THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
JUNE 2017
MATH2018 / MATH2019
ENGINEERING MATHEMATICS 2D/E
(1) TIME ALLOWED – 2 hours
(2) TOTAL NUMBER OF QUESTIONS – 4
(4) THE QUESTIONS ARE OF EQUAL VALUE
(5) ANSWER EACH QUESTION IN A SEPARATE BOOK
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
(7) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER
MAY BE USED
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
JUNE 2017 MATH2018 / MATH2019 Page 2
TABLE OF LAPLACE TRANSFORMS AND THEOREMS
g(t) is a function defined for all t ≥ 0, and whose Laplace transform
G(s) = L(g(t)) =
∫ ∞
0
e−stg(t)dt
exists. The Heaviside step function u is defined to be
u(t− a) =

0 for t < a
1
2
for t = a
1 for t > a
g(t) G(s) = L[g(t)]
1
1
s
t
1
s2
tν , ν > −1 ν!
sν+1
e−αt
1
s+ α
sin ωt
ω
s2 + ω2
cos ωt
s
s2 + ω2
u(t− a) e
−as
s
f ′(t) sF (s)− f(0)
f ′′(t) s2F (s)− sf(0)− f ′(0)
e−αtf(t) F (s+ α)
f(t− a)u(t− a) e−asF (s)
tf(t) −F ′(s)
Please see over . . .
JUNE 2017 MATH2018 / MATH2019 Page 3
FOURIER SERIES
If f(x) has period p = 2L, then
f(x) = a0 +
∞∑
n=1
(
an cos
(npi
L
x
)
+ bn sin
(npi
L
x
))
where
a0 =
1
2L
∫ L
−L
f(x)dx
an =
1
L
∫ L
−L
f(x) cos
(npi
L
x
)
dx
bn =
1
L
∫ L
−L
f(x) sin
(npi
L
x
)
dx
LEIBNIZ’ RULE
d
dx
∫ v
u
f(x, t)dt =
∫ v
u
∂f
∂x
(x, t)dt+ f(x, v)
dv
dx
− f(x, u)du
dx
MULTIVARIABLE TAYLOR SERIES
f(x, y) = f(a, b) + (x− a)∂f
∂x
(a, b) + (y − b)∂f
∂y
(a, b) +
+
1
2!
(
(x− a)2∂
2f
∂x2
(a, b) + 2(x− a)(y − b) ∂
2f
∂x∂y
(a, b) + (y − b)2∂
2f
∂y2
(a, b)
)
+ · · ·
Please see over . . .
JUNE 2017 MATH2018 / MATH2019 Page 4
SOME BASIC INTEGRALS∫
xndx =
xn+1
n+ 1
+ C for n 6= −1∫
1
x
dx = ln |x|+ C∫
ekxdx =
ekx
k
+ C∫
axdx =
1
ln a
ax + C for a 6= 1∫
sin kx dx = −cos kx
k
+ C∫
cos kx dx =
sin kx
k
+ C∫
sec2 kx dx =
tan kx
k
+ C∫
cosec2kx dx = −1
k
cot kx+ C∫
tan kx dx =
ln | sec kx|
k
+ C∫
sec kx dx =
1
k
(ln | sec kx+ tan kx|) + C∫
1
a2 + x2
dx =
1
a
tan−1
(x
a
)
+ C∫
1√
a2 − x2dx = sin
−1
(x
a
)
+ C∫
1√
x2 + a2
dx = sinh−1
(x
a
)
+ C∫
1√
x2 − a2dx = cosh
−1
(x
a
)
+ C∫ pi
2
0
sinn x dx =
n− 1
n
∫ pi
2
0
sinn−2 x dx∫ pi
2
0
cosn x dx =
n− 1
n
∫ pi
2
0
cosn−2 x dx
Please see over . . .
JUNE 2017 MATH2018 / MATH2019 Page 5
Answer question 1 in a separate book
1. a) A metal cylinder contains a volume of liquid given by
V = pir2h,
where r is the radius of the cylinder and h is the height of the cylin-
der. Small variations in the manufacturing process can result in errors
in the cylinder radius of 1% and the cylinder height of 2%. What is the
maximum percentage error in the volume of the cylinder?
b) Suppose the temperature in a region of space is given by the scalar field
T (x, y, z) = x4 + y4 + z4.
i) Calculate the gradient of T at the point P (1, 1, 1).
ii) Find the rate of change of temperature with respect to distance at
the point P (1, 1, 1) in the direction i + j.
iii) Write down the equation of the tangent plane to the surface T (x, y, z) =
3 at the point P (1, 1, 1).
c) You are given the following integral,∫ ∞
0

xe−tx dx =

pi
2t3/2
.
Use Leibniz’ rule to evaluate∫ ∞
0
x3/2e−tx dx.
d) You are given the function f(x) = ax2 + y2 − 2y, where a is a constant
not equal to zero. This function has one critical point.
i) Find the critical point of the function.
ii) Find the value of the function at the critical point.
iii) State whether the critical point can be a maximum, a minimum, or
a saddle point. Write down the values of a (if they exist) for each
case.
Please see over . . .
JUNE 2017 MATH2018 / MATH2019 Page 6
Answer question 2 in a separate book
2. a) A charged particle moves in an electric field given by
F(x, y, z) = 3yi− 3xj.
Let C denote the path taken by the particle travelling anticlockwise
around the unit circle, starting at (1, 0) and ending at (0, 1).
i) Write down a vector function r(θ) that describes the path C and give
the values of θ at the start and the end of the path.
ii) Calculate the work done on the particle as it moves along the path
C by evaluating the line integral∫
C
F · dr.
b) Use the substitution v =
y
x
to solve the ordinary differential equation
x2
dy
dx
= 2x2 + xy + 2y2.
c) Use the method of undetermined coefficients to solve the second order
differential equation
y′′ + 9y = 6 cos 3t+ 5et.
d) Use double integration to find the area bounded by y = x and y = x2.
Please see over . . .
JUNE 2017 MATH2018 / MATH2019 Page 7
Answer question 3 in a separate book
3. a) Find:
i) L{t u(t− 2)}.
ii) L−1
{
3s
s2 − 2s+ 10
}
.
b) The function g(t) is given by
g(t) =
{
1 for 0 ≤ t < 1,
e−t+1 for t ≥ 1.
i) Sketch the function g(t) for 0 ≤ t ≤ 3.
ii) Write g(t) in terms of the Heaviside step function u(t− a).
iii) Hence, or otherwise, show that the Laplace transform of g(t) is
L{g(t)} = 1
s
− e−s
(
1
s
− 1
s+ 1
)
.
c) Use the Laplace transform method to solve the initial value problem
y′′ − y′ = g(t), y(0) = −1, y′(0) = 0,
where g(t) is the function from question 3b).
d) Consider the set of differential equations
dx
dt
= −x+ y,
dy
dt
= x− y,
with initial conditions x(0) = 1, y(0) = 0.
i) Express this set of differential equations in the form
dx
dt
= Ax, where x =
(
x
y
)
.
and find the eigenvalues and eigenvectors of the matrix A.
ii) Hence, or otherwise, write down the solution for the problem using
the initial conditions.
Please see over . . .
JUNE 2017 MATH2018 / MATH2019 Page 8
Answer question 4 in a separate book
4. a) Define the function f on the interval [0, pi) as
f(x) =
x
2
, on 0 ≤ x < pi.
i) Sketch the odd periodic extension of f on the interval −4pi ≤ x ≤ 4pi.
ii) Show that the Fourier sine series of f is given by
f(x) =
∞∑
n=1
1
n
(−1)n+1 sin (nx) .
iii) To what value does the Fourier sine series of f converge at x = pi?
b) Consider the differential equation
d2y
dx
+ 5y = f(x),
where f(x) is the function defined in part (a).
i) Find the homogeneous solution to the differential equation. What is
the natural (or fundamental) frequency of the system?
ii) Find the particular solution of the differential equation in the form
of a Fourier series.
iii) Write out the first four non-zero terms of this particular solution and
comment on which term dominates the solution.
c) Consider the one-dimensional wave equation,
∂2u
∂t2
= 9
∂2u
∂x2
,
where u(x, t) is the displacement at position x and time t. D’Alembert’s
solution to this wave equation is
u(x, t) = φ(x+ 3t) + ψ(x− 3t),
for arbitrary functions φ and ψ. If the initial displacement of the wave is
u(x, 0) = g(x) and the initial velocity is ut(x, 0) = 0, prove that
u(x, t) =
1
2
[g(x+ 3t) + g(x− 3t)] . 