BC 3
Taylor Quiz
Name:
Calculator Allowed
Clearly show all appropriate work for full credit. NO magically calculator "leaps of faith," please.
1 (1 pt each) The function y  f ( x) is approximated near x = 0 by the third degree Taylor Polynomial
1
P3 ( x)  3 x  x 2  x 3 .
3
Find.
f (0) 
f (0) 
f (0) 
f (0) 
2 (5 pts) Determine all values of x (exact) for which x 2  x3  x 4  x5  x6  L L L  1 .
3(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.

32 34 36 38
  

2! 4! 6! 8!
1 1 1 1
1      
3 5 7 9
IMSA
=
=
4(6 pts)
Suppose g  x   xe
 x2
.
a. Find the Maclaurin series for g  x  .
b. Evaluate g
(100)
 0  and g (101)  0  , the 100th and 101st derivatives of g at
5(8 pts) Let f  x  be a function such that f
 n
 0  (1)n n2
x = 0.
for all n ≥ 0. Thus, f  0  0,
f     1, f   2  4 , and so on.
a. Find the MacLaurin series for f  x  . Write out the first 4 non-zero terms along with the
general term.
b. Given that the MacLaurin series converges at x  1 , how many non-zero terms are needed to
approximate f 1 with an error of at most 0.001? Clearly show analysis.
IMSA

6. (2 pts) Evaluate the series
n0
IMSA
4
 (4n)! .