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University of British Columbia
MIDTERM TEST 2 for MATH 110
Date: February 9, 2011
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
6
3
8
4
8
5
5
6
5
7
2 (bonus)
Total
40
• 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. Compute the derivatives of the following functions.
(a) [2 marks] f (x) = 3x
(b) [3 marks] g(x) = ln
3x + 3
x2 + 1
(c) [3 marks] h(x) = ln (x · x2 · x3 · x4 · x5 · x6 ) + ex·x
2
2 ·x3 ·x4 ·x5 ·x6
2. HOP-62 and NCI-H226 are two cell lines maintained by the National Cancer Institute
in the United States. Under ideal laboratory conditions, a population of HOP-62 cells
doubles every 39 days, and a population of NCI-H226 cells doubles every 61 days.
(a) [3 marks] How long does it take for a population of HOP-62 cells to triple?
(b) [3 marks] If you start with 200 HOP-62 cells and 10 NCI-H226 cells, after how many
days are the populations equal?
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3. (a) [6 marks] Below are three different assumptions which can be made about a population P . For each, write an expression for P (t), where t is the time in years from the
present.
(i) The population grows by 5 individuals every month.
(ii) The population at the end of each year is always 102% of the population at
the beginning of the year.
(iii) At any given time, the population is decreasing at a rate equal to 0.1P (t).
(b) [2 marks] Suppose the current population is 1000. Which of the assumptions in part
(a) leads in the long run to the greatest population? Justify your answer.
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4. In this question, you will analyze the burning area of two types of wildfire which originate
from a single ignition point.
(a) [4 marks] On flat, windless grassland, a wildfire spreads in the shape of a circle. Suppose the front of the fire moves away from the ignition point at 20 km/h. Determine
how quickly the wildfire area is growing when it has a radius of 3 km.
(a) [4 marks] Wind can cause the fire to spread in
the shape of an ellipse, shown to the right. Suppose the axis a grows at a rate of 20 km/h and
the axis b grows half as fast. Determine how
quickly the wildfire area is growing 30 minutes
after it begins to spread from its ignition point.
(You may use the fact that the ellipse has area
πab.)
5
b
a
5. [5 marks] As a woman walks away from a streetlight, her shadow lengthens. Prove that
it does so at a rate which depends on her speed but not on her position.
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6. (a) [3 marks] Use an appropriate linear approximation to estimate
√
48.9.
(b) [2 marks] Is your estimate in part (a) greater than, less than, or equal to
Justify your answer.
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√
48.9?
7. [2 bonus marks] Find the equation of the line which is tangent to y = 3x and passes
through the origin.
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This page may be used for rough work. It will not be marked.
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