Student

Number

Semester 1 Assessment, 2018

School of Mathematics and Statistics

MAST20029 Engineering Mathematics

Writing time: 3 hours

Reading time: 15 minutes

This is NOT an open book exam

This paper consists of 5 pages (including this page)

Authorised Materials

• Mobile phones, smart watches and internet or communication devices are forbidden.

• Calculators and written or printed material related to the subject are not permitted.

Instructions to Students

• You must NOT remove this question paper at the conclusion of the examination.

• There are 10 questions on this exam paper.

• All questions may be attempted.

• Marks for each question are indicated on the exam paper.

• Start each question on a new page.

• Clearly label each page with the number of the question that you are attempting.

• There is a separate 5 page formula sheet accompanying the examination paper, that you

may use in this examination.

• The total number of marks available is 120.

Instructions to Invigilators

• Students must NOT remove this question paper at the conclusion of the examination.

• Initially students are to receive the exam paper, the 5 page formula sheet, and two 14

page script books.

This paper may be held in the Baillieu Library

Blank page (ignored in page numbering)

MAST20029 Semester 1, 2018

Question 1 (8 marks)

Let R be the region in R2 satisfying x2 + y2 ≤ 2 and y ≥ x2.

(a) Sketch the region R.

(b) Evaluate the double integral ∫∫

R

x2y dydx

Question 2 (11 marks)

Let F be the vector field

F(x, y, z) = [xy sinh(x) + y cosh(x)]i+ [x cosh(x) + 3e3zy2]j+ 3e3zy3k,

and let C be the curve from (3, 4, 16) to (3, 0, 0) along the intersection of the surfaces z = y2

and x = 3.

(a) Write down a parametrisation for C in terms of an increasing parameter t.

(b) Show that F is a conservative vector field.

(c) Find a scalar function φ such that F = ∇φ.

(d) Evaluate ∫

C

F · dr

Question 3 (14 marks)

Consider the vector field

F(x, y, z) =

(

6x− x

2

2

)

i+

(

xy − x

2

2

)

j+ (z − 3)2k

Let V be the region in R3 that is the part of the cylindrical region x2 + y2 ≤ 1 bounded above

by 2z−y = 6 and bounded below by the cone z =

√

x2 + y2. Let S be the surface of V oriented

with outward unit normal, and let S1 be the part of S lying on the plane 2z − y = 6 .

(a) Sketch the region V .

(b) Evaluate the flux integral ∫∫

S1

F · nˆ dS

(c) Using Gauss’ theorem, evaluate the flux integral∫∫

S

F · nˆ dS

Page 2 of 5 pages

MAST20029 Semester 1, 2018

Question 4 (14 marks)

(a) Using eigenvalues and eigenvectors, find the real general solution for the following system

of differential equations

dx

dt

= x− 2y, dy

dt

= x+ 3y

(b) The system of first order ordinary differential equations

dx

dt

= 3x− 2y, dy

dt

= 3x− 4y

has the following general solution(

x(t)

y(t)

)

= α1

(

1

3

)

e−3t + α2

(

2

1

)

e2t

i. Sketch a phase portrait near the critical point at the origin, showing any straight

line orbits and at least 4 other orbits.

To sketch the phase portrait, determine:

• any straight line orbits;

• the asymptotic behaviour of the orbits as t→∞ and as t→ −∞;

• the slopes at which the orbits cross the x and y axes.

ii. What is the type and stability of the critical point at the origin?

Question 5 (9 marks)

(a) Consider the function

y(t) =

8, 0 ≤ t < 2

t3, t ≥ 2

i. Represent y using step functions.

ii. Using part (a)i., find the Laplace transform of y.

(b) Compute the inverse Laplace transform of

F (s) =

s2 + e−s

s2 + 1

(c) Compute the Laplace transform of

g(t) = cos(2t) sinh(t)

Page 3 of 5 pages

MAST20029 Semester 1, 2018

Question 6 (8 marks)

Use Laplace transforms to solve the second order ordinary differential equation

x′′ − 3x = 2 cos(t)

subject to the initial conditions x(0) = −2 and x′(0) = 0.

Question 7 (8 marks)

In this question you must state if you use any standard limits, continuity, l’Hoˆpital’s rule or

any convergence tests for series.

(a) Consider the sequence

an =

(

n+ 2

n+ 5

)3n

, n ≥ 1

i. Determine the limit as n→∞ if it exists, or explain why the sequence diverges.

ii. Determine whether the series

∞∑

n=1

an converges or diverges.

(b) Determine whether the following series converges or diverges:

∞∑

n=1

cos2(n)

n3/2

Question 8 (13 marks)

In this question you must state if you use any standard limits, continuity, l’Hoˆpital’s rule or

any convergence tests for series.

(a) Find the radius of convergence and the interval of convergence for the power series

∞∑

n=1

3(x− 1)n

5n

(b) Consider the function f(x) =

√

x.

i. Find the second order Taylor polynomial for f about x = 25.

ii. Use the polynomial in part (b)i. to approximate

√

26.

iii. Determine an upper bound for the error of the approximation in part (b)ii.

Page 4 of 5 pages

MAST20029 Semester 1, 2018

Question 9 (18 marks)

Consider

f(t) =

{

t, 0 < t ≤ 1

1, 1 < t < 2

(a) Sketch fe(t), the even periodic representation of f , in the range −4 ≤ t ≤ 4.

(b) Determine the first four non zero terms for the Fourier series representation of fe.

(c) Determine the first four non zero terms for the series representation of the energy density

of fe.

(d) What value does the Fourier series for fe converge to when t = 0?

(e) What value does the Fourier series for fe converge to when t = 2?

Question 10 (17 marks)

Consider the second order wave equation

∂2φ

∂t2

= 16

∂2φ

∂x2

for 0 < x < pi and t > 0, subject to the boundary and initial conditions

φ(0, t) = 0

φ(pi, t) = 0

φ(x, 0) = 3 sin(4x)

∂φ

∂t

(x, 0) = 8 sin(2x)

(a) Using the method of separation of variables, show that the wave equation reduces to two

ordinary differential equations (ODE’s) of the form:

X ′′(x)− λX(x) = 0

T ′′(t)− 16λT (t) = 0

where λ is the separation constant.

(b) By solving the ODE’s in part (a) for the case λ < 0, determine the solution of the wave

equation subject to the given initial and boundary conditions.

You may assume that solving the ODE’s in part (a) for the cases λ > 0 and λ = 0 leads

to trivial solutions. You do not need to work through these two cases.

End of Exam—Total Available Marks = 120

Page 5 of 5 pages

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