BC 3
Taylor Quiz
Name:
Calculator Allowed
Clearly show all appropriate work for full credit. NO magically calculator "leaps of faith," please.
#1 (6 pts) Suppose P2 ( x)  a  b( x  3)  c( x  3) 2 is the second degree Taylor polynomial for f centered
at x  3 . Determine the signs (positive, negative or zero) of a,b, and c if the graph of f is shown below.
Explain briefly.
2(5 pts) Find an upper bound for the error made in approximating the value of e
non-zero terms in the Maclaurin series for e x
3
using the the first 7
BC 3
Taylor Quiz
3(6 pts) Look at the function f ( x) 
1
Name:
.
1 x
a. Use the Binomial Theorem [the series for (1  x) p ] to write the first three non-zero terms of the
Maclaurin polynomial for f.
2
b. Find f (4) (0)
4(6 pts) Let f  x   tan( x) . Find the Taylor polynomial of degree 3 approximating f centered at x   .
BC 3
Taylor Quiz
Name:
5(3 pts each) Find the value of each series by recognizing the function and the point at which it is
evaluated. [Exact answer please.] Indicate clearly the function and the value of x you use.
1
2
22
23
24
25
 2
 3
 4
 5
  =
3 3  2! 3  3! 3  4! 3  5!
1
2 3 4 5 6
     
3 32 33 34 35
=
(1)n1 (n  1)2
1
1
1
6. (2 pts) Evaluate the series 
 22  1   32  3   42  5  
2 n 1
2
2 
2 
2 
n 1
1
d  1 
2 x
 1  x2  x4  x6 
 

 2 x  4 x 3  6 x 5 
2
2
2 
2
1 x
dx  1  x  1  x 

2 x 2
 2 x 2  4 x 4  6 x 6 
1  x2
d  2 x 2  4 x( x 2  1)
 

 22 x  42 x 3  62 x 5 
2
2 
2
dx  1  x  1  x 

.