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AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 1
Page 1 of 22
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
Page 2 of 22
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
Page 3 of 22
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
Page 4 of 22
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
Page 5 of 22
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
Page 6 of 22
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
Page 7 of 22
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
Page 8 of 22
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
Page 9 of 22
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
Page 10 of 22
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
Page 11 of 22
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
Page 12 of 22
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
Page 13 of 22
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
Page 14 of 22
22
For all if ( ) ∑( )
( )
then ( )
A) ∑( )
( )
B) ∑( )
( )
C) ∑( )
( )
D) ∑( )
( )
E) ∑( )
( )
ANSWER
AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 15
Page 15 of 22
23
If ( ) ∑( )
( )
then ( )
A) ∑( )
( )
B) ∑( )
( )
C) ∑ ( )
D) ∑( )
( )
E) ∑ ( )
ANSWER
AP Calculus BC Review Binder Power, Taylor, & Maclaurin Series Page 16
Page 16 of 22
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
Page 17 of 22
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
Page 18 of 22
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
Page 19 of 22
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
Page 20 of 22
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
Page 21 of 22
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
Page 22 of 22
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
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