Name _____________________________ Date _______________________________
AP Calculus Test #10 - Series Test #2 (Taylor & Maclaurin) – B
You may not use your calculator on this part of the assessment.
Multiple-Choice: Please choose the best answer to each of the following questions. Show all relevant work
although the questions are multiple-choice.
1. The graph of the function 𝑓 is shown below.
Which of the following could be a portion of the graph of the Taylor polynomial of degree 13 for 𝑓 about
𝑥 = 0?
A. B.
C. D.
2. Values of a function 𝑓 and its first three derivatives at 𝑥 = 3 are given in the table below.
𝑥 𝑓 (𝑥 ) 𝑓′(𝑥 ) 𝑓′′(𝑥 ) 𝑓′′′(𝑥 )
3 2 −1 8 24
What is the third degree Taylor polynomial for 𝑓 about 𝑥 = 3?
A. 2 − (𝑥 − 3) + 4(𝑥 − 3)2 + 4(𝑥 − 3)3
B. 2 − (𝑥 − 3) + 4(𝑥 − 3)2 + 8(𝑥 − 3)3
C. 2 − (𝑥 − 3) + 8(𝑥 − 3)2 + 24(𝑥 − 3)3
D. 2 − 𝑥 + 4𝑥 2 + 4𝑥 3
(𝑥+4)3𝑛
3. The interval of convergence for the series ∑∞
𝑛=1 is
𝑛∙8𝑛
A. −6 ≤ 𝑥 < −2
B. −6 < 𝑥 < −2
C. −12 ≤ 𝑥 < 4
D. −12 < 𝑥 < 4
𝑥2 𝑥3 𝑥4 𝑥5
4. The first six nonzero terms of the Maclaurin series for 𝑓 are −1 − 𝑥 + + − − . Which of the
2! 3! 4! 5!
following could be an expression for 𝑓 (𝑥 )?
A. −𝑒 −𝑥
B. sin 𝑥 − cos 𝑥
C. cos 𝑥 − sin 𝑥
D. − cos 𝑥 − sin 𝑥
5. The power series
∞
∑ 𝑎𝑛 (𝑥 − 4)𝑛
𝑛=0
diverges at 𝑥 = −1. Which of the following must be true?
A) The series diverges at 𝑥 = 0.
B) The series converges at 𝑥 = 7.
C) The series diverges at 𝑥 = 9.
D) The series converges at 𝑥 = 9.
E) The series diverges at 𝑥 = 10.
𝑛+1
6. The 𝑛th derivative of a function at 𝑥 = 0 is given by 𝑓 (𝑛) (0) = (−1)𝑛+1 (𝑛+2)2𝑛 for all 𝑛 ≥ 0. Which
of the following is the Maclaurin series for 𝑓?
1 1 3 1
A. − 2 + 3 𝑥 − 16 𝑥 2 + 10 𝑥 3 + ⋯
1 1 3 1
B. − 2 − 3 𝑥 − 32 𝑥 2 − 60 𝑥 3 + ⋯
1 1 3 1
C. − 2 + 3 𝑥 − 32 𝑥 2 + 60 𝑥 3 + ⋯
1 32
D. − 2 + 3𝑥 − 𝑥 2 + 60𝑥 3 + ⋯
3
1 1 3 1
E. − 3 𝑥 + 32 𝑥 2 − 60 𝑥 3 + ⋯
2
Open-Ended: Please show all work to each of the following questions.
3𝑥
7. Let 𝑓 be the function given by 𝑓 (𝑥 ) = .
1+𝑥 2
a) Show that the Taylor series for 𝑓 about 𝑥 = 0 is given by
3𝑥 − 3𝑥 3 + 3𝑥 5 − 3𝑥 7 + ⋯ + 3(−1)𝑛 𝑥 2𝑛+1 + ⋯
b) Find the interval of convergence for the series found in part (a). Show the work that leads to your
answer.
c) Write the first four nonzero terms and the general term of the Taylor series for 𝑓′ about 𝑥 = 0.
1
d) Use the first three terms of the series found in part (c) to find an approximation for 𝑓′ (2). Show that the
7 1
approximation found in part (c) is within 16 of the true value of 𝑓′ (2).
2 𝑥
8. Let 𝑓 and 𝑔 be the functions given by 𝑓(𝑥 ) = 𝑥𝑒 𝑥 and 𝑔(𝑥 ) = ∫0 𝑓 (𝑡) 𝑑𝑡. The graph of 𝑓 (5), the fifth
1 1
derivative of 𝑓, is shown below for − 2 ≤ 𝑥 ≤ 2.
a) Write the first four nonzero terms and the general term of the Taylor series for 𝑒 𝑥 about 𝑥 = 0.
Write the first four nonzero terms and the general term of the Taylor series for 𝑓 about 𝑥 = 0.
b) Write the first four nonzero terms of the Taylor series for 𝑔 about 𝑥 = 0.
c) Find the value of 𝑔(8) (0).
d) Let 𝑃5 (𝑥 ) be the fifth-degree Taylor polynomial for 𝑔 about 𝑥 = 0. Use the Lagrange Error Bound
1 1
along with information from the given graph to find an upper bound on |𝑃5 (2) − 𝑔 (2)|.
