UNIVERSITY OF NEW SOUTH WALES
MATH2018 ENGINEERING MATHEMATICS 2D
WRITTEN TEST SAMPLE PAPER 1
This sheet must be filled in and stapled to the front of your answers
Student’s Surname Initials Student Number
Tutorial Code Tutor’s Surname
Time allowed: 50 minutes 7 questions Attempt all questions
Approved calculators are permitted
Leibniz Rule for Differentiation of Integrals
d
dx
∫ v
u
f(x, t)dt =
∫ v
u
f
x
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)
)
+ · · ·
1. [5 marks]
Consider the function
z = f(x, y) = x2y3 + 5x 6y.
(a) Find
f
x
and
f
y
.
(b) Verify that
2f
y x
=
2f
x y
.
2. [5 marks]
The volume V of a cone with radius r and perpendicular height h is given by
V =
1
3
pir2h. Determine the maximum percentage error in calculating V given
that r = 1 cm and h = 2 cm to the nearest millimetre.
3. [5 marks]
Use Leibniz’ theorem to find
d
dt
∫ t2
1
ln(1 + x6)dx.
4. [5 marks]
Find and classify the critical points of
f(x, y) = 2×3 9×2 + 12x+ 3y2 18y + 4.
Also give the function values at the critical points.
5. [5 marks]
A particle moves along a curve with parametric equations
x(t) = et, y(t) = sin(t), z(t) = 3t,
where t is time.
Determine the magnitude of its acceleration vector at t = 0.
6. [5 marks]
Suppose that the atmospheric pressure in a certain region of space is given by
φ(x, y, z) = y2z + 6x.
(a) Calculate φ at the point (1, 1, 2).
(b) Find the rate of change of the pressure with respect to distance at the
point (1, 1, 2) in the direction of the vector b = i + 2j 2k.
7. [5 marks]
A fluid’s velocity field in a turbine of a hydroelectric generator is given in the
plane by
F(x, y) = 2yi + 2xj.
Let C be the unit circle with centre at (0,0) parametrized by
r(t) = (cos t, sin t) from t = 0 to t = 2pi.
Calculate the circulation of F around C by computing∮
C
F · dr.
ANSWERS
1) a) 2xy3 + 5, 3x2y2 6, b) Proof 2) 12.5% 3) 2t ln(1 + t12)
4) Local min at (2, 3, 19), Saddle Point at (1, 3, 18) 5) 1
6) a)
64
1
, b) 4 7) 4pi.
