School

McGill UniversityDepartment

Mathematics and StatisticsCourse Code

MATH 141Professor

AllStudy Guide

MidtermThis

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McGill University — Faculty of Science

FINAL EXAMINATION

MATHEMATICS 141 2011 01 CALCULUS 2

EXAMINER: Professor S. W. Drury DATE: Friday April 15, 2011

ASSOCIATE EXAMINER: Professor N. Sancho TIME: 9 am. to noon

FAMILY NAME:

GIVEN NAMES:

MR, MISS, MS, MRS, &c.: STUDENT NUMBER:

If you expect to graduate in Spring 2011 put an ×in this box:

Instructions

1. Fill in the above clearly. Enter your name as it appears on your student card.

2. Do not tear pages from this book; all your writing — even rough work — must be handed in. You

may do rough work anywhere in the booklet except in the box below and in the answer boxes.

3. This is a closed book examination. Calculators are not permitted, but regular and translation

dictionaries are permitted.

4. This examination consists of two parts, Part A and Part B. READ CAREFULLY THE MORE

DETAILLED INSTRUCTIONS AT THE START OF EACH PART.

5. Part A has 20 multiple choice questions each of which is worth 2 points for a total of 40 points.

It is recommended that in the ﬁrst instance, you do not spend more than 4 minutes per question

or 75 minutes in total on Part A.

6. Part B has ﬁve questions worth a total of 46 points which should be answered on this question

paper. The questions in part B are of two types:

•In BRIEF SOLUTIONS questions, each answer will be marked right or wrong. Within a

question, each answer has equal weight.

•In SHOW ALL YOUR WORK questions a correct answer alone will not be suﬃcient unless

substantiated by your work. Partial marks may be awarded for a partly correct answer.

In Part B, you are expected to simplify all answers wherever possible.

7. This examination booklet consists of this cover, Pages 1–4 containing questions in Part A, Pages

5–10 containing questions in Part B and Pages 11–14 which are blank continuation pages.

8. A TOTAL OF 86 POINTS ARE AVAILABLE ON THIS EXAMINATION.

PLEASE DO NOT WRITE INSIDE THIS BOX

code 21 22 BS total

/8 /8 /16

23 24 25 SA total

/10 /10 /10 /30

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Calculus 2 MATH 141 Final Exam Version 1

Part A: MULTIPLE CHOICE QUESTIONS

Each of the following 20 questions is worth 2 points. Half of a point will be subtracted for

each wrong answer. The maximum number of points you may earn on these multiple choice

questions is 40 points. There is only one correct answer expected for each question. The

questions are to be answered on the answer card provided. Be sure to enter on the answer

card:

◦Your student number. ◦Your name.

◦The check code, i.e. the ﬁrst two letters of your family name.

Fill in the disks below your student number, check code and verify that the

marked version number matches the version number on the question paper. If

it does not, please notify an invigilator.

It is recommended that in the ﬁrst instance, you do not spend more than 4 minutes per

question or 75 minutes in total on Part A.

Please note:

The Examination Security Monitor Program detects pairs of students with un-

usually similar answer patterns on multiple-choice exams. Data generated by

this program can be used as admissible evidence, either to initiate or corrob-

orate an investigation or a charge of cheating under Section 16 of the Code of

Student Conduct and Disciplinary Procedures.

1. (2 points) Evaluate the Riemann sum for Z4

−2

x2dx which uses three intervals of equal

length and evaluation points at the midpoints of these intervals.

(a) 16, (b) 22, (c) 11, (d) 56

3, (e) 40.

2. (2 points) Find lim

n→∞

1

n

n

X

k=1

ln µ2n

n+k¶.

(a) ln(2) −1

2, (b) 1, (c) 1−ln(2)

2, (d) 1 −ln(2), (e) 2 ln(2) −1.

3. (2 points) Let F(x) = Z√x

0p1 + t4dt. Then F′(1) equals

(a) 2, (b) 1, (c) √2, (d) 1

√2, (e) 1

2.

1

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Calculus 2 MATH 141 Final Exam Version 1

4. (2 points) Let the function be deﬁned by

f(x) =

sin(x) if x < 0,

1−cos(x) if x≥0.

Then for x≥0, Zx

−π

f(t)dt =

(a) −2 + x−sin(x), (b) x−sin(x), (c) −2 + x+ sin(x), (d) x+ sin(x),

(e) x+ cos(x) + sin(x).

5. (2 points) Let f(x) = 2x

1 + x2. The Integral Mean Value Theorem applied to Z3

1

f(t)dt

asserts the existence of a number awith 1 ≤a≤3 satisfying

(a) f′(a) = −1

4, (b) f(a) = ln(5)

2, (c) f(a) = ln(5), (d) f′(a) = −6

25,

(e) f(a) = 4

5.

6. (2 points) Zπ

π

2

sin(x) + cos(x)

sin(x)−cos(x)dx =

(a) ln(2), (b) 1, (c) 0, (d) −ln(2), (e) not deﬁned.

7. (2 points) Zπ

π

2

sin(x)−cos(x)

sin(x) + cos(x)dx =

(a) 1, (b) ln(2), (c) 0, (d) −ln(2), (e) not deﬁned.

8. (2 points) The area of the region bounded by the parabola x=y(y−3) and the line

y=xis

(a) 37

6, (b) 9, (c) 32

3, (d) 8, (e) 5.

9. (2 points) Z1

−1

arctan(x)dx =

(a) π−ln(4)

2, (b) π−ln(4)

4, (c) π, (d) 0, (e) π−ln(2)

4.

10. (2 points) The volume obtained by rotating the region x > 0, 0 ≤y < x−1e−xabout

the y-axis is

(a) π

2, (b) π, (c) 2π, (d) 4π, (e) inﬁnite.

2

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