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FEDERAL UNIVERSITY OF TECHNOLOGY, MINNA

CENTRE FOR OPEN DISTANCE AND e-LEARNING

CODeL

MTH 121MTH 121MTH 121

Differential And

Integral Calculus

FEDERAL UNIVERSITY OF TECHNOLOGY MINNA,

NIGER STATE, NIGERIA.

CENTRE FOR OPEN DISTANCE

AND e-LEARNING (CODeL)

B.TECH. COMPUTER SCIENCE

PROGRAMME

COURSE TITLE

DIFFERENTIAL AND INTEGRAL

CALCULUS

COURSE CODE

MAT 121

ii

Course Development Team

Subject Matter Expert(s) Abdulhakeem YUSUF (Ph.D.)

Department of Mathematics,

Federal University of Technology,

Minna, Nigeria.

Course Coordinator Bashir MOHAMMED (Ph.D.)

Department of Computer Science

FUT Minna, Nigeria.

Instructional Designers Oluwole Caleb FALODE (Ph.D.)

Bushrah Temitope OJOYE (Mrs.)

Centre for Open Distance & e-Learning,

Federal University of Technology,

Minna, Nigeria

ODL Experts Amosa Isiaka GAMBARI (Ph.D.)

Nicholas Ehikioya ESEZOBOR

Language Editors Chinenye Priscilla UZOCHUKWU (Mrs.)

Mubarak Jamiu ALABEDE

Centre Director Abiodun Musa AIBINU (Ph.D.)

Centre for Open Distance & e-Learning

FUT Minna, Nigeria.

iii

MAT 121 Study Guide

Introduction

MAT 121 Differential and Integral Calculus is a 3- credit unit course for students studying

towards acquiring a Bachelor of Technology in Mathematics and Statistics and other related

disciplines. The course is divided into 5 modules and 11 study units. It will first take a brief

review of Function of a real variable, limits and idea of continuity. This course will then go

ahead to deal with the derivative, as limit of change and techniques of differentiation. The

course went further to deal with extreme curve sketching, Integration as an inverse of

differentiation and methods of integration. The course concluded by dealing with definite

integrals, application to area and volume.

The course guide therefore gives you an overview of what the course; MAT 121 is all about,

the textbooks and other materials to be referenced, what you expect to know in each unit, and

how to work through the course materials.

Recommended Study Time

This course is a 3-credit unit course having 15 study units. You are therefore enjoined to spend

at least 3 hours in studying the content of each study unit.

What you are about to learn in this Course

The overall aim of this course, MAT 121 is to introduce you to the basic concepts of differential

and integral calculus and to enable students to have basic knowledge of differential and

integral calculus as it applied to their disciplines. This course highlights different methods of

solving differential and integral calculus problems:

The differentiation of inverse trigonometric, curve sketching method, the substitution methods,

method of integration by parts and further integration by parts, the use of reduction formula,

integration using method of trigonometric substitution, integration using partial fractions and

application of integration such as area under a curve, length of a curve and volume of

revolution.

Course Aims

The aim of this course is to introduce students to the basic concepts of differential and integral

calculus systems. It is believed that the knowledge will enable students understand the

functionalities and capabilities of differential and integral calculus because calculus is a

versatile branch of Mathematics employed as a very useful tool in the study of functions.

Several information about functions and the quantity they represent can be obtained by

techniques in calculus. The application of calculus to physical problems depends very much

on expressing physical quantities in terms of functions whose analysis gives the required

information about the quantities of interest. This makes the study of the theory of functions

essential in calculus. This subject of calculus itself is classified into two distinct parts;

(a) Differential calculus

(b) Integral calculus

Differential calculus is the study of rate of change of functions with respect to change in the

independent variable while Integral calculus is associated with summation of aggregate value

of functions as in the study of area and volume.

v

Unit 2 Integration as an Inverse of Differentiation

Module Five

Unit 1 Method of Integration

Module Six Unit 1 Definite Integrals

Unit 2 Application to Area and Volume

Recommended Texts

The following texts and Internet resource links will be of enormous benefit to you in learning

this course:

1. BLAKEY, J Intermediate Pure Mathematics, 5

th

Edition. Macmillan Press Limited.

London

2. BUNDAY, B.D Pure Mathematics for Advanced Level, Second Edition. Heinemann

Educational Books Limited, 1988. London

3. CLARKE, L.H Pure Mathematics at Advanced Level, Metric Edition. Heinemann

Educational Books Limited, 1977.London

4. (4) STROUD, K.A Engineering Mathematics, 4

th

Edition. Macmillan Press Limited, 1995.

London

5. STROUD, K.A Further Engineering Mathematics, 3rd Edition. Macmillan Press Limited,
   1995. London
6. (TRANTER, C.J And LAMBE, C.G Advanced Level Mathematics, Pure and Applied, 4

th

Edition Holder & Stoughton, 1979. Great Britain.

Assignment File

The assignment file will be given to you in due course. In this file, you will find all the details of

the work you must submit to your tutor for marking. The marks you obtain for these

assignments will count towards the final mark for the course. Altogether, there are tutor

marked assignments for this course.

Presentation Schedule

The presentation schedule included in this course guide provides you with important dates for

completion of each tutor marked assignment. You should therefore endeavour to meet the

deadlines.

Assessment

There are two aspects to the assessment of this course. First, there are tutor marked

assignments; and second, the written examination. Therefore, you are expected to take note

of the facts, information and problem solving gathered during the course. The tutor marked

assignments must be submitted to your tutor for formal assessment, in accordance to the

deadline given. The work submitted and an online test will count for 30% of your total course

mark.

At the end of the course, you will need to sit for a final written examination. This examination

will account for 70% of your total score. TUTOR-MARKED ASSIGNMENT (TMA)

There are TMAs in this course. You need to submit all the TMAs. When you have completed

each assignment, send them to your tutor as soon as possible and make certain that it gets to

vi

your tutor on or before the stipulated deadline. If for any reason you cannot complete your

assignment on time, contact your tutor before the assignment is due to discuss the possibility

of extension. Extension will not be granted after the deadline, unless on extraordinary cases.

Final Examination and Grading

The final examination for MAT 121 will last for a period of 3 hours and have a value of 70% of

the total course grade. The examination will consist of questions which reflect the Self-

Assessment Exercises and tutor marked assignments that you have previously encountered.

Furthermore, all areas of the course will be examined. It would be better to use the time

between finishing the last unit and sitting for the examination, to revise the entire course. You

might find it useful to review your TMAs and comment on them before the examination. The

final examination covers information from all parts of the course.

Practical Strategies for Working through this Course

1. Read the course guide thoroughly

2. Organize a study schedule. Refer to the course overview for more details. Note the time

you are expected to spend on each unit and how the assignment relates to the units.

Important details e.g. details of your tutorials and the date of the first day of the semester

are available. You need to gather together all this information in one place such as a

diary, a wall chart calendar or an organizer. Whatever method you choose, you should

decide on and write in your own dates for working on each unit.

3. Once you have created your own study schedule, do everything you can to stick to it. The

major reason that students fail is that they get behind with their course works. If you get

into difficulties with your schedule, please let your tutor know before it is too late for help.

4. Turn to Unit 1 and read the introduction and the objectives for the unit.
5. Assemble the study materials. Information about what you need for a unit is given in the

table of content at the beginning of each unit. You will almost always need both the study

unit you are working on and one of the materials recommended for further readings, on

your desk at the same time.

6. Work through the unit, the content of the unit itself has been arranged to provide a

sequence for you to follow. As you work through the unit, you will be encouraged to read

from your set books

7. Keep in mind that you will learn a lot by doing all your assignments carefully. They have

been designed to help you meet the objectives of the course and will help you pass the

examination.

8. Review the objectives of each study unit to confirm that you have achieved them.

If you are not certain about any of the objectives, review the study material and consult your

tutor.

9. When you are confident that you have achieved a unit’s objectives, you can start on the

next unit. Proceed unit by unit through the course and try to pace your study so that you

can keep yourself on schedule.

10. When you have submitted an assignment to your tutor for marking, do not wait for its

return before starting on the next unit. Keep to your schedule. When the assignment is

viii

Table of Contents

Course Development Team ................................................................................................ ii MAT 121 Study Guide ........................................................................................................ iii

• Module One……………………………………………………………………………………..…… Table of Contents viii
• Unit 1: Function Theory……………………………………………………………………………….
• Unit 2: Graphs…………………………………………………………………………………………
• Module Two………………………………………………...……………………………………….1
• Unit 1: Limit of a Function……………………………………………………………………….….1
• Unit 2: Differential Calculus………………………………………………………………………...1
• Module Three…………………………………………………………………………...…………..2
• Unit 1: Further Problems in Differentiation………………………………………………………..2
• Unit 2: Inverse and Parametric Function………………………………………………………….
• Module Four………………………………………………………………………………..……….
• Unit 1: Extreme Curve Sketching………………………………………………………………….
• Unit 2: Integration as an Inverse of Differentiation………………………………………………..
• Module Five………………………………………...……………………………………………….6
• Unit 1: Method of Integration……………………………………………………………………….6
• Unit 2 : Definite Integrals…………………………………………………………………………….
• Unit 3 : Application to Area and Volume……………………………………………………………
• Answers to Self-Assessment Exercises…………………………………………………………..

1

Module 1

Unit 1 Function Theory

Unit 2 Graphs

3

1.0 Introduction

It is very important to have the idea of what a function is in Mathematics before we can

undertake the study of calculus.

2.0 Learning Outcomes

At the end of this unit you should be able to:

1. Give the definition of a function.
2. Determine if a function is continuous or otherwise.
3. Know when a function tends to a certain limit say L.

3.0 Learning Content

3.1 Definition of a Function

Consider any two set X and Y. If there exist a mapping that mapped the set X to any subset A

of Y in which every element in X has an image in A and every element in A has a pre-image

in X, then the set X is the Domain of the map and Y is the co-domain. The set A constitute the

range of the function. This functional relation is formally represented as

y=f(x).....................................................................................................................................1.

That is, y is a function of the variable x called the independent variable and y the dependent

variable. If the function is a bijective mapping in which every element in Y has a pre-image in

X then there exists a unique mapping called the inverse mapping (function) given as

g(y)=x....................................................................................................................................1.

3.2 Continuous Function and Its Properties

The function f ( ) x defined in any domain D is said to be continuous at any point x

0

in D if given

 >0 then ( )  such that

0

f x f x Whenever − 

0

x x .........................................................................1.

The following are some of the common properties exhibited by continuous functions:

Suppose

f ( ) x

and

g ( ) x

are any two functions of x that are defined and continuous in any

domain D then the following are true.

( ) i ( f + g )( ) x = f ( ) x + g ( ) x

( ) ii For any two arbitrary constants  and 

( f + g )( ) x = f ( ) x   g ( ) x

( iii )( f  g )( ) x = f ( ) x  g ( ) x

( iv )

( ) =

















x

g

f

g ( ) x

f x

4

( ) v ( f • g )( ) x = f ( g ( ) x )

NOTE:

In the case of property (iv) above the function is undefined at the zero of g(x). And so a

restriction is always placed in the domain of definitions for all rational functions so as to isolate

the singularities from the domain of the resulting quotient function.

3.3 Limit of a Function

A function f(x) is said to tend to the limit L as x tends to a point x 0

if given

 0 

such

that

f ( ) x − L  

Whenever

0

x x

It is instructive from the above definitions that for any function

f ( ) x

to be continuous at

appoint x 0

the function must tend to f ( ) x

as x tends to the point x o.

it must however be noted

that the existence of a limit at appoint does not imply continuity at that point. If the point x o

is

a point of discontinuity of the function

f ( ) x

then the limit of the function ceases to be unique.

The limit thus becomes directional in the sense that the value of the limit now depends on the

direction we take it. The limit as we move from left to right differs from that obtained while

moving in the opposite direction. Hence, we have the left-hand limits and the right-hand limits.

NOTE:

If in a domain D a function is continuous at every point throughout the domain we say that the

function is continuous in D.

3.4 Examples of Functions

Below we consider some few examples to illustrate the above concepts:

Let ( ) 1

3

fu = u + and g ( ) u =cos 5 u ,then we have the following

( ) i ( )( ) ( ) ( ) 1 cos 5.

3

f  g u = f u  gu = u +  u

ii ( f g )( ) u f ( u ) g ( u ) ( u 1 )cos 5 u

3

( iii )

( )

( )

( )

( )

( )

( u ) u

u

u

gu

f u

u

g

f

1 sec 5

cos 5

1

3

3

= +

= =

















(Valid for cos5u 0, i.e. u p Z

p

 )

( iv )

( )( ) ( ( )) ( cos 5 ) 1 cos 1

3

3

f • g u = f gu = u + = u +

( ) v

( )( ) ( ( )) cos 5 ( 1 )

3

g • f u = f gu = u +

The examples in (iv) and (v) above could be used to establish that composition of functions

as earlier defined is not in general commutative.

Recall that a binary operator • defined over a set is said to be commutative if given any two

elements and  of  such that • =•

6

Unit 2

Graphs

Contents

1.0 Introduction

2.0 Learning Outcomes

3.0 Learning Content

3.1 Graphs

3.2 Linear Functions

3.3 Quadratic Functions

3.4 Intercepts

3.5 Slope(Gradient/Tangent)

3.6 Symmetry

3.7 Limiting Values

4.0 Conclusion

5.0 Summary

6.0 Tutor-Marked Assignments

7.0 Reference/Further Reading

7

1.0 Introduction

In other to undergo the study of calculus in Mathematics, possession of the knowledge of

graphs such as that of linear and quadratic is something that can never be over emphasize.

So therefore, in this unit, you will be introduced to graphs and some of its components such

as slope and intercept.

2.0 Learning Outcomes

At the end of this unit you should be able to identify:

1. the graph of a linear function.
2. the graph of a quadratic function.

3.0 Learning Content

3.1 Graphs

This is a diagram showing the relationship between a dependent variable y and independent

variable x.

3.2 Linear Functions

The general expression for a linear function of a variable x is give

y =  x +

Where  is a

constant referred to as the gradient (slope) of the function

y

and

the intercept of the graph

on the y axis.

We note here that  corresponds with the value of y when x = 0 .This gives the intercept of

the graph on y-axis. Hence the equation of y-axis is given as x = 0. whereas the equation of

x-axis is given as

y = 0.

Now for any given point( )

0 0

x , y to lie on the straight line we must have that

−

−

0

0

x x

y y

.

Hence the equation of the line that passes through the point ( ) 0 0

x , y with gradient m is given

as ( )

0 0

y = x + y −  x

Hence the intercept of the line on the y-axis is given as ( )

o o

y = y −  x

We note here that the intercept of any graph on the x-axis corresponds with the zero of the

function that is the roots of the equation obtained by setting the function to zero.

3.2 Quadratic Functions

The function

f x is said to be a quadratic function of x if

f x is of the form

2

f x =  x + x + 

9

a) A Graph- is a diagram showing the relationship between a dependent variable

y

and

independent variable x.

b) A Linear function is y =  x +Where is a constant referred to as the gradient (slope)

of the function y and  the intercept of the graph on the y axis.

c) Quadratic functions as ( ).

2

f x =  x + x + 

d) Intercepts.

e) Slope.

f) Symmetry.

g) Limiting value.

6.0 Tutor-Marked Assignments

1. Check if the following function are symmetry or not:

a.

2

y = x b.

2

y =− x

2. Define the following in your own way:

a. Graph

b. Gradient

c. Symmetry

d. Limiting values.

7.0 Reference/Further Reading

BLAKEY, J Intermediate Pure Mathematics, 5

th

Edition. MacMillan Press Limited.

London

BUNDAY, B.D Pure Mathematics for Advanced Level, Second Edition. Heinemann

Educational Books Limited, 1988. London

CLARKE, L.H Pure Mathematics at Advanced Level, Metric Edition. Heinemann Educational

Books Limited, 1977.London

10

Module 2

Unit 1 Limit of a Function

Unit 2 Differential Calculus