AP CALCULUS BC Test Booklet

1.
The power series converges conditionally at x = 5. Which of the following statements about n=0

convergence of the series at x = −4 is true?

(A) The series converges absolutely at x = −4.
(B) The series converges conditionally at x = −4.
(C) The series diverges at x = −4.
(D) There is not enough information given to determine convergence of the series at x = −4.

2.
The series converges to S. Based on the alternating series error bound, what is the least number

of terms in the series that must be summed to guarantee a partial sum that is within 0.03 of S ?
(A) 34
(B) 333
(C) 1111
(D) 9999

3.
The Taylor series for a function f about x = 0 is given by and converges to f for all real

numbers x. If the fourth-degree Taylor polynomial for f about x = 0 is used to approximate alternating series
error bound?
(A)
(B)
(C)
(D)

4.
The Taylor series for ln x, centered at x=1, is . Let f be the function given by the sum of the

first three nonzero terms of this series. The maximum value of for is
(A) 0.030
(B) 0.039
(C) 0.145
(D) 0.153
(E) 0.529

5.
The power series has radius of convergence 2. At which of the following values of x can the

alternating series test be used with this series to verify convergence at x ?

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(A) 6
(B) 4
(C) 2
(D) 0
(E) -1

6. Suppose and for all . Which of the following statements must be true?

(A) diverges.

(B) converges.

(C) converges.

(D) converges.

7. The alternating series test can be used to show convergence of which of the following alternating series?

I.

II.

III.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

8.
Which of the following statements are true about the series , where ?

I. The series is alternating.

II. for all

III.

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(A) None
(B) I only
(C) I and II only
(D) I and III only
(E) I, II, and III

9. The Taylor polynomial of degree 100 for the function f about x = 3 is given by
. What is the value of f (30)
(3) ?
(A)
(B)
(C)
(D)
(E)

10. Which of the following series are conditionally convergent?

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only

11. Which of the following series converges for all real numbers x ?

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(A)

(B)

(C)

(D)

(E)

12. Which of the following series converge?

I.

II.

III.

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

13. Let f be the function given by f(x)=ln(3-x) . The third-degree Taylor polynomial for f about x=2 is

(A)

(B)

(C)

(D)

(E)

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14. The third-degree Taylor polynomial for a function f about is . What is the
value of ?
(A)
(B)
(C)
(D)
(E)

15.
If the infinite series is approxiately by , what is the least value of k for

which the alternating series error bound guarantees that ?

(A) 64
(B) 66
(C) 68
(D) 70

16. Let be the fourth-degree Taylor polynomial for the function f about . What is the
value of ?
(A) 0
(B)
(C) 6
(D) 24
(E) 144

17. Which is the best of the following polynomial approximations to cos 2x near x = 0?
(A) 1+x/2
(B) 1+x
(C) 1−x2/2
(D) 1 - 2x2
(E) 1 - 2x + x2

18.
What are all values of x for which the series converges?

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(A)
(B) only
(C) only
(D) and only
(E) and

19.
If the power series converges at and diverges at , which of the following must be

true?

I. The series converges at .

II. The series converges at .

III. The series diverges at .

(A) I only
(B) II only
(C) I and II only
(D) II and III only

20.
The interval of convergence of is

(A)
(B)
(C)
(D)
(E)

21.
What are all values of x for which the series converges?

(A) All x except x = 0

(B) |x| = 3
(C)
(D) |x| > 3
(E) The series diverges for all x.

22.
The radius of convergence for the power series is equal to 1. What is the interval of convergence?

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(A) −4≤x<−2
(B) −1<x<1
(C) −1≤x<1
(D) 2<x<4
(E) 2≤x<4

23.
Which of the following is the interval of convergence for the series ?

(A) -4 < x < 0

(B)
(C) -2 < x < 0
(D)

24.
What is the interval of convergence of the power series ?

(A)
(B)
(C)
(D)
(E)

25.
The power series converges at . Which of the following must be true?

(A) The series diverges at .

(B) The series diverges at .
(C) The series converges at .
(D) The series converges at .
(E) The series converges at .

26.
What are all values of x for which the series converges?

(A)
(B)
(C)
(D)
(E)

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27.
What are all values of x for which the series converges?

(A)
(B) -3<x<3
(C)
(D)
(E)

28.
Consider the series and . Which of the following statements is true?

(A) Both series converge absolutely.

(B) Both series converge conditionally.

(C) converges absolutely, and converges conditionally.

(D) converges conditionally, and converges absolutely.

29.
For what values of is the series conditionally convergent?

(A)
(B)
(C) only
(D) only

30.
Which of the following statements is true about the series ?

(A) The series converges conditionally.

(B) The series converges absolutely.
(C) The series converges but neither conditionally nor absolutely.
(D) The series diverges.

31. Which of the following series is conditionally convergent?

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(A)

(B)

(C)

(D)

32.
Consider the series . Which of the following statements is true?

(A) The series converges absolutely.

(B) The series converges conditionally.
(C) The series diverges.
(D) It cannot be determined whether the series converges or diverges from the information given.

33.
Consider the series . Which of the following statements is true?

(A) The series converges, and .

(B) The series converges, and .

(C) The series converges, and .

(D) The series diverges.

34. Which of the following series converges?

(A)

(B)

(C)

(D)

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35. Which of the following statements is true?

(A) The series diverges by the alternating series test.

(B) The series converges by the alternating series test.

(C) The series converges by the alternating series test.

(D) The series converges by the alternating series test.

36.
Consider the series , where for all and . Which of the following statements

is true?
(A) The series converges conditionally.
(B) The series converges absolutely.
(C) The series converges, but there is not enough information to determine if the series converges absolutely.
(D) There is not enough information to determine if the series converges or diverges.

37. The alternating series test can be used to show convergence for which of the following series?

I. , where
II. , where
III. ,

where

(A) I only
(B) II only
(C) I and II only
(D) I and III only

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38.

Let be the function defined by . The derivative of is . The

graph of the function defined by is shown above for . Let

for all integers . Which of the following statements about the series is true?

(A) The series converges by the alternating series test.

(B) The alternating series test cannot be used to determine convergence because the series is not alternating.
(C) The alternating series test cannot be used to determine convergence because .

(D) The alternating series test cannot be used to determine convergence because the terms are not decreasing.

39. Which of the following series is conditionally convergent?

(A)

(B)

(C)

(D)

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40. Which of the following series are conditionally convergent?

I.

II.

III.

(A) I only
(B) II only
(C) II and III only
(D) I, II, and III

41.
Which of the following statements about the series is true?

(A) The series can be shown to diverge by comparison with .

(B) The series can be shown to diverge by limit comparison with .

(C) The series can be shown to converge by comparison with .

(D) The series can be shown to converge by the alternating series test.

42. Which of the following series converge?

I.

II.

III.

(A) I only
(B) II only
(C) I and II only
(D) I, II, and III

43.
Which of the following is the interval of convergence for the series ?

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(A)
(B)
(C)
(D)

44.
What are all values of for which the series converges?

(A) only
(B) only
(C) only
(D)

45.
The power series converges at . Which of the following must be true?

(A) converges at .

(B) diverges at .

(C) converges at .

(D) diverges at .

46.

Which of the following statements about the power series above is true?
(A) The series diverges for all .
(B) The series converges for only.
(C) The series converges for only.
(D) The series converges for all .

47.
What are all positive values of for which the series will converge?

(A)
(B) only
(C) only
(D) There are no positive values of for which the series will converge.

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48.
If the ratio test is applied to the series , which of the following inequalities results, implying that the

series converges?

(A)

(B)

(C)

(D)

49. If for all and , which of the following series converges?

(A)

(B)

(C)

(D)

50. Let be the third-degree Taylor polynomial for about . Which of the following statements
is true?
, and provides a good approximation for
(A)
only for values of that are close to .
, and provides a good approximation for
(B)
for all real numbers .

(C) , and provides a good approximation for

only for values of that are close to .

(D) , and provides a good approximation for

for all real numbers .

51.

Selected values of a function and its first four derivatives are shown in the table above. What is the approximation
for the value of obtained by using the third-degree Taylor polynomial for about ?

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(A)
(B)
(C)
(D)

52. Let be the function defined by . What is the approximation for the value of obtained by using the
second-degree Taylor polynomial for about ?
(A)
(B)
(C)
(D)

53. Which of the following series converge?

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

54. Let f be a function that has derivatives of all orders for all real numbers, and let P3(x) be the third-degree Taylor
polynomial for f about x = 0. The Taylor series for f about x = 0 converges at x = 1, and , for
and all values of x. Of the following, which is the smallest value of k for which the Lagrange error
bound guarantees that ?
(A)
(B)
(C)
(D)
(E)

55.
Consider the series . If the ratio test is applied to the series, which of the following inequalities results,

implying that the series converges?

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(A)

(B)

(C)

(D)

(E)

56. Let be the fifth-degree Taylor polynomial for the function f about x = 0. What
is the value of ?
(A) -30
(B) -15
(C) -5
(D)
(E)

57. The nth derivative of a function f at x=0 is given by for all n≥0 . Which of the following
is the Maclaurin series for f ?
(A)
(B)
(C)
(D)
(E)

58. Let f be a function with f(3) = 2, and f′(3) = −1, f″(3) = 6, and f‴(3) = 12. Which of the following is the third-degree
Taylor polynomial for f about x = 3?
(A) 2 − (x − 3) + 3(x − 3)2 + 2(x − 3)3
(B) 2 − (x − 3) + 3(x − 3)2 + 4(x − 3)3
(C) 2 − (x − 3) + 6(x − 3)2 + 12(x − 3)3
(D) 2 − x + 3x2 + 2x3
(E) 2 − x + 6x2 + 12x3

59. The third-degree Taylor polynomial about x = 0 of ln(1 - x) is

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(A)
(B)
(C)
(D)
(E)

60. Let f be a function with , , and . Which of the following could be the
graph of the second-degree Taylor polynomial for f about ?

(A)

(B)

(C)

(D)

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61.

The figure above shows the graph of a function f. Which of the following could be the second-degree Taylor
polynomial for f about x = 2 ?
(A)
(B)
(C)
(D)
(E)

62. The function f has derivatives of all orders for all real numbers with f(0) = 3, f'(0) = -4, f''(0) = 2, and f'''(0) = 1. Let
g be the function given by . What is the third-degree Taylor polynomial for g about x = 0?

(A)
(B)
(C)
(D)
(E)

63. Let f be a function having derivatives of all orders for such that , , , and
. Which of the following is the third-degree Taylor polynomial for f about ?
(A)
(B)
(C)
(D)
(E)

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64.

The function f has derivatives of all orders for all real numbers, and f(4) (x)=esinx . If the third-degree Taylor
polynomial for f about x=0 is used to approximate f on the interval [0,1] , what is the Lagrange error bound for the
maximum error bound for the maximum error on the interval [0,1] ?
(A) 0.019
(B) 0.097
(C) 0.113
(D) 0.399
(E) 0.417

65. What is the radius of convergence of the Maclaurin series for ?

(A) 1/2
(B) 1
(C) 2
(D) infinite

66.
What is the radius of convergence of the series ?

(A)
(B) 3
(C)
(D)
(E) 0

67.
The coefficients of the power series (x-2)n satisfy and for all n≥1. The radius

of convergence of the series is

(A) 0
(B)
(C)
(D) 2
(E) infinite

68.
What is the radius of convergence for the power series ?

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(A)
(B)
(C) 3
(D) 4
(E) 6

69.
What is the radius of convergence for the series ?

(A) 1/2
(B) 1
(C) 2
(D) 5

70. Let P be the second-degree Taylor polynomial for about x=3. What is the slope of the line tangent to the graph
of P at x =3?
(A)
(B)
(C)
(D)
(E)

71.
The series converges to and for all . If is the

partial sum of the series, which of the following statements must be true?
(A)

(B)
(C)
(D)

72.
If the series is approximated by the partial sum with 15 terms, what is the alternating series

error bound?
(A)
(B)
(C)
(D)

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73.
The Taylor series for a function about is given by and converges to for

. If the third-degree Taylor polynomial for about is used to approximate , what is the
alternating series error bound?
(A)
(B)
(C)
(D)

74.
If the series is approximated by the partial sum , what is the

least value

of for which the alternating series error bound guarantees that ?

(A) 31
(B) 32
(C) 33
(D) 34

75.
The alternating series converges to . If the sum of the first six terms of the series is used to

approximate , what is the alternating series error bound?

(A)
(B)
(C)
(D)

76.
The series converges to . Based on the alternating series error bound, what is the least number of

terms in the series that must be summed to guarantee a partial sum that is within of ?
(A) Two
(B) Three
(C) Four
(D) Five

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77.

Let be the fifth-degree Taylor polynomial for a function about . Information about the maximum of
the absolute value of selected derivatives of over the interval is given in the table above. Of the
following, which is the smallest value of for which the Lagrange error bound guarantees that
?
(A) 0.082
(B) 0.174
(C) 0.244
(D) 0.293

78. Let be a polynomial function with nonzero coefficients such that .

is the third-degree Taylor polynomial for about such that
. Based on use of the Lagrange error bound,
must equal which of the following?
(A) 0
(B)
(C)
(D)

79. The function has derivatives of all orders for all real numbers. It is known that and
for . Let be the fourth-degree Taylor polynomial for about . The Taylor
series for about converges at . Of the following, which is the smallest value of for which the
Lagrange error bound guarantees that ?
(A)
(B)
(C)
(D)

80. The function has derivatives of all orders for all real numbers with , , , and
. Using the third-degree Taylor polynomial for about , what is the approximation of ?
(A)
(B)
(C)
(D)

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81.
The Maclaurin series for the function is given by . What is the value of ?

(A)
(B)
(C)
(D)

82.
What is the interval of convergence of the power series ?

(A)
(B)
(C)
(D)

83. A function has a Maclaurin series given by , and the Maclaurin series converges to
for all real numbers . If is the function defined by , what is the coefficient of in the
Maclaurin series for ?
(A)
(B)
(C)
(D)

84.
For what values of does the series converge?

(A) only
(B) only
(C) only
(D) The series converges for all real numbers .

85.
Which of the following statements about the power series is true?

(A) The series does not converge for any real number .
(B) The series converges for only.
(C) The series converges on the interval only.
(D) The series converges for all real numbers .

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86.

The table above gives values of a function and its derivatives at selected values of . Which of the following is
the third-degree Taylor polynomial for about ?
(A)
(B)
(C)
(D)

87. Let be a function with third derivative . What is the coefficient of in the fourth-
degree Taylor polynomial for about ?
(A)
(B)
(C)
(D) 18

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88.

The function has derivatives of all orders. Shown above is the graph of , the 25th-degree Taylor
polynomial for about . Which of the following could be the graph of ?

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(A)

(B)

(C)

(D)

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89. Which of the following is the fourth-degree Taylor polynomial for about ?
(A)
(B)
(C)
(D)

90. What is the coefficient of in the Taylor series for about ?

(A) -2
(B) -1
(C) 0
(D) 1
(E) 2

91. Let f be a function with second derivative f''(x) = . The coefficient of x3 in the Taylor series for f about x
= 0 is
(A)
(B)
(C)
(D)
(E)

92.
If , then

(A)

(B)

(C)

(D)

93. If , which of the following is the Taylor series for f aboutx=0 ?

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(A)

(B)

(C)

(D)

(E)

94. What is the coefficient of in the Taylor series for about ?

(A)
(B)
(C)
(D)
(E)

95.

(A)

(B)

(C)

(D)

96. For , the power series converges to which of the

following?
(A)
(B)
(C)
(D)
(E)

97. A function f has Maclaurin series given by . Which of the following is an

expression for f(x) ?

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(A)
(B)
(C)
(D)
(E)

98.
Let be the function with and derivative . Which of the following is the Maclaurin
series for ?
(A)
(B)
(C)
(D)

99. The second-degree Taylor polynomial for about is

(A)
(B)
(C)
(D)

100. Let be the function defined by . Which of the following is the Maclaurin series for ?

(A)

(B)

(C)

(D)

101. Let be the function defined by . Which of the following is the Maclaurin series for , the derivative
of ?
(A)
(B)

(C)

(D)

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102. Which of the following is a power series expansion of ?

(A)

(B)

(C)

(D)

103. What is the sum of the series ?

(A)
(B)
(C) 2
(D)
(E) The series diverges.

104. A function f has Maclaurin series given by . Which of the following is an

expression for
(A)
(B)
(C)
(D)
(E)

105. The series converges to which of the following?

(A)
(B)
(C)
(D)

106. The sum of the series is

(A) ln 2
(B) e2
(C) cos 2
(D) sin 2
(E) nonexistent

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107. Which of the following is the Maclaurin series for ?

(A)
(B)
(C)
(D)
(E)

108. A series expansion of is

(A)

(B)

(C)

(D)

(E)

109.
The Maclaurin series for the function f is given by . What is the value of ?

(A) -3
(B)
(C)
(D)
(E) 4

110. What is the coefficient of x2 in the Taylor series for about x = 0?

(A)
(B)
(C) 1
(D) 3
(E) 6

111. The coefficient of x6 in the Taylor series expansion about x=0 for is

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(A)
(B) 0
(C)
(D)
(E) 1

112. Let f be a function such that and What are the first three nonzero terms of the
Maclaurin series for f ?
(A)
(B)
(C)

(D)

113. Which of the following is the Maclaurin series for ?

(A)

(B)

(C)

(D)

(E)

114.
The Maclaurin series for is . Which of the following is a power series expansion for

(A)
(B)
(C)
(D)
(E)

115. Let f be the function given by . What is the coefficient of x3 in the Taylor series for f about x = 0 ?

(A) -3/8
(B) -1/8
(C) -1/16
(D) 1/24
(E) 1/16

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116. The Maclaurin series for is . Which of the following is a

power series expansion for ?
(A)
(B)

(C)

(D)

117.

What are the first three nonzero terms of the Maclaurin series for the piecewise function defined as shown?
(A)
(B)

(C)

(D)

118. Let be the function with and derivative given by . Which of the following is the
Maclaurin series for ?

(A)

(B)

(C)

(D)

119. Which of the following are the first four nonzero terms of the Maclaurin series for the function defined by
?
(A)
(B)
(C)
(D)

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120. Which of the following is the Maclaurin series for the function defined by ?
(A)
(B)
(C)
(D)

121. What is the sum of the infinite series ?

(A) 0
(B)
(C) 1
(D)

122.
The series is the Maclaurin series for what function?

(A)
(B)
(C)
(D)

123. Which of the following series converges to ?

(A)

(B)

(C)

(D)

124. Which of the following is the Maclaurin series for ?

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(A)

(B)

(C)

(D)

125. Let be a function with derivatives of all orders. If is the third-degree Taylor polynomial for about
, which of the following statements about must be true?
(A)
(B)

(C) One of the terms of is .

(D) One of the terms of is .

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