chapter9_testbank

chapter9_testbank - AP Calculus Testbank (Chapter 9) (Mr....

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Unformatted text preview: AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions ÔÒ ½ ½ 1. The series ½·Ô ¼ Ò ·Ò·½ Ò (A) Ô (B) Ô ½ ½ 2. The series ½ Ò ¾ ½ Ò Ò will converge, provided that ½ (C) Ô ´ ½µÒ¾Ò Ò¾ diverges because I. The terms are not all positive. ¬¬¬ ¬¬¬ ½ II. The terms do not tend to 0 as Ò tends to III. Ð Ñ Ò Ò·½ Ò ½. (A) I only (B) II only (C) III only (D) I and II only (E) II and III only (D) Ô ½. (E) Ô ¼ 3. Which of the following series converge? ½ I. Ò ¾Ò . Ò·½ ½ ½ II. Ò ¿ ½Ò ½ III. Ò Ó× ¾Ò . Ò¾ ½ (A) I only (B) II only (C) III only (D) I and III only (E) II and III only ½ 4. The interval of convergence for the series Ò ½ ¿ ½ (B) ¿ ½ (C) ¿ ½ (D) ¿ Ü (E) 5. If Ò ½ ½ ½ ¼ is ½ Ü Ò¾ ½ Ü ½ ¾µÒ·¾ ½ Ü (A) ´¿Ü Ò ´Ü ½ ¿ Ü µÒ is a Taylor series that converges to ´Üµ for every real number Ò, then (A) 0 (B) ½ ¼¼ ´ (C) ¾ ¾ µ ½ (D) Ò ¼ Ò´Ò ½µ Ò´Ü ¾ µÒ ½ (E) Ò ¼ Ò 6. The graph of the function represented by the Taylor series ½ Ò ¼ ´ ½µÒ´Ü (A) ½µÒ intersects the graph of Ý ¾ (B) 0.567 Ü at (C) 0.703 Ü (D) 0.773 ½ 7. The graph of the function represented by the Taylor series ½ intersects the graph of Ý ½µÒ Ò Ü ¼ (E) 1.763 (A) at no values of Ü (B) at Ü ¼ (C) at Ü ¼ ¼¿ (D) at Ü ¼ ¿ (E) at Ü ½ ¿ 8. Using the fifth-degree Maclauren polynomial Ý ¾ , this estimate is (A) 7.000 (B) 7.267 (C) 7.356 9. The interval of convergence of the series ¿ (B) ¿ (C) (D) (E) (A) ½ ´Ü · ¾µÒ Ô is Ò Ü¿ Ü¿ ܽ ܽ ܽ Ü to estimate (D) 7.389 Ò ½ Ò Ò¿ ´ ½µÒÒ´Ü (E) 7.667 10. What is the approximation of the value of Ó× ¾ obtained by using the sixth-degree Taylor polynomial about Ü ¼ for Ó× Ü? (A) ½ ¾ ¾· ¿ ½ · . ¾ ¾¼ ½ ½ ½ · ¾¾ ¾¼ (B) ½ · ¾ · (D) ¾ · ¿½ ¿½ (C) ½ (E) ¾ · · ¿¾ ½¾ · ½¾¼ ¼¼ ½ ´¾Ü · ¿µÒ ¾ (B) ¾ (C) ¾ (D) ¾ (E) ¾ (A) Ü ½ Ü ½ Ü ½ Ü ½ ܽ Ò ½ 12. The interval of convergence of the series ¾µ Ü ´ · ¾µ (B) ´ ¾µ Ü ´ · ¾µ (C) Ü ´ ¾µ or Ü ¾. (D) ´ ¾µ Ü ´ ¾µ (E) ´ ¾µ Ü ´ ¾µ ÔÒ converges is ½ 11. The set of all values of Ü for which Ò ¼ is ´Ü · ¾µÒ Ò½ (A) ´ 13. What is the sum of the Maclaurin series ´ ½µÒ ¾Ò·½ ´¾Ò · ½µ (A) 1 · (B) 0 ¡¡¡? (C) ½ (D) ¿ ¿ · · ¡¡¡ · (E) This is divergent ½ 14. If ´Üµ ¼ ¾ (A) ´ Ó×¾ ܵ , then (B) ½ (C) 0 15. The Maclauren series expansion of ܾ (D) 1 (E) 2 ½ is ½ · ܾ ¾ · Ü Ü · ¡¡¡ (B) ½ ܾ · Ü Ü · ¡ ¡ ¡ ܾ Ü Ü · · · ¡¡¡ (C) ½ · ¾ (D) ½ · ܾ · Ü · Ü · ¡ ¡ ¡ ܾ Ü Ü (E) ½ · ½¾ · ¡ ¡ ¡ (A) ½ 16. Which of the following series converge(s)? ½ I. Ò ´ ½µÒ Ò ½ ½ II. Ò ½Ò ½ III. Ò Ô½ ¿ ½ Ô½ ¾ ¿ Ò (A) I only (B) II only (C) I and II (D) I and III (E) I, II, and III 17. Which of the following gives a Taylor polynomial approximation about Ü ¼ for × Ò ¼ , correct to four decimal places? ´¼ µ¿ ´¼ µ · (A) ¼ · ¿ ´¼ µ¿ ´¼ µ (B) ¼ · ¿ ´¼ µ¿ ´¼ µ (C) ¼ · ¿ ´¼ µ ´¼ µ¾ ´¼ µ¿ ´¼ µ · · · (D) ¼ · ¾ ¿ ¾ ´¼ µ¿ ´¼ µ ´¼ µ ´¼ µ (E) ¼ · · ¾ ¿ Part II. Free-Response Questions 1. The function has derivatives of all orders for all real numbers ¼¼´¾µ ¿ and ¼¼¼´¾µ Ü. Assume ´¾µ ¿ ¼ ´¾µ (a) Write the third-degree Taylor polynomial for and use it to approximate ´½ µ about Ü ¾ (b) The fourth derivative of satisfies the inequality ´ µ ´Üµ ¿ for all Ü in the closed interval ½ ¾ . Use the Lagrange error bound on the approximation to ´½ µ found in part (a) to explain why ´½ µ (c) Write the fourth-degree Taylor polynomial, È ´Üµ, for ´Üµ ´Ü¾ · ¾µ about Ü ¼. Use È to explain why must have a relative minimum at Ü ¼. 2. The Taylor series about Ü for a certain function converges to ´Üµ for all Ü in the interval of convergence. The Òth deriva´ ½µÒÒ ½ ´Òµ ´ µ is given by , and ´ µ tive of at Ü ¾Ò ´Ò · ¾µ ¾ (a) Write the third-degree Taylor polynomial for about Ü . (b) Find the radius of convergence of the Taylor series for about Ü . (c) Show that the sixth-degree Taylor polynomial for ½ Ü approximates ´ µ with error less than . ½¼¼¼ about 3. A function is defined by ¾ ¿ ½ · ¾ Ü · ¿ ܾ · ¿¿ ¿ ´Üµ ¡ ¡ ¡ · Ò Ò· ½ ÜÒ · ¡ ¡ ¡ ¿ ·½ for all Ü in the interval of convergence of the given power series. (a) Find the interval of convergence for this power series. Show the work that leads to your answer. (b) Find Ð Ñ Ü ¼ ´Üµ Ü ½. ¿ (c) Write the first three nonzero terms and the general term for an infinite series that represents ¾ ¼ ´Üµ Ü. (d) Find the sum of the series determined in part (c). 4. The Maclaurin series for the function ´Üµ ½ ´¾ÜµÒ·½ Ò·½ Ò¼ ¾Ü · ܾ ¾ · is given by Ü¿ ½ Ü ¿ · · ¡¡¡ ´¾ÜµÒ·½ · · Ò·½ ¡¡¡ on its interval of convergence. (a) Find the interval of convergence of the Maclaurin series for . Justify your answer. (b) Find the first four terms and the general term for the Maclaurin series for ¼ ´Üµ. (c) Use the Maclaurin series you found in part (b) to find the ½ value of ¼ ¿ 5. The function ´Üµ ½ is defined by the power series ´ ½µÒÜ¾Ò ´¾Ò · ½µ Ò¼ ½ ܾ ¿ · Ü Ü · ¡¡¡ ´ ½µ¾ÒÜ¾Ò · · ´¾Ò · ½µ ¡¡¡ for all real numbers Ü. (a) Find ¼ ´¼µ and ¼¼´¼µ. Determine whether has a local maximum, a local minimum, or neither at Ü ¼. Give a reason for your answer. ½ (b) Show that ½ approxmates ´½µ with error less than ½¼ ¾. ´Üµ is a solution to the differential equation (c) Show that Ý ¼ · Ý Ó× Ü. ÜÝ 6. The function has a Taylor series about Ü ¾ that converges to ´Üµ for all Ü in the interval of convergence. The Òth derivative ´Ò · ½µ of at Ü ¾ is given by ´Òµ ´¾µ for Ò ½, and ´¾µ ¿Ò ½. (a) Write the first four terms and the general term of the Taylor series for about Ü ¾. (b) Find the radius of convergence for the Taylor series for about Ü ¾. Show the work that leads to your answer. (c) Let be a function satisfying ´¾µ ¿ and ¼ ´Üµ ´Üµ for all Ü. Write the first four terms and the general term of the Taylor series for about Ü ¾. (d) Does the Taylor series for as defined in part (c) converge at Ü ¾? Give a reason for your answer. 7. Let be the function given by ´Üµ × Ò Ü · , and let È ´Üµ be the third-degree Taylor polynomial for about Ü ¼. (a) Find È ´Üµ. (b) Find the coefficient of ܾ¾ in the Taylor series for Ü ¼. ½ (c) Use the Lagrange error bound to show that ½¼ ½ . ½¼¼ ¬¬¬ ¬ (d) Let be the function given by ´Üµ third-degree Taylor polynomial for Ü ¼ about È ½ ½¼ ´Øµ Ø. Write the about Ü ¼. 8. Let be a function having derivatives of all orders for all real numbers. The third-degree Taylor polynomial for about Ü ¾ is given by Ì ´Üµ (a) Find ´¾µ and ¼¼´¾µ. ´Ü ¾µ¾ ¿´Ü ¾µ¿ (b) Is there enough information given to determine whether has a critical point at Ü ¾? If not, explain why not. If so, determine whether ´¾µ is a relative maximum, a relative minimum, or neither, and justify your answer. (c) Use Ì ´Üµ to find an approximation for ´¼µ. Is there enough information given to determine whether has a critical point at Ü ¼? If not, explain why not. If so, determine whether ´¼µ is a relative maximum, a relative minimum, or neither, and justify your answer. ¬¬ ¬¬ (d) The fourth derivative of satisfies the inequality ´ µ ´Üµ for all Ü in the closed interval ¼ ¾ . Use the Lagrange error bound on the approximation to ´¼µ found in part (c) to explain why ´¼µ is negative. ¬¬¬ ¬ 9. The Taylor series about Ü ¼ for a certain function converges to ´Üµ for all Ü in the interval of convergence. The Òth derivative of at Ü ¼ is given by ´Òµ ´¼µ The graph of . ´ ½µÒ·½´Ò · ½µ for Ò Ò ´Ò ½µ¾ ¾ has a horizontal tangent line at Ü ¼, and ´¼µ (a) Determine whether has a relative maximum, a relative minimum, or neither at Ü ¼. Justify your answer. about Ü (b) Write the third-degree Taylor polynomial for ¼. (c) Find the radius of convergence of the Taylor series for about Ü ¼. Show the work that leads to your answer. 10. The function ´Üµ is defined by the power series ½ · ´Ü · ½µ · ´Ü · ½µ¾ · ¡ ¡ ¡ · ½ Ò ¼ ´Ü · ½µÒ for all real numbers for which the series converges. (a) Find the interval of convergence of the power series for . Justify your answer. (b) The power series above is the Taylor series for Ü ½. Fnd the sum of the series for . (c) Let be the function defined by ´Üµ value of ½ ¾ be determined. Ü ½ , if it exists, or explain why about ´Øµ Ø. Find the ½ ¾ ¡ cannot (d) Let be the function defined by ´Üµ ܾ ½ . Find the first three nonzero terms and the general term of the Taylor ½ series for about Ü ¼, and find the value of . ¾ ...
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This note was uploaded on 01/23/2010 for the course MT AP taught by Professor Sukowsi during the Spring '10 term at 4.1.

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