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.
1
1
1
1
1
1  2
 3
 4
 5
 
=
3 3  2! 3  3! 3  4! 3  5!

4

2 
3
43  3

5
45  5

7
47  7

9
49  9
 
=
1
1
1
1
1
 4  3  6  5  8  7  10  9   =
2
2
2
2
2
#2(6 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 2  tan 1 ( x) =
sin( x)  x
=
x3
#3(6 pts) For a differentiable function ƒ, f(0) = 1, f ( n ) (0) 
| f ( n ) ( x) | 
n
, for n  1 , and
2n  1
1
for all values of x and n  1 .
2x 1
a. Find the Maclaurin series for ƒ. 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.
#4(6 pts) Use the MacLaurin series for tan 1 ( x 2 ) to approximate  0 tan 1 ( x 2 )dx with an
error less than 0.001. Show clearly how many terms must be used.
.7
Graph of y  f (5) ( x)
#5(9 pts) Let f ( x)  ln(1  x 2 )  tan 1 ( x) . The graph of y  f (5) ( x) is shown above.
a. Write the first six terms of the MacLaurin Series for ln(1  x 2 ) .
b. Write the first six terms of the MacLaurin Series for f ( x) .
c. Find the value of f (6) (0) .
d. Let P4 ( x) be the fourth degree MacLaurin Polynomial for f. Using the graph
of y  f (5) ( x) shown above, find an upper bound for the error in
approximating f(.4) with P4 (.4) . That is, find a bound on f (.4)  P4 (.4) .

#6(2 pts) Evaluate the series
4
 (4n)! .
n0