BMTC - 131
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
1
ESSENTIAL PRELIMINARY CONCEPTS
Course Introduction 3
Block Introduction 5
Notations and Symbols 7
UNIT 1
Sets and Operations on Them 9
UNIT 2
Functions 39
UNIT 3
2D Coordinate Systems 58
UNIT 4
Complex Numbers 77
UNIT 5
Polynomial Equations and Their Solutions 107
Miscellaneous Exercises 131
,
,COURSE INTRODUCTION
This is the first mathematics course you will be studying in the Bachelor’s Degree
Programme. The aim of this course is to develop an understanding of basic
mathematical concepts and techniques that you will require for studying other
mathematics courses of the programme, as well as any further study and work you
undertake in mathematics.
Calculus is divided into two broad areas, differential calculus and integral calculus.
Broadly speaking, differential calculus is the study of change and integral calculus is
about adding up the parts. Differential calculus helps you to find, for example, the
effect of changing conditions on a system being investigated, and hence to gain control
over the system. The process of this mathematical investigation uses the powerful
technique of modelling the phenomena concerned. The models usually involve
differential equations. Differential calculus is useful in formulating models and integral
calculus is used to solve the differential equations associated with the model. Apart
from well known applications in physics, mathematical models based on calculus are
used for the study of population ecology, cybernetics, management practices,
economics and medicine.
In this course we shall focus on integral calculus after we discuss differential calculus.
However, it was integral calculus that developed first historically. This has its origins in
the need for measuring lands for the purpose of revenue collection. It is said that the
Egyptian river Nile changed its course often, and the lands near it, with differently
curved boundaries, were required to be measured again and again for levying taxes.
This led to the development of mensuration in the Egyptian civilisation. We can regard
mensuration as the forerunner of integral calculus. Indeed, one meaning of the word
‘quadrature’, used in integration, is computing the area.
The modern development of calculus began with the work of the famous 17th century
mathematicians, Newton and Leibnitz, in developing differential calculus. One of the
early successes of calculus was the prediction of the period of Halley’s comet. As you
can see, calculus provides a powerful tool for the study of not only such natural
phenomenon, but also artificial entities like the stock market. Over the centuries, many
European mathematicians like Euler, Lagrange, the Bernoullis, Gauss, Cauchy and
Riemann contributed to the development of this subject.
Now, a few words about how this course unfolds. In the first block of this course, you
will be introduced to two basic building blocks of mathematics, namely, sets and
functions. In the process you would recall a lot of related mathematics you studied in
school, including two important coordinate systems for representing and studying
two-dimensional spaces. Next, you will get more than a glimpse of the world of
complex numbers, C . Finally, you will study ways of solving certain polynomial
equations over R .
In the second block, you will begin by studying the properties of real numbers that you
will need again and again. You will also study the concept of limits and continuity,
which play a central role in calculus and, more generally, in mathematics.
In the third block, you will begin your study of differential calculus with the definition of a
derivative and its basic properties. You will study several formulae for the derivatives
of some functions which are used often, like polynomial functions and trigonometric
functions. This block ends with a discussion on higher order derivatives and the
Leibnitz rule for finding higher order derivatives.
3
, In the fourth block, you will find some applications of differentiation. You shall study
how derivatives can help to get information about various geometrical properties of
curves. This block ends with a discussion on the tracing of different types of curves.
In the fifth, and last, block, your focus will be on integral calculus. You will study
concepts of ‘integral of a function’ and ‘primitive of a function’. You will also study the
integrals of common functions like polynomial functions and trigonometric functions.
You will get an opportunity to apply some techniques of integration like the substitution
method, integration by parts and reduction formulae. The block ends with some
applications of integral calculus for measuring lengths, areas and volumes.
Now a word about our notation. Each block has units and each unit is divided into
sections, which may be further divided into sub-sections. These sections/sub-sections
are numbered sequentially, as are the exercises and important equations in a unit.
Since the material in the different units is heavily interlinked, there will be a lot of
cross-referencing. For this we will be using the notation Sec. x.y to mean Section y
of Unit x.
Throughout this course the emphasis will be on techniques rather than on theory. So
you may not find many proofs here. (You will be able to find the proofs of many of the
theorems you study and apply here in our third semester course, Real Analysis.)
Another compulsory component of this course is its assignment, which covers the
whole course. Your academic counsellor at the study centre will evaluate it, and
return it to you with detailed comments. Thus, the assignment is meant to be a
teaching as well as an assessment aid. Further, you will not be allowed to take the
exam of this course till you submit your assignment response at your study centre. So
please submit it well in time.
The course material that we have sent you is self-sufficient. If you have a problem in
understanding any portion, please ask your academic counsellor for help. Also, if you
feel like studying any topic in greater depth, you may consult.
A word of friendly advice here is that to learn the various techniques presented in this
course, you will need to put in a lot of practice in solving problems given in the material.
You should attempt to solve all the exercises in the block as you go along, before you
look up the solution. As a part of the tutorials, we have added miscellaneous examples
and exercises at the end of each block. You should also attempt these exercises. In
addition, you may also like to look up some other books in the library of your study
centre, and try to solve some exercises from these books.
Some useful books and websites are the following:
1. Essential Calculus, by James Stewart, Cengage Publication.
2. Calculus, by Anton, Bivens and Davis, Wiley Publications.
3. https://brilliant.org/courses/calculus-done-right
4. https://www.mathisfun.com/calculus
5. www.math.mit.edu/Ndjk/calculus_beginners
We have also prepared a video programme, which will be available at your study
centre, called “Limits”, based on the material in Block 2.
Wishing you a happy learning experience,
The Course Team
4
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
1
ESSENTIAL PRELIMINARY CONCEPTS
Course Introduction 3
Block Introduction 5
Notations and Symbols 7
UNIT 1
Sets and Operations on Them 9
UNIT 2
Functions 39
UNIT 3
2D Coordinate Systems 58
UNIT 4
Complex Numbers 77
UNIT 5
Polynomial Equations and Their Solutions 107
Miscellaneous Exercises 131
,
,COURSE INTRODUCTION
This is the first mathematics course you will be studying in the Bachelor’s Degree
Programme. The aim of this course is to develop an understanding of basic
mathematical concepts and techniques that you will require for studying other
mathematics courses of the programme, as well as any further study and work you
undertake in mathematics.
Calculus is divided into two broad areas, differential calculus and integral calculus.
Broadly speaking, differential calculus is the study of change and integral calculus is
about adding up the parts. Differential calculus helps you to find, for example, the
effect of changing conditions on a system being investigated, and hence to gain control
over the system. The process of this mathematical investigation uses the powerful
technique of modelling the phenomena concerned. The models usually involve
differential equations. Differential calculus is useful in formulating models and integral
calculus is used to solve the differential equations associated with the model. Apart
from well known applications in physics, mathematical models based on calculus are
used for the study of population ecology, cybernetics, management practices,
economics and medicine.
In this course we shall focus on integral calculus after we discuss differential calculus.
However, it was integral calculus that developed first historically. This has its origins in
the need for measuring lands for the purpose of revenue collection. It is said that the
Egyptian river Nile changed its course often, and the lands near it, with differently
curved boundaries, were required to be measured again and again for levying taxes.
This led to the development of mensuration in the Egyptian civilisation. We can regard
mensuration as the forerunner of integral calculus. Indeed, one meaning of the word
‘quadrature’, used in integration, is computing the area.
The modern development of calculus began with the work of the famous 17th century
mathematicians, Newton and Leibnitz, in developing differential calculus. One of the
early successes of calculus was the prediction of the period of Halley’s comet. As you
can see, calculus provides a powerful tool for the study of not only such natural
phenomenon, but also artificial entities like the stock market. Over the centuries, many
European mathematicians like Euler, Lagrange, the Bernoullis, Gauss, Cauchy and
Riemann contributed to the development of this subject.
Now, a few words about how this course unfolds. In the first block of this course, you
will be introduced to two basic building blocks of mathematics, namely, sets and
functions. In the process you would recall a lot of related mathematics you studied in
school, including two important coordinate systems for representing and studying
two-dimensional spaces. Next, you will get more than a glimpse of the world of
complex numbers, C . Finally, you will study ways of solving certain polynomial
equations over R .
In the second block, you will begin by studying the properties of real numbers that you
will need again and again. You will also study the concept of limits and continuity,
which play a central role in calculus and, more generally, in mathematics.
In the third block, you will begin your study of differential calculus with the definition of a
derivative and its basic properties. You will study several formulae for the derivatives
of some functions which are used often, like polynomial functions and trigonometric
functions. This block ends with a discussion on higher order derivatives and the
Leibnitz rule for finding higher order derivatives.
3
, In the fourth block, you will find some applications of differentiation. You shall study
how derivatives can help to get information about various geometrical properties of
curves. This block ends with a discussion on the tracing of different types of curves.
In the fifth, and last, block, your focus will be on integral calculus. You will study
concepts of ‘integral of a function’ and ‘primitive of a function’. You will also study the
integrals of common functions like polynomial functions and trigonometric functions.
You will get an opportunity to apply some techniques of integration like the substitution
method, integration by parts and reduction formulae. The block ends with some
applications of integral calculus for measuring lengths, areas and volumes.
Now a word about our notation. Each block has units and each unit is divided into
sections, which may be further divided into sub-sections. These sections/sub-sections
are numbered sequentially, as are the exercises and important equations in a unit.
Since the material in the different units is heavily interlinked, there will be a lot of
cross-referencing. For this we will be using the notation Sec. x.y to mean Section y
of Unit x.
Throughout this course the emphasis will be on techniques rather than on theory. So
you may not find many proofs here. (You will be able to find the proofs of many of the
theorems you study and apply here in our third semester course, Real Analysis.)
Another compulsory component of this course is its assignment, which covers the
whole course. Your academic counsellor at the study centre will evaluate it, and
return it to you with detailed comments. Thus, the assignment is meant to be a
teaching as well as an assessment aid. Further, you will not be allowed to take the
exam of this course till you submit your assignment response at your study centre. So
please submit it well in time.
The course material that we have sent you is self-sufficient. If you have a problem in
understanding any portion, please ask your academic counsellor for help. Also, if you
feel like studying any topic in greater depth, you may consult.
A word of friendly advice here is that to learn the various techniques presented in this
course, you will need to put in a lot of practice in solving problems given in the material.
You should attempt to solve all the exercises in the block as you go along, before you
look up the solution. As a part of the tutorials, we have added miscellaneous examples
and exercises at the end of each block. You should also attempt these exercises. In
addition, you may also like to look up some other books in the library of your study
centre, and try to solve some exercises from these books.
Some useful books and websites are the following:
1. Essential Calculus, by James Stewart, Cengage Publication.
2. Calculus, by Anton, Bivens and Davis, Wiley Publications.
3. https://brilliant.org/courses/calculus-done-right
4. https://www.mathisfun.com/calculus
5. www.math.mit.edu/Ndjk/calculus_beginners
We have also prepared a video programme, which will be available at your study
centre, called “Limits”, based on the material in Block 2.
Wishing you a happy learning experience,
The Course Team
4
