Name (print):
Student number:
University of British Columbia
MIDTERM TEST: Science One Mathematics
Date: February 28, 2013
Time: 8:30 a.m. to 9:30 a.m.
Number of pages: 7 (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
5
2
6
3
5
4
4
5
5
Total
25
• 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.
1. (a) [2 marks] Define what it means for a function f to be integrable on the interval [0, 1].
(b) [3 marks] Let
f (x) =
0 if x 6= 1/2
.
1 if x = 1/2
Explain, using your definition in part (a), why this function is integrable on the interval
[0, 1].
2
2. [6 marks] Find the volume of the solid S whose base is the region
n
√ o
(x, y)| 0 ≤ y ≤ (4 − x2 )1/4 , 0 ≤ x ≤ 2
(i.e. the region above the x-axis and below y = (4 − x2 )1/4 , for 0 ≤ x ≤
cross-sections perpendicular to the x-axis are squares.
3
√
2), and whose
3. [4 marks] The average speed v̄ of molecules with molecular mass M in an ideal gas at
temperature T is
3/2 Z ∞
M
4
2
v 3 e−M v /(2RT ) dv,
v̄ = √
π 2RT
0
where R is the ideal gas constant. Determine an expression for v̄ in terms of M , R and
T by evaluating the improper integral and simplifying the result. You may use without
proof the fact that
2
lim v p e−αv = 0
v→∞
for any constants p, α > 0.
4
4. [5 marks] The three water tanks illustrated below√are of equal height H. The two cones
are of radius R, while the cylinder is of radius R/ 3 (thus all three tanks have the same
volume). Which tank requires the most work to empty by pumping all of the water over
the top? Which tank requires the least work? The tanks are small enough that the force
of gravity may be assumed to be constant.
Please note that you must justify your answers.
A
B
5
C
Z
5. [5 marks] Evaluate the indefinite integral
6
2x3 + 3x2 + 3x − 5
dx.
x2 (x2 + 2x + 5)
This page may be used for rough work. It will not be marked.
7