Date: April 23, 2015
Time: 12:00 noon to 2:30 p.m.
Number of pages: 13 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action:
• Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners;
• Speaking or communicating with other candidates;
• Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.
Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
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6
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For examiners’ use only
Question Mark Possible marks
1
2
18
5
8
9
Total
5
6
5
5
4
6
6
60

Note that your answers must be in “calculator-ready” form, but they do not have to be simplified.
In general, you may use any result proven in class or on assignments. You may find the following trigonometric identities helpful.
sin
2
( x ) + cos
2
( x ) = 1 sin(2 x ) = 2 sin( x ) cos( x ) cos(2 x ) = cos
2
( x ) − sin
2
( x )
This page may be used for rough work. It will not be marked.
2

1. (a) [2 marks] Explain what it means for a function f ( x ) defined on an interval [ − a, a ] to be integrable on that interval.
(b) [2 marks] Evaluate the integral
Z
∞
1
1 t 3 dt .
(c) [3 marks] Evaluate the integral
Z
π/ 6 sin
8
( t ) cos
5
( t ) dt .
0
3

(d) [3 marks] Write down (but do not evaluate) an integral describing the volume of the solid obtained by rotating the circle ( x − 3) 2 + y 2 = 1 about the y -axis.
(e) [2 marks] Does the series
X e
1 /n !
converge? Justify your answer.
n ≥ 1
4

(f) [3 marks] Does the series
X
√ n 2 − 2 n
3 n 2 + 4 n n ≥ 2 converge? Justify your answer.
(g) [3 marks] Determine the radius of convergence of the power series
X n ( x + 3) n
.
2 n n ≥ 0
5

2.
[5 marks] Recall the error function erf( x ) =
2
√
π
Z x e
− t
2 dt.
0
Use a technique of integration to compute
Z
2 erf( x ) dx . (Your answer may depend on
0 erf(2).)
6

3.
[5 marks] Consider two full tanks of water. As pictured below, one is an inverted cone and the other is a cylinder. Both tanks are of height H , and have the same volume.
H
R
Suppose the tanks are emptied by pumping water out of a spout at the top. In which case is more work done?
7

4. (a) [1 mark] Define what it means for a sequence { a n
} to converge to a limit L .
(b) [3 marks] Suppose { a n
} converges. Prove that {| a n
|} also converges.
(c) [2 marks] Suppose {| a n
|} converges. Can we conclude that { a n
} also converges?
Why or why not?
8

5. The Cantor set is a fractal constructed as follows.
(i) Begin with the line segment I
0
= [0 , 1] .
(ii) Remove the open middle third
1
3
,
2
3
, to get I
1
= 0 ,
1
3
∪ 2
3
, 1 .
(iii) Remove the open middle third from every remaining line segment, to get
I
2
= 0 ,
1
9
∪
2
9
,
1
3
∪
2
3
,
7
9
∪
8
9
, 1 .
(iv) Repeat the process ad infinitum .
A few iterations of this construction are illustrated below.
(a) [4 marks] By considering a suitable series, show that the total length of all of the intervals removed is equal to 1.
(b) [1 mark] Despite the fact that an infinite number of intervals of total length 1 have been removed, there are points in the interval [0 , 1] that will never be removed. What is one such point?
9

6. (a) [2 marks] Use an appropriate linear approximation to estimate sin(0 .
1).
(b) [1 mark] Let L (0 .
1) be your estimate in part (a). Is L (0 .
1) greater than, less than, or equal to sin(0 .
1)? Justify your answer.
(c) [2 marks] What is a good upper bound on | L (0 .
1) − sin(0 .
1) | ? (Hint: use the Lagrange form of the remainder.)
10

7.
[4 marks] Write down a series that converges to log( π ). The series must have an infinite number of nonzero terms, but the terms may include π . (Hint: note that π = 3 1 + π − 3
3
.)
11

8.
[6 marks] Evaluate
Z
1
0
1 − e
− t dt . Your solution may be written as a series.
t
12

9. The differential equation dy dx
+
1 x y = − x
2 e x y
2
(1) is known as a Bernoulli differential equation . Clearly y = 0 is a solution. In this question you will find a family of nonzero solutions.
(a) [2 marks] By making the substitution w =
1 y
, show that equation (1) may be rewritten dw
− dx
1 x w = x
2 e x
.
(2)
(b) [4 marks] Solve the differential equation (2).
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