THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
NOVEMBER 2018
MATH2018 / MATH2019
ENGINEERING MATHEMATICS 2D/E
(1) TIME ALLOWED - 2 hours
(2) TOTAL N UMBER OF QUESTIONS - 4
(3) ANSWER ALL QUESTIONS
(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 MAYBE 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 2018 MATH2018 / MATH2019
TABLE OF LAPLACE TRANSFORMS AND THEOREMS
g(t) is a function defined for all t ::C: 0, and whose Laplace transform
G(s) = L(g(t)) = 1= e-''g(t)dt
exists. The Heaviside step function u is defined to be
{o u(t-a) = t
g( t)
1
t
t",v > -1
e-at
sin wt
I
cos wt
u(t -a)
f'(t)
f"(t)
e-at J(t)
J(t -a)u(t -a)
tj(t)
fort< a
fort= a
fort> a
G(s) = L[g(t)]
1
s
l
s2
v!
8v+l
1 - -s+a
w
s2 + w2
s
s2 + w2
e-a
s
s
sF(s) - f(O)
s2F(s) - sf(O) � f'(O)
F(s + a)
e-a, F(s)
-F'(s)
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NOVEMBER 2018 MATH2018 / MATH2019
FOURIER SERIES
If f(x) has period T = 2 L, then
where
lL 1
L f(x)dx 2 -L
± 1: f(x) cos("; x) dx
1 1L (mr )
L -L
f(x) sin L x d.x
LEIBNIZ RULE FOR DIFFERENTIATING INTEGRALS
d j" j" of dv du dx u f(x, t)dt = u ox dt + f(x, v) dx - f(x, v.) dx.
f(x,y)
MULTIVARIABLE TAYLOR SERIES
of of f ( a, b) + ( X - a)
OX
(
a, b) + (y - b) oy ( a, b)
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+- (x - a)2-(a, b) + 2(x - a)(y - b)- -(a, b) + (y - b)2-(a, b) + .. · 1( o2f o2f o2f) 2! ox2 oxoy oy2
VARIATION OF PARAMETERS
Suppose that the second order differential equation
y" + p(x)y' + q(x)y = f(x)
has homogeneous solution Yh = Ay1 (x) + By2 (x). Then a particular solution is given by
J Y2(x)f(x) J y,(x)f(x) yp(x) = -y1(x) W(x) dx + y2 (x) W(x) dx
( y,(x) Y2(x) ) where W(x) = dot y\(x) y;(x)
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NOVEMBER 2018 MATH2018 / MATH2019
SOME BASIC INTEGRALS
J xn+l xndx = -- + C for n # -l n+l
J �dx = In lxl + C
J ekxdx = ekx + C k
Jaxdx= -11 a"+C forac/ 1 na
J coskx sin kxdx = --k-" - + C
J k d. sinkx cos "X X = ---;;- + C
J ? tankx sec kx dx = - - + C k
J cosoc2kx dx = -i cot kx + C
J k d In I sec kxl C tan x x = + k
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J sec kx dx = ion I seckx + tan kxl) + C
J 2 1 2dx = ! tai1-1 (::.) + C a +x a a
J l dx = sin-1 (::.) + C -../a2- x2 a
J 1 dx = sinh-1 (::.) + C
-../x2 + a2 a
J l dx = cosh-1 (:'.) + C -../x2- a2 a
1¥ n -111f sinn xdx=-- sinn-2 x dx o n o
11[ n -11!f cosn x dx = - - cosn-z x dx o n o
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NOVEMBER 2018 MATH2018 / MATH2019 Page 5
Answer question 1 in a separate book
1. i) Suppose that the temperature T, at a point (x, y, z) in space is given by
T(x, y, z) = z·- x2 - y2 .
a) Sketch the level surface of all points with a temperature of zero. b) Find grad(T). c) Calculate the rate of change of the temperature T at the point P(l, 1, 0) in the direction of the vector b = 3i + 4j + 12k. ii) Use the method of Lagrange multipliers to find the extreme values of
f(x, y) = 12 + 3x + 4y
subject to the constraint
g(x,y) = x2 +y2 -1 = 0.
iii) The volume V of a circular cylinder with radius r and perpendicular height h is given by V = 1rr2h. Use a linear approximation to estimate the maximum percentage error in calculating V given that r = 30 metres and h = 20 metres, to the nearest metre.
iv) You are given that 100 1 7f
2 2 dx=-a-1. o a + X 2
Use Leibniz' theorem to find the following integral in terms of a
>) l.eC /(,) � { :
100 1 ( 2 ')' dx.
0 a +x
7f Oa) Sketch the odd periodic extension of f over the domain -1r :S x :S 1r. b) Calculate the half range Fourier sine series of f.
7f c) To what value does the series in b) converge at x = :(
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NOVEMBER 2018 MATH2018 / MATH2019 Page 6
Answer question 2 in a separate book
2. i) Consider the following differential equation describing a vibrating system:
d2y dt2 + 4y = 8cos(21rwt) .
a) Find the solution Yh to the homogeneous equation. b) For which value(s) of w will the system exhibit resonance? Give reasons for your answer. (Note that you are not being asked to find the particular solution Yr) ii) Consider the double integral
I= 14 f 2 lOx dydx.
0 yX
a) Sketch the region of integration. b) Evaluate I with the order of integration reversed.
iii) A quadratic curve is given by the equation 7x2 + Gxy + 7y2 = 200. a) Express the curve in the form
xT Ax= 200
whore x = (:), and A is a 2 x 2 symmetric matrix.
b) Find the eigenvalues and eigenvectors of the matrix A in part a). c) Hence, or otherwise, find the shortest distance between the curve and the origin.
iv) Use the substitution v = 11_ to solve
X
xy' = y + 2x3 cos2 (�)
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NOVEMBER 2018 MATH2018 / MATH2019 Page 7
Answer question 3 in a separate book
3. i) Let !1 be the semi-circular region bounded by y = v'l - x2 and y = 0. The region !1 is of uniform density and has centroid (x, y).
a) Sketch the region !1 and write down its area. b) Explain why x = 0. c) Find y by evaluating an appropriate double integral expressed in polar coordinates.
ii) Find a) L'.{sin(3t)},
b) L'.{e-7'sin(3t)},
) c-1 { 4s - 28 } c
(s-l)(s-9) ·
iii) The function g ( t) is defined for t 2 0 by
( ) {t2 ' g t = Zt e ' 0 � t < 1, t 2 1.
a) Express g(t) in terms of the Heaviside function.
b) Hence, or otherwise, show that the Laplace transform of g(t) is
2
(
2 2 1) e2-,
G(s) = --,--e-' - + - + - + - -s3 s3 s2 s s - 2
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NOVEMBER 2018 MATH2018 / MATH2019
Answer question 4 in a separate book
4. i) Consider the vector field
F = yz2i + xz2j + (2xyz + 3)k.
a) Calculate div(F). b) Show that F is conservative by evaluating curl(F).
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c) Tho path C in Jf!.3 starts at the point (3, 4, 7) and subsequently travels anticlockwise four complete revolutions around the circle x2 +y2 = 25 within the plane z = 7, returning to the starting point (3, 4, 7). Using
part b) or otherwise, evaluate the work integral 1 F · dr,
ii) A vibrating string of length 1r metres satisfies the wave equation
82u 82n 8t2 = 25 8x2 '
whcro u(x, t) is the transverse displacement of the string, at position x and time t. The ends of the string arc held fixed so that
n(O, t) = u( 1r, t) = 0, for all time t.
a) Assuming a solution of the form n(x, t) = F(x)G(t), show that
for some constant k.
1 d2G 1 d2F 25G dt2 = F dx2 = k
b) You may assume that only k < 0 yields non-trivial solutions and set
k = -(p2) for some p > 0. Applying the boundary conditions, show that p = n, n = 1, 2, 3, ... and that possible solutions for F( x) are
where Bn are constants. c) Find all possible solutions Gn (t) for G(t). d) If the initial displacement and velocity of the string are
u(x, 0) = 2 sin(x) - sin(2x)
find the general solution n(x, t).
and u,(x, 0) = 0,
c) Hence determine the maximal transverse displacement of the string
• 1f at tnnc t = 15 seconds.
END OF EXAMINATION