AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 1

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Directions:

Solve the following problems using the available space for scratchwork. Indicate your answers on the front

page. Do not spend too much time on any one problem.

Note: Let ln(x) denote the natural logarithm of x with base e.

01.

07.

13.

19.

25.

31.

02.

08.

14.

20.

26.

32.

03.

09.

15.

21.

27.

33.

04.

10.

16.

22.

28.

34.

05.

11.

17.

23.

29.

35.

06.

12.

18.

24.

30.

36.

01

If

∑

, what is ∑ ?

A)

B)

C)

D)

( )

E)

( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 2

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02

If ∑

, then ∑

( )

must be

A)

B)

C)

D)

E) None of the above

ANSWER

03

If ∑( )

, then ∑

( )

is

A)

B)

C)

D) ( )

E) ( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 3

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04 What are the first four nonzero terms in the power series expansion of about ?

A)

B)

C)

D)

E)

ANSWER

05

The first three terms of the Maclaurin series of ∫

are

A)

B)

C)

D)

E) None of the above

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 4

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06

The coefficient of in the Taylor series for ( ) about is

A)

B)

C)

D)

E)

ANSWER

07

The coefficient of in the Taylor series for ( )

about is

A)

B)

C)

D)

E)

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 5

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08 The coefficient of in the Taylor series for ( ) about is

A)

B)

C)

D)

E) 2

ANSWER

09

Which is the approximate value of (

) obtained by using a fourth-degree

Taylor polynomial for centered at ?

A)

B)

C)

D)

E)

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 6

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10

The Taylor series for ( )

centered at is

A) ∑( )

( )

B) ∑( )

( )

C) ∑( )

( )

D) ∑( )

( )

E)

∑

( )

( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 7

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11

The figure above shows the graph of ( ) and ( ) where ( ) is the Taylor polynomial for ( ) centered at zero. Which of the following statements is true?

I. ( ) is a good approximation for ( )

II. ( ) is a good approximation for ( )

III. ( ) ( )

A) I only

B) II only

C) III only

D) I and III only

E) I, II, and III

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 8

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12

The graph above shows a function f with a relative minimum at The approximation of ( ) near by the second-degree Taylor polynomial centered about is given by ( ) ( ) Which of the following is true about and

A)

B)

C)

D)

E)

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 9

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13 Let be a function that is everywhere differentiable. The table below provides information about ( ) and its first, second, and third derivatives for selected values of

x ( ) ( ) ( ) ( ) 0 4.00 2.00 1.00 0.50 1 5.15 2.50 1.25 0.75 2 7.20 3.50 1.75 0.85 3 10.50 5.25 2.10 1.00

Which of the following best approximates ( )

A) ( ) ( )

B) ( )

( )

C) ( ) ( ) ( )

D) ( )

( )

( )

E) ( )

( )

( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 10

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14 The first three nonzero terms in the Maclaurin series about of are

A)

B)

C)

D)

E)

ANSWER

15

The Maclaurin series for a function is given by ∑

What is the value of

( )( ) the fourth derivative of at

A) 1

B) 2

C) 3

D) 4

E) 5

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 11

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16 The graph of the third-degree Maclaurin polynomial that approximates intersects the graph of at how many points.

A) None

B) One

C) Two

D) Three

E) Four

ANSWER

17 What is the approximation of the value of obtained by using a fourth-degree Taylor polynomial about for ?

A)

B)

C)

D)

E)

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 12

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18

The Maclaurin series for

is ∑

Which of the following is a power

series expansion for

?

A)

B)

C)

D)

E)

ANSWER

19 The graph of the function represented by the Taylor series, centered at ( ) ( ) ( ) ( ) ( ) intersects the graph of at

A)

B)

C)

D)

E)

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 13

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20 Let be the first four nonzero terms of the Maclaurin polynomial used to approximate the value of Determine the area bounded by the graph and the x-axis for

A)

B)

C)

D)

E)

ANSWER

21

For if ( ) ∑( )

then ( )

A) ∑ ( )

B) ∑ ( )

C) ∑ ( )

D) ∑ ( )

E) ∑ ( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 14

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22

For all if ( ) ∑( )

( )

then ( )

A) ∑( )

( )

B) ∑( )

( )

C) ∑( )

( )

D) ∑( )

( )

E) ∑( )

( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 15

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23

If ( ) ∑( )

( )

then ( )

A) ∑( )

( )

B) ∑( )

( )

C) ∑ ( )

D) ∑( )

( )

E) ∑ ( )

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 16

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24

If ( ) ∑( )

for all real then ( )

A)

B)

C)

D)

E)

ANSWER

25 The Taylor series of a function ( ) about is given by

( ) ( )

( )

( )

( )( )

What is the value of ( )

A) 0

B) ̅̅̅̅

C)

D)

E) 7

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 17

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26

Let ( ) be a function that is continuous and differentiable for all The derivative of this function is given by the power series

( )

If ( ) then ( )

A)

B)

C)

D)

E)

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 18

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27

Let ( ) ∑( )

Be the Taylor series for a function What is the

value of ( )( ) the tenth derivative of at ?

A)

B)

C)

D)

E)

ANSWER

28 ∑

converges for in

A) [ ]

B) [ )

C) ( ]

D) ( )

E) Cannot be determined

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 19

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29

What are all the values of for which the series ∑( )

converges?

A) ( ]

B) [ )

C) [ ]

D) [ ]

E) ( )

ANSWER

30

What are all the values of for which the series

( )

converges?

A)

B)

C)

D)

E) All values of x

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 20

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31

What are all the values of for which the series ∑( )

converges?

A) [ ]

B) *

+

C) (

+

D) *

)

E) (

)

ANSWER

32 The power series converges for what values of ?

A) only

B)

only

C) only

D)

E) All real numbers

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 21

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33

The power series

converges for all real numbers.

For values in the interval *

+, what is the minimum number of terms of the

power series necessary to approximate the value of with an error whose absolute value is less than 0.0001?

A)

B) C)

D) 7

E) 8

ANSWER

34

Let E be the error when the Taylor polynomial ( )

is used to

approximate ( ) at Which of the following is true?

A) | |

B) | |

C) | |

D) | |

E) | |

ANSWER

AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 22

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35

Let be a function whose seventh derivative is ( )( ) If is the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at will approximate ( ) with an error of not more than

A)

B)

C)

D)

E)

ANSWER

Let ( ) be a function whose Taylor series converges for all If | ( )( )|

where ( )( ) is the nth derivative of ( ) what is the minimum number of terms of the Taylor series centered at necessary to approximate ( ) with an error less than 0.00001?

A) Three

B) Four

C) Five

D) Six E) Ten

36

ANSWER