Name_______________________________
Calculus
Period________
Chapter 9 Test Review
1. Know the Maclaurin series for the following functions. Be sure to recognize the expanded form and sigma
form.
a. e x =
b. sin x 
c. cos x 
d.
1

1 x
2. Write the 3rd degree Taylor polynomial for f x   3 6 x  1 about x  0 .
Write the first 4 terms in each series.
3. 2 cos( x  1)
4.
1
2  4x
Write the function represented by

 1n x n
5. 
2  n!
n 0
6.
3x 
3x 3  3x 5  3x 7
3!
5!
7!
 ...
7. Write the first three terms for g x  
ex 1
.
x
Determine if the series converges or diverges. Be sure to show the test used to determine your answer.



3n
n  n  1 !
n 1
n
8.   1
9.   1
10.   1
n
 n  1!
n
n 1
n 1
n 1
 n  1
11. Determine the interval of convergence for each of the following. Include endpoints and also write the
radius of convergence.
a)
2 n  x  3

n
n 1
b)
(1) n 2 x 

(n  1)!
n0


n
n
12a) Write the 3rd degree Taylor polynomial for f  x   x about x  9 .
12b) Use the first three terms to estimate 11 . What is the magnitude of this error?
 x
13a) Write the first three non-vanishing terms and general term in h x   5 sin   .
2
13b) What are the minimum number of terms necessary to approximate h2 with an error less than 0.1?
14a) Write the first three non-vanishing terms and general term for h  x   5sin  2 x  .
14b) What is the largest error of approximation when using the polynomial in part a to approximate h .4  .
14c) Write the first three terms for h( x ) .
14d) Is h( x) increasing or decreasing at x  0?
14e) Is h( x) concave up or concave down at x  0?