Name (print):
Student number:
University of British Columbia
MIDTERM TEST: Science One Mathematics
Date: February 27, 2014
Time: 8:30 a.m. to 9:20 a.m.
Number of pages: 8 (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;
For examiners’ use only
Question Mark Possible marks
1
18
2
5
3
4
4
4
5
4
Total
35
• 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.
Please note that your answers must be in “calculator-ready” form, but they do not have to
be simplified. Make sure to justify all of your answers.
This page may be used for rough work. It will not be marked.
2
1. Evaluate the following. Make sure to show all your work.
Z ∞
xe−2x dx
(a) [6 marks]
1
√
Z
(b) [6 marks]
0
Z
(c) [6 marks]
3/2
√
x2
dx
3 − 4x2
x2 + x − 4
dx
x3 + 4x
(You may continue your solutions to this question on the following page.)
3
(You may continue your solutions on this page.)
4
2. (a) [2 marks] Write down, but do not evaluate, the Riemann sum for f (x) = x1 on the
interval [1, 2] with 10 subintervals of equal size, taking the sample points to be right
endpoints.
(b) [3 marks] Explain carefully why the following inequality is true:
10
1 10 10 10
+
+
+ ··· +
> ln 2.
10 10 11 12
19
(Hint: how is the sum on the left hand side different from your answer for part (a)?)
5
3. [4 marks] The cosine integral function is defined to be
Z x
cos t − 1
Ci(x) = γ + ln x +
dt,
t
0
where γ is a constant known as the Euler-Mascheroni constant. Determine where this
function has local maxima on the interval (0, ∞).
6
4. [4 marks] A spherical tank of diameter 4 m is filled to a depth of 3 m with a fluid which
has density ρ = 1200 kg · m−3 . Write down a definite integral which describes the work
done to pump all the fluid out the top of the tank, assuming that the acceleration of any
mass due to gravity is g = 10 m · s−2 . Give correct units, but do not evaluate the integral.
7
5. [4 marks] Prove that a right circular cone of radius r and height h has volume 13 πr2 h.
8