Name (print, with surname first):
Student number:
University of British Columbia
DECEMBER EXAM: Science One Mathematics
Date: December 12, 2012
Time: 12:00 noon to 2:30 p.m.
Number of pages: 11 (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 must conduct themselves honestly and in accordance
with established rules for an examination. Should dishonest behaviour be observed, pleas of accident or forgetfulness shall not
be received.
Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be subject to disciplinary action:
• Speaking or communicating with other candidates;
• Purposely exposing written papers to the view of other candidates or imaging devices;
• Purposely viewing the written papers of other candidates;
• Having visible at the place of writing any books, papers or
memory aid devices;
• Using or operating electronic devices — electronic devices
must be completely powered down if present at the place of writing.
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 invigilator.
For examiners’ use only
Question Mark Possible marks
1
8
2
11
3
5
4
5
5
9
6
4
7
8
Total
50
1. Determine whether each of the following statements is true or false. If it is true, provide
justification. If it is false, provide a counterexample.
(a) [2 marks] A function with a horizontal asymptote never crosses it.
(b) [2 marks] ex > ln x for all x > 0.
(c) [2 marks] If a function f is continuous on the interval (0, 1), then there exists a point
c in (0, 1) such that f (c) ≤ f (x) for all x in (0, 1).
(d) [2 marks] If a function f is not continuous on the interval [a, b], then there is no
(a)
.
point c in (a, b) such that f 0 (c) = f (b)−f
b−a
2
2. Let f (x) = ln(x2 − 1). In this question you will sketch the graph of f (x). You may use
the following page as well as this page to write down your answers.
(a) [1 mark] Find the domain of f (x).
(b) [2 marks] Find the x- and y-intercepts of f (x).
(c) [2 marks] Determine if f (x) has any vertical or horizontal asymptotes.
(d) [2 marks] Find f 0 (x), and determine where the graph of f (x) is increasing or decreasing.
(e) [2 marks] Find f 00 (x), and determine where the graph of f (x) is concave up or concave
down.
(f) [2 marks] Make a large sketch of the graph of f (x), including all the features determined in parts (a) through (e).
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4
3. (a) [3 marks] Use the linear
approximation of f (x) = tan x at x =
1
π
value of tan 4 + 40 . Your answer must be a number.
π
4
to approximate the
(b) [2 marks] Is your approximation
in part (a) smaller than, larger than, or equal to
π
1
the actual value of tan 4 + 40 ? Justify your answer.
5
4. (a) [3 marks] Verify, by differentiating, substituting and simplifying, that
y = C1 e−t + C2 e4t ,
where C1 and C2 are constants, is a solution of the differential equation
y 00 − 3 y 0 − 4 y = 0.
(b) [2 marks] Find the solution of the above differential equation that satisfies the initial
conditions
y(0) = 3, y 0 (0) = 2.
6
5. A freshly poured cup of coffee, initially 93◦ C, is placed in a room which is maintained at
a constant temperature of 20◦ C. According to Newton’s Law of Cooling, the temperature
u(t) (in ◦ C) of the coffee at time t (in minutes) satisfies the differential equation
u0 = −k(u − 20),
where k is a positive constant.
(a) [2 marks] Find all equilibrium temperatures (solutions of the differential equation of
the form u(t) = m, where m is a constant).
(b) [3 marks] Sketch, on the u-axis, the vector field of the differential equation; and
then sketch, on the (t, u)-plane, the graph of the temperature u(t) of the coffee as a
function of t.
(c) [1 mark] Let y(t) = u(t) − 20. Using the original differential equation above, write
down the differential equation satisfied by y(t).
(d) [3 marks] If after one minute in the room the coffee has cooled to 88◦ C, determine
when the the temperature of the coffee reaches 63◦ C.
Hint: You should be able to solve the differential equation for y(t) obtained in part
(c), and then use the relationship y(t) = u(t) − 20 to solve for u(t).
7
6. The growth of certain populations in a restricted environment may be modelled by the
logistic equation, a differential equation of the form
y(t)
0
y(t),
y (t) = r 1 −
K
where y(t) is the population at time t, and r and K are positive constants.
(a) [2 marks] Write down an expression for y 00 (t) in terms of y(t).
(b) [2 marks] At what population y(t) in the interval [0, K] is the population growth
rate maximized? Justify your answer.
8
7. Imagine a forest of dead trees. The dead trees decay naturally, from seasonal and biological
events. Decayed trees eventually become humus, which itself decays, but at a slower rate.
Let x(t) denote the biomass of humus at time t, and y(t) denote the biomass of dead trees
at time t. Then x and y satisfy the following system of differential equations:
x0 = −x + 3y
y 0 = −3y.
(a) [6 marks] On the following page, sketch on a large (x, y)-plane the following items,
indicating them clearly. You should include negative values of x and y as well as
positive ones, even though these will not have a physical interpretation.
(You may do your rough work below this question or on the back of this page. Do
your rough work neatly, as it will be used to determine part marks in the event of an
error in your sketch.)
(i) The x- and y-nullclines, indicating the direction of crossing trajectories.
(ii) The lines through the origin representing both sets of eigenvectors, indicating
the direction of trajectories.
(iii) A number of representative trajectories, including at least one trajectory starting
in each quadrant, indicating the direction of trajectories.
(b) [2 marks] Under what initial ratio of dead trees to humus does the biomass of humus
initially increase? Describe your answer in two or three sentences.
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This page may be used for rough work. It will not be marked.
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