Date: April 21, 2012
Time: 12:00 noon to 2:30 p.m.
Number of pages:
Exam type:
Aids:
13 (including cover page)
Closed book
No calculators or other electronic aids
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
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
3
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For examiners’ use only
Question Mark Maximum mark
1
2
12
12
4
6
6
6
7
8
9
Total
6
6
6
64

You do not have to simplify your answers; in particular, numerical answers can be left unsimplified in a “calculator-ready” form. You should show your calculations and your solutions should be clear. Good luck!
This page may be used for rough work. It will not be marked.
2

1. Evaluate the following integrals.
1.
(a) [3 marks]
Z x
2
√
2 − x 3 dx
1.
(b) [4 marks]
Z x p
4 − ( x + 1) 2 dx
3

1.
(c) [5 marks]
Z
0
−∞ xe x dx
4

2. Determine whether the following series converge conditionally, converge absolutely, or diverge.
2.
(a) [3 marks]
∞
X
√ n n =1
1
√ n + 1 +
√ n
2.
(b) [4 marks]
∞
X
2 k k !
k =1 k k
5

2.
(c) [5 marks]
∞
X sin n nπ
2 n =1
6

3.
[4 marks] Let f ( x ) =
− 3 if x is rational sin(3 x ) if x is irrational
.
Prove that the right Riemann sum on any regular partition of [0 , 2] with n subintervals is equal to − 6.
7

4.
[6 marks] Find the first four nonzero terms of the Maclaurin series for f ( x ) = e x
2
, and determine the interval of convergence of the series.
8

5.
[6 marks] Find the first four nonzero terms of the Taylor series for f ( x ) = ln( x ) centred at a = 2, and determine the interval of convergence of the series.
9

6.
[6 marks] Sketch, and then calculate, the area enclosed by the curves x = y ( y − 4) and x = y (2 − y ).
10

7.
[6 marks] Find the volume of a solid whose base is the triangle with vertices (0 , 0), (3 , 0) and (0 , 3) and whose cross sections perpendicular to the base and parallel to the y -axis are squares.
11

8. A tank is shaped like an inverted cone (i.e. with the tip pointing downward) with height
2 m and radius 1 m. Assume an acceleration due to gravity of g m/s 2 .
8.
(a) [4 marks] If the tank is full of a liquid of uniform density ρ kg/m 3 , how much work is required to pump all the liquid to the top of the tank and out of the tank?
8.
(b) [2 marks] If the tank is filled to half its depth with the same liquid as in part (a), does it take half as much work to pump the liquid out of the tank as in part (a)?
Justify your answer.
12

9.
[6 marks] Solve the logistic equation dy dt y
= ky 1 −
L for y ( t ), given the initial condition y (0) = constants, and that 0 < y ( t ) < L .)
L
. (You may assume that k and L are positive
2
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