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) = 2x3 ? 9x2 + 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.