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University of British Columbia
MIDTERM TEST 2 for MATH 110
Date: February 15, 2012
Time: 6:00 p.m. to 7:30 p.m.
Number of pages: 9 (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
8
2
5
3
7
4
5
5
5
6
5
7
1 (bonus)
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.
Your answers must be “calculator-ready”, but they do not have to be simplified.
This page may be used for rough work. It will not be marked.
2
1. (a) [2 marks] Find y 0 , given the ellipse
x2
2
+ y 2 = 1.
1. (b) [2 marks] Find y 0 , given the hyperbola 2x2 − 2y 2 = 1.
1. (c) [2 marks] Let (a, b) be a point where the ellipse
2x2 − 2y 2 = 1 intersect. Show that a2 = 2b2 .
2
x2
2
+ y 2 = 1 and the hyperbola
1. (d) [2 marks] Prove that the ellipse x2 +y 2 = 1 and the hyperbola 2x2 −2y 2 = 1 intersect
at right angles. You may use the result of part (c) even if you have not proven it.
3
2. Consider a spring which hangs from the
ceiling with an iron weight attached to
its end, as in the picture to the right.
The weight is pulled down from its equilibrium position and then released. Here
is a proposed model for the displacement of the weight:
equilibrium position
Let h be the distance of the weight from
its original position at rest, measured in
centimetres. Then h(t) = cos t, where t
is the time, measured in seconds.
h
2. (a) [2 marks] Describe, in a few sentences, one major problem with the proposed model.
2. (b) [3 marks] Propose a model which addresses the problem you described in part (a).
Explain why your new model is better.
4
3. Crystals formed in a salt solution are cube-shaped. The side-length x of a single crystal
satisfies the following differential equation:
dx
= k(x30 − x3 ),
dt
where k and x0 are positive constants.
3. (a) [3 marks] Suppose you have one small crystal and one big crystal suspended in the
same solution. Which crystal is growing faster? Justify your answer.
3. (b) [2 marks] Find an expression for the rate of change of the volume of a crystal.
3. (c) [2 marks] What do you think the constant x0 represents? Explain your answer in a
few sentences.
5
4. [5 marks] The biological half-life of a drug introduced into the blood stream is defined
to be the time required for the concentration of the drug to reduce by one-half.
Morphine has a biological half-life of 2 hours. How long does it take a concentration of
morphine administered intravenously to reduce by one-third?
6
5. [5 marks] Termites of the species Macrotermes bellicosus build giant cone-shaped nests
whose height is 2.5 times the radius. The height of a nest increases at a constant rate of
50 cm/year for the first ten years. How fast is the volume increasing when the height is
100 cm?
7
6. [5 marks] Atmospheric pressure decreases as altitude increases; for example, the pressure
at the top of Mount Everest is roughly one-third of the pressure at sea level. Up to an
altitude of 15000 metres, the relationship between pressure p (in millibars) and height h
(in metres) is described by the following equation:
p = 1000e−h/10000 .
Consider a skydiver free falling at a rate of 56 m/s. How fast is the pressure on him
increasing when he is at a height of 5000 metres?
8
7. [1 bonus mark] Question 3 referred to a model for the side-length of a cubic salt crystal;
namely, the side-length x satisfies the differential equation
dx
= k(x30 − x3 ),
dt
where k and x0 are positive constants. Sketch the graph of
justify the shape of your graph.
9
dx
dt
with respect to x, and