SAMPLE EXAMINATION 2013

DUBLIN CITY UNIVERSITY
SAMPLE EXAMINATION 2013
MODULE:
MS205
Calculus of Several Variables
QUAL:
B.Sc. in Actuarial Mathematics
Common Entry into Actuarial, Financial
and Mathematical Sciences
YEAR OF STUDY:
2
EXAMINERS:
F. Surname
J. Quinn (ext. 8901)
TIME ALLOWED:
2 hours
INSTRUCTIONS:
Answer all parts of Section A and any two questions
in Section B.
Requirements for this paper
Please mark (X) as appropriate
X
Log Tables
Graph Paper
Dictionaries
Statistical Tables
Thermodynamic Tables
Actuarial Tables
MCQ only – Do not publish on Web
PLEASE DO NOT TURN OVER THIS PAGE UNTIL YOU ARE INSTRUCTED TO DO SO
The use of programmable or text storing calculators is expressly forbidden.
Please note that where a candidate answers more than the required number of questions, the examiner will mark all
questions attempted and then select the highest scoring ones.
MS205
Academic Year 2012/2013
PAGE 1 OF 4
Sample Exam
Section A
QUESTION 1
(a) Find the equation of the plane perpendicular to the path
~x : R → R3 : t 7→ (sin(2t), cos(3t), et )
at t = π.
[5 marks]
(b) Determine the equation of √
the tangent planepto the surface in R3 with equation
2
2
2
2
cos(x y ) + sin(x z ) = 1/ 2 at the point ( π/4, 2, 1).
[6 marks]
(c) Find the directional derivative of
2
f : R3 → R : (x, y, z) 7→ xz 2 e(x+z ) + yey+z
(1)
at (1, 1, 2) in the direction of ~h = (2, 0, 5).
[5 marks]
(d) Show that the limit of
xy(x+y)
x3 +xy2
as (x, y) → 0 does not exist. Then explain why
(
f (x, y) =
xy(x+y)
,
x3 +xy2
α
(x, y) 6= (0, 0)
,
(x, y) = (0, 0)
is continous for all (x, y) 6= (0, 0), whatever α ∈ R is chosen
[8 marks]
(e) Let f (x, y) be a given differentiable function. Consider the function F (u, v) = f (sin(uv), cos(uv))
i.e. the function f under the substitution x = sin(uv), y = cos(uv). Prove that
1 ∂F
1 ∂F
∂f
∂f
=
=y
−x .
v ∂u
u ∂v
∂x
∂y
[7 marks]
(f) Find the length of the following path to one decimal place
√
~x : [0, 5] 7→ R3 : t 7→ 41 15t2 , 41 t2 ,
2 3
3t
.
[5 marks]
(g) Sketch the region of integration for the integral
Z
1
e3
Z
y
e3
1
1−x
ln(x)
y
2
dxdy,
and evaluate the integral by changing the order of integration.
[5 marks]
3
~
(h) Let F~ : R3 7→
(x2 yez , y 2 − cos(z), z 2 − yez ). Evaluate
R R be the vector field given by F (x, y, z) =
the integral C F~ · d~s, where C is the path ~σ (t) = (−1, e2t , t) for 0 6 t 6 π/2.
[8 marks]
Section B
QUESTION 2
(a) Consider the function
f : R2 7→ R : (x, y) 7→


 2−
x3 −y 3
x2 +y 2


2
(x, y) 6= (0, 0)
.
(x, y) = (0, 0)
(i) Write down the general definition of differentiability for a function of n variables.
(ii) Investigate if f is differentiable at (0, 0).
[15 marks]
(b) Find the maximum and minimum of the function f (x, y) = x2 + y 2 over the ellipse 4x2 + 2y 2 = 1.
[10 marks]
QUESTION 3
(a) Compute the second order Taylor approximation of f (x, y) = x2 y 2
(1, 1).
p
1 + x2 + y 2 around the point
[10 marks]
(b) By transforming to spherical coordinates (ρ, θ, φ), where x = ρ cos(θ) sin(φ), y = ρ sin(θ) sin(φ)
and z = ρ cos(φ), evaluate the integral
ZZZ p
z x2 + y 2 dxdydz,
D
where D = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 6 4, x, y, z > 0}.
[12 marks]
QUESTION 4
Consider the region D in the first quadrant bounded by the curves y = 0, y = cos(x) and y = sin(x)
for 0 6 x 6 π/4 and its perimeter C, orientated in the counterclockwise sense
R
(a) Evaluate the line integral F~ · d~s where F is the vector field
C
F~ : R2 7→ R2 : (x, y) 7→ (xy, xy − x).
[16 marks]
(b) If P (x, y) and Q(x, y) are two differentiable functions defined over the region D with perimeter C
defined in part (a), then write down the double integral for which Green’s Theorem states that
Z P
· d~s
Q
C
is equivalent to in terms of D,
∂Q
∂x
and
∂P
∂y .
[4 marks]
(c) Use Green’s Theorem, to convert the line integral from part (a) into the double integral you stated
in part (b) and evaluate the double integral.
[5 marks]