The University of British Columbia
Science One Mathematics
Exam #2
19 April 2009
Time: 150 minutes
Full Name:
Student # :
Signature:
This Examination paper consists of 11 pages (including this one). Make sure you have all 11.
Instructions:
No memory aids allowed. No calculators allowed. No communication devices allowed.
Rules:
– Each candidate must be prepared to produce, upon request, a UBC card for identification.
– Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities
in examination questions.
– 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; and
- 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.
marking:
Q1
/7
Q2
/5
Q3
/ 13
Q4
/ 10
Q5
/5
Q6
/ 10
TOTAL
Names of Instructors: Eric, Fok, Mark
/ 50
Science One Mathematics — Exam #2 — 19 April 2009 — p. 2 of 11
Q1
[7 marks]
2
Let f (x) = e−x .
(a) Write down a function F (x) with derivative F 0 (x) = f (x), and such that f (x) = −f (−x)
for all x. [2 marks]
(b) Write down a Maclaurin series representation of f (x). [2 marks]
(c) Find a Maclaurin series representation of an antiderivative of f (x). [3 marks]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 3 of 11
Q2
[5 marks]
The Cantor set is formed as follows:
(i) Begin with the interval [0,1].
(ii) Of all the intervals that remain, remove the open middle third.
(iii) Go back to step (ii).
For example, after the first iteration, we are left with
0,
2
1
∪ ,1 ;
3
3
after the second, we are left with
0,
1
2 1
2 7
8
∪ ,
∪ ,
∪ ,1 .
9
9 3
3 9
9
The first six iterations are illustrated below.
The Cantor set is what remains after repeating the process ad infinitum.
(a) The measure of the Cantor set is equal to 1 − L, where L is the sum of the lengths of all
the intervals that are removed. Prove that the Cantor set has measure 0. [4 marks]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 4 of 11
(Q2 continued)
(b) Is the Cantor set empty? Justify your answer. [1 mark]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 5 of 11
Q3
[13 marks]
In the following, you may assume previous parts of the question. (For example, you may assume
part (a) and part (b) when answering part (c)).
(a) Show that ln(n!) >
Z n
ln t dt. [4 marks]
1
(b) Integrate by parts to conclude from part (a) that ln(n!) > n ln n − n + 1. [3 marks]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 6 of 11
(Q3 continued)
(c) Show that the radius of convergence of the series
1
1+
n
proof, the fact that n→∞
lim
n
n!xn
is e. (You may use, without
n
n≥1 n
X
= e.) [4 marks]
(d) Using part (b), show that the series diverges at the endpoints of its interval of convergence. [2 marks]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 7 of 11
Q4
[10 marks]
A simple model for the concentration of oxygen in the atmosphere (isothermal and at equilibrium) is
C(r) = C0 e
−
r−R0
h0
where C0 is the concentration at sea level (in units of tonnes/km3 ), r is the distance to the
centre of the earth (in units of km), R0 is the radius of the earth (in units of km), and h0 is
roughly 8000 km.
(a) Identify a shape of thin volume within which the concentration of oxygen is approximately
constant. [3 marks]
(b) Use this to set up an integral whose value gives the total mass of oxygen in the atmosphere. [5 marks]
(c) What technique of integration would you use to solve this integral? You do not have to
solve it. (It might come in handy to know that the volume and surface area of a sphere
are 34 πr3 and 4πr2 , respectively.) [2 marks]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 8 of 11
Q5
[5 marks]
Match the phase plane to the sketch of x(t) and y(t) by filling in the blank with the appropriate
letter.
A.
B.
3
2
x(t)
y(t)
x(t)
y(t)
1.5
2
1
0.5
1
0
0
−0.5
−1
−1
−1.5
−2
0
0.2
0.4
0.6
0.8
−2
0
1
0.2
t
0.4
0.6
0.8
1
t
Letter:
Letter:
1
1.5
1
0.5
0.5
y(t)
y(t)
0
0
−0.5
−0.5
−1
−1.5
−2
−1
−1
0
1
x(t)
Briefly justify your answer.
2
3
−1.5
−2
−1
0
x(t)
1
2
Science One Mathematics — Exam #2 — 19 April 2009 — p. 9 of 11
Q6
[10 marks]
The following (non-dimensional) system of equations is referred to as a “sniffer”:
X0 = S − X
R0 = aS − XR
where a and S are positive constants. S is referred to as a signal, X(t) is referred to as an
intermediate, and R(t) is referred to as a response.
(a) Sketch the phase plane. What are the coordinates of the steady state? Determine its
classification and stability (graphically if possible, analytically if necessary). [6 marks]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 10 of 11
(Q6 continued)
(b) Imagine you have control of the parameter S. For the first half of an experiment, you
hold S at a low level S1 , allowing the system to settle down. For the second half of the
experiment, you suddenly increase the value of S to a higher level S2 . Treating the steady
state reached in the first half of the experiment as an initial condition for the second half,
sketch X(t) and R(t) as functions of time. [3 marks]
(c) This system is called a sniffer by analogy with the human sense of smell where S can be
thought of as the concentration of substance being smelled. What feature(s) of that sense
does the experiment above demonstrate? [1 mark]
Science One Mathematics — Exam #2 — 19 April 2009 — p. 11 of 11
This page may be used for rough work. It will not be marked.
Q7