NAMES:
MATH 152
April 22, 2015
QUIZ 10
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
1. (3 points) Find the sum of the series
∞
X
(−2)n
n=0
Solution: Recall that ex =
P∞
n!
xn
n=0 n! . Setting
∞
n
X
n=0
x = −2 gives
(−2)
= e−2 .
n!
√
2. (3 points) Find the degree 2 Taylor polynomial, T2 (x), for f (x) = x centered at a = 9.
1
Solution: First, f 0 (x) = 2√1 x and f 00 (x) = − 4x13/2 so f (9) = 3, f 0 (3) = 16 and f 00 (3) = − 108
.
Thus
1
1
T2 (x) = 3 + (x − 9) −
(x − 9)2 .
6
2! · 108
NAMES:
MATH 152
April 22, 2015
3. (3 points) Find the Taylor series about a = 2 for f (x) = xex . Solution:Computing the first
few derivatives
n
0
f (n) (x)
xex
1
ex (1 + x)
2
ex (2 + x)
3
ex (3 + x)
we can see that in general f (n) (x) = ex (n + x). So f (n) (2) = e2 (n + 2) and the Taylor series
of f about a = 2 is
∞
X
f (n) (a)
n=0
n!
n
(x − a) =
∞
X
e2 (n + 2)
n=0
n!
(x − 2)n .