MAST20029 Engineering Mathematics 2021
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Semester 1 Assessment, 2021
School of Mathematics and Statistics
MAST20029 Engineering Mathematics
Question 1 (11 marks)
(a) Consider the double integral
4 2
cosh(y3 ) dydx.
0 ′x
(i) Sketch the region of integration.
(ii) Evaluate the double integral by changing the order of integration.
(b) Let W be the solid region that is inside both
x2 + y2 + z2 = 9 and z = ′ .
(i) Sketch W , labelling any points of intersection.
(ii) Write down a triple integral in cylindrical coordinates to calculate the volume of W .
Do not evaluate the triple integral.
Question 2 (7 marks)
Let C be the part of the ellipse 4x2 + 9y2 = 36 from (0, _2) to (3, 0) traversed clockwise. Let F be the force field
F(x, y) = 2yi + xj.
(a) Is F is a conservative vector field? Justify your answer.
(b) Determine the work done by F to move a particle along C .
Question 3 (12 marks)
Let S be the closed surface oriented with outward unit normal, consisting of the hemisphere S1 : z = _ ←4 _ x2 _ y2
and the disk
S2 : x2 + y2 < 4, z = 0.
Let F be the vector field
F(x, y, z) = (2x2 , 3y, z _ 4).
(a) Find the flux of F across S2 in the direction of the upward normal.
(b) Using Gauss’ theorem, find the flux of F across S in the direction of the outward normal.
(c) Using parts (a) and (b), determine the flux of F across S1 in the direction of the outward normal.
Question 4 (12 marks)
Consider the system of non-linear ordinary differential equations
dx 2
= _2y + xy.
(i) Determine the linearised system.
(ii) Using eigenvalues and eigenvectors, find the general solution for the linearised system.
(iii) Discuss whether the linearised system can be used to approximate the behaviour of
the non-linear system near this critical point.
Question 5 (9 marks)
The system of differential equations given by
dx
dt
has the solution
┐ = α 1 ┌ _11┐ e一t + α2 ┌ 1t(_) t┐ e一t , α 1 , α2 e R.
(a) Sketch a phase portrait of the system near the critical point at the origin.
You should include the following in your sketch, and explain your reasoning to justify your conclusions:
· any straight line orbits and their directions;
· at least 2 other orbits and their directions, showing the asymptotic behaviour as
t → o and t → _o;
· the slopes at which the orbits cross the x and y axes.
(b) What is the type and stability of the critical point for this system?
Question 6 (12 marks)
(a) Solve the integral equation
y(t) = 3e2t +
t
y(τ )e2(t一τ)dτ,
0
t 2 0.
(b) Use Laplace transforms to solve the differential equation
x′′ + 3x = f (t)
where
f (t) = , 6(0)
if x(0) = 0 and x′ (0) = 1.
0 < t < 2,
t 2 2,
Question 7 (9 marks)
Use Laplace transforms to solve the system of differential equations
dx
= _x _ 2y
dt
subject to the initial conditions x(0) = 2 and y(0) = _1.
Question 8 (11 marks)
In this question you must state if you use any standard limits, continuity, L’Hˆopital’s rule or any convergence tests for series.
(a) Determine whether the following series converges or diverges:
& ′n cos2 (n)
2n2 _ n .
(b) Consider the power series
& (_1)n (x + 2)n
nn .
(i) Find the radius of convergence.
(ii) Find the interval of convergence.
Question 9 (7 marks)
Consider the function
f (x) = (x _ 1) sinh(x).
(a) Find the cubic Taylor polynomial for f (x) about x = 1.
(b) Find an upper bound for the error R3(x) on the interval 1 < x < 2.
Note: You can express your upper bound in terms of hyperbolic functions.
Question 10 (16 marks)
Consider the function
f (t) =
0 < t < π,
π < t < 2π.
(a) Sketch fe(t), the even extension of f (t), for _4π < t < 4π . (b) Determine the first 3 non-zero terms for the Fourier cosine series for fe .
(c) What value does the Fourier series in (b) converge to if
(i) t = _ , (ii) t = π?
Question 11 (5 marks)
Determine a particular solution for the differential equation
d2y
where
g(t) = 0 & ┌ _ ┐ sin(ωt) dω.
Question 12 (14 marks)
Consider a plucked violin string released from rest. The displacement of the string φ(x, t) is
given by the wave equation
∂2 φ ∂2 φ
where 0 < x < 4 and t > 0.
The boundary and initial conditions are
φ(0, t) = 0,
φ(4, t) = 0,
(x, 0) = 0,
∂t
φ(x, 0) = sin ╱ 、_ sin(πx).
(a) Using the method of separation of variables, show that wave equation reduces to two
ordinary differential equations of the form:
X ′′ (x) _ λX(x) = 0
T′′ (t) _ λT (t) = 0
where λ e R is the separation constant.
(b) Using part (a), solve the wave equation subject to the given boundary and initial condi-
tions.
In your solution, you may assume that the cases λ > 0 and λ = 0 lead to trivial solutions. You do not need to work through these cases.
2022-02-08