Name (print):
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001
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University of British Columbia
FINAL EXAMINATION for MATH 110
Date: April 28, 2010
Time: 12:00 noon to 2:30 p.m.
Number of pages: 16 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
For examiners’ use only
Question Mark Possible marks
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
1
8
Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
2
8
3
10
4
12
5
8
6
16
7
6
8
6
9
6
10
4 (bonus)
Total
80
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;
• 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. State whether each of the following statements is true. If it is, explain why. If it is not,
explain why not (for example, by providing a counterexample).
1. (a) [2 marks] If f (x) is a cubic function, then f (x) = 0 has a solution.
1. (b) [2 marks] A function can have two different horizontal asymptotes.
1. (c) [2 marks] The derivative of an even function is odd. (Recall that f is said to be even
if f (x) = f (−x) for all x, and odd if f (x) = −f (−x) for all x.)
1. (d) [2 marks] If f 0 (c) = 0, then f has a local maximum or minimum at c.
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2. Let f (x) =
√
x. Sketch each of the following curves, making sure to indicate intercepts.
2. (a) [1 mark] y = f (x)
2. (b) [2 marks] y = f (x + 1) + 2
2. (c) [2 marks] y = f −1 (x)
2. (d) [3 marks] y = F (x), where F is an antiderivative of f passing through (0, 1)
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3. Evaluate the following limits.
3. (a) [2 marks] lim +
x→−3
x+2
x+3
1 − 2x2 − x4
x→−∞ 5 + x − 3x4
3. (b) [2 marks] lim
4
1 − cos x
x→∞
x2
3. (c) [3 marks] lim
1 − cos x
x→0
x2
3. (d) [3 marks] lim
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4. For each of the following curves, find y 0 . (You do not have to simplify your answer.)
4. (a) [2 marks] y =
x3
1 − x2
4. (b) [3 marks] x2 + xy − y 2 = 4
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4. (c) [4 marks] y = xcos x
4. (d) [3 marks] y = sin (cos (tan x))
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5. Let f (x) =
√
x.
5. (a) [3 marks] Use the limit definition of derivative to find f 0 (4).
5. (b) [2 marks] Find the equation of the tangent line to y = f (x) at x = 4.
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5. (c) [2 marks] Use a linear approximation to estimate
√
4.01. Simplify your answer.
5. (d) [1 mark]√Is your estimate in part (c) greater than, less than, or equal to the actual
value of 4.01? Justify your answer.
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6. Let f (x) = xex . Consider the curve y = f (x).
6. (a) [2 marks] Find the x-intercepts and y-intercepts of the curve.
6. (b) [2 marks] Find the horizontal and vertical asymptotes.
6. (c) [4 marks] Determine where f is increasing and where it is decreasing, and find its
local maxima and minima.
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6. (d) [4 marks] Determine where the curve is concave up and where it is concave down,
and find its inflection points.
6. (e) [4 marks] Sketch the curve, showing the properties determined in parts (a) through
(d).
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7. [6 marks] Caesium-137, or 137 Cs, is a radioactive isotope used to treat certain types of
cancers. Its biological half-life, the time it takes a living human being to eliminate half of
a given substance, is 70 days.
Determine the proportion of 137 Cs remaining in a cancer survivor one year after radiation
therapy has been stopped.
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8. The Space Shuttle launches vertically. In the initial part of its flight, its altitude in
kilometres after t seconds is given by
h(t) =
t2
.
300
The launch is supervised from the Launch Control Centre (LCC), 5 kilometres away from
the launch pad.
8. (a) [2 marks] Determine the vertical speed of the Space Shuttle 30 seconds after liftoff.
8. (b) [4 marks] Determine the rate at which the distance between the LCC and the Space
Shuttle is increasing 30 seconds after liftoff.
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9. [6 marks] The word “threat” was originally pronounced “threet”. In linguistics, this
shift in pronunciation is called a language change. It is theorized that any given language
change spreads through a population according to a logistic differential equation such as
the one described below.
Consider a population of 1000. Let y(t) denote the number of people who have, as of
time t, adopted the language change. Then
y 0
,
y (t) = ky 1 −
1000
where k is a positive constant. The graph of y(t) is below:
y
1000
t
(Note that y 0 (t) is always positive.) What is y(t) when y 0 (t) is maximal?
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10. [4 bonus marks] Let (a, a2 ) be a point on the parabola y = x2 . Let L1 be the line
passing through (a, a2 ) and perpendicular to the tangent line at (a, a2 ). Let L2 be the
horizontal line passing through (a, a2 ). Prove that the distance between the y-intercepts
for these two lines is 21 , for any nonzero a.
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This page may be used for rough work. It will not be marked.
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