Name:
Directions: This quiz is worth a total of 10 points. To receive full credit, all work must be shown.
1. For approximately what values of x can you replace sin x by x − x3 /6 with an error of magnitude no
greater that 5 × 10−4 .
Solution. Let f (x) = sin x and P3 (x) = x − x3 /6. The error E(x) in the approximation is given by
(4)
f (α) 4 E(x) = |f (x) − P3 (x)| = x 4!
where α is some number between 0 and x. Note that |f (4) (α)| ≤ 1. Therefore, it suffices to find the
values of x for which
|x|4
≤ 5 × 10−4 .
4!
This inequality holds for x-values such that
|x| ≤ 4! × 5 × 10−4
1/4
≈ 0.33.
ln 1 + x3
.
2. Use series to evaluate the limit lim
x→0 x sin x2
Solution. Recall that
ln(1 + x) =
Therefore
∞
X
(−1)n+1 n
x
n
n=1
and
sin x =
∞
X
(−1)n 2n+1
x
.
(2n + 1)!
n=0
ln 1 + x3
x3 − x6 /2 + x9 /3 − · · ·
lim
=
lim
= 1.
x→0 x sin x2
x→0 x3 − x7 /3! + x11 /5! − · · ·