Some MacLaurin Series we should know: (You may tear this page off the exam.)
x 2 x3
xn
e  1 x 
 L  L
2! 3!
n!
for x  (–, )
x3 x5 x 7
x 2n1
n
sin  x   x     L   1 
L
3! 5! 7 !
 2n  1 !
for x  (–, )
x
cos  x   1 
x 2 x 4 x6
x 2n
n
   L   1 
L
2! 4! 6!
 2n  !
for x  (–, )
1
 1  x  x 2  x3  L  x n  L
1 x
for x  (–1, 1)
1
 1  x  x 2  x3  L  (1) n x n  L
1 x
for x  (–1, 1)
ln(1  x)  x 
1
1 x
x 2 x3
xn
  L  (1) n1
L
2
3
n
 1  x  x  x  L  (1) x
2
2
4
6
n 2n
L
x3 x5 x 7
x 2n1
n
tan  x   x     L   1 
L
3 5 7
 2n  1
1
for x  (–1, 1]
for x  (–1, 1)
for x  [–1, 1]
For p a constant,
p( p  1) x 2 p( p  1)( p  2) x 3
p ( p  1)( p  2) L ( p  (n  1)) x n
(1  x)  1  p  x 

L 
L
2!
3!
n!
for x  (–1, 1)
p
BC Calc 3
Taylor Exam
Name:
Calculator allowed.
You must show enough work so that I can recreate your results.
#1(9 pts) 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.
(.5)2
(.5)4
(.5)6
.5 


 .............. =
2!
4!
6!
1
1
1
1
1
 3
 4
 5
 6
  =
3  2! 3  3! 3  4! 3  5! 3  6!
2
1
1
1
1
1
1
 2  3  4  5  6   =
3 3  2 3 3 3  4 3 5 3 6
#2(8 pts) Find the Maclaurin series for each of the following – use known series to help
with this. Show four terms, the general term, and the interval of convergence.
x
=
4 x
1
ex
=
#3(6 pts) For a differentiable function ƒ, f(1) = 1, f ( n ) (1) 
n
, for n  1 , and
2n  1
| f ( n ) ( x) |  4  n for all values of x and n  1 .
a. Find the Taylor series for ƒ centered at x0  1 . Show the first three terms
along with the general term.
b. If terms through n = 5 are used to approximate ƒ(1/2), find an upper bound for
the error.
 sin x
, if x  0

#4(9 pts) Let f ( x)   x
.
 1, if x  0
a. Find the eighth degree MacLaurin polynomial for f.
b. What are the possible values for x if we want the error in approximating f ( x)
with the MacLaurin polynomial of degree 8 to be less than 0.1?
c. The graphs of f and P8 are shown together below. Is your result from b.
consistent with the graph? Explain briefly, but carefully.
#5(5 pts) Use the MacLaurin series for e x to approximate
2
than .005. Show clearly how many terms must be used.

#6(3 pts) Find the Maclaurin series for
ex
.
1 x
0
.8
ex dx with an error less
2