Name (print):
Student number:
University of British Columbia
MIDTERM TEST: Science One Mathematics
Date: October 23, 2012
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
4
4
6
5
4
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] Explain, in terms of δ and ε, what the equation “lim f (x) = 0” means.
x→0
(b) [3 marks] Let
f (x) =
n if x = 101n where n = 1, 2, 3, . . .
.
0 otherwise
Explain, in terms of δ and ε, why lim f (x) 6= 0.
x→0
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2. Find
dy
given each of the equations below (simplification is not required).
dx
tan x
.
4 − x2
√
(b) [3 marks] y 3 − xy = x2 + 3x + 20. (Your answer may be in terms of both x and y.)
(a) [3 marks] y =
3
3. [4 marks] The model used in class to describe the salmon population in a branch of the
Fraser river is an example of a discrete population model. These models are described by
functions of the form
f : population Pt in a given year → population Pt+1 in the following year.
Given a discrete population model, it is possible to graph Pt+1 on the vertical axis versus
Pt on the horizontal axis. Suppose the graph of a given population model lies entirely
beneath the line of slope 1 through the origin. Describe the long-term population trend
predicted by the model. Justify your answer.
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4. A mass of 1 kg is attached to a spring which obeys Hooke’s Law with a spring constant of 9
N·m−1 , and moves along a straight line without friction. Let x(t) denote the displacement
of the mass (in m) from the equilibrium position at time t (in s).
(a) [3 marks] Show that x(t) = C1 cos(3t) + C2 sin(3t) is a solution of the differential
equation that describes the motion of the mass (write down the differential equation),
for any constants C1 and C2 .
(b) [3 marks] If initially at t = 0 the displacement of the mass is −1 m and the velocity
is −1 m·s−1 , find the displacement of the mass for any t ≥ 0.
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5. [4 marks] The biological half-life of a drug introduced by intramuscular injection is the
time required for any amount of the drug, which decays exponentially, to reduce by onehalf.
Morphine has a biological half-life of 120 minutes. Suppose a patient is given an initial
dose of 10 mg, and after 45 minutes, another dose of 10 mg. How much morphine, in total,
remains 5 minutes after the second dose?
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This page may be used for rough work. It will not be marked.
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