chapter9_testbank

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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¿ Ü½ Ü½ Ü½ 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 Ü½ 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 Üµ , then (B) ½ (C) 0 15. The Maclauren series expansion of Ü¾ (D) 1 (E) 2 ½ is ½ · Ü¾ ¾ · Ü Ü · ¡¡¡ (B) ½ Ü¾ · Ü Ü · ¡ ¡ ¡ Ü¾ Ü Ü · · · ¡¡¡ (C) ½ · ¾ (D) ½ · Ü¾ · Ü · Ü · ¡ ¡ ¡ Ü¾ 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 · ¿ Ü¾ · ¿¿ ¿ ´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 · Ü¾ 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 Ü¾ ¿ · 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 Ü¾¾ 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 Ü¾ 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 ...
View Full Document

## 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.

Ask a homework question - tutors are online