Math 1AA3/1ZB3 Sample Test 1, Version #1

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1 Math 1AA3/1ZB3 Sample Test 1, Version 1 Name: (Last Name) (First Name) Student Number: Tutorial Number: This test consists of 20 multiple choice questions worth 1 mark each (no part marks), and 1 question worth 1 mark (no part marks) on proper computer card filling. All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Only the McMaster standard calculator Casio fx-991 is allowed. 1. Evaluate the following integral, % (a) (b) (c) (d) 1 (e) % 1 0 ( % B sin B.B 2Þ Which of the following series converge? / (i) (ii) Ðln Ñ œ œ 3. What is the minimum number of terms needed in order to estimate the sum correct to within.001? (a) (b) (c) (d) % (e) & Ð &Ñ œ % 4. Evaluate the following improper integral. % &! (a) (b) (c) (d)! (e) Divergent & % B B/.B

2 5. Evaluate the following integral. B* B B!.B (a) lnlb&l lnlbl (b) lnlb&llnlbl (c) lnlb&l lnlbl (d) lnlb&l lnlbl (e) lnlb&llnlbl 6. Which of the below integrals is equal to B %.B B (with an appropriately defined ))? (a) & cos ) sin ).) (b) cos ) sin ).) (c) cos ) sin ).) (d) cos ). (e) cos ) ).) sin ) sin ) 7. Using the comparison theorem, which of the following integrals is convergent? Bsin B.B B (i).b (ii) (iii).b ( B B B/ B (a) (i) only (b) (ii) only (c) (i) and (ii) only (d) (i) and (iii) only (e) (ii) and (iii) only. Consider the sequence defined by + œ ß + œ Ð+ Ñ. Which of the following statements is correct? (a) Ö+ is increasing and bounded above by 3 (b) Ö+ converges to 5 (c) Ö+ is increasing and bounded above by & (d) Ö+ is increasing and bounded above by (e) Ö+ diverges 9. Determine whether the following sequences are convergent or divergent. When convergent, find the limit. (i) + œ ÐÑ (ii) + œ sinð1ñ (a) diverges, diverges (b) diverges, 0 (c) 0, 0 (d) 1, diverges (e) 1, 0

3 10. If you were to use use Mathematical Induction to show that 3 œ ÐÑÐÑ, 3œ which of the following would be the second step? 5 5 (a) Assume 5Ð5ÑÐ5Ñ 3 œ, and show that Ð3Ñ œ 3œ 3œ 5 5 5Ð5ÑÐ5Ñ, and show that 3œ 3œ 5 5 5Ð5ÑÐ5Ñ, and show that 3œ 3œ 5 5 5Ð5ÑÐ5Ñ, and show that 3œ 3œ 5 5 5Ð5ÑÐ5Ñ, and show that 3œ 3œ (b) Assume 3 œ 3 œ (c) Assume 3 œ 3 œ (d) Assume 3 œ 3 œ (e) Assume 3 œ 3 œ Ð5Ñ5Ð5Ñ Ð5Ñ5Ð5Ñ Ð5ÑÐ5ÑÐ5Ñ Ð5ÑÐ5ÑÐ5Ñ Ð5ÑÐ5ÑÐ5Ñ 11. Find the sum of the following series sin B sin B sin B sin B (a) (b) (c) (d) (e) sin B sin B sin B œ >2 12. If the partial sum of the series + is = œ, find +. % (a) (b) (c) (d) (e) divergent % ln œ! œ! 13. Which of the following series converge? sin (i) (ii) œ œ ( %%

4 14. What is the minimum number of terms needed in order to estimate the following sum to within.001? (a) (b) (c) % (d) & (e) ÐÑ Ð Ñx œ 15. Which of the following series converge? (i) (ii) & ( â ÐÑ ÐÑ ÐÑx œ œ 16. Which of the following series are absolutely convergent? (i) ÐÑ ÐÑ (ii) œ œ 17. Find the interval of convergence of ÐB Ñ (a) Ðß Ñ (b) Ò ß Ó (c) Ò ß Ñ (d) Òß Ó (e) Òß Ñ œ 1. If - is convergent, what can you conclude about the convergence of the following œ! series? (ii) -ÐÑ œ! œ! (i) -ÐÑ (a) convergent, nothing (b) convergent, convergent (c) nothing, divergent (d) nothing, nothing (e) convergent, divergent 19. Find the radius of convergence of (a) (b) (c) (d) (e)! Ð ÑxÐB Ñ œ

5 20. Find the radius of convergence of the following power series. ÐÑ œ! (a) (b) (c) (d)! (e) % ( â ÐÑ x B

6 Math 1AA3/1ZB3 Sample Test 2, Version 2 Name: (Last Name) (First Name) Student Number: Tutorial Number: This test consists of 20 multiple choice questions worth 1 mark each (no part marks), and 1 question worth 1 mark (no part marks) on proper computer card filling. All questions must be answered on the COMPUTER CARD with an HB PENCIL. Marks will not be deducted for wrong answers (i.e., there is no penalty for guessing). You are responsible for ensuring that your copy of the test is complete. Bring any discrepancy to the attention of the invigilator. Only the McMaster standard calculator Casio fx-991 is allowed. 1. Evaluate the following integral, (a) (b) (c) % (d) (e) 1 ln 1 B B sin lnð1îñ / Ð/ Ñ.B 2. Evaluate the following integral. B *B.B *B *B B B * B *B *B B B B G (e) G *B (a) sec B G (b) G (c) tan B G (d) 3. Consider the sequence defined by + œ ß + œ +. To show that this sequence is monotonic using induction, which of the following would be the second step? (a) Assume that and show that (b) Assume that and show that (c) Assume that + 5 Ÿ + 5 and show that + Ÿ (d) Assume that + 5 Ÿ + 5 and show that + Ÿ (e) Assume that Ÿ and show that + Ÿ

7 4. Find the values of : for which the series is convergent. Ðln Ñ œ : (a) : (b) :! (c) :Ÿ! (d) : (e) :Ÿ 5. If we use the partial sum =! to approximate the sum of the series œ estimate the error involved in the approximation. (a) (b) (c) (d) (e)!!! &!!!!!!!!!! %% % 6. Evaluate the following improper integral B.B (a) ln (b) ln (c)! (d) divergent (e) 7. Evaluate the following improper integral (a)! (b) (c) ln (d) (e) Divergent ln B B.B. Evaluate the following sum: Ð Ñ œ% (a) divergent (b) (c) (d) (e) % &

8 9. Determine whether the following sequences are convergent or divergent. When convergent, find the limit. (i) + œ lnð Ñ lnðñ (ii) + œ sinðîñ (a) divergent, divergent (b) divergent,! (c) ln,divergent (d) ln ß! (e) ln, 10. A sequence + is defined by + œ and + œ +. Assuming Ö+ is convergent, find its limit. & & (a) (b) (c)! (d) (e) 11. Find the sum of the following series % % % ( (a) (b) (c) (d) % (e) * â % % 12. Find the values of B for which the series converges ÐB &Ñ (a) Ð ß Ñ (b) Ð)ß Ñ (c) Ðß Ñ (d) Ð ß Ñ (e) Ðß %Ñ œ! & & & 13. Which of the following series converge? (i) â (ii)! % ) œ 14. Which of the following series converge? & arctan (i) (ii) œ! œ

9 15. Which of the following series converge? (i) % & ( ) â (ii) ÐÑ ln ( ) *! œ 16. Which of the following series are conditionally convergent? (i) (ii) ÐÑ œ œ 17. Which of the following series converge? % (i) (ii) Î ÐÑ / Ð Ñx œ œ 1. Find the interval of convergence of ÐÑ œ B (a) Òß Ó (b) Ð ß Ó (c) Ò ß Ó (d) Ðß Ó (e) Òß Ó 19. If - ÐB &Ñ converges when B œ, what can you say about the convergence of the œ! following series? œ! œ! (i) -ÐÑ (ii) - (a) convergent, nothing (b) divergent, nothing (c) convergent, convergent (d) nothing, nothing (e) convergent, divergent 20. Find the radius of convergence of (a) (b) (c) (d) (e) ÐB Ñ ÐÑ œ

10 Answers for 1 st Sample Test 1 1. a 2. c 3. a 4. c 5. b 6. a 7. c. d 9. b 10. d 11. d 12. e 13. a 14. d 15. c 16. a 17. e 1. d 19. b 20. c. Answers for 2 Sample Test 1 1. c 2. b 3. d 4. b 5. a 6. d 7. b. d 9. e 10. d 11. a 12. e 13. d 14. c 15. b 16. a 17. a 1. d 19. a 20. a