Mathematics for Computer Science I (Calculus).

Faculty of Information and Communication Technology, Universiti Teknikal Malaysia Melaka,

Karung Berkunci 1200, 75450 Melaka,

Malaysia.

© FTMK, KUTKM 2008

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STUDENT NAME:

MATRIC NUMBER:

SEMESTER:

COURSE:

YEAR / SEMESTER SESSION:

MATHEMATICS FOR

COMPUTER

SCIENCE II

(CALCULUS)

Version 2.0

Ahmad Fadzli Nizam Abdul Rahman

Burhanuddin Mohd Aboobaider

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



Have enough knowledge and

fundamental concepts of Calculus



Be able to apply Calculus concept and

techniques to solve problems

particularly in computer science area

MATHEMATICS FOR

Dedication

TABLE OF CONTENTS

Dedication_______________________________________________________________iii

Table of Contents__________________________________________________________iv

1

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

2

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

3

Lecture 3: Techniques of Differentiation___________________________________85

TABLE OF CONTENTS

4

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

5

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

6

Lecture 6: Techniques of Integration_____________________________________147

6.1 Integration by substitution______________________________________________148 6.2 Integration by Pats_____________________________________________________159

7

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