The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 4
2011
Assumed Knowledge Sketching curves of simple functions. Integrals of simple functions
such as xn (including 1/x), sin x, cos x, ex .
Objectives
(3a) To
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 3
2011
Assumed Knowledge Integrals of simple functions such as xn (including 1/x), sin x,
cos x, ex . The simple properties of denite integrals.
Objectives
(2a
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 2
2011
Assumed Knowledge Sigma notation for sums. The ideas of a sequence of numbers and
of the limit of a sequence. Sketching a representative graph of a func
The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Tutorial 1 Week 2
1.
Which of the following strings of brackets are balanced? In each case, explain
carefully why the string is, or is not, balanced:
(i )
(ii )
(iii )
()()
()()
1
oscillations_03
Damped Oscillations
Forced Oscillations and Resonance
SHM
shm_v.avi
Review: SHM
2
mass/spring system
x = xmax cos ( t )
v = xmax sin ( t )
a = xmax 2 cos ( t )
a = 2 x
2
= 2 f =
T
k
2
=
m
1212
PE = k x = k xmax cos2 ( t )
2
2
1
1
12
2
2
oscillations_02
1
Oscillations
Time variations that repeat themselves at regular
intervals - periodic or cyclic behaviour
Examples: Pendulum (simple);
heart (more complicated)
Terminology:
Amplitude: max displacement from
equilibrium position [m]
shm_v.av
MATERIALS CIVL2110
FRICTION fundamentals and its relevance to soil and rock
Introduction
Friction is important in many engineering applications particularly in mechanical
engineering where friction between moving parts, and the wear that this causes, are
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 5
2011
Assumed Knowledge Sketching curves of simple functions. Integrals of simple functions
such as xn (including 1/x), sin x, cos x, ex .
Objectives
(4a) To
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 6
2011
Assumed Knowledge Simple properties of the functions ln x and ex , including their
derivatives.
Objectives
(5a) To know and be able to use the propertie
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 7
2011
Assumed Knowledge Proportionality and inverse proportionality. Integration techniques.
Taylor series expansion.
Objectives
(6a) Given a verbal descripti
1
waves_03
TRANSVERSE WAVES
ON STRINGS
Animations courtesy of Dr. Dan Russell, Kettering University
ves_03: MINDMAP SUMMARY - TRANSVERSE WAVES ON STRINGS
Travelling transverse waves, speed of propagation, wave function,
string tension, linear density, ref
waves_02
1
SUPERPOSITION OF WAVES
Two waves passing through the same region will
superimpose - e.g. the displacements simply add
Two pulses travelling in opposite directions will pass
through each other unaffected
While passing, the displacement is sim
waves_01
1
WAVES
What is a wave?
A disturbance that propagates
Examples
Waves on the surface of water
Sound waves in air
Electromagnetic waves
Seismic waves through the earth
Electromagnetic waves can propagate through a vacuum
All other waves propa
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 13
2011
Assumed Knowledge
Objectives
(12a) To be able to rewrite two coupled rst-order dierential equations as a single secondorder dierential equation.
(12b)
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 12
2011
Assumed Knowledge Finding the roots of quadratic equations.
Eulers formula ei = cos + i sin .
Objectives
(11a) To be able to write down the auxiliary (
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 11
2011
Assumed Knowledge Integration techniques.
Objectives
(10a) To be able to solve dierential equations that are separable, linear or both.
(10b) To be abl
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 10
2011
Assumed Knowledge Integration techniques.
Objectives
(9a) To be able to distinguish between separable and linear rst-order dierential equations.
(9b) T
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 9
2011
Assumed Knowledge Integration techniques.
Objectives
(8a) To be able to write a rational expression as a sum of partial fractions.
(8b) To be able to so
The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 8
2011
Assumed Knowledge Factorisation of expressions. Simple techniques of integration.
Objectives
(7a) To be able to recognise a dierential equation as a sep
The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Tutorial 12 Week 13
1.
(i )
Show that xn = 6 2n 4 is a solution to the recurrence relation
xn = 3xn1 2xn2 .
(ii ) Show that xn = (3n+1 1)/2 is a solution to the recurrence relati
The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Tutorial 11 Week 12
1.
Find a closed form for the generating function of the sequence a0 , a1 , . . . ,
where a0 = 1, a1 = 1 and
an = 5an1 6an2 .
2.
Let a0 , a1 , a2 , . . . , be
The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Tutorial 10 Week 11
Use induction to prove the following propositions.
For each of the given propositions, we let S (n) be the given proposition, which is
to be proved true for a
The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Tutorial 9 Week 10
1.
Draw truth tables for the following propositions.
(i ) (p q ).
(ii ) p q .
(iii ) p (p q ).
2.
Show that the following pairs of propositions are equivalent: