Mathematics for Computer Science I (Calculus).
MATHEMATICS FOR
COMPUTER
SCIENCE II
(CALCULUS)
Version 2.0
Ahmad Fadzli Nizam Abdul Rahman
Burhanuddin Mohd Aboobaider
Zuraida Abal Abas
Siti Azirah Asmai
Faculty of Information and Communication Technology,
Universiti Teknikal Malaysia Melaka,
Karung Berkunci 1200,
75450 Melaka,
Malaysia.
© FTMK, KUTKM 2008
All rights reserved. No part of this publication may be
reproduced stored in a retrieval system, or transmitted in any
form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without either the prior written permission
of the publisher.
The programs in this book have been included for their
instructional value. They have been tested with care but are not
guaranteed for any particular purpose. The author or publisher
does not offer any warranties or representations nor does it
accept any liabilities with respect to the programs.
STUDENT NAME:
MATRIC NUMBER:
SEMESTER:
COURSE:
YEAR / SEMESTER SESSION:
This module covers one of the disciplines of mathematics that will be taught in
Mathematics for Computer Science students, which is Calculus. Topics for Calculus
include functions, derivatives, integrals, definite integrals, exponential, natural logarithm
functions and functions of several variables. Student will use this practical module during
tutorial session throughout the semester.
Upon completion of this module the students
will
MATHEMATICS FOR
COMPUTER SCIENCE
II
Have enough knowledge and
fundamental concepts of Calculus
Be able to apply Calculus concept and
techniques to solve problems
particularly in computer science area
Dedication
We would like to thank our colleagues and friends for their helpful comments and
suggestions for this practical module
TABLE OF CONTENTS
Dedication_______________________________________________________________iii
Table of Contents__________________________________________________________iv
1
2
3
Lecture 1: Functions____________________________________________________1
1.1
FUNCTIONS AND THEIR GRAPH________________________________________2
1.2
SOME IMPORTANT FUNCTIONS________________________________________8
1.3
THE ALGEBRA OF FUNCTIONS________________________________________12
1.4
ZEROS OF FUNCTIONS – THE QUADRATIC FORMULA FACTORING______17
1.5
EXPONENTS AND POWER FUNCTIONS_________________________________25
1.6
FUNCTIONS AND GRAPHS IN APPLICATION____________________________34
Lecture 2: The Derivative_______________________________________________38
2.1
THE SLOPE OF A STRAIGHT LINE______________________________________39
2.2
THE SLOPE OF A CURVE AT A POINT___________________________________43
2.3
THE DERIVATIVE_____________________________________________________45
2.4
LIMIT AND THE DERIVATIVE__________________________________________56
2.5
DIFFERENTIABILITY & CONTINUITY__________________________________67
2.6
SOME RULES FOR DIFFERENTIATION_________________________________72
2.7
MORE ABOUT DERIVATIVES___________________________________________76
2.8
THE DERIVATIVE AS A RATE OF CHANGE______________________________81
Lecture 3: Techniques of Differentiation___________________________________85
3.1
THE PRODUCT AND QUOTTIENT RULES_______________________________86
3.2
THE CHAIN RULE AND THE GENERAL POWER RULE___________________90
3.3
IMPLICIT DIFERENTIATION AND RELATED RATES_____________________93
TABLE OF CONTENTS
4
5
6
7
8
Lecture 4: The Exponential and Natural Logarithm Functions________________98
4.1
EXPONENTIAL FUNCTION____________________________________________99
4.2
THE EXPONENTIAL FUNCTION ex_____________________________________101
4.3
DIFFERENTIATION OF EXPONENTIAL FUNCTIONS____________________103
4.4
THE NATURAL LOGARITHM FUNCTIONS_____________________________107
4.5
THE DERIVATIVE OF ln x_____________________________________________108
4.6
PROPERTIES OF THE NATURAL LOGARITHM FUNCTION______________110
Lecture 5: The Definite Integral_________________________________________112
5.1
ANTI DIFFERENTIATION_____________________________________________113
5.2
DEFINITE INTEGRALS AND THE FUNDAMENTAL THEOREM___________120
5.3
AREAS IN THE XY-PLANE____________________________________________131
5.4
APPLICATIONS OF THE DEFINITE INTEGRAL_________________________136
5.5
Improper Integrals_____________________________________________________144
Lecture 6: Techniques of Integration_____________________________________147
6.1
Integration by substitution______________________________________________148
6.2
Integration by Pats_____________________________________________________159
Lecture 7: Functions of Several Variables_________________________________165
7.1
Examples of Functions of several variables_________________________________166
7.2
Partial Derivatives_____________________________________________________168
7.3
Maxima and minima of functions of several variables________________________175
7.4
Double Integrals________________________________________________________180
Lab Sheet___________________________________________________________191
COMPUTER
SCIENCE II
(CALCULUS)
Version 2.0
Ahmad Fadzli Nizam Abdul Rahman
Burhanuddin Mohd Aboobaider
Zuraida Abal Abas
Siti Azirah Asmai
Faculty of Information and Communication Technology,
Universiti Teknikal Malaysia Melaka,
Karung Berkunci 1200,
75450 Melaka,
Malaysia.
© FTMK, KUTKM 2008
All rights reserved. No part of this publication may be
reproduced stored in a retrieval system, or transmitted in any
form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without either the prior written permission
of the publisher.
The programs in this book have been included for their
instructional value. They have been tested with care but are not
guaranteed for any particular purpose. The author or publisher
does not offer any warranties or representations nor does it
accept any liabilities with respect to the programs.
STUDENT NAME:
MATRIC NUMBER:
SEMESTER:
COURSE:
YEAR / SEMESTER SESSION:
This module covers one of the disciplines of mathematics that will be taught in
Mathematics for Computer Science students, which is Calculus. Topics for Calculus
include functions, derivatives, integrals, definite integrals, exponential, natural logarithm
functions and functions of several variables. Student will use this practical module during
tutorial session throughout the semester.
Upon completion of this module the students
will
MATHEMATICS FOR
COMPUTER SCIENCE
II
Have enough knowledge and
fundamental concepts of Calculus
Be able to apply Calculus concept and
techniques to solve problems
particularly in computer science area
Dedication
We would like to thank our colleagues and friends for their helpful comments and
suggestions for this practical module
TABLE OF CONTENTS
Dedication_______________________________________________________________iii
Table of Contents__________________________________________________________iv
1
2
3
Lecture 1: Functions____________________________________________________1
1.1
FUNCTIONS AND THEIR GRAPH________________________________________2
1.2
SOME IMPORTANT FUNCTIONS________________________________________8
1.3
THE ALGEBRA OF FUNCTIONS________________________________________12
1.4
ZEROS OF FUNCTIONS – THE QUADRATIC FORMULA FACTORING______17
1.5
EXPONENTS AND POWER FUNCTIONS_________________________________25
1.6
FUNCTIONS AND GRAPHS IN APPLICATION____________________________34
Lecture 2: The Derivative_______________________________________________38
2.1
THE SLOPE OF A STRAIGHT LINE______________________________________39
2.2
THE SLOPE OF A CURVE AT A POINT___________________________________43
2.3
THE DERIVATIVE_____________________________________________________45
2.4
LIMIT AND THE DERIVATIVE__________________________________________56
2.5
DIFFERENTIABILITY & CONTINUITY__________________________________67
2.6
SOME RULES FOR DIFFERENTIATION_________________________________72
2.7
MORE ABOUT DERIVATIVES___________________________________________76
2.8
THE DERIVATIVE AS A RATE OF CHANGE______________________________81
Lecture 3: Techniques of Differentiation___________________________________85
3.1
THE PRODUCT AND QUOTTIENT RULES_______________________________86
3.2
THE CHAIN RULE AND THE GENERAL POWER RULE___________________90
3.3
IMPLICIT DIFERENTIATION AND RELATED RATES_____________________93
TABLE OF CONTENTS
4
5
6
7
8
Lecture 4: The Exponential and Natural Logarithm Functions________________98
4.1
EXPONENTIAL FUNCTION____________________________________________99
4.2
THE EXPONENTIAL FUNCTION ex_____________________________________101
4.3
DIFFERENTIATION OF EXPONENTIAL FUNCTIONS____________________103
4.4
THE NATURAL LOGARITHM FUNCTIONS_____________________________107
4.5
THE DERIVATIVE OF ln x_____________________________________________108
4.6
PROPERTIES OF THE NATURAL LOGARITHM FUNCTION______________110
Lecture 5: The Definite Integral_________________________________________112
5.1
ANTI DIFFERENTIATION_____________________________________________113
5.2
DEFINITE INTEGRALS AND THE FUNDAMENTAL THEOREM___________120
5.3
AREAS IN THE XY-PLANE____________________________________________131
5.4
APPLICATIONS OF THE DEFINITE INTEGRAL_________________________136
5.5
Improper Integrals_____________________________________________________144
Lecture 6: Techniques of Integration_____________________________________147
6.1
Integration by substitution______________________________________________148
6.2
Integration by Pats_____________________________________________________159
Lecture 7: Functions of Several Variables_________________________________165
7.1
Examples of Functions of several variables_________________________________166
7.2
Partial Derivatives_____________________________________________________168
7.3
Maxima and minima of functions of several variables________________________175
7.4
Double Integrals________________________________________________________180
Lab Sheet___________________________________________________________191