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MODULE HANDBOOK
Mathematics II
TEACHING PLAN (2012-13) Semester 1
Name Tom Lunney Room MG121D
Ext 75388
Module Mathematics II (COM420; CRN 46532)
Semester 1
Course BEng(Hons) Computing & Electronics/BEng(Hons) Computer Games
Lecturer(s): Dr Tom Lunney;
Arrangements for Lectures/Tutorials/Practicals: Classes occur on a weekly basis.
Venue and times: Lectures:
Tuesday 10.15- 12.05 (Room MF131);
Tutorials:
Tuesday 13.15- 15.05 (Room MB104);
Practicals: Tuesday 9.15- 10.05 (Room MF116_PC);
Presentation Schedule:
Week 1/2 – Differentiation
Basic ideas and definitions
Differentiation of a function of one variable
The product quotient and function of a function rules
Rates of change, maximum and minimum
Week 3/4/5 – Integration
Integration as the inverse of differentiation
Definite and indefinite integration
Integration by inspection, substitution, partial fractions, by parts
Week 6 – Numerical Integration
Introduction to Numerical Integration
Trapezium Rule
Simpson’s Rule
Week 7/8 – Differential Equations
First order differential equations
Solution by
direct integration
separating the variables
use of an integration factor
Standard linear equations
1
Week 9 – Complex Numbers
Addition, subtraction, multiplication of complex numbers
Powers of complex numbers
Graphical representation of complex numbers
Polar form of a complex number, phase angle and modulus calculations
Week 10 – Vectors
Introduction to three dimensional vectors
Addition, subtraction of vectors
Dot (scalar) products of vectors
Cross (vector) product of vectors
Week 11 – Laplace Transforms
Explanation and Definition
Simple transforms
Inverse transforms
Application to the solution of differential equations
Week 12 – Revision and Examination Preparation
TEACHING AND LEARNING METHODS
Lectures will be used to expose the students to new material.
Tutorials will be used to help students become familiar with material delivered in the
lectures and will take a variety of forms including tutor-led problem-solving sessions,
solving assigned problems in groups and singly in order to allow the student to engage
with the tutor and other class mates.
Additional backup online support material is provided to help with student understanding
and provide context for the different topics covered during the module. This material will
be introduced during the laboratory sessions.
Students will be directed to read relevant text indicated below in order to gain more
practice on topics if required.
Students will be expected to complete Tutorials and Class Tests as appropriate.
The module is web blended and all material relating to the module will be available
online.
ASSESSMENT METHODS
2
Coursework Activity(Typical)
Coursework 1 (75% of coursework marks)
Biweekly Multiple-choice Tests are administered during the module to facilitate timely
feedback and contribute to the coursework marks. Each test will last approximately one
hour.
Coursework 2 (25% of coursework marks)
This will typically be a two hour open-book written test occurring towards the latter
stages of the module, where students will be required to develop full mathematical
solutions. This coursework aims to give students a good indication of their overall
progress in the module.
Examination:
A written examination lasting three hours is completed by the student at the end of the
semester (during the January examination period) and this contributes 75% of the
overall marks for the module (see Appendix A).
READING LIST
Essential
STROUD, K.A., BOOTH, D.J, 2007, Engineering Mathematics, (6th ed.), Palgrave
Macmillan
Required
BIRD, J.O., 2005, Basic Engineering Mathematics, Oxford: Newnes
BIRD, J.O., MAY, A.J.C., 1999, Mathematical Formulae, (3rd ed), Pearson Longham.
BOOTH, D.J., 1998, Foundation Mathematics, (3rd ed), Addison-Wesley Longman
Ltd.
CROFT, A., DAVISON, R., 2003, Foundation Mathematics, (3rd ed), Pearson
Education Ltd.
CROFT, A., DAVISON, R., 2004, Mathematics for Engineers: A Modern Interactive
Approach, Pearson Education Ltd.
CROFT, A., DAVISON, R., HARGREAVES, M., 2001, Engineering Mathematics: a
Foundation for Electronic, Electrical, Communications and Systems Engineers, (3rd ed),
Pearson Education Ltd.
GREENBERG, M. D., 1998, Advanced Engineering Maths, (2nd ed), Prentice Hall Inc
JAMES, G., 2008, Modern Engineering Mathematics, (4th ed), Pearson Education
3
Ltd.
JEFFERY, A., 2005, Essentials of Engineering Mathematics: Worked Examples and
Problems, (2nd ed), Chapman and Hall
JORDAN, D.W., SMITH, P., 2002, Mathematical Techniques: An Introduction for the
Engineering, Physical and Mathematical Sciences, (3rd ed), Oxford University Press.
THOMPSON, S.P., GARDINER, M., 1998, Calculus made easy, Palgrave
TUMA, J., 1998, Engineering Mathematics Handbook, (4th ed), McGraw Hill
SUMMARY DESCRIPTION
This module introduces students to the essential mathematics required for embarking
on further study in engineering, computing or a related discipline. It develops the
students mathematical skills required to solve problems that arise in the context of their
undergraduate study. The module content is introduced in a pragmatic way and then
related to real world problems, which enhances understanding and makes the concepts
more meaningful and relevant for the student. The module also aims to generate in the
student a spirit of mathematical investigation and discovery leading to the development
of mathematical confidence.
Also note that the full set of Lecture Notes, Tutorials, Support Material, Module
Specification and other associated module material is available via the link
(http://www.infm.ulst.ac.uk/~tom/)
4
Appendix A – Sample Examination Paper
UNIVERSITY OF ULSTER
UNIVERSITY EXAMINATIONS
Semester One
Module Code: COM420
CRN:
Title: Mathematics II
Time Allowed: 3 hrs
Use of Dictionaries: Dictionaries are not permitted
Examination Aids: Candidates may use any programmable graphics calculator
Instructions to Candidates:
Candidates should read this section carefully
before commencement of the examination.
There are two sections to this paper.
 Section A contains ten compulsory questions each worth 4 marks.
 Section B contains five questions of which the candidates should answer three.
Students are advised to write their registration number and desk number only on
any attachment, e.g. graph paper, or any other documentation being submitted
with their examination script book(s).
[insert name(s) of Module Co-ordinator(s)]
5
Section A – Compulsory Section – Answer all questions
(Each question is worth 4 marks)
1.
Differentiate y  10 cos ecx
2.
Using the quotient rule, differentiate y 
3.
d 2y
3
2
Find the second derivative
of y  2x  x  10 x
2
dx
4.
Find
5.
Using the substitution method, find the integral 
6.
Using integration by parts, find the integral  x cos( x )dx
7.
Using direct integration, find the general solution of the differential
dy
 6x 4  3
equation x
dx
8.
By separating the variables, solve the differential equation
9.
Find 3  j 42  j 5
10.
 sin x  2
x
1 x

x  1 dx
dx
2x  1
dy 2 x

dx
y
Given the vectors p = 2i + 3j + 5k and q = 4i +j+6k find the scalar product p.q
of the two vectors.
6
Section B – Answer any 3 questions
(Each question is worth 20 marks).
Question 1
a) Differentiate y  4 x 3 sin x using the product rule.
(4 marks)
 x 5 
b) Differentiate y  sin
 using the chain rule followed by the quotient rule.
 2x  4 
(10 marks)
c) The volume v of a sphere of radius r is given by v 
4 3
r . If the radius of a soap
3
bubble is increasing by 0.1cm per second, find the rate of increase of its volume
when the radius is 2cm.
(6 marks)
Question 2
2
1
a) Find the definite integral  cos xdx
2
0
b) After resolving
2x
2x
dx
into partial fractions, find  2
x x 2
x x 2
2
(6 marks)
(14 marks)
Question 3
OA  a  i  j  4k
Given
OB  b  8i  2k
OC  c  5i  2 j  11k
Find:
a)
c-b
b)
b-a
c)
the magnitude of the vector OB , i.e. OB
(2 marks)
(2 marks)
d)
the magnitude of the vector OC , i.e. OC
(4 marks)
e)
f)
the scalar product
the scalar product
(4 marks)
(4 marks)
a.c
b.c
(4 marks)
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Question 4


a)
Determine the Laplace transform of 2e  t  t , i.e. find L 2e t  t
b)
Determine the Laplace transform of e 3t (t 2  4) , i.e. find L e 3t (t 2  4)
(4 marks)
c)
d)

 8 
Find the inverse Laplace transform for F (s )   2
,
 s  64 
 8 
L1  2
i.e. find

 s  64 
 9s  8 
L1  2

 s  2s 
Determine
(2 marks)

(4 marks)
(10 marks)
Question 5
2
a)
1
 cos 2 xdx
Evaluate
using the Trapezium rule with 8 intervals to 3 decimal
0
places.
(10 marks)
 x
2
b)
Evaluate
3

 x 2 dx using Simpson's rule with 4 intervals to 3 decimal
1
places.
(10 marks)
8
MATHEMATICAL FORMULAE
Standard Derivatives:
Standard Integrals:
y
dy
dx
y
 ydx
f(x)
xn
axn
sin ax
a cos ax
sin ax
cos ax
 a sinax
cos ax
tan ax
a sec 2 ax
tan ax
sec ax
a sec ax tan ax
sec 2 ax
cot ax
 a cos ec 2ax
cos ec 2ax
cos ecax
 a cos ecax cot ax
sec ax tan ax
eax
aeax
sec ax tan ax
ln x
1
x
cos ecax cot ax
a n 1
x
n 1
1
 cos ax
a
1
sin ax
a
1
ln(sec ax )
a
1
tan ax
a
1
 cot ax
a
1
sec ax
a
1
sec ax
a
1
 cos ecax
a
1 ax
e
a
sin1
x
a
cos 1
x
a
1
a x
1
2

2
a2  x 2
eax
1
x
ln x
cos x
sin x
1
1 y2
1
a

2
ln sin x
tan 1 y
 x2
1
a
2

 x2
1
x 2  a2
arcsin

x
a
arccos
x
a
1
x
arctan
a
a
9
Laplace Transforms
Function f ( t )
1
a
t
tn
eat
te at
sinat
cos at
sinh at
cosh at
u( t  c )  f ( t  c )
Transform F( s)
1
s
a
s
1
s2
n!
sn 1
1
sa
1
(s  a)2
a
2
s  a2
s
2
s  a2
a
2
s  a2
s
2
s  a2
e csF(s)
Inverse Laplace Transforms
Function F( s)
a
s
1
sa
n!
sn  1
1
sn
a
s  a2
Transform f ( t )
a
e at
tn
t n1
(n  1)!
sinat
2
10
s
2
s  a2
a
2
s  a2
s
2
s  a2
cos at
sinh at
coshat
Product Rule for Differentiation
If y  uv where u and v are functions of x, then
dy
du
dv
v
u
dx
dx
dx
Quotient Rule for Differentiation
u
If y  where u and v are functions of x, then
v
dy

dx
v
du
dv
u
dx
dx
2
v
Chain Rule for Differentiation
If u is a function of x and y is a function of u, then
dy dy du

.
dx du dx
‘Integration by Parts’ Rule for Integration
If u and v are functions of x, then
dv
du
 u dx dx  uv   v dx dx
Trapezium Rule for Numerical Integration
b
 ydx  2 y 0  2y1  2y 2  ......  2y n1  y n  .
h
a
Simpson’s Rule for Numerical Integration
b
h
 ydx  3 y 0  4( y1  y 3  y 5  .......)  2( y 2  y 4  y 6  .......)  y n  .
a
11