winter 2012 cal 3 .pdf

10 Pages

Mathematics & Statistics (Sci)
Course Code
MATH 222
Christa Scholtz

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McGill University April 2012 Faculty of Science Final examination VERSION number 1 Calculus 3 Math 222 Tuesday, April 24, 2011 Time: 6pm-9pm Examiner: Prof. J. Loveys Associate Examiner: Prof. W. Jonsson Student name (last, ▯rst) Student number (McGill ID) INSTRUCTIONS This is a closed book exam. Calculators are not permitted. Use of a regular dictionary is not permitted. Use of a translation dictionary is permitted. PART 1 of this exam consists of 10 multiple choice problems. They are to be answered on the machine readable sheets (scantrons) provided. PART 2 consists of 4 problems to be answered on the exam itself. If you require the extra pages at the end of the exam (which may also be used for scrap work), please indicate there which problem(s) you are continuing. This exam comprises the cover page, three pages of 10 multiple choice ques- tions, numbered 1 to 10, 4 pages of written questions, and 2 extra pages. The Examination Security Monitor Program detects pairs of students with unusually similar answer patterns on multiple-choice exams. Data generated by the program can be used as admissible evidence, either to initiate or corroborate an investigation or a charge of cheating under Section 16 of the Code of Student Conduct and Disciplinary Procedures. (Part 2) Problem 1 2 3 4 Total Multiple choice Mark Out of 12.5 12.5 12.5 12.5 50 50 Math 222 Final Exam Page 2 April 24, 2012 PART 1 (Multiple choice.) BEFORE EVEN LOOKING AT THESE PROBLEMS, MAKE SURE YOU HAVE COMPLETELY AND ACCURATELY FILLED IN THE NECES- SARY INFORMATION ON YOUR SCANTRON. IN PARTICULAR, BE SURE YOU HAVE FILLED IN THE CORRECT VERSION NUMBER IN BOTH APPROPRIATE PLACES, AND YOUR STUDENT NUMBER IN BOTH APPROPRIATE PLACES. IF THIS IS NOT DONE CORRECTLY, YOU WILL RECEIVE 0% FOR THIS PART OF THE EXAM. THIS IS VERSION 1. Each of these question is worth 5 marks. 1. Let (n ) be the sequence de▯ned by p (lnn) ▯ n + 2n an= : n The limit lin!1 an (a) is ▯ 2: (b) is ▯ 1: (c) is 0: (d) is 1: (e) does not exist: 2. Let (n!) an= (2n)! for n = 0;1;2;:::. The interval of convergence of X1 an(x + 2) is n=0 (a) (▯2;6): (b) [▯6;2): (c) [▯2;6): (d) [▯6;2]: (e)(▯6;2): 3. Let ‘ and ‘ be two lines in 3-space, de▯ned by 1 2 ‘1: (x;y;z) = (2;3;4) + th1;1;1i2‘ : (x;y;z) = (▯2;1;0) + sh2;2;3i: The shortest distance between these lines is p p (a) 6: (b) 0: (c) 2: (d) 6: (e) 2: 16 4. The arc length from (0;0;0) to3( 2;16;2) along the curve de▯ned by 8 2~ ~ 1 ~ ~r(t) =3t i + 8tj +2t k is q 2852 488 (a) 18: (b) 10: (c) 9 : (d) 3 (e) 40: Math 222 Final Exam Page 3 April 24, 2012 5. Let 2 2 ▯xy f(x;y) = (x + y )e : The graph of f has (a) 1 local maximum, no local minimums, and 2 saddle points. (b) no saddle points, no local maximums, and 1 local minimums. (c) 2 saddle points, 1 local minimum, and no local maximums. (d) 2 saddle points, no local minimums, and 1 local maximum. (e) 1 local maximum,
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