MATH 137 : Calculus 1 for Honours Mathematics. Online Assignment #5. Limits and Continuity of Functions

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1 1 Instructions: MATH 137 : Calculus 1 for Honours Mathematics Online Assignment #5 Limits and Continuity of Functions Due by 9:00 pm on WEDNESDAY, June 13, 2018 Weight: 2% This assignment includes the coverage of the topics from your reading assignments in Chapter 2 up to and including Arithmetic Rules for Continuous Functions from the Course Notes and the corresponding online lectures. You will require this information to complete the following questions. Ensure you have also completed the Introduction to Limits of Functions in Maple and the Asymptotes, Limits at Infinity, and Continuity in Maple labs found in the Weekly Tasks. Read and complete the following assignment. IMPORTANT: Keep a record of the answers you are going to submit on the printed copy of your assignment so that you can check your answers against the posted solutions. ***Completion of this assignment must be done by the student who submits it; additional help with this assignment is only permitted in accordance with the University of Waterloo s Academic Integrity rules. By submitting this assignment you understand and agree to the University s policies regarding academic honesty. Any violation of the University of Waterloo s Policy 71 Student Academic Discipline Policy (see on academic honesty will result in an academic penalty. To submit your solutions: Go to Waterloo LEARN course management site at learn.uwaterloo.ca Enter your QUEST Username and Password in the space provided and click Login. Click on the link for MATH 137 Online. Once inside the course website, click on the Syllabus icon, in the left-hand column click the Activities and Assignments link, and then click the Weekly Online Assignments link in the centre panel of the page. Read the guidelines and instructions about how to submit your assignments. When you are ready to submit your answers, click on the Submit tab near the top of the Math 137 home page in LEARN, and then click on Quizzes. Under the Submit Weekly Online Assignments section, click on Weekly Online Assignment 5. Read and follow any instructions provided. An answer key for this assignment will appear where you can fill-in your solutions. Please your instructor immediately if you encounter any problems (baforres@uwaterloo.ca). You should submit your solutions well in advance of the due date/time to avoid problems. Click on the SUBMIT button when you are done. You have only 1 attempt to submit your solutions. Any assignment submitted after 9:00 pm (Waterloo, Ontario time) on the due date will be considered LATE and will NOT be counted toward your final grade (no eceptions). Part 1: Multiple Choice: Choose the best answer. (1 mark each unless otherwise indicated) The following questions are a review of material from High School mathematics. You may find the High School Function Review available from the Weekly Tasks helpful for this section. If you have any difficulty with this section, please contact your instructor for etra review eercises = ( ) 2.

2 < (5 2 ) 3 = > 2 for all < ln( p ) = p ln(). 6. ln( 1 ) = ln(). 7. If, y > 0, then ln( y) = ln() + ln(y). 8. The following represents the graph of: a) f() = 1 b) f() = 3 c) f() = e d) f() = ln() 9. The following represents the graph of: a) f() = 1 b) f() = 1 1 c) f() = e d) f() = ln()

3 3 The following questions are based on material in your course notes and course lectures. 10. Calculate lim 1 ( ). a) 4 b) 3 c) 2 d) 1 e) 8 f) None of the above 11. Calculate lim 3 a) 1 2 c) 0 d) 2 e) f) None of the above Calculate lim a) 1 b) 1 c) 1 2 d) 1 2 e) Does not eist Calculate lim 2 2. a) 1 b) 1 c) 1 2 d) 1 2 e) Does not eist 14. Calculate lim ( ). (Hint: you will need to do some algebra and factoring.) a) 4 c) 1 2 d) 1 4 e) Does not eist

4 4 Use the following graph for multiple choice questions 15-22: 15. For the function f() whose graph is given above, state lim f(). 3 + h) Does not eist 16. For the function f() whose graph is given above, state lim f(). 3 h) Does not eist 17. For the function f() whose graph is given above, state lim f(). 3 h) Does not eist 18. For the function f() whose graph is given above, state f(3). h) Does not eist

5 5 19. For the function f() whose graph is given above, state lim 2 f(). h) Does not eist 20. For the function f() whose graph is given above, state lim 2 + f(). h) Does not eist 21. For the function f() whose graph is given above, state lim 2 f(). h) Does not eist 22. For the function f() whose graph is given above, state f( 2). h) Does not eist 23. Let g() = { + 1 if if < 1 Determine c) 2 lim g(). 1 d) Does not eist 24. Using the same definition of g() as in the previous question, determine lim 1 + g(). c) 2 d) Does not eist

6 6 25. Using the same definition of g() as in the previous question, what is lim 1 g()? c) 2 d) Does not eist 26. Assume that f() = Also assume that lim 1 f() eists. What is the value of b? a) 1 b) 0 c) 1 d) Cannot be determined 27. At what value of does the function (+1)2 2 1 a) = 3 b) = 3 c) = 2 d) = 1 e) = 1 f) None of the above 28. At what value of does the function (+1)2 2 1 a) = 3 b) = 3 c) = 2 d) = 1 e) = 1 f) None of the above { b + 1 if if < 1 have a removable discontinuity? have an infinite jump discontinuity? 29. At what value of does the function a) = 3 b) = 3 c) = 1 2 d) = 1 2 e) = 1 f) None of the above have a finite jump discontinuity? 30. Find lim f() if 4 1 < f() < b) 3 c) 4 d) Cannot be determined

7 7 Use the following graph of f() for the net five multiple choice questions: 31. The function f() has a removable discontinuity at: a) = 3 b) = 1 c) = 1 d) = The function f() has a finite jump discontinuity at: a) = 3 b) = 1 c) = 1 d) = The function f() has an infinite jump discontinuity at: a) = 3 b) = 1 c) = 1 d) = The function f() has an oscillatory discontinuity at: a) = 3 b) = 1 c) = 1 d) = The function f() is continuous at: a) = 3 b) = 1 c) = 1 d) = 2

8 8 36. A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Using this information, it can be shown that the concentration of salt after t minutes (in grams per liter) is C(t) = 30t t Using the theory of limits that you have studied, what is the concentration of salt as t? a) The concentration approaches 3 20 g/l salt b) The concentration approaches 30 g/l salt c) The concentration approaches 100 % salt d) The concentration cannot be determined sin() 37. What is lim 0 =? (Do you know the special name of this limit?) c) 1 2 d) The limit does not eist. 38. What is lim 0 sin() =? c) 1 2 d) The limit does not eist. 39. What is lim 0 sin(3) tan(7) =? b) 3 7 c) 7 3 d) The limit does not eist. 40. What is lim 0 cos() =? c) The limit does not eist. 41. What is lim 0 cos() =? c) The limit does not eist.

9 9 42. Assume that 2 2 f() for all > 0. Then a) lim 0 + f() = 0. b) lim 0 + f() = 1. f() c) lim 0 + does not eist. 43. What is lim 0 b) -1 e 2 =? c) d) The limit does not eist and the function has a vertical asymptote at = What is lim 1 b) -1 e 2 =? c) d) The limit does not eist and the function has a vertical asymptote at = Which of the following imply that a function f is continuous at = a? a) lim f() = f(a). a b) lim f() = f(a). a + c) lim f() = f(a). a d) All of the above. 46. Which of the following imply that a function f is continuous at = a? a) For every ɛ > 0 there eists a δ > 0 such that if a < δ, then f() f(a) < ɛ. b) lim a f() = f(a) c) For every sequence { n } with lim n n = a, we have lim n f( n) = f(a). d) All of the above. 47. In applying the ɛ δ definition of continuity to the function f() = 8 5 at = 1, given ɛ > 0, we can choose a) δ = 8 ɛ b) δ = ɛ 8 c) δ = 8ɛ

10 In applying the ɛ δ definition of continuity to the function f() = 2 at = 7, given ɛ = 0.1, we can always assume that δ 1, a) hence 7 < 1 6 < < 8 and we can choose δ = min{ 0.1 6, 1} b) hence 7 < 1 6 < < 8 and we can choose δ = min{ 0.1 8, 1} c) hence 7 < 1 6 < < 8 and we can choose δ = min{ , 1} d) None of the above 49. In applying the ɛ δ definition of continuity to the function f() = 2 at = 7, given ɛ = 0.1, a) choose δ = b) choose δ = c) both of the above values for δ will work d) None of the above 50. The function f() = a) has a removable discontinuity at = 1 and an essential discontinuity at = 1. b) has a removable discontinuity at = 1 and an essential discontinuity at = 1. c) has an essential discontinuity at = 1 and at = The function { sin( 1 f() = ) if 0, 0 if = 0 a) has a removable discontinuity at = 0. b) has an essential discontinuity at = 0 which corresponds to a vertical asymptote. c) has an essential discontinuity at = 0 which is an oscillatory discontinuity. 52. The function { sin( 1 f() = ) if 0, 0 if = 0 a) has a removable discontinuity at = 0. b) has an essential discontinuity at = 0 which corresponds to a vertical asymptote. c) has an essential discontinuity at = 0 which is an oscillatory discontinuity.

11 11 Sequential Characterization of Limits of Functions: A Very Strange Function is rational is irrational 53. Recall that Q is the set of all Rational numbers and R is the set of all Real numbers. Let { 2 if Q, f() = 2 if R \ Q, Which of the following statements is true: a) The sequence n = n converges to 2 and {f( n)} converges to 4. b) The sequence y n = n converges to 2 and {f(y n)} converges to 4. c) The function f has an essential discontinuity at = 2. d) All of the above. 54. Let f() = { 2 if Q, 2 if R \ Q, Which of the following statements is true: a) 2 f() 2 for all R. b) lim 0 f() = 0 c) The function f is continuous at = 0. d) All of the above. Part 2: True and False: Indicate whether the following statements are true (T) or false (F). (1 mark each) 55. If lim f() = L, then both the left-hand and right-hand limit must eist at = a and these two limits must a be equal to each other (i.e., lim f() = lim f()). a a A function can have more than one limit. 57. If 1, then ln() If > 0, then ln() <. ln() 59. lim = 0 ln() 60. lim = 0 when p is any Real number p ln( 61. lim 100 ) =

12 lim 3 f() = means the limit eists. 63. The function f() = The function f() = 1 1 Part 3: Maple Lab Questions has no vertical asymptotes. has no horizontal asymptotes. (1 mark each) Answer the following questions by using Maple CLASSIC version. Ensure that you have completed the Introduction to Limits of Functions in Maple and the Asymptotes, Limits at Infinity, and Continuity in Maple labs before answering the following questions. 65. Choose the function represented by the following MAPLE graph: a) f() = sin( 1 ) b) f() = 2 sin( 1 ) c) f() = cos( 1 ) d) f() = 2 cos( 1 ) 66. Choose the function represented by the following MAPLE graph: a) f() = sin( 1 ) b) f() = 2 sin( 1 ) c) f() = cos( 1 ) d) f() = 2 cos( 1 ) 67. Choose the function represented by the following MAPLE graph: a) f() = sin( 1 ) b) f() = cos( 1 ) c) f() = sin() d) f() = cos()

13 Choose the function represented by the following MAPLE graph: a) f() = sin( 1 ) b) f() = cos( 1 ) c) f() = sin() d) f() = cos() 69. Find lim 0 sin( 1 ). c) d) If lim f() and lim f() eist and equal the limit L, then lim f() = L. a + a a sin( ) 71. Find lim 0. d) Does not eist sin( ) 72. Find lim 0 +. d) Does not eist sin( ) 73. Find lim 0. d) Does not eist END OF ASSIGNMENT: Submit your answers according to the instructions on page 1.

MATH 1A MIDTERM 1 (8 AM VERSION) SOLUTION. (Last edited October 18, 2013 at 5:06pm.) lim

MATH 1A MIDTERM 1 (8 AM VERSION) SOLUTION. (Last edited October 18, 2013 at 5:06pm.) lim MATH A MIDTERM (8 AM VERSION) SOLUTION (Last edited October 8, 03 at 5:06pm.) Problem. (i) State the Squeeze Theorem. (ii) Prove the Squeeze Theorem. (iii) Using a carefully justified application of the