CALCULUS BC
WORKSHEET 1 ON TAYLOR POLYNOMIALS
Work the following on notebook paper. Use your calculator only on problem 8(b) , 9(b), and 10(a). Show all
work.
1. Find a fourth-degree Maclaurin polynomial for f  x   e3 x .
2. Find a sixth-degree Maclaurin polynomial for f  x   cos x.
3. Find a fifth-degree Maclaurin polynomial for f  x  
1
x 1
.
4. Find a third-degree Taylor polynomial for f  x   sin x, centered at x 
5. Find a fifth-degree Taylor polynomial for f  x  

6
1
, centered at x  2 .
1 x
 x  4
6. Find a third-degree Taylor polynomial for f  x   e
, centered at x = 4.
7. Find a fifth-degree Taylor polynomial for f  x   ln  x  1 , centered at x = 2.
_________________________________________________________________________________________
8. (a) Find a Taylor polynomial of degree n = 4 for f  x   e 2 x centered at x  3.
f  3.31  P4  3.31 ?
(b) Find P4  3.31 . What is the value of f  3.31 and the value of
___________________________________________________________________________________________
9. Let g be a function which has derivatives of all orders for all real numbers. Assume
g  5  3, g   5   2, g   5   7, g   5   3.
(a) Write the Taylor polynomial of degree 3 for g centered at x = 5.
(b) Use the polynomial that you found in part (a) to approximate g  4.9  .
___________________________________________________________________________________________
10. (1998 BC 3)
Let f be a function that has derivatives of all orders for all real numbers. Assume
f  0   5, f   0    3, f   0   1, f   0   4.
(a) Write the third-degree Taylor polynomial for f about x = 0, and use it to approximate f  0.2  .
 
(b) Write the fourth-degree Taylor polynomial for g, where g  x   f x 2 , about x = 0.
(Hint: Substitute x 2 in place of x in your answer to (a).)
(c) Write the third-degree Taylor polynomial for h, where h  x  
 f  t  dt , about
x
0
x = 0.
(d) Let h be defined as in part (c). Given that f 1  3, either find the exact value of h 1 or explain why it
cannot be determined.
Answers to Worksheet 1 on Taylor Polynomials
9 x2
27 x3 81x 4
x2
x4
x6
1. 1  3x 
2. 1 




2!
3!
4!
2!
4!
6!
x  


1
3
 
6

4. 
x   
2
2 
6
2  2!
2
3. 1  x  x 2  x3  x 4  x 5


3x

6
3


2  3!
5. 1   x  2    x  2    x  2    x  2    x  2 
2
6. 1   x  4  
3
4
5
 x  4 2  x  4 3

2!
8. (a) e6  2e6  x  3 
7.  x  2  
3!
4e6  x  3
2!
2
8e6  x  3
3

3!

 x  2 2  x  2 3  x  2 4  x  2 5

2
16e6  x  3
3

4

5
4
4!
(b) P4  3.31 = 749.602
f  3.31 = 749.945
f  3.31  P4  3.31 = 0.343
9. (a) 3  2  x  5  
7  x  5
2!
2

3  x  5
3
3!
(b) 3.236
10. (a) 5  3 x 
1
2
(b) 5  3 x 2 
(c) 5 x 
3
2
(d) h 1 
x2 
1
2
x2 
2
3
x3 ,
f  0.2  4.425
x4
1
6
x3
 f  t  dt .
1
0
for t = 0 and t = 1.
The exact value of h 1 cannot be determined because f  t  is known only
CALCULUS BC
WORKSHEET 2 ON TAYLOR POLYNOMIALS
Work the following on notebook paper. Use your calculator only on problem 9(b). Show all work.
1. Suppose the function f  x  is approximated near x = 5 by a third-degree Taylor
polynomial P3  x    3  7  x  5  2  x  5 . Give the value of:
2
3
(a) Give the value of: f  5 , f   5  , f   5  , and f   5  . .
(b) Does f have a local maximum, a local minimum, or neither at x = 5? Justify your answer.
____________________________________________________________________________________________
For problems 2 – 5, suppose that P2  x   a  bx  cx 2 is the second-degree Taylor polynomial for f about x = 0.
What are the signs of a, b, and c if f has the graph pictured on the below? Explain your reasoning.
2.
3.
4.
5.
____________________________________________________________________________________________
6. Find a fifth–degree Maclaurin polynomial for f  x   sin  3x  .
7. Find a fifth–degree Taylor polynomial for f  x   ln  x 1 centered at x = 2.
 x  4 centered at x = 4.
8. Find a fourth–degree Taylor polynomial for f  x   e
____________________________________________________________________________________________
9. (Modified form of 2000 BC 3)
The Taylor series about x = 5 for a certain function f converges to f  x  for all x in the interval of
n
convergence. The nth derivative of f at x = 5 is given by f    5  
(a) Write the third-degree Taylor polynomial for f about x = 5.
(b) Use your answer to (a) to approximate the value of f  6  .
 1n n !
2n  n  2 
1
and f  5   .
2
Answers to Worksheet 2 on Taylor Polynomials
1. (a) f  5   3, f   5  0, f   5  7  2! or 14, f   5  2  3! or  12
(b) Since f   5   0 and f   5  is positive, f has a local minimum at x = 5 by the Second
Derivative Test.
2. f  0   a , the constant in the Taylor polynomial. Since the y-intercept is negative, f  0    .
The graph of f is decreasing so f   0   . The graph of f is concave up so f   0   .
3. Since the y-intercept is positive, f  0  . The graph of f is decreasing so f   0  .
The graph of f is concave down so f   0   .
4. Since the y-intercept is negative, f  0  . The graph of f is increasing so f   0   .
The graph of f is concave up so f   0   .
5. Since the y-intercept is positive, f  0  . The graph of f is increasing so f   0   .
The graph of f is concave down so f   0   .
27 x3 243x5

6. 3x 
3!
5!
7.  x  2 
 x  2

8. 1   x  4  
2
2
 x  2

3
3
 x  4
2!
 x  2

4
 x  4
2

3!
3

 x  5
1
x  5  x  5
9. (a)
 1
 2
 3
2 2  3
2  4
2 5
2
(b) 0.371
3
4
 x  2

5
5
 x  4
4!
4
CALCULUS BC
WORKSHEET ON RADIUS AND INTERVAL OF CONVERGENCE
Work the following on notebook paper.
Find the radius and interval of convergence for each of the following series. All work must be shown.
(Note: Every time you are asked to find the interval of convergence, you must check to see if the endpoints
are included in the interval.)
1.

 1n  x  3n
n 1
n


2.

n 1

3.
5.
 1n  x  5n
n 1
n2

n
1 x 2 n

6. 
 2n  !
n 1

xn
n!
 n! x  2

n
7.
n 1

 2 x  3 n
n 1
n5n

n
x  4

4.  n
n  0 3  n  1

Answers to Worksheet on Radius and Interval of Convergence
1. Radius = 1, Interval: 2  x  4
2. Radius =  , Interval:   x  
3. Radius = 0, Interval:
 2
4. Radius = 3, Interval:  7  x   1
5. Radius = 1, Interval: 4  x  6
6. Radius =  , Interval:   x  
5
7. Radius = , Interval: 1  x  4
2
CALCULUS BC
WORKSHEET 1 ON POWER SERIES
Work these on notebook paper, except for problem 1.
1. Derive the Taylor series formula for f  x  by filling in the blanks below.
Let
f  x   a0  a1  x  c   a2  x  c   a3  x  c   a4  x  c   a5  x  c   ...  an  x  c   ...
2
3
4
5
n
Find f  c  and solve for a0 .
f  c   __________
so a0  __________
Now differentiate f  x  to find f   x  , and then find f   c  and solve for a1. .
f  x 
f   c   __________
so a1  __________
Differentiate again, this time to find f   x  , and then find f   c  and solve for a2.
f   x  
f   c   __________
so a2  __________
Now find f   x  , and then find f   c  and solve for a3.
f   x  
f   c  = __________
Do you see a pattern?
so a3  __________
n
f    c   ______________
so an  ______________
Now substitute your results into
f  x   a0  a1  x  c   a2  x  c   a3  x  c   a4  x  c   a5  x  c   ...  an  x  c   ...
2
3
4
5
n
f  x   ____  ____  x  c   _______  x  c   _______  x  c   ...  _______  x  c   ...
2
3
n
________________________________________________________________________________
On problem 2, find a Taylor series for f  x  centered at the given value of c. Give the first four nonzero terms
and the general term for the series.
2. f  x   e2 x , c  3
_________________________________________________________________________________
On problem 3 - 4, find a Taylor series for f  x  centered at the given value of c. Give the first four nonzero
terms. (You do not need to give the general term.)


3. f  x   sin  2 x 

, c  0
3
4. f  x   cos x, c 
2
3
TURN->>>
On problems 5 – 8, find a Maclaurin series for f  x  . Give the first four nonzero terms and the general term for
each series.
 
5. f  x   sin x3
cos  3x 
x
2 x
7. f  x   x e
6. f  x  
8. f  x   sin 2 x
1  cos  2 x  

2
.
 Hint: Use the fact that sin x 
2


________________________________________________________________________________
Answers to Worksheet 1 on Power Series
f   c 
f   c 
f  n  c 
, ac 
, an 
2!
3!
n!
2
3
n
f   c  x  c 
f   c  x  c 
f  n   c  x  c 
f  x   f  c   f   c  x  c  

 ... 
 ...
2!
3!
n!
1. a0  f  c  , a1  f   c  , a2 
4e6  x  3 8e6  x  3
2n e6  x  3
2. e  2e  x  3 

 ... 
 ...
2!
3!
n!
2
6
3.
3
n
6
3
2 3x 2 4 x3
x

 ...
2
2!
3!
2 
2 


x
3 x 



1
3
2  
3 
3 

x



 ...
4.  


2 2 
3 
2  2!
2  3!
2
3
1 x 6 n 3

x9 x15 x 21


 ... 
 ...
5. x 
3! 5! 7!
 2n  1!
n
3
1 32 n x 2 n 1

1 9 x 81x3 729 x5


 ... 
 ... where x  0
6. 
x 2!
4!
6!
 2n  !
n
1 x n  2

x 4 x5
  ... 
 ...
7. x  x 
2! 3!
n!
n
2
3
 1 22 n 1 x 2 n  ...
2 x 2 8 x 4 32 x 6 128 x8



 ... 
8.
2!
4!
6!
8!
 2n  !
n 1
CALCULUS BC
WORKSHEET 2 ON POWER SERIES
Work the following on notebook paper. Do not use your calculator. Show all work.
1. (a) Find a Maclaurin series for f  x   cos x . Give the first four nonzero terms and the general term.
(b) Use your answer to (a) to find lim
cos x  1
.
x2
___________________________________________________________________________________________
1
. Give the first four nonzero terms and the general term.
2. (a) Find a Maclaurin series for f  x  
1  2x
f  x 1
(b) Use your answer to (a) to find lim
.
x 0
x
___________________________________________________________________________________________
3. (a) Find a Maclaurin series for f  x   sin x . Give the first four nonzero terms and the general term.
x0
1
sin t
dt so that the error in your
(b) Use your answer to (a) to approximate the value of 

0
approximation is less than
t
1
. Justify your answer.
500
____________________________________________________________________________________________
On problems 4 - 5, find a series for the given function. Give the first four nonzero terms and the general term for
the series.
 x  2
 x  2
centered at x  0
centered at x   2
4. f  x   e
5. g  x   e
___________________________________________________________________________________________
6. (a) Let f  x   sin x 2 . Write the first four nonzero terms of the Taylor series for sin x 2 about x = 0.
 
 
(b) Let g  x   cos  x  . Write the first four nonzero terms of the Taylor series for cos  x  about x = 0.
(c) Let h  x   sin  x 2   cos  x  . Write the first four nonzero terms of the Taylor series for h about x = 0.
__________________________________________________________________________________________
7. (a) Let f  x   sin x 2 . Write the first four nonzero terms and the general term of the Taylor series
 
 
for sin x 2 about x = 0.
(b) Let g   x   sin  x 2  . Given that g  0   1, write the first five nonzero terms and the general term
of the Taylor series for g  x  about x = 0.
__________________________________________________________________________________________
1
.
8. (1990 BC 5) Let f be the function defined by f  x  
x 1
(a) Write the first four terms and the general term of the Taylor series expansion of f  x  about x = 2.
(b) Use the result from part (a) to find the first four terms and the general term of the series expansion
about x = 2 for ln x  1 .
(c) Use the series in part (b) to find an approximation for ln
than
3
so that the error in your approximation is less
2
1
. How many terms were needed? Justify your answer.
20
Answers to Worksheet 2 on Power Series
 1 x 2n
 ... 
 ...
6!
 2n  !
(b) 
2. (a) 1  2 x  4 x2  8x3  ...   2 x   ...
(b) 2
1. (a) 1 
x2
2!

x4
4!

n
x6
n
x3
3. (a) x 
(b)
3!

x5
5!

x7
7!
1
2
 1 x2n 1
 ...
 2n 1!
n
 ... 
17
. Since the terms of the series are alternating, decreasing in magnitude, and having a limit of 0 and the
18
approximation is made by using the first two terms, the error will be less than the absolute value of the
1
1

third term, so Error 
.
600 500
4. e2  e2 x 
e2 x 2 e2 x3
e2 x n

 ... 
 ...
2!
3!
n!
5. 1   x  2  
6. (a) x 2 
 x  2
2
2!
 x  2
3

 ... 
3!
 x  2
n
 ...
n!
x6 x10 x14


 ...
3!
5!
7!
(b) 1 
x2 x4 x6


 ...
2! 4! 6!
1
x4  1 1  6
x 2 x 4 121x6

(c) 1  1   x2 
    x  ...  1 


 ...
4!  3! 6! 
2! 4!
6!
 2! 
 1 x 4 n  2  ...
x 6 x10 x14


 ... 
3!
5!
7!
 2n  1!
n
7. (a) x 2 
 1 x 4 n  3  ...
x3
x7
x11
x15



 ... 
3 7  3! 11  5! 15  7!
 4n  3 2n  1!
n
(b) 1 
8. (a) 1   x  2    x  2    x  2   ...   1
2
(b)  x  2  
(c) ln 2 
3
 x  2 2
2

3
 x  2 3
3

n
 x  2 4
4
 x  2 n  ...
 ... 
 1n  x  2 n 1
n 1
 ...
. Two terms are needed. Since the terms of the series are alternating, decreasing in
8
magnitude, and having a limit of 0 and the approximation is made by using the first two terms,
1
1

 0.05 .
the error will be less than the absolute value of the third term, so Error 
24 20
CALCULUS BC
WORKSHEET 3 ON POWER SERIES
Work the following on notebook paper. No calculator except on 6(c).
On problems 1 – 3, find a power series for the given function, centered at the given value of c, and find its interval
of convergence. Give the first four nonzero terms and the general term of the power series.
6x
24
1
, c0
, c 1
1. f  x  
2. f  x  
3. f  x  
, c0
2
x2
4 x
1 x
________________________________________________________________________________________

4. (a) Find the first four nonzero terms of the power series for f  x   sin x, centered at x  .
6
(You do not need to write the general term.)



(b) Find the first four nonzero terms of the power series for g  x   sin  x   , centered at x  .
6
6

(You do not need to write the general term.)
(c) Could the answers to (a) and (b) be found by substitution, or is it necessary to find derivatives and use
the Taylor formula? Explain.
________________________________________________________________________________________
5. (1986 BC 5)
(a) Find the first four nonzero terms of the Taylor series about x = 0 for f  x   1  x .
(b) Use the results found in part (a) to find the first four nonzero terms in the Taylor series about x = 0
for g  x   1  x 3 .
(c) Find the first four nonzero terms in the Taylor series expansion about x = 0 for the function h such
that h  x   1  x 3 and h  0   4.
________________________________________________________________________________________
6. (1994 BC 5)
Let f be the function give by f  x   e 2 x .
2
(a) Find the first four nonzero terms and the general term of the power series for f  x  about x = 0.
(b) Find the interval of convergence of the power series for f  x  about x = 0. Show the analysis that
leads to your conclusion.
(c) Let g be the function given by the sum of the first four nonzero terms of the power
series for f  x  about x = 0. Show that f  x   g  x   0.02 for  0.6  x  0.6.
__________________________________________________________________________________________
7. (Modification of 1996 BC 2)
x x 2 x3
xn
The Maclaurin series for f  x  is given by 1  

 ... 
 ...
2! 3! 4!
 n  1!
17
(a) Find f  0 , f  0 , f  0 , and f   0 .
 
 
 
 
(b) For what values of x does the given series converge? Show your reasoning.
(c) Let g  x   x f  x  . Write the Maclaurin series for g  x  , showing the first three nonzero terms and
the general term.
(d) Write g  x  in terms of a familiar function without using series. Then, write f  x  in terms of the same
familiar function.
Answers to Worksheet 3 on Power Series
1. 1  x 2  x 4  x 6  ...   1 x 2 n  ...
n
2. 3x 
3. 8 
4. (a)
1
2
3x 2

2
3x3
4
8  x  1
3




 ... 
8
8  x  1
3x
2

3x 4
2
9


6

 1
n
Int. of conv.:  1  x  1
3x n 1
2n
8  x  1
 ...
3

 ... 
27
x 


6


2
8  x  1
n
3n
Int. of conv.:  2  x  4
 ...


6
 ...
3


3x
2  2!
3
Int. of conv.:  2  x  2
2  3!
5
7
x 
x 
x 






6
6
6
  
 
 
 ...
(b)  x   
6
3!
5!
7!

(c) The answer to (a) must be found by taking derivatives and using the Taylor formula. Since none of the

derivatives of the function on (a) give a value of 0 when evaluated at , substituting into the Maclaurin
6
series for sin x will not give the correct series. The answer to (b) can be found by substituting into the

Maclaurin series for sin x because all of the derivatives of g  x  when evaluated at
will give the
6
same values that the derivatives of sin x gives when evaluated at 0.
5. 1986 BC 5
1
1
1 3
x  ...
(a) 1  x  x 2 
2
8
16
1
1 7
1 10
x 
x  ...
(c) 4  x  x 4 
8
56
160
6. 1994 BC 4
 2x 

2 2
(a) 1  2 x
 2x 

2 3
 1
(b) 1 
n
1
2
x3 
1
8
x6 
1
16
x 9  ...
 2x 
2 n
 ... 
 ...
(b) Converges for all x,    x  
2!
3!
n!
(c) Since f is an alternating series whose terms decrease in magnitude and have a limit
of 0, Error  5th term so f  x   g  x   0.0112  0.02 by the Alternating Series Remainder.
2
7. Modification of 1996 BC 2
1!
1
2!
1
3!
1
17!
1
17 
or ; f   0  
or ; f   0  
or ; f
or
(a) f   0  
 0 
2!
2
3!
3
4!
4
18!
18
2
3
x
x
xn
  ...   ...
(b) Converges for all x,    x  
(c) x 
2! 3!
n!
x
(d) g  x   e  1
 ex 1
if x  0

f  x   x
 1
if x  0
\
CALCULUS BC
WORKSHEET 4 ON POWER SERIES
Work the following on notebook paper. Do not use your calculator.
Find the sum of each of the following convergent series.
1. 1 
2. 1 
2
1!
1

4
2!

1


8
3!
2n
 ... 
1
n!
 ...
 1n
3
n
x2
x3
 ...
4. 1 
100

10, 000
 ... 
 1n 102n
 2n  !
 ...
3! 5! 7!
2!
4!
 2n  1!
_________________________________________________________________________________________
5. Let f  x   x 
 ... 
2
1 1
1
3. 1         ...     ...
4 4 4
4
1
 1n 1 xn
x4


 ... 
 ... .
2
3
4
n
(a) For what values of x does the series for f  x  converge?
(b) Find the first four nonzero terms and the general term of the power series for g  x   f   x  .
1
(c) Use the series found in part (b) to find the value of g   . Show the steps that lead to your answer.
5
_________________________________________________________________________________________
1
6. The function f is defined by f  x  
.
1  x2
(a) Write the Maclaurin series for f. Give the first four nonzero terms and the general term. For what values
of x does the series converge?
(b) Find the first three nonzero terms and the general term for the Maclaurin series for f   x  .
1
(c) Use your results from part (b) to find the sum of the infinite series 

 ...   2n   
3 27 243
3
2
4
6
2 n 1
 ...
if possible. Show the steps that lead to your answer.
(d) Use your results from part (b) to find the sum of the infinite series
2 n 1
4
 ...   2n   
 ... if possible. If it isn’t possible, explain.
3
27
243
3
_________________________________________________________________________________________
1
7. Let f be the function defined by f  x  
.
1  x2
(a) Write the Maclaurin series for f. Give the first four nonzero terms and the general term. For what values
of x does the series converge?
8

256

6144
(b) Use your answer to (a) to find the first four nonzero terms and the general term of the Maclaurin series for
x
 0 f t  dt .
For what values of x does this series converge? (Remember that you are always supposed to
check to see if the endpoints are included in your interval. This is understood when you are asked to “find the
values of x for which this series converges.”)
1 1 1
 1
(c) Use your answer to (b) to find the value of 1     ... 
 ... . Show the steps that lead to your
3 5 7
2n  1
answer.
n
TURN->>>
8. The Maclaurin series for f is given by 1  7 x 
72 x2

73 x 3
 ... 
7n xn
 ...
2!
3!
n!
(a) For what values of x does the series for f converge?
(b) Find the first three nonzero terms and the general term for the Maclaurin series for f   x  .
(c) Use your answer to (b) to find the value of f  1 . Show the steps that lead to your answer.
_________________________________________________________________________________________
Answers to Worksheet 4 on Power Series
1. e 2
2. sin 1
3.
5. (a) Converges for 1  x  1 .
4
3
4. cos 10
n 1 n 1
(b) g  x   1  x  x2  x3  ...   1
x  ...
n
1
1
1
1
5
1
 1


 ...      ... , the sum is
 .
(c) g    1 
5
24
125
 1 6
5
 5
1   
 5
6. Modification of 2006, Form B, BC 6
(a) 1  x 2  x 4  x6  ...  x 2 n ... . Converges for 1  x  1 .
(b) 2 x  4 x3  6 x5  8 x7  ...  2nx 2 n 1  ... . Converges for 1  x  1 .
1
3
(c) The series is f    . f   x 
1
x
3

2x
1  x 
1
2 2 x
3

27
or 0.844
32
(d) Since the limit of the terms is not zero, the series diverges so it is not possible to find the sum.
(OR since this is the series from (b) with x replaced by
4
4
, and
lies outside the interval of
3
3
convergence, it is not possible to find the sum.)
7. (a) 1  x 2  x 4  x 6  ...    1 x 2 n  ... . Converges for 1  x  1 .
n
2 n 1
x3 x5
x7
n x


 ...    1
 ... , Converges for 1  x  1 .
(b) x 
3
5
7
2n  1
x 1

x
(c) This is g 1 . g  x   
dt  arctan t 0  arctan x so g 1  arctan1  .
0 1 t2
4
8. Modification of 1983, BC 5
(b) 7  72 x 
(a) Converges for all real numbers
 c  f  1  7  72

73
2!
 ... 
7n
 n  1 !
 ...


72
7 n 1
 7  1  7 
 ... 
 ... 
2!
 n  1 ! 

 7e7
73 x 2
2!
 ... 
7 n x n 1
 n  1!
 ...
CALCULUS BC
WORKSHEET ON POWER SERIES AND LAGRANGE ERROR BOUND
Work the following on notebook paper. Use your calculator on problem 1 only.
1. Let f be a function that has derivatives of all orders for all real numbers x Assume that
4
f  5  6, f   5  8, f   5  30, f   5  48, and f    x   75
for all x in the interval 5, 5.2 .
(a) Find the third-degree Taylor polynomial about x = 5 for f  x  .
(b) Use your answer to part (a) to estimate the value of f  5.2 . What is the maximum possible error in
making this estimate? Give three decimal places.
(c) Use your answer to (b) to find an interval [a, b] such that a  f  5.2  b . Give three decimal places.
(d) Could f  5.2 equal 8.254? Show why or why not.


2. Let f be the function given by f  x   cos  2 x 

 and let P  x  be the third-degree Taylor polynomial
6
for f about x = 0.
(a) Find P  x  .
(b) Use the Lagrange error bound to show that
1
1
1
f    P  
.
 10 
 10  12, 000
__________________________________________________________________________________________
3. Find the first four nonzero terms of the power series for f  x   sin x centered at x 
3
.
4
__________________________________________________________________________________________
Find the first four nonzero terms and the general term for the Maclaurin series for each of the following, and find
the interval of convergence for each series.
 
5. g  x  
4. f  x   x cos x3
x2
1 x
____________________________________________________________________________________________
Find the radius and interval of convergence for:

6.

 1  x  2 
n
n
3 n
n 0
n

7.
2
  2n ! x  5
n
n 1
____________________________________________________________________________________________
Multiple Choice. Show your supporting work.
 
8. The coefficient of x 6 in the Taylor series expansion about x = 0 for f  x   sin x 2 is
A

1
6
 B
0
C
1
120
D
1
6
E 1
____________________________________________________________________________________________
 
9. If f is a function such that f   x   sin x 2 , then the coefficient of x 7 in the Taylor series
for f  x  about x = 0 is
A
1
7!
 B
1
7
C 0
D 
1
42
E

1
7!
Answers to Worksheet on Power Series and Lagrange Error Bound
1. (a) 6  8  x  5   15  x  5   8  x  5 
2
3
(b) f  5.2  P3  5.2  8.264
R3  5.2   0.005
(c) 8.259  f  5.2  8.269
(d) No, f  5.2 can’t equal 8.254 because 8.254 does not lie in the interval found in part (c).
2. (a)
3
2 3 2 4 3
x
x  x
2
2!
3!
4
1
 1 
16  
24  4 4 
1
1
1
1
 10    2  5   1 
(b) R3   


4
4!
4!
5  4! 625  24 15, 000 12, 000
 10 
3.
2
2
3 
2 
3  2
2 
3  3

x

x
 
x
  ...
2
2 
4  2  2! 
4 
2  3! 
4 
4. x 
6 n 1
x7
x13 x19
n x


 ...   1
 ... .
2!
4!
6!
 2n !
5. x 2  x3  x 4  x 5  ...   1 x n  2  ... .
n
6. Radius = 3; interval: 1  x  5
7. Converges only if x = 5. Radius = 0.
8. A
9. D
Converges for all real numbers.
Converges for 1  x  1 .
CALCULUS BC
WORKSHEET ON SERIES AND ERROR
1. (Calc) Let f be a function that has derivatives of all orders. Assume f  3  1,
f   3 
1
2
, f   3  
1
3
4
, f   3   , and the graph of f    x  on [3, 4]
4
8
4
is shown on the right. The graph of f    x 
(4, 6)
is increasing on [3, 4].
(a) Find the third-degree Taylor polynomial about x = 3 for the function f.
(b) Use your answer to part (a) to estimate the value of f  3.7  .
(3, 2)
4
(c) Use information from the graph of y  f    x  to show
4
Graph of f    x 
that f  3.7   P  3.7   0.08.
(d) Could f  3.7  equal 1.283? Show why or why not.
____________________________________________________________________________________________
2. (Calc) Let f be the function defined by f  x  
x.
(a) Find the second-degree Taylor polynomial about x = 4 for the function f.
(b) Use your answer to part (a) to estimate the value of f  5.1 .
(c) Use the Lagrange error bound to find a bound on the error for the approximation in part (b).
(d) Find the value of f  5.1  P2  5.1 .
____________________________________________________________________________________________
3. (Calc) Find the maximum error incurred by approximating the sum of the series
1
2
3
4
n 1  n  1 
1



 ...    1 
  ... by the sum of the first five terms. Justify your answer.
2! 3! 4! 5!
 n! 
____________________________________________________________________________________________


4. Let f be the function given by f  x   cos  3 x   and let P  x  be the fourth-degree Taylor polynomial
6

for f about x = 0.
1
1
1
(a) Find P  x  .
(b) Use the Lagrange error bound to show that f    P   
.
6
 6  3000
____________________________________________________________________________________________
1  x2
1
e dx so that the error is less than
5. Use series to find an estimate for
. Justify your answer.
0
200
____________________________________________________________________________________________
6. (Calc) Suppose a function f is approximated with a fourth-degree Taylor polynomial about x = 1. If the

5
maximum value of the fifth derivative between x = 1 and x = 3 is 0.01, that is, f    x   0.01 , find the
maximum error incurred using this approximation to compute f  3 .
____________________________________________________________________________________________
7. The Taylor series about x = 5 for a certain function f converges to f  x  for all x in the interval of
n
convergence. The nth derivative of f at x = 5 is given by f    5  
 1n n !
2n  n  2 
1
and f  5   .
2
(a) Write the third-degree Taylor polynomial for f about x = 5.
(b) Show that the third-degree Taylor polynomial for f about x = 5 approximates f  6  with an error less
than 0.02.
Answers to Worksheet on Series and Error
3  x  3
x  3  x  3


2
4  2!
8  3!
2
1. (a) 1 
(b) 1.310
(c) Since f
 4
Error 
2
 x  is increasing on [3, 4], f 4  x   6 on [3, 3.7] so
4
6  3.7  3
 0.060  0.08 .
4!
(d) Yes, 1.250  f  3.7   1.370 so f  3.7  could equal 1.283.
x  4  x  4
2. (a) 2 

4
32  2!
2
(b) 2.256
(c) The maximum value of the third derivative f   x  
3
8x
f   4  
3
8  4
5
2
5
on [4, 5.1] is
2
3
3
 5.1  4 
3
so Error  256
 0.003 .

3!
256
(d) 0.002
3. The series has terms that are alternating in sign, decreasing in magnitude, and having a limit of 0
so the error is less than the absolute value of the first truncated term by the Alternating Series
Remainder.
5
or 0.012 .
6!
3 3 x 9 3 x 2 27 x3 81 3 x 4




4. (a) P  x  
2
2
2  2!
2  3!
2  4!
Error  6th term so Error 
f 5  z  x  0 
(b) R4  x  
5!
5
5
243x5
1
1
1
 1   243   1 

so R4    


  
5
5!
120 32  3000
 6   5!   6  5!2
5. The series has terms that are alternating in sign, decreasing in magnitude, and having a limit of 0
so the error is less than the first truncated term by the Alternating Series Remainder.
1  x2
0 e
dx  1 
1
1
1
43
1
1




. Error 
.
3 10 42 105
216 200
6. 0.003
 x  5
1
x  5  x  5
7. (a)
 1
 2
 3
2 2  3
2  4
2 5
2
3
Since the series has terms that are alternating, decreasing in magnitude, and having a limit of 0.
the error involved in approximating f  6  with the third-degree Taylor polynomial is less
than the absolute value of the fourth-degree term so
 6  5
 4
2  6
4
Error

1
1

by the Alternating Series Remainder.
96 50
CALCULUS BC
REVIEW SHEET 1 ON SERIES
Work the following on notebook paper. Use your calculator only on problems 7 and 9(a) and (b).
On problems 1 and 2, find the radius of convergence and the interval of convergence.

1.

n 1
 x  5
n

2.
n3n

n 1
 x  2
n
n!
________________________________________________________________________________
 1  ...
1
1
1


 ... 
3. Find the sum of 1 
3! 5! 7!
 2n  1!
n
3n 1
 n ?
n 0 4

4. What is the value of
x 4 x 6 x8
   ... What is f  x  ?
3! 5! 7!
6. If f is a function such that f   x   cos  x3  , find the coefficient of the x 7 term in the Taylor polynomial
5. A function f has a Maclaurin series given by x 2 
for f  x  about x = 0.
7. The graph of the function represented by the Maclaurin series
 1 x n  ... intersects the graph of y  x3 at x = ?
x 2 x3
1 x 

 ... 
2! 3!
n!
n
__________________________________________________________________________________________
8. A function f is defined by f  x  
1
2
3
n 1
 2 x  3 x 2  ...  n 1 x n  ...
4 4
4
4
for all x in the interval of convergence of the power series.
(a) Find the interval of convergence for this power series. Show the work that leads to your answer.
f  x 
(b) Find lim
x 0
x
1
4.
(c) Write the first three nonzero terms and the general term for an infinite series that represents
0 f  x  dx .
1
(d) Find the sum of the series determined in part (c).
__________________________________________________________________________________________
9. The function f has derivatives of all orders for all real numbers x. Assume f  2  3,
f   2   4, f   2  1, and f   2  5.
(a) Write the third-degree Taylor polynomial for f about x = 2, and
use it to approximate f 1.2 .
(b) The graph of the fourth derivative of f on the interval 1.2, 2]
 4
is shown on the right. The graph of f  x  has a horizontal
tangent at x = 1.7. Use the Lagrange error bound on the
approximation of f 1.2 to explain why f 1.2  5.3.
(c) Write the fourth-degree Taylor polynomial, P  x  , for


g  x   f x 2  2 about x = 0. Use P to explain whether
g has a relative maximum, relative minimum, or neither at x = 0.
y
(1.2, 3)
(2, 2)
(1.7, 1)
4
Graph of f    x 
x
Answers to Review Sheet 1 on Series
1. 3; 2  x  8
2.  ; converges for all x
3. sin 1
4. 12
5. x sin x
6. 
1
14
7. 0.773
8. (a)  4  x  4
1
8
1
1
1
1
1
(c) 


 ...  n1  ...
4 16 64 256
4
1
(d)
3
(b)
9. (a) 3  4  x  2 
2
x  2


2!
5  x  2

; 5.453
3!
3
3 1.2  2 
 0.0512 so 5.402  f 1.2  5.505. Therefore f 1.2   5.3.
(b) R3 1.2  
4!
4
x4
2!
Since g  0  0 and g   0    8 , g must have a relative maximum at x = 0 by the
(c) 3  4 x 2 
Second Derivative Test.
CALCULUS BC
REVIEW SHEET 2 ON SERIES
Work the following on notebook paper. Do not use your calculator.
1. Which of the following is a term in the Taylor series about x = 0 for the function f  x   cos  2 x  ?
(A) 
1
2
(B) 
x2
4
3
x3
(C)
2
3
x4
(D)
1
60
x5
(E)
4
45
x6
___________________________________________________________________________________________
 x  2
 n 3 n
n 1
 
n

2. Find the values of x for which the series
(B) 1  x  5
(A) x = 2
converges.
(C) 1  x  5
(D) 1  x  5
(E) All real numbers
___________________________________________________________________________________________
3
3. The series x  x 

(A) x ln 1  x 2

x5
2!

x7
3!
 ... 

(B) x ln 1  x 2
x 2n1

n!
 ... is the Maclaurin series for
(C) e x
2
(D) xe x
2
2
(E) x2e x
___________________________________________________________________________________________
4. The coefficient of x 3 in the Taylor series for e 2x at x = 0 is
1
1
2
4
8
(A)
(B)
(C)
(D)
(E)
6
3
3
3
3
___________________________________________________________________________________________

3n
 2 
5. Let f  x     cos x  . Evaluate f 
.
 3 
n 1
6. Find the sum of the geometric series
9
8

3
4

1
2

1
3
 ...
7. Find the Taylor polynomial of order 3 at x = 0 for f  x   1  x .
___________________________________________________________________________________________
(2003 Form B, Problem 6)
8. The function f has a Taylor series about x = 2 that converges to f  x  for all x in the interval of
 n  1 !
n
convergence. The nth derivative of f at x = 2 is given by f    2  
for n  1, and f  2   1.
3n
(a) Write the first four terms and the general term of the Taylor series for f about x = 2.
(b) Find the radius of convergence for the Taylor series for f about x = 2. Show the work that leads to your
answer.
(c) Let g be a function satisfying g  2   3 and g   x   f  x  for all x. Write the first four terms and the
general term of the Taylor series for g about x = 2.
(d) Does the Taylor series for g as defined in part (c) converge at x  2? Give a reason for your answer.


9. Let f be the function given by f  x   cos  3 x 
3 
 and let P  x  be the third-degree Taylor
4 
polynomial for f about x = 0.
(a) Find P  x  .
(b) Use the Lagrange error bound to show that
1
1
1
f    P  
.
6
 6  300
(c) Let G  x  be the function given by G  x    f  t  dt . Write the third-degree Taylor polynomial, T  x  ,
x
0
for G about x = 0.
___________________________________________________________________________________________
10. Suppose that the function f  x  is approximated near x = 5 by a fourth-degree Taylor
polynomial P4  x   3  7  x  5   2  x  5   9  x  5  .
4
(a) Find the value of f  5 , f   5 , f   5 , f  5 , and f   5 .
3
2
4
(b) Does f have a local maximum, a local minimum, or neither at x = 5? Justify your answer.
___________________________________________________________________________________________
Answers to Review Sheet 2 on Series
1. C
2. C
3. D
4. D
2
3
1
27
x x
x
5. 
6.
7. 1  

9
40
2
8
16
8. (2003 Form B, Problem 6)
2  x  2 3 x  2
4  x  2
 n  1 x  2   ...
(a) 1 


 ... 
2
3
3
3
3
3n
2
3
n
(b) Radius = 3
c
 x  2
1 x 
3
or 3   x  2 
2
 x  2

3
9
 x  2

2
 x  2

27
 x  2

3
4
 x  2
 ... 
3n
 x  2

4
n 1
 ...
 x  2
 ... 
n 1
 ...
3n
(d) No, x  2 is outside the interval of convergence, which is 1  x  5.
3
9
27
2 3 2 x 9 2 x 2 27 2 x3



9. (a) P  x   
2
2
2  2!
2  3!
3 


 4
(b) Since f
 x   81cos  3x 

3
 and 1  cos  3x 
4 
4

1

  1 for all x in the interval o  x  ,
6

4
R3  x  
f  4  z  x  0 
4!
4

81x 4
4!
1
81 
34
1
1
6
1
so R3      


4
4
4!
6
 4!  3  2   24 16  300
2
3 2 2 3 2 3
x
x 
x
2
4
4
 4
10. (a) f  5  3, f   5  0, f   5  14, f   5   12, f  5   216
(c) 
(b) Since f   5  0 and f   5  14 and 14  0 , f must have a relative minimum at x = 5 by the
Second Derivative Test.