Lecture notes ENGINEERING COURSES (MATE1A1)

1. ALGEBRA

INTRODUCTION


When conducting a mathematical investigation, it is a normal practice to assign an algebraic
symbol to the quantity whose value is sought, either numerically or as an explicit algebraic
expression. Thereafter, we apply a combination of known laws, consistency conditions and
given constraints to derive equations satisfied by the unknown quantity. These rules and
equations may take many forms, ranging from simple polynomial equations, binomials to
complex systems of equations and differential equations. In an attempt to solve these, one
may use the theory of matrices and determinants to solve them. The binomial expansion is
used when considering functions containing powers of the sum or difference of two terms

e.g. a  b  . The power series is then written as a polynomial as a sum of terms each of
n



which contains powers of a and b separately, as opposed to a power of their sum or
difference.




LEARNING OUTCOMES


On completion of this chapter, you will be able to:

 Define functions.
 Draw free-hand sketches of straight lines and conical sections such as the parabola,
hyperbola, the circle and also combinations of these on one set of axes.

 Raise a binomial of the form a  bn , where n is a positive or negative integer.
 Solve systems of equations, comprising two or three variables, using determinants.




COMPILED BY T. PAEPAE

,TABLE OF CONTENTS

1. ALGEBRA ..................................................................................................................... 0


1.1 FUNCTIONS ........................................................................................................... 2
1.1.1 Defining Functions by Equations ...................................................................... 3

1.1.2 Functional Notation and Substitution................................................................ 8

1.1.3 Operations on Functions; Composition .......................................................... 11

1.1.4 Inverse Functions .......................................................................................... 17


1.2 ALGEBRAIC GRAPHS ........................................................................................ 23
1.2.1 Cartesian Plane ............................................................................................. 24

1.2.2 The Straight Line ........................................................................................... 25

1.2.1 Conic Sections ............................................................................................... 34


1.3 THE BINOMIAL THEOREM ................................................................................. 59
1.3.1 The Binomial Series ....................................................................................... 60

1.3.2 The Binomial Theorem for a Natural Number Exponent ................................. 60

1.3.3 The Binomial Theorem for Rational and Negative Exponents ........................ 73


1.4 THE THEORY OF MATRICES AND DETERMINANTS ........................................ 77
1.4.1 Matrix Notation............................................................................................... 78

1.4.2 Second-Order Determinants .......................................................................... 79

1.4.3 Third-Order Determinants .............................................................................. 80

1.4.4 Using Cramer’s Rule to Solve Systems of Equations ..................................... 85




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,1.1 FUNCTIONS



Why it is important to understand: Functions

“Mathematics is concerned with relationships between things, and it is through the generality
of these relations that applications arise. Engineers, whether electrical, mechanical,
chemical, metallurgy, etc. are concerned with expressing relationships between physical
quantities clearly and unambiguously. This might be the relationship between the
displacement of an oscillating object and time, or perhaps the amplitude of an AC voltage
and time. Functions can be used to describe the way quantities change; hence, we need
functions to handle practical problems analytically. This unit is about how we can represent
such relationships in mathematical terms”. Bird, J., 2017. Higher engineering mathematics.
Routledge.



SPECIFIC OUTCOMES


On completion of this study unit, you will be able to:

 Define a function.
 Define functions by equations.
 Substitute into a function.
 Apply functional notation to evaluate combinations of functions.
 Apply functional notation to answer practical questions.
 Formulate the inverse of a function.




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, INTRODUCTION


One of the most basic and important ways that mathematics can be applied to other areas of
study is the establishment of correspondence among various types of phenomena. In many
cases, once a correspondence is known, it can be used to make important decisions and
predictions. The following expression x( x  1) contains a single variable x , but many
everyday life relations involve two or more variables. Common examples include:


 The area of a circle ( A ) depends on the radius of that circle ( r )
 The time taken for a particular journey ( t ) depends on the average speed ( s )
 Shoe size ( y ) depends on how big the foot is ( x )
 Your final mark ( m ) depends on how well you study ( n )


In these examples, the two variables in each case are related and we shall define a function
as a special type of relation between two variables, and this will lead to the powerful and
convenient functional notation which plays an important role in mathematics.


1.1.1 Defining Functions by Equations

Suppose driving a car that averages 100 km h . The distance travelled is determined by the

time travelled.

Distance (D)  Speed (S)  Time (t)

This relation can be expressed by the equation:

D  100 t

For t  2 hours, the distance travelled will be:

D  100 (2)  200 km

 For each specific value of t  0 , the equation produces exactly one value for D
 This correspondence between the distance D and the time t is an example of a
functional relationship.
 Being more specific, we say that the equation D  100 t defines D as a function of
t because for each chosen value of t , there is exactly one corresponding value
for D .



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