Calculus Maximus Review: Taylor Series & Polynomials
Taylor Series & Polynomials MC Review
Select the correct capital letter. NO CALCULATOR unless specified otherwise.
2345 _____ 1. Let Tx5 3573 x x x x be the fifth-degree Taylor polynomial for the function f about x 0 . What is the value of f 0 ?
5 1 (A) 30 (B) 15 (C) 5 (D) (E) 6 6
kn n 1 k _____ 2. For what integer k, k 1, will both and converge? n 1 n n 1 4
(A) 6 (B) 5 (C) 4 (D) 3 (E) 2
n n 1 x 1 _____ 3. (Calculator Permitted) The Taylor series for ln x , centered at x 1, is 1 . Let f n 1 n be the function given by the sum of the first three nonzero terms of this series. The maximum value of ln xfx for 0.3 x 1.7 is which of the following?
(A) 0.030 (B) 0.039 (C) 0.145 (D) 0.153 (E) 0.529
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x 2 n _____ 4. What are the values of x for which the series converges? n 1 n
(A) 31x (B) 31x (C) 31x (D) 11x (E) 11x
_____ 5. (Calculator Permitted) The graph of the function represented by the Maclaurin series n xx23 1 xn 1 x intersects the graph of yx 3 at x 2! 3!n !
(A) 0.773 (B) 0.865 (C) 0.929 (D) 1.000 (E) 1.857
_____ 6. Which of the following sequences converge? 5n en en I. II. III. n 21n n 1 e
(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III
Page 2 of 10 Calculus Maximus Review: Taylor Series & Polynomials _____ 7. What is the approximation of the value of sin1 obtained by using the fifth-degree Taylor polynomial about x 0 for sin x ?
11 11 11 11 11 (A) 1 (B) 1 (C) 1 (D) 1 (E) 1 224 24 35 48 6120
_____ 8. Which of the following series converge? n cos n 1 I. II. III. n 1 n 2 n 1 n n 1 n
(A) None (B) II only (C) III only (D) I and II only (E) I and III only
b dx _____ 9. If lim is finite, then which of the following must be true? b p 1 x
1 1 1 (A) converges (B) diverges (C) converges p p p 2 n 1 n n 1 n n 1 n 1 1 (D) converges (E) diverges p 1 p 1 n 1 n n 1 n
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n _____ 10. If axn is a Taylor series that converges to fx for all real x, then f 1 n 0
n 1 (A) 0 (B) a1 (C) an (D) nan (E) nan n 0 n 1 n 1
2n 1 _____ 11. What is the value of ? n n 1 3
(A) 1 (B) 2 (C) 4 (D) 6 (E) The series diverges
1 _____ 12. The Maclaurin series for is xn . Which of the following is a power series expansion 1 x n 0 x2 for ? 1 x2
(A) 1 xxxx2468 (B) xxxx2345 (C) xxxx2345 234
(D) xxxx2468 (E) xxxx2468
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xxx456 xn 3 _____ 13. A function f has a Maclaurin series given by . Which of the 2! 3! 4! n 1 ! following is an expression for fx ?
(A) 3sinxxx 32 (B) cos x2 1 (C) xxx22cos 2 (D) xe232x x x (E) exx 2 1
_____ 14. Which of the following series diverge? n sin 2 1 en I. II. III. 3 n n 0 n 1 n n 1 e 1
(A) III only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III
1 _____ 15. What is the coefficient of x2 in the Taylor series for about x 0 ? 1 x 2 1 1 (A) (B) (C) 1 (D) 3 (E) 6 6 3
Page 5 of 10 Calculus Maximus Review: Taylor Series & Polynomials 39 2781 _____ 16. The sum of the infinite geometric series is 2161281024
(A) 1.60 (B) 2.35 (C) 2.40 (D) 2.45 (E) 2.50
x 2 n _____ 17. What are all values of x for which the series converges? n n 1 n3
(A) 33x (B) 33x (C) 13x (D) 15x (E) 15x
xx35 _____ 18. The Taylor series for sin x about x 0 is x . If f is a function such that 3! 5! fx sin x2 , then the coefficient of x7 in the Taylor series for fx about x 0 os
1 1 1 1 (A) (B) (C) 0 (D) (E) 7! 7 42 7!
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1 n xn _____ 19. is the Taylor series about zero for which of the following functions? n 0 n!
(A) sin x (B) cos x (C) ex (D) e x (E) ln 1 x
_____ 20. For what values of x does the series 12 xxx 3 4 n x converge?
(A) No values of x (B) x 1 (C) x 1 (D) x 1 (E) All values of x
x 1 k _____ 21. The complete interval of convergence of the series is 2 k 1 k
(A) 02 x (B) 02 x (C) 20x (D) 20x (E) 20x
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1 n 1 x21n _____ 21. For 11x , if fx , then fx n 1 21n
(A) 1 n 1 x22n (B) 1 n x22n (C) 1 2n x2n (D) 1 n x2n (E) 1 n 1 x2n n 1 n 1 n 1 n 1 n 1
_____ 22. The coefficient of x3 in the Taylor series fo e3x about x 0 is
1 1 1 3 9 (A) (B) (C) (D) (E) 6 3 2 2 2
i 1 _____ 23. in 3
n n n n n 1 31 31 31 21 21 (A) (B) 1 (B) (D) (E) 23 23 23 33 33
Page 8 of 10 Calculus Maximus Review: Taylor Series & Polynomials _____ 24. Which of the followign series converge? n n 1 1 13 1 I. 1 II. III. n 1 21n n 1 n 2 n 2 nnln
(A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III
100 5 n 5n _____ 25. If s , to what number does the sequence s converge? n n 1100 n 5 4 n
100 1 5 5 (A) (B) 1 (C) (D) (E) The sequence does not converge 5 4 4
k _____ 26. (Calculator Permitted) If fx sin2 x , then f 1 is k 1
(A) 0.369 (B) 0.585 (C) 2.400 (D) 2.426 (E) 3.426
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_____ 27. Let the function given by fx ln 3 x . The third-degree Taylor polynomial for f about x 2 is
23 23 xx 22 xx 22 23 (A) x 2 (B) x 2 (C) xx 22 x 2 23 23
xx 22 23 xx 22 23 (D) x 2 (E) x 2 23 23
_____ 28. (Calculator Permitted) Suppose a function f is approximated with a fourth-degree Taylor polynomial about x 1. If the maximum value of the fifth derivative between x 1 and x 3 is 0.01, that is, fx 5 0.01, then the maximum error incurred using this approximation to compute f 3 is
(A) 0.054 (B) 0.0054 (C) 0.26667 (D) 0.02667 (E) 0.00267
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