McGill University Faculty of Science
May 2013 Supplemental/Deferred Examination
Calculus 2 Math 141 May, 2013 Time: 3 hours
Examiner: Antonio Lei
Associate Examiner: Axel Hundemer
Student name (last, first)
Student number (McGill ID)
INSTRUCTIONS 1. Please write your answers in this exam booklet. 2. You should simplify all your answers wherever possible. 3. This is a closed book exam. Translation dictionary is permitted. Calculators are not permitted. 4. This exam booklet consists of this cover, Pages 2 through 9 containing questions; and Pages 10 through 12, which are blank. 5. There are eight questions, divided into two categories: • ANSWERS ONLY For these you are expected to write down your final answers in the boxes provided only; you may use the available space for your rough work. • SHOW ALL YOUR WORK For these you must write down full solution in the space provided on the page where the question is printed. If the space is exhausted, you may also write on the facing page and the blank pages at the end of the booklet, provided that you indicate clearly any continuation on the page where the question is printed. 6. A total of 100 marks are available on this exam.
Q1
DO NOT WRITE INSIDE THE BOXES BELOW! Q2 Q3 Q4 Q5 Q6 Q7 Q8
Total
Math 141 Suppl./Deferred Exam
Page 2
May, 2013
1. ANSWERS ONLY Evaluate the Riemann sum for the function y = 2 cos(x) − 1 by dividing [0, π] into 3 sub-intervals and using right-end points. [4 marks] ANSWER
Find the limit lim
n→∞
n X k=1
9n2 . (n + 2k)3
[4 marks]
ANSWER
Z
ex
2 +1
Find the derivative of the function F (x) = e−2x
ANSWER
p ln(t) + 3dt.
[4 marks]
Math 141 Suppl./Deferred Exam
Page 3
May, 2013
2. ANSWERS ONLY Find, in terms of polar coordinates, the intersections between the polar curves r = 2 + 2 sin(θ) and r = 1. [4 marks] ANSWER
Find the area of the region that lies inside r = 2 + 2 sin(θ), but outside r = 1.
[4 marks]
ANSWER
Let C be the parametric curve x = 3t2 , y = t3 − 3t + 2. Find the two values of t at which the self-intersection of C occurs. [4 marks] ANSWER
Find the area of the loop formed by C. ANSWER
[4 marks]
Math 141 Suppl./Deferred Exam
Page 4
May, 2013
3. SHOW ALL YOUR WORK Let R be the region bounded by x + y = 3 and y + 1 = (x − 2)2 . (a) Find the area of R.
[6 marks]
(b) Find the volume of the solid obtained by rotating R around y = 3.
[6 marks]
Math 141 Suppl./Deferred Exam
4.
Page 5
May, 2013
SHOW ALL YOUR WORK Find the volume of the solid obtained by rotating the region 1 , 0 ≤ x ≤ 2 around x = 2. [12 marks] 0≤y≤ (3x + 2)2 (2x + 1)
Math 141 Suppl./Deferred Exam
Page 6
5. SHOW ALL YOUR WORK Let C be the curve y =
May, 2013 1 cosh(2x) − cosh(1), −2 ≤ x ≤ 1. 2
(a) Find the arc-length of C.
[4 marks]
(b) Find the area of the surface obtained by rotating C around the y-axis.
[8 marks]
Math 141 Suppl./Deferred Exam
Page 7
May, 2013
6. SHOW ALL YOUR WORK For each of the following improper integral, determine whether it converges or diverges. If it converges, find its value. ∞
Z (a) 5
Z (b)
dx √ . x2 x2 − 9
π/2
csc(x)dx. 0
[8 marks]
[4 marks]
Math 141 Suppl./Deferred Exam
Page 8
May, 2013
7. SHOW ALL YOUR WORK For each of the following two sequences, determine whether it converges or diverges as n → ∞. If it converges, find its limit. √1 (a) arctan (n2 + 1) n
(b)
1 × 3 × · · · × (2n − 1) n!
[6 marks]
[6 marks]
Math 141 Suppl./Deferred Exam
Page 9
May, 2013
8. SHOW ALL YOUR WORK (a) Show that the series
∞ X
n
(−1) ln
n=1
n+3 n
is conditionally convergent.
[8 marks]
n X k+3 k (b) Let S be the limit of the series above. For n ≥ 1, let Sn be the partial sum . (−1) ln k k=1 How large should n be for the inequality |S − Sn | ≤ 0.01 to hold? Justify your answer. [4 marks]
Math 141 Suppl./Deferred Exam
Page 10
CONTINUATION PAGE FOR QUESTION NUMBER
May, 2013
Math 141 Suppl./Deferred Exam
Page 11
CONTINUATION PAGE FOR QUESTION NUMBER
May, 2013
Math 141 Suppl./Deferred Exam
Page 12
CONTINUATION PAGE FOR QUESTION NUMBER
May, 2013
