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Catalogue

Basic Applied Mathematics Form 6


About the book

It is based on the 2009 MOEVT syllabus. The book seeks to demonstrate real life application of the mathematical theories for example: application of probability in research, business planning and gaming; application of trigonometry in measuring difficult distances and music; application of exponential and logarithmic functions in compound interest problems, population growth and rate of loss o f temperature of bodies; use of matrices in solving systems of linear equations and optimization o f use o f scarce resources through use of linear programming. This book is a product of highly skilled scholars on the subject. Author: Septine I Sillemu

Table of contents

 

TABLE OF CONTENTS

 

Acknowledgments........................................................................... vii

 

Probability ......................................................................................... 1

Permutations and Combinations............................................. 1

Arranging Objects.................................................................... 1

Factorial Notation..................................................................... 3

Permutations.............................................................................. 5

Combination............................................................................... 7

Probability of an Event................................................................ 9

Combined Events....................................................................... 11

Mutually and Non-Mutually Exclusive Events.................... 14

Independent and Dependent Events.................................... 17

Independent............................................................................. 17

Dependent Events.................................................................. 19

Application of Probability........................................................ 21

Trigonometry................................................................................... 25

Trigonometric Ratios................................................................ 25

The Angles Rotated in a Circle........................................... 26

Rotation by Angle Less Than 90°.................................. 26

Rotation by Angle Greater Than 90° But Less

Than 180°............................................................................. 27

Rotation by Angle Greater 180° But Less

Than 270°..............................................................................27

Rotation by Angle Greater 270°  But Less

Than 360°............................................................................. 28

Reciprocal of Trigonometric Functions............................32

Evaluating Trigonometric Functions Using

Spreadsheets.......................................................................... 33

Sine and Cosine Rules.............................................................. 36

The Sine Rule........................................................................... 36

The Cosine Rule...................................................................... 38

Trigonometric Identities........................................................... 41

Analogy of Trigonometric Ratios to Pythagorean Theorem................................................................................ 41

Other Trigonometric Identities....................................... 41

The Angle Formulae............................................................... 42

Double Angle Formulae........................................................ 46

The Double Angle Formula for  tan 2A........................ 47

Graphs of Trigonometric Functions....................................... 48

The Graph of y = sin x........................................................... 49

The Graph of y = tan x.......................................................... 49

Periodicity and Amplitude of Trigonometric Functions................................................................................ 51

Periodicity and Amplitude of Trigonometric Functions in Real Life............................................................................. 55

Calculus of Trigonometric Functions................................... 57

Derivatives of Trigonometric Functions and their Reciprocals................................................................................ 57

Derivatives of the Functions of the Forms sin f(x), cos f(x) and tan f(x) ....................................................................... 59

Integrals of trigonometric functions.................................. 61

The Integral of sin x.......................................................... 61

The Integral of  sec2 x....................................................... 62

Integrals of sin f(x), cos f(x) and sec2 f(x), Where f(x) is a Linear Function............................................................ 62

Exponential and Logarithmic Functions................................... 67

Exponential Functions................................................................ 67

Basic Properties of Exponential Functions....................... 68

The graphs of  y = ex ...................................................... 68

Calculus of Exponential Functions f(x) = ex....................... 70

Differentiation of Exponential Functions.......................... 71

Derivative of  f(x) = ex .................................................... 71

Derivative of  g(x) = ax.................................................... 71

Derivative of  y = eg(x) where g(x) is another function................................................................................... 71

Integration of Exponential Functions................................. 73

Logarithmic Functions................................................................ 75

Common Logarithms......................................................... 75

Natural Logarithms............................................................. 76

Graphs of Logarithmic Functions........................................ 79

Properties of Logarithmic Functions.............................. 80

Differentiation of Logarithmic and Exponential Functions .......................................................................................................... 81

Logarithmic Functions....................................................... 81

Exponential Functions ...................................................... 83

Integration of Functions of the Form f(x) = 1/(ax + b) ......................... 84

Application of Log and Exponential Functions............................ 87

Compound Interest........................................................................ 87

Use of Exponential Functions for Compound Interest Problems............................. 90

Depreciation................................................................................. 90

Use of Logarithms in Problems Related to Population Growth and Decay.................. 93

Use of Logarithms in Problems Related Cooling Processes ................................. 93

 

Matrices......................................................................................... 97

Introduction to Matrices............................................................... 97

The Size (Order) of Matrices......................................................... 98

Naming the Elements in a Matrix.................................................. 99

Types of Matrices....................................................................... 100

Operations with Matrices............................................................ 102

Addition and Subtraction of Matrices............................................ 102

Commutative and Associative Properties .................................................. 102

Multiplication of Matrices by a Scalar.......................................... 104

Multiplication of Matrix by another Matrix..................................... 105

Multiplying a 1x2 Matrix by a 2x1 Matrix .............................................. 105

Multiplying a 1x3 Matrix by a 3x1 Matrix .............................................. 106

Multiplying a 2x3 Matrix by a 3x1 Matrix.............................................. 106

Multiplying a 1x3 Matrix by a 3x2 Matrix.............................................. 106

Multiplying a 3x2 Matrix by a 2x1 Matrix .............................................. 107

Multiplying a 3x2 Matrix by 2x2 a Matrix............................................... 107

Multiplying a 3x3 Matrix by 3x1 a Matrix .............................................. 108

Multiplying a 3x3 Matrix by 3x2 a Matrix............................................... 108

Multiplying a 3x3 Matrix by a 3x3 Matrix .............................................. 109

No Commutative Properties in Multiplication ............................................... 109

Transpose of a Matrix................................................................. 112

Determinant of a Matrix............................................................. 113

Minors and Cofactors of a Matrix................................................. 115

Ad joint of a matrix.................................................................... 118

Identity and Inverse of a Matrix  .............................................. 118

Inverse of a Matrix..................................................................... 119

Identity of a Matrix...................................................................... 120

Matrix Solution of Systems of Linear Equations....................... 121

Representing Real Life Problems in Matrix Form.......................... 121

Solving Systems of Linear Equations using Inverse Matrix Method 124

Cramer’s Rule............................................................................. 126

Linear Programming................................................................. 133

Background................................................................................. 133

Common Terms Used in Linear Programming.............................................. 133

Real Life Problems for Linear Programming............................. 134

Mathematical Formulation of Linear Programming Problem (LPP).. 134

Choosing the Decision Variables........................................................... 135

Formulating the Constraints................................................................. 135

Formulating the Objective Function......................................................... 135

Graphs of Linear Inequalities..................................................... 137

Solutions to Linear Programming Problems............................... 144

Answers to the Exercises........................................................ 151

Index............................................................................................ 166

Additional References ............................................................ 171