Name (print, with surname first):
Student number:
University of British Columbia
APRIL EXAM: Science One Mathematics
Date: April 19, 2013
Time: 12:00 noon to 2:30 p.m.
Number of pages: 12 (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
6
2
8
3
13
4
4
5
4
6
6
7
6
8
6
9
7
Total
60
You do not have to simplify your answers; in particular, mumerical answers can be left
unsimplified in a “calculator-ready” form. You should show your calculations and your
solutions should be clear. Good luck!
This page may be used for rough work. It will not be marked.
2
1. [6 marks] Find the volume of the solid S whose base is the region
n
√ o
(x, y)| 0 ≤ y ≤ (4 − x2 )1/4 , 0 ≤ x ≤ 2
(i.e. the region that is above the x-axis and below y = (4 − x2 )1/4 , for 0 ≤ x ≤
whose cross-sections perpendicular to the x-axis are squares.
3
√
2), and
2. Evaluate the following integrals.
Z
1
dx
(a) [4 marks]
x − x2
Z λ
nπ (b) [4 marks]
x sin
x dx, where λ > 0 is real and n > 0 is an integer
λ
0
4
3. Determine, with justification, whether each of the following series converges or diverges.
(a) [4 marks]
(b) [4 marks]
(c) [5 marks]
∞
X
en
10n−2
n=1
∞
X
100n n100
n=1
∞
X
n=2
n!
1
n(ln n)
5
4. [4 marks] Solve the differential equation
dy
cos x
= cos x −
y.
dx
sin x
6
5. [4 marks] Determine, with justification, whether the following integral converges or
diverges:
Z ∞
64 sin4 x
dx.
16 + x2
8
7
6. (a) [3 marks] Find the first four nonzero terms of the Maclaurin series for f (x) = cos(x3 ),
and determine the radius of convergence of the series.
1 − x6 − cos(x3 )
(b) [3 marks] Find lim
or determine that the limit does not exist.
x→0
x6
8
cn+1
= 0 for any constant c.
n→∞ (n + 1)!
7. (a) [2 marks] Explain why lim
(b) [4 marks] Use Taylor’s Remainder Theorem and the fact in part (a) to prove that
c
e =
∞
X
cn
n=1
n!
for any constant c. (Note that you are not being asked to prove that the Maclaurin
series converges; rather, you are being asked to prove that it converges to ec .)
9
8. Let y0 (t) = 1, −∞ < t < ∞ and for n = 1, 2, 3, . . . define
Z t
yn−1 (s) ds.
yn (t) = 1 +
0
Assume that limn→∞ yn (t) = y∞ (t) exists and is a continuous function for t belonging to
some open interval containing 0.
(a) [2 marks] Show that y∞ (t) solves the integral equation
Z t
y(t) = 1 +
y(s) ds.
0
(b) [4 marks] Find y1 (t), y2 (t), y3 (t) and then find y∞ (t) in terms of elementary functions.
Give justification for y∞ (t).
10
9. (a) [3 marks] It is well known that in all exams involving mathematics, the examination
room initially contains 100 L of aura mathematica, or “math air”. At the beginning
of this exam, 5 kg of materia mala, or “anxiety”, was thoroughly mixed into the math
air. Since the beginning of the exam, we have been breathing into the room 5 L/min
of pure math air. We have also been filtering out 5 L/min of the thoroughly combined
mix of anxiety and math air. Come up with an expression describing the amount A
of anxiety in the room as a function of the minutes t since the start of the exam.
(b) [3 marks] The physics exam started with the same initial conditions of 5 kg of anxiety
mixed into 100 L of math air. In an energetic but naive effort, James and Mark
breathed into the room 7 L/min of pure math air, and filtered out 5 L/min of the
thoroughly combined mix. Come up with an expression describing the amount of
anxiety in the physics exam room as a function of minutes since the start of that
exam.
(c) [1 mark] Make a physical argument for why the mathematicians’ strategy is a superior
one. Then speculate briefly on the psychological reasons for the physicists’ error.
11
This page may be used for rough work. It will not be marked.
12