SSCM 1803
QUESTION 1 (20 MARKS)
(a) Given z =
p
25 − x2 − y 2.
(i) Find the domain and range of the function.
(2 marks)
(ii) Sketch the level curves projected on the xy-plane for z = 1, z = 3 and
z = 5.
(3 marks)
(iii) Sketch the surface of the function.
(2 marks)
∂z
and
(b) z = f(x, y) is defined implicitly. If 6x + 2yexz + (xy)3 + 4z = 0, find
∂x
∂z
.
(6 marks)
∂y
(c) If w = 7x + cos 2y + 4z 2 where x = (t + 2s) − ln 4r, y = rst2 and z = te3rs, find
∂w
∂w
and
.
(7 marks)
∂r
∂t
QUESTION 2 (20 MARKS)
(a) Find the equation for the tangent plane to the graph of f(x, y) = 3x4 + 2y 4 at
the point (1, 2, 35).
(5 marks)
(b) The flow of nanofluid in a tube is given by
πP R4
N=
,
8kL
where L is the length of the tube, R is the radius, P is the difference in pressure
berween the two ends of the tube which is a constant and k is constant. Find
the maximum percentage error in calculating the flow of the nanofluid N if the
maximum error in measuring in L and R are 1% and 2%, respectively.
(6 marks)
(c) Given that f(x, y) = 2x4 + y 2 − 8xy.
(i) Find all critical points of f(x, y).
(4 marks)
(ii) Use the second derivatives test to each of the critical points to determine
whether it is a local maximum, a local minimum or a saddle point.
(5 marks)
2
SSCM 1803
QUESTION 3 (20 MARKS)
(a) Consider the integral
Z
1
0
Z
4−2x
dydx.
2
Sketch the region of integration and write an equivalent double integral with
the order of integration reversed. Hence evaluate the integral.
(5 marks)
(b) Find the area of the region enclosed by the parabola y = x2 and the line
y = x + 2.
(5 marks)
(c) Use polar coordinate to evaluate
Z Z
2ydA
R
where R is the region in the first quadrant bounded above by (x − 1)2 + y 2 = 1
and below by the line y = x.
(10 marks)
QUESTION 4 (20 MARKS)
(a) Find the volume of the solid bounded by the coordinate planes and the plane
3x + 6y + 4z = 12.
(7 marks )
(b) Transform the integral
Z
0
2
Z
0
√
4−x2
Z
2
√
(x2 + y 2 + z 2 )dzdydx
x2 +y2
into the following coordinates systems
(i) cylindrical,
(5 marks)
(ii) spherical.
(4 marks)
Hence, evaluate the integral using only one of the coordinates systems.
(4 marks)
3
SSCM 1803
QUESTION 5 (20 MARKS)
(a) A lamina of density δ(x, y) = x2 occupies a region R bounded by the parabola
y = 2 − x2 and the line y = x. Find the moment of mass of the lamina about
the x-axis.
(6 marks )
(b) Find the mass of p
a solid with density δ(x, y, z) = x2 +py 2 + z 2 and bounded
by the cone z = x2 + y 2 and two hemispheres z = 1 − x2 + y 2 and z =
p
4 − x2 + y 2 .
(7 marks )
(c) Find the moment of inertia about the z-axis, for the solid that is bounded
below by the paraboloid z = x2 + y 2 and above by the plane z = 1, if the
density is 1.
(7 marks )
4