The University of the West Indies
St. Augustine
Examinations of July 2012
Code and Name of Course: MATH2230 Engineering Mathematics II
Paper:
Date and Time:
Duration: 3 hours
INSTRUCTIONS TO CANDIDATES: This paper has 6 pages and 7 questions
ATTEMPT ANY FIVE (5) QUESTIONS
Only non-programmable calculators are allowed
A table of Laplace transforms is provided
c
The
University of the West Indies
Course Code MATH2230
2011/2012
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner
and/or the External Examiner. Where the examination does not require a University Examiner, the form
must be signed by the First and Second Examiners. Completed forms should be handed to the Assistant
Registrar (Examinations). The EXTERNAL EXAMINER is requested to sign the question paper and return
it with comments, if any, (on a separate sheet), to the Assistant Registrar (Examinations).
...............................................
First Examiner
.............................................
University Examiner
Date: 20...../......../..........
Date: 20...../......../..........
...............................................
Second Examiner
.............................................
External Examiner(where applicable)
Date: 20...../......../..........
Date: 20...../......../..........
Page 2
(1) (a) Find the length of the parametric curve
r(t) = 1 + 3t2 i + 4 + 2t3 j
0 ≤ t ≤ 1.
[5]
(b) Find an equation of the tangent plane to the surface
z = ex
2 −y 2
at the point (1, −1, 1) .
[5]
(c) Let u(x, y), v(x, y) be differentiable functions and let a, b be constants. Show that
(i) O (au + bv) = aOu + bOv
[2]
(ii) O (uv) = uOv + vOu
[3]
(2) (a) Consider the line integral
Z
C
(ex sin y i + ex cos y j) · dr
where C is the straight line segment from (0, 0) to (1, π/2).
R
(i) Give a condition for a line integral C F · dr to be independent of path.
[1]
(ii) Show that the given line integral is independent of path and hence evaluate this
line integral.
[4]
(b) Consider the following double integral
Z 1 Z √1−x2 p
1 − y 2 dy dx.
0
0
(i) Sketch the region of integration D of the above integral.
[1]
(ii) Express the region D as a Type II region and hence evaluate the given integral
by reversing the order of integration.
[4]
(c) Use Green’s theorem to evaluate the line integral
Z
√ √
y i + x j · dr
Question 2(c)
is continued on
the next page
C
c
The
University of the West Indies
Course Code MATH2230
2011/2012
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner
and/or the External Examiner. Where the examination does not require a University Examiner, the form
must be signed by the First and Second Examiners. Completed forms should be handed to the Assistant
Registrar (Examinations). The EXTERNAL EXAMINER is requested to sign the question paper and return
it with comments, if any, (on a separate sheet), to the Assistant Registrar (Examinations).
...............................................
First Examiner
.............................................
University Examiner
Date: 20...../......../..........
Date: 20...../......../..........
...............................................
Second Examiner
.............................................
External Examiner(where applicable)
Date: 20...../......../..........
Date: 20...../......../..........
Page 3
where C is the positively oriented, piecewise smooth, simple closed curve that encloses
2
the region D that is bounded by the lines y = 0, x = 2 and the parabola y = x /2. [5]
(3) (a) Use p
a surface integral to show that the surface area A of the part of the hemisphere
z = a2 − x2 − y 2 that lies between the horizontal planes z = h1 , z = h2
(where 0 ≤ h1 < h2 ≤ a) is given by A = 2πa(h2 − h1 ).
[8]
(b) Evaluate the surface integral (do not use the Divergence Theorem)
ZZ
x2 i + xyj + zk · n dS
S
where S is the surface of the solid that is bounded by the paraboloid z = 4 − x2 − y 2
and the plane z = 0. Assume that S has an outward orientation.
[7]
(4) (a) Use the Laplace transform to solve
y 00 + 2y 0 + 2y = e−t + 5δ(t − 2)
with initial conditions y(0) = 0, y 0 (0) = 1.
(b) Solve the integral equation
y(t) − t = e −
t
c
The
University of the West Indies
Z
t
0
[9]
y(β) cosh(t − β)dβ
[6]
Course Code MATH2230
2011/2012
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner
and/or the External Examiner. Where the examination does not require a University Examiner, the form
must be signed by the First and Second Examiners. Completed forms should be handed to the Assistant
Registrar (Examinations). The EXTERNAL EXAMINER is requested to sign the question paper and return
it with comments, if any, (on a separate sheet), to the Assistant Registrar (Examinations).
...............................................
First Examiner
.............................................
University Examiner
Date: 20...../......../..........
Date: 20...../......../..........
...............................................
Second Examiner
.............................................
External Examiner(where applicable)
Date: 20...../......../..........
Date: 20...../......../..........
Page 4
(5) Consider the piecewise function
f (x) =
(
0 0≤x<2
1 2≤x≤4
that is defined on the interval 0 ≤ x ≤ 4.
(a) Show that the half-range cosine expansion of f (x) is
∞ X
(−1)k
(2k + 1)πx
1 2
cos
−
2 π
2k + 1
4
k=0
and that the half-range sine expansion is

 ∞
∞ X 1 (2k + 1)πx X
2
1
(2k + 1)πx 
sin
−
sin
π
2k + 1
4
2k + 1
2
k=0
k=0
[11]
(b) Explain why the Fourier sine series of part (a) converges to a periodic function; denote
this function as g(x). Sketch the function g(x) for at least two periods.
[4]
(6) A thin metal rod of length 20 cm has an initial uniform temperature of 25 ◦ K. The ends of
the rod that correspond to x = 0 and x = 20 are then kept at temperatures 0 ◦ K and 60 ◦ K
respectively for all time t > 0. The temperature u(x, t) of the rod satisfies
∂2u
∂u
= k2 2
∂t
∂x
where k is constant depending on the physical properties of the rod.
(a) Show that us (x) = 3x is a steady state solution of the above heat equation that satisfies
the given boundary conditions.
[2]
(b) Consider the substitution u(x, t) = v(x, t) + us (x). Show that v(x, t) satisfies a heat
equation with homogeneous boundary conditions. Determine the corresponding initial
condition for this heat equation problem for v(x, t).
[4]
(c) Hence show that the temperature of the rod is given in Kelvin by
∞ X
nπx 50 + 70(−1)n
−(knπ)2 t/400
u(x, t) = 3x +
e
sin
nπ
20
n=1
c
The
University of the West Indies
Course Code MATH2230
[9]
2011/2012
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner
and/or the External Examiner. Where the examination does not require a University Examiner, the form
must be signed by the First and Second Examiners. Completed forms should be handed to the Assistant
Registrar (Examinations). The EXTERNAL EXAMINER is requested to sign the question paper and return
it with comments, if any, (on a separate sheet), to the Assistant Registrar (Examinations).
...............................................
First Examiner
.............................................
University Examiner
Date: 20...../......../..........
Date: 20...../......../..........
...............................................
Second Examiner
.............................................
External Examiner(where applicable)
Date: 20...../......../..........
Date: 20...../......../..........
Page 5
(7) Use Laplace’s equation in polar coordinates
∂ 2 u 1 ∂u
1 ∂2u
+
=0
+
∂r 2
r ∂r
r2 ∂θ 2
to find the steady state temperature of a thin metal disc of radius a > 0 which has boundary
temperature specified by
u(a, θ) = 1 + cos(2θ)
− π < θ ≤ π.
[15]
END OF PAPER
c
The
University of the West Indies
Course Code MATH2230
2011/2012
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner
and/or the External Examiner. Where the examination does not require a University Examiner, the form
must be signed by the First and Second Examiners. Completed forms should be handed to the Assistant
Registrar (Examinations). The EXTERNAL EXAMINER is requested to sign the question paper and return
it with comments, if any, (on a separate sheet), to the Assistant Registrar (Examinations).
...............................................
First Examiner
.............................................
University Examiner
Date: 20...../......../..........
Date: 20...../......../..........
...............................................
Second Examiner
.............................................
External Examiner(where applicable)
Date: 20...../......../..........
Date: 20...../......../..........
Page 6
y(t)
L[y(t)]
n!
tn
sn+1
(n = 0, 1, 2, . . .)
Γ(x + 1)
sx+1
tx
eat
eat y(t)
cos(kt)
sin(kt)
cosh(kt)
sinh(kt)
H(t − t0 )
(x > −1)
1
s−a
Y (s − a)
s
2
s + k2
k
2
s + k2
s
2
s − k2
k
2
s − k2
e−st0
s
H(t − t0 )y(t − t0 )
e−st0 Y (s)
δ(t − t0 )
e−st0
y 0 (t)
sY (s) − y(0)
y 00 (t)
s2 Y (s) − sy(0) − y 0 (0)
Z
t
0
f (t − β)g(β) dβ
F (s)G(s)
Table of Laplace transforms
c
The
University of the West Indies
Course Code MATH2230
2011/2012
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University Examiner
and/or the External Examiner. Where the examination does not require a University Examiner, the form
must be signed by the First and Second Examiners. Completed forms should be handed to the Assistant
Registrar (Examinations). The EXTERNAL EXAMINER is requested to sign the question paper and return
it with comments, if any, (on a separate sheet), to the Assistant Registrar (Examinations).
...............................................
First Examiner
.............................................
University Examiner
Date: 20...../......../..........
Date: 20...../......../..........
...............................................
Second Examiner
.............................................
External Examiner(where applicable)
Date: 20...../......../..........
Date: 20...../......../..........