Name (print):
ID number:
Science One number:
University of British Columbia
MIDTERM TEST 2: MATHEMATICS
Date: February 22, 2011
Time: 8:30 a.m. to 9:30 a.m.
Number of pages: 6 (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.
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;
For examiners’ use only
Question
Mark
Possible marks
1
11
• Speaking or communicating with other candidates;
2
6
• Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
3
7
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.
4
5
5
7
Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Total
36
1. Determine which of the following series converge.
X 2n + 3 n
(a) [3 marks]
3n + 2
n≥1
(b) [4 marks]
X
n≥1
(c) [4 marks]
an , where a1 = 1 and an =
1
2
an−1 +
3
2
for n ≥ 2
X (x + 2)n n
, at the left endpoint of its interval of convergence
3n+1
n≥0
2
X
an+1
an converges.
∈ [0, 1). Prove that
n→∞ an
2. [6 marks] Let {an } be a positive sequence satisfying lim
n≥1
3
3. The differential equation
v 0 = −kv + A sin(ωt)
represents the motion of a piston embedded in thick fluid (drag coefficient mk) and driven by a motor
with frequency ω and force-amplitude mA where A, k and ω are all positive constants. The velocity
of the piston is given by the function v(t). To solve this equation, rewrite it as
ekt v 0 + kekt v = Aekt sin(ωt).
(a) [4 marks] Take antiderivatives of both sides with respect to t and continue to solve for the
function v(t).
(b) [2 marks] If the piston starts at rest (v = 0) at time t = 0, what is the appropriate solution v(t)?
(c) [1 mark] Give an approximate expression for v(t) in the case that k is much bigger than ω.
4
4. [5 marks] The Fresnel integral
Z
f (x) =
x
sin t2 dt
0
appears in descriptions of certain diffraction phenomena. Derive an expression that will allow you to
calculate f (1) to any desired degree of accuracy.
5
5. A total charge of Q is distributed uniformly on a thin rod of length 2L along the y-axis, centered at
the origin, giving the rod a uniform charge density of ρ = Q/(2L). The electric field at a point (a, 0)
on the x-axis is therefore horizontal (by symmetry) and has an x-component given by
Z
L
Ex (a) = kρa
−L
dy
.
(a2 + y 2 )3/2
(a) [4 marks] Simplify this expression by carrying out the integration.
(b) [3 marks] What happens to the electric field at (a, 0) as the size of the rod goes to ∞ (maintaining
the same charge density ρ)?
6