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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