Name (print):
Student number:
University of British Columbia
APRIL EXAM: Science One Mathematics
Date: April 22, 2014
Time: 12:00 noon to 2:30 p.m.
Number of pages: 13 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
Rules governing formal examinations:
For examiners’ use only
Question Mark Possible marks
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
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8
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.
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12
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10
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:
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4
• 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;
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• Speaking or communicating with other candidates;
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6
• Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
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Total
65
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.
Please note that your answers must be in “calculator-ready” form, but they do not have to
be simplified.
This page may be used for rough work. It will not be marked.
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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. You may quote results from lectures
or the textbook without proof.
(a) [2 marks] If f is not differentiable on an interval [a, b], then f is not integrable on
the interval [a, b].
X
X 1
diverges.
(b) [2 marks] If
an converges, then
a
n≥0
n≥0 n
X
(c) [2 marks] There exist coefficients an such that
an (x + 1)2n converges everywhere.
n≥1
dy
(d) [2 marks] The differential equation dx
= 1 + x2 + y 2 has at least one solution y(x)
that is decreasing on an open interval.
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2. Evaluate the following. (You may continue your solutions on the following page.)
Z b√
(a) [4 marks]
b2 − x2 dx, where a and b are constants with 0 < a < b
a
Z
(b) [4 marks]
x2 + 3x + 2
dx
x3 + x
Z
(c) [4 marks]
x arctan(x) dx
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You may continue your solutions on this page.
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3. Determine whether each of the following series converges. (You may continue your solutions on the following page.)
(a) [3 marks]
X sin2 (n)
n≥2
(b) [3 marks]
X 1 + 4n
n≥1
(c) [4 marks]
n2 − 1
5n−1
X n!
2n!
n≥1
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You may continue your solutions on this page.
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4. [4 marks] The Euler-Mascheroni constant γ is defined to be
!
n
X
1
γ = lim
− ln(n) .
n→∞
i
i=1
Explain why 0 < γ < 1. (Hint: consider the area under y =
into n subintervals.)
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1
x
from 1 to n, partitioned
5. The Gauss error function, defined to be
Z
erf(x) =
x
2
e−t dt,
0
is used to describe various solutions in probability and differential equations.
(a) [1 mark] What is the domain of erf(x)? Justify your answer.
(b) [2 marks] On what interval(s) is the graph of erf(x) increasing? On what interval(s)
is it decreasing?
(c) [3 marks] Write down the first three nonzero terms of the Maclaurin series for erf(x).
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6. In the following two questions, assume acceleration due to gravity to be g m/s2 .
(a) [3 marks] A rope of linear mass density δ kg/m and length L m is hanging over the
edge of a tall building. Find the work done pulling half the rope to the top of the
building (leaving half the rope hanging over the edge).
(b) [3 marks] A spherical tank of radius R m is full of a liquid that has spatial mass
density ρ kg/m3 . Find the work done pumping half of the liquid to the top of the
tank (leaving the tank half full).
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7. Gabriel’s horn is the solid of revolution formed by taking f (x) =
and rotating it about the x-axis.
1
x
on the interval [1, ∞)
(a) [4 marks] Find the volume of Gabriel’s horn.
(b) [3 marks] The surface area of a solid of revolution formed by rotating a positive,
differentiable curve y = g(x) on an interval [a, b] about the x-axis is given by
Z b
q
2πg(x) 1 + (g 0 (x))2 dx.
a
Explain why the surface area of Gabriel’s horn is infinite.
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8. [6 marks] Experiments show that the reaction H2 + Br2 → 2HBr satifies the rate law
Rate =
d[HBr]
= k[H2 ][Br2 ]1/2 ,
dt
where k is a positive constant (for a fixed temperature). If the initial concentrations are
[H2 ](0) = a moles/L and [Br2 ](0) = b moles/L, then the concentration of the reaction
product y(t) = [HBr](t), in moles/L, satisfies the differential equation
dy
= k(a − y)(b − y)1/2 .
dt
Find y explicitly as a function of t in the case where a = b and y(0) = 0.
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9. Consider the differential equation
dy
= µy − y 3 = y(µ − y 2 )
dt
−∞<y <∞
where µ is a (constant) parameter. You are not asked to find y(t) explicitly.
(a) [1 mark] Assume µ < 0.
i. Find all steady states (equilibria).
ii. Sketch phase portraits in the phase line (y-axis).
iii. Sketch graphs of representative solutions (t versus y).
(b) [1 mark] Do the same as in part (a), but assuming µ = 0.
(c) [2 mark] Do the same as in part (a), but assuming µ > 0.
(d) [2 marks] Sketch a bifurcation diagram (µ versus y), showing the steady states as
functions of µ. Indicate on the diagram which steady states are stable and which are
unstable.
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