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1 Elementary algebra 1

Introduction 1

1.1 Algebraic expressions and equations 1

1.2 The addition and subtraction of algebraic forms 3

1.3 Products of positive and negative real numbers 6

1.4 Expansion of bracketed terms 7

1.5 Fractions 10

1.6 Exponents 13

1.7 Negative exponents 18

1.8 Cancelling out terms 19

1.9 The order and hierarchy of operations 20

1.10 Factorization 22

1.11 Degree of an expression 27

1.12 Perfect squares 28

1.13 Applications 29

Additional examples 32

2 Solving equations 35

Introduction 35

2.1 Drawing graphs 36

2.2 Straight-line or linear functions 36

2.3 Quadratic functions 38

2.4 Cubic fUnctions 41

2.5 Algebraic solution of equations 43

2.6 Equations involving fractions 44

2.7 QUadratic equations 45

2.8 Formula for solving quadratic equations 46

2.9 Solution of cubic equations 48

2.10 APPlications 50

Additional examples 54

3 Simultaneous equations and inequalities 57

Introduction 57

3.1 Simple equations with one variable 58

3.2 Pairs of equations 58

3.3 Using a set of equations as a model 61

3.4 Sets of three or more equations 64

3.5 Independent and dependent equations 69

3.6 Linear and non-linear equations 70

3.7 Inequalities 73

3.8 Simultaneous inequalities 79

3.9 Applications of inequalities 81

Additional examples 86

4.1 Arithmetic progression(AP) 89

4 Series 89

Introduction 89

4.2 The sigma notation for summation 93

4.3 Sum of terms of an AP 96

4.4 Geometric progression(GP) 100

4.5 Sum of terms of a GP 101

4.6 Notation for interest calculations 103

4.7 Compound interest 104

Additional examples 107

5 Logarithms and exponentials 109

Introduction 109

5.1 Logarithms and exponents 109

5.2 How logarithms work 112

9.2 Rules for integration 115

5.3 Rules for combining logarithms 118

5.4 The exponential function and continuous compounding 119

5.5 Nominal interest rates and effective interest rates 123

5.6 Negative growth 124

5.7 Application 129

Additional examples 132

6 Matrices 134

Introduction 134

6.1 Matrix notation 135

6.2 Equality, addition and subtraction of matrices 138

6.3 Multiplication of matrices 140

6.4 Transposing matrices 148

6.5 Matrix formulation of simultaneous equations 151

6.6 The identity matrix and the inverse 154

6.7 Determinants 157

6.8 The inverse of a 2×2 matrix 159

6.9 Summary 161

Additional examples 162

7 Differentiation 164

Introduchon 164

7.1 The slope of a straight line 166

7.2 Finding the equation of a straight line 168

7.3 A numerical method for finding the slope of a curve 169

7.4 The general method of differentiation 174

7.5 Rules for derivatives 175

7.6 The derivative of the reciprocal of a function 178

Additional examples 181

8.1 The second and higher derivatives 183

Introduction 183

8 More about differentiation 183

8.2 Alternative notation for the derivative 186

8.3 Maxima and minima 187

8.4 Points of inflexion 192

8.5 The function of a function rule 194

8.6 The product rule 199

8.7 Mixing the function of a function and product rules 203

8.8 Differentiating expressions containing fractions 204

8.9 Continuous functions 206

8.10 Partial derivatives 208

Additional examples 209

9 Integration 212

Introduction 212

9.1 Integration as the reverse of differentiation 213

9.3 The definite integral 217

9.4 The integral as the area between the curve and the x-axis 219

9.5 A general remark on integration and differentiation 223

Additional examples 224

10 The application of mathematics 228

10.1 Mathematical style 228

10.2 Tackling mathematical examination questions 229

10.3 Formulating real-life problems 230

10.4 Solving real-life problems 230

Appendix The Greek alphabet 232

Solutions to ad■ examples 233

Index 263