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Final

# Winter Final 2011.pdf

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McGill University

Mathematics & Statistics (Sci)

MATH 222

Christa Scholtz

Fall

Description

Final Examination April 18, 2011 Mathematics 222 Group 1 1
STUDENT NAME:
STUDENT NUMBER:
FACULTY OF SCIENCE
FINAL EXAMINATION
MATHEMATICS 222
CALCULUS III
Examiner: W. Jonsson Date: Thursday, April 18, 2011
Associate Examiner: N. Sancho Time: 9:00 AM - 12:00 PM
Instructions
1. Total number of points: 92. Your total will be pro-rated to be out of 85 for the ﬁnal grade calculation.
2. No books, calculators or notes are allowed for the exam. Do not rip pages from the examination book.
3. There are 4 versions of this examination. This version belongs to Group 1.
4. Answers to the questions in Part I, each of which is worth 3 points, are to be entered on the machine readable
sheet with a soft lead pencil. There are fourteen such questions for a total of 42 points
5. Answers to the questions in part II, each of which is worth 10 points, are to be written in the space provided
on the examination paper. There are ﬁve such questions for a total of ﬁfty points.
6. You1=2nswers ma▯3=2ntain π or other expressions that cannot be computed without a calculator, e.g. ln2,
300 + 13 ▯ 150 .
7. All material (question papers, machine readable sheets) must be turned in.
8. Name, Student number and group number of your examination MUST be entered on the question paper and
on the machine readable sheet.
Score Table
Part I
Multiple Choice
Part II
Problems Points
1.
2.
3.
4.
5.
Total:
This exam comprises eight pages, including the cover and four blank pages for further work.
The Examination Security Monitor Program detects pairs of students with unusually similar answer patterns on
multiple-choice exams. Data generated by this program can be used as admissible evidence, wither to initiate
or corroborate and investigation or a charge of cheating under section 16 of the Code of Student Conduct and
Disciplinary procedures. Final Examination April 18, 2011 Mathematics 222 Group 1 2
PART I. Multiple choice questions
1. Of the two inﬁnite series
∑1 ∑ √
A : en sin n B : 1 1 ▯1
n n
n=1 n=1
(a) both are divergent
(b) both are absolutely convergent
(c) A is divergent and B is convergent
(d) A is convergent and B is divergent
(e) A is absolutely convergent and B is conditionally convergent
4 1
2. The coeﬃcient of x in the Maclaurin series 1 ▯ sin(x ) is
(a) ▯ 1/2, (b) 0, (c) 1/2, (d) 1, (e) ▯1
n
1 1 1 (▯1)
3. Let s denote the sum of the alternating serie3 + 5 ▯ 7 + ▯▯▯ +2n + 1 + ▯▯▯ then
(a) 3 < s < 7/8, (b) s < 0, (c) 0 < s < 2/3, (d) s > 1, (e) 7/8 < s < 1.
6n
∑ 1 (x + 1)
4. The power series n=1 n 8n has interval of convergence
p p
(a) 1 ▯ 2 ▯ x ▯ 1 + 2
p p
(b) ▯1 ▯ 2 < x < ▯1 + 2
p p
(c) ▯1 ▯ 2 ▯ x ▯ 1 ▯ 2
p p
(d) ▯1 ▯ 2 ▯ x ▯ ▯1 + 2
(e) ▯1 ▯ p2 ▯ x ▯ 1 +p 2
( )
5. The arc length L of the curve R = 2t ,t,t from the origin to its intersection with the plane x = 18
3
satisﬁes
(a) 23.5 ▯ L ▯ 24.5
(b) 22.5 ▯ L < 23.5
(c)21.5 ▯ L < 22.5
(d) 20.5 ▯ L < 21.5
(e)19.5 ▯ L < 20.5
3 2 ∂z ∂z
6. Let z(x,y) be deﬁned implicitly by z + z x + zy = 5. Then∂x + ∂y at the point (3,1,1) has the value
(a) ▯ 2, (b) ▯0.2, (c) 0, (d) 0.2, (e) ▯20.
( )
2
7. The unit tangent T to the curve R = t ,t,t when t = 3 is parallel to
3
(a) 12i + 2j + 6kk
(b) 18i + j + 6kk
(c) 18i + 6k
(d)24i + 4j + 9k
(e) 18i + 2j + 3k
( )
2
8. The curvature at (0,0,0) on the curve R = t ,t,t has the value
3
(a) 2, (b) 1, (c) 0, (d) 3, (e) 6. Final Examination April 18, 2011 Mathematics 222 Group 1 3

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