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Catalogue

Advanced Mathematics VI


About the book

The book is based on the 2009 syllabus (as reprinted in 2010 and 2013).

It is prepared based on the competence approach as required by the MOEVT. In line with the syllabus, this volume covered: coordinate geometry, vectors, statistics, probability, hyperbolic functions, complex numbers, differential equations and numerical methods. 

There are 186 worked out examples and 1077 questions that seek to ensure that the student understands each aspect and tricks regarding examination questions. At the end of the book, answers to non-textual questions are given to allow self-evaluation.

Table of contents

TABLE OF CONTENTS

 

Acknowledgments...................................................................................... xi

 

Coordinate Geometry ......................................................................... 1

Conic sections..................................................................................... 1

Defining a Conic Section ..................................................................... 1

Locating the Conic Sections in a Cone ............................................... 2

Listing the Conic Sections ................................................................... 3

The parabola............................................................................................ 4

Derivation of an Equation of a Parabola............................................... 4

Sketching the Graph of a Parabola...................................................... 6

The length of Latus rectum of a Parabola ....................................... 9

Translated Parabolas..................................................................... 11

Equation of the Tangent to a Parabola.............................................. 14

Equation of the Normal to a Parabola................................................ 17

Parametric Equation of a Parabola .................................................. 19

The Ellipse........................................................................................... 19

The Equations of Directrix and Foci of the Ellipse ........................... 20

Derivation of the Equation of an Ellipse ........................................... 21

Sketching the Graph of Ellipse ........................................................ 23

The Translated Ellipse.................................................................. 24

Equations of Tangent and Normal to an Ellipse............................... 26

Parametric Equations of an Ellipse.................................................. 28

The Hyperbola..................................................................................... 29

Derivation of an Equation of a Hyperbola........................................ 29

The Translated Hyperbola .......................................................... 30

Sketching the Graphs of Hyperbolas ............................................. 32

Equations of the Tangent and Normal to a Hyperbola................ 33

Parametric Equations of a Hyperbola............................................. 35

Equations of Asymptotes to a Hyperbola....................................... 36

Polar Coordinates............................................................................. 38

Describing the Polar Coordinates of a Point ................................. 38

Relationship between Polar and Rectangular Coordinates........... 39

Changing Polar Equations into Rectangular Equations and vice versa .... 40

Graphs of Polar Equations............................................................. 41

The General Forms of Polar Curves.......................................... 44

 

Vectors..................................................................................................... 49

Vectors Representation....................................................................... 49

The Concepts of a Vector................................................................ 49

Representing Vectors in Two Dimensions....................................... 50

Basic properties of Vectors.......................................................... 50

Vectors in Two Dimension .......................................................... 53

Representing Vectors in Three Dimensional Space........................ 54

Unit vector in 3D Space................................................................... 56

Magnitude of a Vector in 3D Space ............................................ 56

Modulus of Unit Vector in a 3 Dimensional Space...................... 57

Ratio Theorem................................................................................ 59

Deriving the Ratio Theorem for Internal Division of a Line Segment..................... 59

Deriving the Ratio Theorem for External Division of a Line Segment. 61

Dot Product of Vectors............................................................................ 63

Special Properties of Dot Product of Vectors....................................... 63

The Angle Between Two Vectors....................................................... 65

Projection of a Vector onto Another Vector....................................... 66

Work done by a Force....................................................................... 68

Proving the  Rule Using Dot Product of Vectors.................... 70

Cross Product of Vectors...................................................................... 72

Defining the Cross Product of Vectors.............................................. 72

Calculating the Cross Product of Vectors......................................... 73

Special Properties of Cross Product of Vectors............................ 74

Expressing a Vector Product as a Determinant............................ 74

Determining the angle between Two Vectors.................................. 76

Calculating the Areas of Triangles and Parallelograms Using Cross Product of Vectors            78

The Area of a Triangle................................................................. 78

The Area of a Parallelogram............................................................................... 79

Proving the  Rule Using Cross Product of Vectors................... 81

Vector Differentiation and Integration................................................. 82

Differentiation of a Vector ............................................................... 82

Integration of a Vector..................................................................... 84

Expressing Velocity and Acceleration as Derivative of Displacement..... 86

 

Hyperbolic Functions........................................................................ 91

Hyperbolic Cosine and Sine..................................................................... 91

Defining the Hyperbolic Cosine and Sine........................................... 91

Sketching the Graphs of , and  ........................ 92

The Graph of ............................................................ 92

The Graph of .................................................................. 93

Sketching the Graphs of  and ............................. 94

The graph of ............................................................. 94

The Graph of .......................................................... 94

Drawing the Graphs of  and  by Using Computer Package .......... 95

Definition of , ,  and .......................... 95

Graph of ...................................................................... 96

Identities of Hyperbolic Functions.................................................. 97

Proofs of Some Hyperbolic Identities........................................... 98

Converting the Inverse Hyperbolic Functions into Logarithmic Functions 99

Derivative of Hyperbolic Functions....................................................... 101

Differentiating the Hyperbolic Functions........................................... 101

Derivative of ...................................................................... 101

Derivative of ..................................................................... 101

Derivative of Reciprocal of Hyperbolic Functions ........................ 101

Derivative of Inverse of hyperbolic Function ............................... 102

Additional Examples on Differentiating Hyperbolic Functions...... 103

The Series for Hyperbolic Cosine and Sine................................. 105

Defining the Hyperbolic Cosine Using Maclaurin’s Theorem ........ 105

Integration of Hyperbolic Functions.................................................. 106

Integrals of Hyperbolic Function..................................................... 106

Integrals of the forms  and .............................. 108

Standard Forms of the Integrals of the Results of the Derivatives of the Inverse Hyperbolic Functions      109

Integrals of the Inverse of Hyperbolic Functions.......................... 109

Useful Substitutions to Make When Integrating the Standard Forms of the Hyperbolic Expressions           110

Integrals of the Form ................................... 112

Integrals of the Form ......................................................... 113

 

Statistics

Definitions and Data Grouping Concept............................................ 117

Statistics.................................................................................................. 117

Definitions of Important Terms.................................................................... 117

Data Grouping.......................................................................................... 118

Measures of Central Tendency.............................................................. 119

The Mean................................................................................................. 119

Arithmetic Mean by Using the Coding Method....................................... 122

Deriving the Arithmetic Mean Formula.......................................................... 122

The Median............................................................................................... 125

The Mode.................................................................................................. 127

Measures of Dispersion.......................................................................... 130

The Quartiles............................................................................................ 130

The Percentiles......................................................................................... 134

Variance................................................................................................... 136

Mean Deviation ........................................................................................ 136

Formula for Variance.................................................................................. 137 

Standard Deviation.................................................................................. 139

Use of Standard Deviation in Solving Practical Problems................................. 142

 

Probability............................................................................................... 148

Defining Probability.............................................................................. 148

Counting Techniques for Sample Space............................................. 148

Stating the Fundamental Principle of Counting....................................... 149

Applying the Fundamental Principle of Counting in Solving Related Problems            150

Stating the Principle of Permutation........................................................ 151

The Formula for Permutation of  Objects taken  at a Time.................. 152

Applying the Principle of Permutation in Solving Related Problems..... 153

Stating the Principle of Combination....................................................... 155

The Formula for Combination of  Objects taken  at a Time................. 155

Applying the Principle of Combination in Solving Related Problems.... 156

Probability Axioms and Theorems....................................................... 158

Probability Axioms................................................................................... 158

Probability Theorems............................................................................... 158

Proving Probability Theorems Using Probability Axioms...................... 158

Properties of Probability........................................................................ 159

Probability of the Complement of an Event............................................. 159

Proving the Addition Laws of Probability................................................ 161

The First Rule of Addition........................................................................... 161

The Second Rule of Addition........................................................................ 162

Proving the Probability of the Complement of Union and Intersection of Two Events   162

The Rule of Multiplication of Probabilities.............................................. 164

Conditional Probability.......................................................................... 165

Defining the Concept of Conditional Probability.................................... 165

Calculating the Conditional Probability of Some Events......................... 165

Events that are Statistically Independent................................................. 168

Condition for Statistical Independence of Events........................................... 168

Probability Distributions....................................................................... 169

Defining a Discrete Random Variable.................................................... 169

Calculating the Expected Mean of a Discrete Random Variable .......... 169

Calculating the Variance of a Discrete Random Variable...................... 171

Calculating the Standard Deviation of a Discrete Random Variable...... 173

Probability Distribution for a Continuous Random Variable................... 174

Defining a Continuous Random Variable....................................................... 174

Probability Distribution for a Continuous Random Variable............................... 174

Probability Density Function................................................................... 175

Properties of the Probability Density Function................................................ 176

Evaluating Probability based on the Probability Density Function...................... 177

Calculating the Expected Mean and Standard Deviation of a Continuous Random Variable    179

Expected Mean of the Continuous Random Variable....................................... 179

Variance and Standard Deviation of a Continuous Random Variable................. 180

Applying the Probability Density Functions in Solving Related Problems 182

Some Special Probability Distributions.............................................. 185

The Normal Distribution.......................................................................... 185

Properties of the Normal Distribution Curve ................................................. 186

The Standard Normal Distribution................................................................ 187

The use of the standard normal tables.......................................................... 189

Applying the Normal Distribution in Solving Related Problems............ 191

Constructing the Binomial Distribution.................................................... 193

Constructing the Probability Distribution ....................................................... 194

Binomial distribution, when the number of trials is large................................... 195

Use of Binomial Distribution in Solving the Related Problems............... 198

The Poisson Distribution........................................................................... 200

Using the Poisson Distribution in Solving Related Problems.................. 201

 

Complex Numbers............................................................................. 206

Complex Numbers and their Operations ........................................... 206

Defining a Complex Number.................................................................. 206

Representing a Complex Number on the Argand Diagram................... 208

The Complex Conjugate of a Complex Number..................................... 209

The Modulus and Argument of a Complex Number................................ 210

The Modulus of a Complex Number ............................................................. 211

The Argument of a Complex Number ........................................................... 213

Performing the Operations on Complex Numbers.................................. 214

Addition and Subtraction of Complex Numbers.............................................. 214

Multiplication of Complex Numbers............................................................... 215

Division of Complex Numbers..................................................................... 216

Solving Complex Numbers Using Computer Packages......................... 218

Complex numbers in Maple......................................................................... 218

Solving the Complex Numbers by Using Matlab............................................. 219

Solving Equations with Complex Number Solutions................................ 220

Polar Form of a Complex Number........................................................ 222

Writing a Complex Number in Polar Form.............................................. 222

Representing  the  Complex  Number  in  Polar  Form  on  the  Argand Diagram        225

Describing the Loci of Complex Number Expressions Algebraically and by Sketches 226

The de Moivre's Theorem....................................................................... 228

Stating and Proving the De Moivre’s Theorem........................................ 229

Proof of de Moivre’s Theorem...................................................................... 229

Applying the De Moivre’s Theorem in evaluating the  Roots of a Complex Numbers          230

The Root of the Equation ................................................................ 230

Factors and Roots of the Polynomial  over the Complex Field. 231

Proving the Identities and Simplifying Expressions Using De Moivre’s Theorem          233

Application of De Moivre’s theorem in Binomial theorem.................................. 235

Use of De Moivre’s Theorem in Expressing Trigonometric Functions of Multiples of Angles in Terms of Powers      237

Euler’s Formula......................................................................................... 239

Deducing the Euler’s Formula.................................................................. 239

Applying the Euler’s Formula in Proving Identities and Simplifying Expressions           241

 

Differential Equations...................................................................... 245

Introduction to Differential Equations................................................. 245

Formulating a Differential Equation ........................................................ 245

The Order of a Differential Equation........................................................ 246

The Degree of a Differential Equation..................................................... 247

Distinguishing between Linear and Non-linear Differential Equations.. 248

Characteristics of Linear Differential Equations.............................................. 248

Characteristics of Non – Linear Differential Equations..................................... 248

Solution to a Differential Equation...................................................... 249

Verifying a Solution to a Differential Equation......................................... 250

Solving Simple Differential Equations by Direct Integration................... 251

First Order Differential Equations......................................................... 252

Differential Equation in Standard Differential Form................................. 252

First Order Differential Equations with Separable Variables................... 253

First Order Homogeneous Differential Equations..................................... 255

An Exact Differential Equation................................................................. 258

Proof of Exactness.................................................................................... 258

Determining the Integrating Factor......................................................... 259

Solving Non-Exact Differential Equations Using the Integrating factor.. 261

Applying First Order differential Equations in Solving Simple Real Life Problems        262

Application of First Order Differential Equations in Exponential Growth and Decay 262

Application of First Order Differential Equations in Cooling of Substances.......... 264

Application of First Order Differential Equations in Dilution .............................. 266

Second Order Differential Equations................................................... 267

Second Order Differential Equations of the Simple Form  ...... 267

Distinguishing between Homogenous and Non-homogenous Second Order Differential Equations    268

Determining the Auxiliary Equation for a Homogenous Second Order Differential Equation    269

Evaluating the Roots of Auxiliary Equation and Solution for Homogeneous Second Order Differential Equations.................................................................................................................. 271

Solving  the  Differential  Equation  whose  Auxiliary  Equation  has  Two  Distinct  Real Roots    271

Solving the Differential Equation whose Auxiliary Equation has a Unique Root.... 274

Solving the Differential Equation whose Auxiliary Equation has Complex Conjugate Roots           276

Second Order Differentials Reducible to First Order............................... 279

Non-homogeneous Second Order Linear Differential Equations............ 281

Applying  Second  Order  Differential  Equations  in  Solving  Real  Life Problems      283

In Vibrating Springs ................................................................................... 283

In Damped Vibrations ................................................................................ 285

In Electric Circuits .................................................................................... 287

Solving the Differential Equations by Using Computer Packages.......... 290

 

Numerical Methods........................................................................... 294

Errors.......................................................................................................... 294

Describing Types of Errors and their Sources.......................................... 295

Types of Errors ......................................................................................... 295

Sources of Errors ...................................................................................... 296

Distinguishing between Absolute and Relative Errors............................. 297

Effects Of Absolute And Relative Errors On Basic Operations................. 298

Error in Addition ........................................................................................ 298

Error in Subtraction ................................................................................... 298

Error in Multiplication ................................................................................. 299

Error in Division ........................................................................................ 300

Roots by Iterative Methods.................................................................... 301

Newton – Raphson Iterative Concept .......................................................... 302

Deriving the Newton – Raphson Iterative Formula................................ 303

Alternative Approach of Derivation of Newton – Raphson Formula.................... 304

Condition for existence of a root.................................................................. 304

Applying the Newton – Raphson Formula in Approximating the Root of a Function 305

Applying the Newton – Raphson Formula in Approximating the  Root of a Number           309

Applying the Newton – Raphson  Formula  in  Approximating  the Reciprocal  of a Number          310

Advantages of Newton – Raphson Method.................................................... 311

Drawbacks of Newton – Raphson Method.................................................... 311

Deriving the Secant Iterative Formula.................................................... 311

Applying the Secant Iterative Formula in Approximating the Root of a Function          313

Numerical Integration............................................................................ 315

The Trapezium Rule................................................................................. 315

Deriving the Trapezium Rule........................................................................ 315

Applying the Trapezium Rule in Evaluating the Definite Integrals........ 316

The Simpson’s Rule.................................................................................. 318

Deriving the Simpson’s Rule ...................................................................... 318

Applying the Simpson’s Rule in Evaluating the Definite Integrals.......... 320

 

Answers to the Exercises............................................................. 323

 

Normal Distribution Table............................................................. 372

 

Index........................................................................................................... 374

 

Additional References .................................................................. 383