«
»
 

Catalogue

Advanced Mathematics for Form V


About the book

Advanced Mathematics – Form V is undoubtedly the first book addressing the 2009 syllabus (as reprinted in 2010 and 2013) in its entirety. This book covers calculating devices, sets, logic, coordinate geometry, functions, algebra, trigonometry, linear programming, differentiation and integration. There are 354 demonstrated examples. There are also over 1800 questions in different activities and exercises encouraging group discussions and individual evaluations. It is probably the best present to any student aspiring science and business career.  Teachers will find it an extremely useful tool.

Table of contents

TABLE OF CONTENTS

 

Acknowledgments....................................................................................... xiv

 

Calculating Devices ............................................................................. 1

Scientific Calculators................................................................................. 1

Familiarising with the scientific calculators........................................... 2

Scientific Calculator keys................................................................... 2

Working with a scientific calculators..................................................... 6

Making it Ready and Operating a Scientific Calculator...................... 6

Selecting the Appropriate Mode........................................................ 8

Cartesian and Polar Coordinates....................................................... 8

Polynomial Equations...................................................................... 10

Simultaneous Equations.................................................................. 11

Calculations on statistics................................................................. 13

Microsoft Computer Based Scientific Calculator................................. 15

Computer Packages............................................................................. 18

Introduction to computers.................................................................. 18

Excel Spreadsheet............................................................................. 20

Adding up Two or more Cells......................................................... 20

Calculating the Average................................................................. 21

Copying the Formulae................................................................... 22

Graphing the Polynomial Functions ............................................. 24

Statistical Computations by using Excel....................................... 26

The Maple package.......................................................................... 32

Addition and Subtraction using Maple........................................... 33

Multiplication and Division............................................................. 33

Solving Equation using Maple....................................................... 34

Solution of Systems of Equations................................................. 34

Plotting of 2-D plots...................................................................... 35

Evaluating the matrices by using Maple....................................... 37

Solving statistical problems by using Maple................................. 40

Statistical Product and Service Solutions (SPSS) ........................... 41

Working with SPSS....................................................................... 41

Measures of dispersion................................................................. 45

Quartiles and Percentiles.............................................................. 46

MATLAB package............................................................................. 48

Solving the System of Equations using MATLAB Package........... 48

Finding the Determinant and Inverse of the Matrix........................ 49

Solving Polynomial Functions in MATLAB..................................... 51

Plotting Graphs using MATLAB..................................................... 52

Mathematica Package....................................................................... 55

Initializing Mathematica ................................................................. 56

Mathematical Operations with Mathematica ................................. 56

Computing using Mathematica in-built functions ........................... 57

Solving Equations with Mathematica............................................. 58

Plotting of Functions...................................................................... 59

Salient Features of Different Computer Packages In relation to Specific Problems       62

Microsoft Excel  ............................................................................ 62

SPSS............................................................................................. 62

Matlab........................................................................................... 63

Maple............................................................................................ 63

Mathematica................................................................................ 63

Some common features on mathematical packages................... 64

 

Sets........................................................................................................ 66

Set...................................................................................................... 66

Definition of Set.............................................................................. 66

Types of Sets................................................................................. 67

Basic Operations of Sets................................................................... 68

Union of Sets................................................................................ 68

Union of Two Sets.................................................................... 68

Union of Three Sets.................................................................. 69

Intersection of Sets....................................................................... 72

Intersection of Two Sets............................................................ 72

Intersection of Three Sets.......................................................... 73

Complement of Sets...................................................................... 75

Properties of Complementary Sets............................................ 75

Solving Set Problems Involving Inequalities..................................... 76

Simplification of Set Expressions...................................................... 79

The Laws of Algebra of Sets.......................................................... 79

Idempotent Law........................................................................... 80

Commutative Law........................................................................ 80

Associative Law........................................................................... 80

Distributive Law........................................................................... 81

De-Morgan’s Law........................................................................ 82

Identity Law................................................................................. 82

Complement Law ....................................................................... 83

Proving Identities and Simplifying Set Expressions....................... 84

Number of Elements of a Set............................................................. 86

Formula for Finding Number of Elements of Sets.......................... 87

Number of Elements of Two Sets .............................................. 87

Number of Elements of Three Sets ........................................... 89

Using Venn Diagramme to Find Number of Elements of Set......... 91

Venn Diagrammes and Elements of Sets................................... 92

Importance of Venn Diagrammes in Representing Sets............. 97

 

Logic...................................................................................................... 101

Statements......................................................................................... 101

Describing Statements and Sentences.......................................... 101

Validity of Statements and Range of Variables ......................... 103

Defining Statement and Sentence............................................. 103

Distinguishing Between Statements and Sentences..................... 104

Simple and Compound Statements.............................................. 105

Logical Connectives......................................................................... 106

Types of Logical Connectives...................................................... 106

Conjunction.............................................................................. 106

Disjunction............................................................................... 107

Conditional Statement............................................................. 108

Bi Conditional Statement......................................................... 108

Negation of a Statement............................................................. 109

Representing the Logical Connectives in Symbolic Form.......... 110

Symbolizing the Statements with Conjunction........................ 110

Symbolizing the Statements with Disjunction.......................... 111

Symbolizing the Conditional Statement................................... 111

Symbolizing the Bi conditional Statements.............................. 112

Symbolizing the Negation of a Statement................................ 112

Statements with Logical Connectives and Negation of a Statement ................. 113

Constructing Truth Table for Simple and Compound Statements...115

Tabulating the Truth Values........................................................ 115

Adding a Conclusion Column in a Truth Table............................ 116

Tautologies, Contradictions and Equivalent Statements................ 120

Tautology.................................................................................... 120

Contradiction............................................................................... 121

Equivalent Statements................................................................ 123

Relevance of Truth Tables  ........................................................ 125

Converse, Inverse and Contrapositive of a Statement................... 128

Converse of a Statement............................................................. 128

Inverse of a Statement ................................................................ 128

Contra Positive of a Statement..................................................... 128

Laws of Algebra of Proposition............................................................. 129

The Proposition Laws of Algebra...................................................... 129

Commutative law........................................................................... 129

Associative law............................................................................. 129

Idempotent law............................................................................ 130

Distributive law............................................................................ 130

De-Morgan’s law.......................................................................... 130

Identity law................................................................................... 130

Complement law........................................................................... 130

Modulus law.................................................................................. 130

Conditional and Bi-conditional laws.............................................. 131

Simplifying Compound Statements Using Laws of Algebra of Proposition 131

Validity of Argument.............................................................................. 132

Valid and Non-valid Arguments........................................................ 132

Valid Argument ........................................................................... 132

Non – Valid Argument ................................................................. 134

Testing Validity of an Argument.................................................... 135

Electrical Networks................................................................................. 138

Relating the Electrical Switches to Statements........................................ 139

Relating the Series Connected Switches to Logical Connectives........... 140

Relating the Parallel Connected Switches to Logical Connectives......... 142

Representing the Compound Statements Using Electrical Networks..... 143

Simplifying an Electrical Network........................................................... 145

 

Coordinate Geometry 1.................................................................. 153

Rectangular Cartesian Coordinates...................................................... 153

Distance Between the Coordinates............................................................... 154

The Area of a Rectangle in Terms of the Coordinates of Vertices............ 156

Properties of a Parallelogram.................................................................. 157

The Angle between Two Lines................................................................. 159

Perpendicular Distance of a Point from a Line....................................... 161

The Equation of an Angle Bisector........................................................... 164

Solving Locus Problems........................................................................... 167

Application in Real Life............................................................................... 169

Ratio Theorem.......................................................................................... 171

Proof of the Ratio Theorem...................................................................... 173

For internal division of a line segment  by point  in the ratio   ... 173

For external division of a line segment  by point  in the ratio  ... 174

Application of Ratio Theorem in Solving Related Problems................... 177

Circles........................................................................................................ 178

Deriving the General Equation of a Circle............................................. 179

Equation of a Tangent and Normal to a Circle........................................ 182

Points of Intersection of Circles................................................................ 185

Condition for Circles Touching each other Externally  ..................................... 185

Condition for Circles Touching each other Internally....................................... 185

Circles Intersecting at Two Points  ............................................................. 187

Orthogonal Circles.................................................................................... 189

The Length of a Tangent From a Point.................................................... 192

 

Functions................................................................................................ 196

Graphs of Functions................................................................................ 197

Drawing Graphs of Polynomial Functions up to 4th Degree................. 197

Drawing Graphs ....................................................................................... 198

Graphs of Linear Functions ........................................................................ 198

Graphs of Quadratic Functions ................................................................... 200

Graphs of Cubic Functions.......................................................................... 203

Graphs of Quartic Functions........................................................................ 205

Graphs of Rational Functions................................................................... 207

Identification of Asymptotes of Rational Functions......................................... 207

Drawing the Graphs of Rational Functions .................................................... 212

Graphs of Composite Functions................................................................ 217

Graphs of Exponential Functions.............................................................. 220

Graphs of Logarithmic Functions............................................................. 224

 

Algebra...................................................................................................... 232

Indices and Logarithms........................................................................... 232

Index......................................................................................................... 232

Proofs of Laws of Logarithms.................................................................... 234

Solving Equations of Logarithms and Indices........................................... 236

Conversion of Logarithm From one Base to Another................................ 236

Series.......................................................................................................... 238

Finite and Infinite Series........................................................................... 238

Representing a Series in Sigma Notation................................................ 240

The Sum of the First  Squares and Cubic Numbers................................ 242

Sum of terms whose  term is .............................................................. 242

Sum of terms whose  term is ............................................................ 243

Proofs by Mathematical Induction....................................................... 246

The Principle of Mathematical Induction................................................. 247

Using the Principle of Mathematical Induction in Proving the Mathematical Statements         248

Roots of Polynomial Functions.............................................................. 250

Determining the Roots of Polynomial Function......................................... 250

Relationship Between Roots and Coefficients of Polynomials................... 251

Quadratic equation..................................................................................... 251

Cubic equation........................................................................................... 252

Forming Equations From Known Roots...................................................... 253

Quadratic Equations .................................................................................. 253

Cubic Equations ........................................................................................ 254

Remainder Theorem................................................................................ 256

Determining the Remainder of Polynomial Using Remainder Theorem.. 257

Determining the Remainder of Polynomial Using Synthetic Method  ..... 258

Inequalities................................................................................................ 260

Solution to Quadratic Inequality................................................................ 261

Solution to Inequalities Involving Rational Functions................................ 264

Solution to Absolute Value Inequalities...................................................... 267

Solving Absolute value inequalities using a graph ........................................... 269

Matrices...................................................................................................... 272

Determinant of a 33 Matrix.................................................................... 272

Using Cramer’s Rule in Solving Three Equations with Three Unknowns. 274

The Inverse of a 33 Matrix..................................................................... 278

Applying the Knowledge of Inverse of Matrix in Solving Simultaneous Equations with Three Unknowns................................................................................................................... 282

Binomial Theorem.................................................................................. 286

Developing the Pascal’s Triangle............................................................. 286

Using Pascal’s Triangle in Expanding Binomial Expression.................... 287

Using Factorial Notation to Express Binomial Theorem........................... 288

Defining the Factorial Notation.................................................................... 288

Applying Binomial Theorem to Solve Problems....................................... 289

Partial Fractions....................................................................................... 293

Using Partial Fractions in Decomposing a Rational Function.................. 293

Rational functions with Linear factor denominators......................................... 293

Rational Functions with Quadratic Factor Denominators.................................. 295

Rational Functions with Repeated Factor Denominators.................................. 297

Rational  functions  with  Degree  of  denominator  Less Than or Equal to that of Numerator        299

Applying Partial Fraction in Finding the Sum of Some Series.................. 300

 

 

 

Trigonometry......................................................................................... 305

Trigonometric Ratios.............................................................................. 305

The Functions of ,  and ............................................. 305

Reciprocals of Trigonometric Ratios......................................................... 306

Trigonometric Identities....................................................................... 308

Deriving Trigonometric Identities............................................................. 308

Proof of  ............................................................... 308

Proof of ............................................................. 308

Simplifying Trigonometric Identities......................................................... 309

Solving Trigonometric Equations.............................................................. 310

Proving Trigonometric Identities............................................................... 311

Properties of Sine and Cosines.................................................................... 312

Compound Angle Formula.................................................................... 314

Deriving the Compound Angle Formula................................................... 314

Deriving Compound Angle Formula for ....................................... 314

Deriving Compound Angle Formula for ...................................... 315

Proving Trigonometric Identities Involving Compound Angles................ 316

Simplifying Trigonometric Expressions Involving Compound Angles...... 317

Solving Trigonometric Equations Involving Compound Angles............... 318

Double Angle Formula........................................................................... 320

Deriving the Double Angle Formula from Compound Angle Formula.... 320

Using the Double Angle Formula in Proving Trigonometric Identities.... 321

Using the Double Angle Formulae in Simplifying Trigonometric Identities 322

Using the Double Angle Formulae in Solving Trigonometric Equations.. 323

Trigonometric Equations of the Form  ............ 324

Using -Formula in Solving the Equations of the Form  c 325

Expressing  in terms of .................................................................... 325

Expressing  in terms of ................................................................... 325

Expressing  as a Single Trigonometric Function........ 327

Expressing  in the Form ...................... 327

Expressing  in the Form ...................... 329

Using the Form  in Solving Trigonometric Equations of the Form            331

Finding the General Solution to equations of the form , Taking  and  as Sides of the Right angled Triangle......................................................................................... 333

Factor Formulae....................................................................................... 336

Deriving the Factor Formulae................................................................... 337

Proving the Identities Using the Factor Formulae.................................... 338

Simplifying Expressions Using Factor Formulae...................................... 340

Solving Trigonometric Problems Involving Factor Formulae.................. 340

Radians and Small Angles...................................................................... 342

Converting Degrees into Radians.............................................................. 342

Approximating the Small Angles.............................................................. 344

Approximation for  when .......................................................... 344

Approximation for  when .......................................................... 345

Approximation for  when ......................................................... 346

Trigonometric Functions....................................................................... 348

Domain and Range of Trigonometric Functions....................................... 348

Inverse of Trigonometric Functions.................................................... 349

Graphs of Trigonometric Function............................................................ 350

A graph of ........................................................................... 350

A graph of   ......................................................................... 350

A graph of ........................................................................... 351

The Graphs of the Inverse of Trigonometric Functions..................................... 351

Proving Equations Involving Inverse Trigonometric Functions................. 353

Solving Problems Involving Inverse Trigonometric Functions................. 354

 

Linear Programming......................................................................... 357

Common Terms Used in Linear Programming................................................ 357

Linear Programming Problems Formulation.................................... 358

Simple Steps for Formulation of Linear Programming Problems ....................... 359

Graphical Solution................................................................................... 362

Graphical Presentation of a Linear Programming Problem.................... 363

Finding the Maximum and Minimum Values Using the Objective Function 365

Transportation Problems....................................................................... 370

Mathematical Formulation of the Transportation Problem....................... 371

Solving Transportation Problems Graphically......................................... 376

 

Differentiation...................................................................................... 387

Derivatives................................................................................................ 387

Differentiating a Function from the First Principles.................................. 387

Differentiation of a Function................................................................. 391

Derivatives of a Polynomial Function....................................................... 391

Derivative of Product of Polynomials........................................................ 392

Differentiating a Polynomial Using the Chain Rule ......................................... 392

Differentiating a Polynomial Using Product Rule............................................. 393

Derivative of Quotient of Polynomials...................................................... 395

Derivation of the Quotient Rule ................................................................... 395

Derivatives of Trigonometric Functions and their Inverses...................... 398

Derivative of ................................................................................... 398

Derivative of ................................................................................... 400

Derivative of ................................................................................... 403

Derivatives of Reciprocals of Trigonometric Functions.................................... 405

Derivative of Inverse of Trigonometric Functions ........................................... 407

Derivative of Inverse of the Reciprocal of Trigonometric Functions .................. 410

Practicing with Computer Packages in Differentiating Polynomials and Trigonometric Functions        415

Differentiation using Maple ......................................................................... 415

Differentiation and integration using MATLAB ................................................ 416

Working out calculus problems using Mathematica........................................ 418

Derivative of Logarithmic Functions........................................................ 418

Derivative of Common Logarithmic Function ................................................ 419

Derivative of Natural Logarithmic Function ................................................... 420

Derivative of Exponential Functions.......................................................... 422

Expansion of  and its Derivative With Respect to x..................................... 422

Derivative of Exponential Functions of the Form ................................ 424

Application of Differentiation............................................................... 426

Solving Problems Involving Rates of Change........................................... 426

Small Change in Length, Area and Volume of Quantities ................................ 428

Determining the Turning Points and Points of Inflexion........................... 430

Guiding Steps in Determining the Stationary Points ....................................... 431

Sketching the Graphs Using the Intercepts and Turning Points............... 433

Real Life Problems Involving, Maximum and Minimum Values............... 435

Taylor’s Series and Maclaurin’s Series................................................. 439

Deriving the Taylor’s Theorem................................................................. 440

Deducing Maclaurin’s Theorem as a Special case of Taylor’s Theorem. 443

Introduction to Partial Derivatives....................................................... 445

Identifying the Functions of Two variables................................................ 445

Examples of functions of two variables ........................................................ 445

Finding the Partial Derivative of Functions of Two variables................... 445

 

Integration.............................................................................................. 450

Inverse of Differentiation ...................................................................... 450

Anti – derivative in Integral Notation ....................................................... 450

Integrating the Simple Expressions........................................................... 451

Integration of Functions......................................................................... 453

Integrals of Polynomials by Substitution Method...................................... 453

Integrals of Trigonometric Functions ........................................................ 455

Integrals of Exponential Functions ............................................................ 457

Integrals Relating to .............................................................................. 457

Integrals Relating to .............................................................................. 458

Integrals of Functions Using the Method of Integration by Parts ............. 459

Integrals of Logarithmic Functions ........................................................... 460

Integrals of the form ....................................................... 462

Integrals Containing the Powers of  and .................................. 464

Integrals Involving Multiples of Angles of  and  ....................... 465

Integrals Resulting to Inverse Trigonometric Functions ........................... 468

Integrals of the Form ............................................................... 468

Integrals of the Form ................................................................. 469

Integrals of the Form ............................................................. 471

Integrals of the Inverses of Trigonometric Functions................................. 473

Integrals of Functions of the Form , where  ........................... 475

Integrals of Functions Using Partial Fraction Methods  ........................... 477

Integrals of the Form , where  has Non-repeating Linear Factors 478

Integrals of the Form , where  has Repeating Linear Factors...... 479

Integrals of the Form , where  has Quadratic Factors................ 480

Integrals of the Form , where  has a Degree less than or equal to Degree of Numerator     481

Integrals of the form , where  does not Factorize 483

Evaluating Definite Integrals of Polynomials, Trigonometric Functions, Exponential and rational Functions  ................................................................................................................... 485

Application of Integration..................................................................... 489

The Area Under a Curve .......................................................................... 489

The Area Between Two Curves................................................................. 492

Applying the Computer Package - Maple to Solve for Derivatives, Integrals and Area under a Curve         495

Application of MATLAB in Integration............................................................ 496

Volumes of Solids of Revolution................................................................. 497

The Volume of Solid of Revolution About x - axis........................................... 497

The Volume of Solid of Revolution about the y–axis........................................ 499

The Volume of Solid of Revolution about any line............................................ 501

The Length of an Arc................................................................................. 504

The Length of an Arc Through Parametric Equations....................................... 506

The Arc Length in Polar Coordinates............................................................. 507

The Area of a Sector.................................................................................. 509

 

Answers to the Exercises............................................................. 514

 

Index........................................................................................................... 609

 

Additional References ................................................................... 622