The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 2
2014
Assumed Knowledge: Sigma notation for sums. The ideas of a sequence of numbers and
of the limit of a sequence. Sketching a curve given a f
8002A SEMESTER 1 2022 PAGE 16 OF 30
Extended Answer Section
There are three questions in this section, each with a number of ports, Write your answers
in the space provided below each part. If you need more space there are extra pages at the
end of the ex
8002A SEMESTER 1 2009 PAGE 2.8 0F 30
Extended Answer Section
There arethree questions in this section, each with a number ofperts. Write your answers
in the space provided below each part. If you need more space there are extra pages at the
end of the e
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 3
2014
Assumed Knowledge: Integrals of simple functions such as xn (including 1/x), sin x, cos x,
ex . The simple properties of denite integrals.
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 4
2014
Assumed Knowledge: Sketching curves of simple functions. Integrals of simple functions
such as xn (including 1/x), sin x, cos x, ex .
Obje
Attached are questions selected from the MATH1002 examinations for 2005, 2006 and
2007, together with brief solutions. Question 3 in the 2005 and 2006 examinations has
been omitted, since it involves elementary matrices that were not covered in 2007 and
2
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises and Solutions for Week 5
2014
Assumed Knowledge: Sketching curves of simple functions. Integrals of simple functions
such as xn (including 1/x), sin x, cos x, ex .
Obje
0
Ll
lS
)I\
\
I
0
X
'I
1
'
I
li
I
'1St:,
hei ht

IS
',
I
.

I
.
t =I
,
'
I
I
,
'
t;
I

'
'
1
l
\
1f
 . I

_.
l
=\
l
)
0
'
1nn~r
t
\
r
,.~.
I
II
l
'
\
I
\
I
'
I
3
5
I
I

1
1
f


I
r
l
\
I
1f
1I
.
fl/lll/llllll/6
1
.
'
~
1
3
I
=I
'

'
'
.
. ' .
Summer School, 2012 Page 4 of 26
1. [14 marks total]
{a} [2 marks] Find the general solution ydr) of the diffmonrial equation
11; + 9y 2 U. Summer School, 2012 Page 5 01 26
{h} [3 marks] Find a particular solution yplfli) of
I! + 9.1} I sin 1?.
Jla
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 3
Questions marked with the dagger symbol are intentionally more challenging.
1. Find the Laplace transforms of each of the following functions.
(a) t3 e2t
(b) t sin 4t
(c) H (t 3)
(d) e3t sin
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 2
Questions marked with the dagger symbol are intentionally more challenging.
1. Use the method of undetermined coecients to nd a particular solution to each of the
following inhomogeneous dier
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 1
1. Show that the ordinary dierential equation
dx
= x,
dt
= constant
has the general solution
x = A et ,
where A is a constant.
Hence solve
(a) dx/dt = 3x,
x = 2 when t = 0.
(b) dy/dz = 5y,
y
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 6
1. Determine the eigenvalues and corresponding eigenfunctions of the dierential equation
d2
+ = 0
dx2
d
(L) = 0. Analyze the three cases with real :
with boundary conditions (0) = 0 and
dx
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 4
1. An electrical circuit in which a resistor (R) and an inductor (L) are connected in series
with a voltage source (V (t) has a resulting current ow i(t). Using Kirchos Laws, it is
possible t
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 5
1. To practice your partial dierentiation, verify that
u(x, t) = sin (2x) e4
2 kt
is a solution to the heat equation
u
2u
=k 2.
t
x
2. Verify that u(x, t) = sin (x ct) is a solution to the wa
Summer School 2014
Tutorial Questions 9
Questions marked with the dagger symbol are intentionally more challenging.
1. For the following functions, sketch the Fourier series of f (x) on the interval L x L
and determine the Fourier coecients:
(a) f (x) = x
MATH2065: INTRO TO PDEs
Summer School 2014
Tutorial Questions 7
1. Solve Laplaces equation
2 2
+ 2 =0
x2
y
for (x, y ) in the square 0 < x < L, 0 < y < L, subject to the boundary conditions
(x, 0)
(x, L)
(0, y )
(L, y )
=
=
=
=
0 for 0 x L,
0 for 0 x L,
0
Summer School 2014
Tutorial Questions 13
This Tutorial Set contains a selection of questions covering material over the semester.
1. Consider the wave equation
2u
2u
= c2 2
;
t2
x
u(x, 0) = f (x) ,
u
(x, 0) = 0 .
t
< x < ,
(a) Let U (, t) = F cfw_u(x, t)
Summer School 2014
Tutorial 8
There is no new material in this tutorial; Instead, you are being given a chance to revise
previous work.
1. Consider the initial value problem
d
5 = sin t + (t + 1) e5t
dt
;
(0) = 0 ,
where the unknown is (t). In First Year