THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
NOVEMBER 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.
NOVEMBER 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 . . .
NOVEMBER 2017 MATH2018 / MATH2019 Page 3
FOURIER SERIES
If f(x) has period T = 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(x)
u(x)
f(x, t)dt =
∫ v(x)
u(x)
∂f
∂x
(x, t)dt+ f ((x, v(x))
dv
dx
(x)− f (x, u(x)) du
dx
(x)
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 . . .
NOVEMBER 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∫
cosec2(kx) 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 . . .
NOVEMBER 2017 MATH2018 / MATH2019 Page 5
Answer question 1 in a separate book
1. a) i) Calculate the Taylor series expansion of the function f(x, y) = ln (x+ y)
ii) Use your solution to find an approximate value for ln(1.1).
b) Consider the scalar field
φ(x, y, z) = x2 − y2 + z2.
i) Calculate the gradient of φ at the point P (1, 1, 0).
ii) Find the direction and magnitude of the maximum rate of increase
of φ at P (1, 1, 0).
iii) Write down any non-zero vector b that is perpendicular to the gra-
dient of φ at the point P (1, 1, 0).
iv) What is the rate of change of φ at the point P (1, 1, 0) in the direction
b found in part iii)?
c) Find the volume of the solid bounded above by the surface z = 1−x2−y2
and below by the plane z = 0.
d) The temperature in a region of space is given by T (x, y) = x2 + y2. A
sensor measures temperature along a curve given by the equation xy = 1.
i) Why does the sensor measure no maximum value of the temperature?
ii) Use the method of Lagrange multipliers to find the minimum tem-
perature measured by the sensor.
e) You are given the following integral,∫ a
0
1
(x2 + a2)1/2
dx = sinh−1 (1) .
Use Leibniz’ rule to evaluate∫ a
0
1
(x2 + a2)3/2
dx.
Please see over . . .
NOVEMBER 2017 MATH2018 / MATH2019 Page 6
Answer question 2 in a separate book
2. a) Use the method of undetermined coefficients to solve the second order
ordinary differential equation
y′′ − 2y′ − 8y = 8 + 5et cos t.
b) Consider the double integral
I =
∫ 2
0
∫ x
0
x
x2 + y2
dy dx.
i) Sketch the region of integration.
ii) Evaluate I by first changing to polar coordinates.
c) A quadratic curve is given by the equation 2x2 + 4xy − y2 = 1.
i) Express the curve in the form
xTAx = 1, where x =
(
x
y
)
,
and find the eigenvalues and eigenvectors of the matrix A.
ii) Hence, or otherwise, find the closest distance from the curve to the
origin. Write down the coordinates (x, y) of the points on the curve
closest to the origin.
d) A vector field is given by
F(x, y, z) = sinx sin y k.
i) Calculate ∇× F.
ii) Calculate ∇× (∇× F).
iii) Hence, or otherwise, evaluate ∇× (∇× (∇× (∇× F))).
Please see over . . .
NOVEMBER 2017 MATH2018 / MATH2019 Page 7
Answer question 3 in a separate book
3. a) The Laplace transform of a function f(t) is defined for t ≥ 0 by
F (s) = L{f(t)} =
∫ ∞
0
f(t)e−st dt.
Prove directly from the above definition that
L{u(t− a)} = e
−as
s
where a > 0 and u(t− a) is the Heaviside function.
b) Find:
i) L{et u(t− 3)}.
ii) L−1
{
s+ 2
s2 + 2s+ 5
}
.
c) The function g(t) is given by
g(t) =
{
sin pit for 0 ≤ t < 1,
0 for t ≥ 1.
i) Sketch the function g(t) for 0 ≤ t ≤ 2.
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)} = pi
s2 + pi2
(
1 + e−s
)
.
[Hint: You can use sin(A+ pi) = − sinA.]
d) Use the Laplace transform method to solve the initial value problem
y′′ − y′ − 2y = 6u(t− 1), y(0) = 1, y′(0) = 2.
Please see over . . .
NOVEMBER 2017 MATH2018 / MATH2019 Page 8
Answer question 4 in a separate book
4. a) Define the piecewise continuous function f by
f(x) =
 2, 0 ≤ x <
pi
2
0,
pi
2
≤ x < pi.
i) Sketch the even periodic extension of f(x) for −3pi ≤ x ≤ 3pi.
ii) Show that the Fourier cosine series of f(x) is given by
f(x) = 1 +
∞∑
m=1
4
(2m− 1)pi (−1)
m+1 cos(2m− 1)x.
iii) To what value does the Fourier cosine series in (ii) converge at x =
pi
2
?
b) The temperature in a conducting metal bar of length pi is described by
the heat equation
∂u
∂t
= 2
∂2u
∂x2
,
where u(x, t) is the temperature at position x and time t. The ends of
the metal bar are insulated so that
∂u
∂x
(0, t) =
∂u
∂x
(pi, t) = 0, for all t > 0.
You are additionally given that the initial temperature distribution in the
metal bar is
u(x, 0) = U0(x).
i) Assuming a solution of the form u(x, t) = F (x)G(t) show that
G′(t)
2G(t)
=
F ′′(x)
F (x)
= k
for some constant k.
ii) Write down the boundary conditions for F (x).
iii) Separately consider the cases of k > 0, k = 0, and k < 0. Apply the
boundary conditions to find all non-trivial solutions for F (x).
iv) For each non-trivial solution for F (x) found in (iii), find the corre-
sponding solutions for G(t). Hence, write down the general solution
for u(x, t).
v) The initial temperature distribution in the metal bar is
U0(x) = f(x),
where f(x) is the function considered in part (a). Find the solution 