Essential Mathematics for Economics and Business

2013·20.76 MB·English

by  Teresa Bradley

Front Matter Page: ix
Contents Page: ix
Introduction Page: xiii
An Approach to Learning Page: xiii
1. Goals Page: xiii
2. Plan of Action to Understand and Become Competent in the Material Covered Page: xiii
WORKED EXAMPLE 3.12 TAXES AND THEIR DISTRIBUTION Page: xiv
3. Test whether goals were achieved Page: xv
Structure of the Text Page: xv
WileyPLUS Page: xvi
WileyPLUS Tools for Instructors Page: xvi
WileyPLUS Resources for Students within WileyPLUS Page: xvi
Ancillary Teaching and Learning Materials Page: xvii
1 Mathematical Preliminaries Page: 1
Chapter Objectives Page: 1
1.1 Some Mathematical Preliminaries Page: 2
Accuracy: rounding numbers correct to x decimal places Page: 2
1.2 Arithmetic Operations Page: 3
Addition and subtraction Page: 3
WORKED EXAMPLE 1.1 ADDITION AND SUBTRACTION Page: 4
Multiplication and division Page: 4
WORKED EXAMPLE 1.2a MULTIPLICATION AND DIVISION Page: 4
Remember Page: 5
Remember Page: 5
Remember Page: 5
WORKED EXAMPLE 1.2b Page: 6
1.3 Fractions Page: 6
1.3.1 Add/subtract fractions: method Page: 6
WORKED EXAMPLE 1.3 ADD AND SUBTRACT FRACTIONS Page: 7
1.3.2 Multiplying fractions Page: 8
WORKED EXAMPLE 1.4 MULTIPLYING FRACTIONS Page: 8
1.3.3 Dividing by fractions Page: 8
WORKED EXAMPLE 1.5 DIVISION BY FRACTIONS Page: 9
Reducing a fraction to its simplest form and equivalent fractions Page: 9
PROGRESS EXERCISES 1.1 Page: 10
1.4 Solving Equations Page: 11
Methods for solving equations Page: 11
WORKED EXAMPLE 1.6 SOLVING EQUATIONS Page: 11
WORKED EXAMPLE 1.7 SOLVING A VARIETY OF SIMPLE ALGEBRAIC EQUATIONS Page: 13
1.5 Currency Conversions Page: 14
Table 1.1 Euro exchange rates Page: 14
WORKED EXAMPLE 1.8 CURRENCY CONVERSIONS Page: 15
PROGRESS EXERCISES 1.2 Page: 17
1.6 Simple Inequalities Page: 18
Inequality symbols Page: 18
The number line Page: 18
Figure 1.1 Number line, numbers increasing from left to right Page: 18
Figure 1.2 The inequality, x > 2 Page: 19
Intervals defined by inequality statements Page: 19
Manipulating inequalities Page: 19
Solving inequalities Page: 20
WORKED EXAMPLE 1.9 SOLVING SIMPLE INEQUALITIES Page: 20
Figure 1.3 x > 22 Page: 20
Figure 1.4 x < −5 and x > 0 is not possible Page: 20
Figure 1.5 0 < x ≤ 3 Page: 21
1.7 Calculating Percentages Page: 21
WORKED EXAMPLE 1.10 CALCULATIONS WITH PERCENTAGES Page: 22
PROGRESS EXERCISES 1.3 Page: 23
1.8 The Calculator. Evaluation and Transposition of Formulae Page: 24
1.8.1 The Calculator Page: 24
1.8.2 Evaluating formulae using the calculator Page: 24
WORKED EXAMPLE 1.11 EVALUATION OF FORMULAE Page: 25
1.8.3 Transposition of formulae (Making a variable the subject of a formula) Page: 26
WORKED EXAMPLE 1.12 TRANSPOSITION AND EVALUATION OF FORMULAE Page: 27
1.9 Introducing Excel Page: 28
Figure 1.6 Cell reference on a spreadsheet Page: 28
WORKED EXAMPLE 1.13 USING EXCEL TO PERFORM CALCULATIONS AND PLOT GRAPHS Page: 29
Figure 1.7 Data for Worked Example 1.13 entered on a spreadsheet Page: 30
Figure 1.8 a Type the formula = B3*B4 into cell B5 to calculate pay = 165 for J.M. Page: 30
Figure 1.8b Preparing to drag and drop the formula in cell B5 across row 5 Page: 31
Figure 1.9 Highlight the data required for graph plotting, then click on ‘Insert’ on the main menu bar: select ‘Column’ to plot a bar or column graph Page: 31
Figure 1.10 Basic bar chart without titles Page: 31
Figure 1.11 Chart tools: use these to add titles and format your graph Page: 31
Figure 1.12 A very basic plot of weekly pay for seven members of staff Page: 31
PROGRESS EXERCISES 1.4 Page: 33
www.wiley.com/college/bradley Page: 34
TEST EXERCISES 1 Page: 34
2 The Straight Line and Applications Page: 37
Chapter Objectives Page: 37
Table 2.1 Price and quantity observations Page: 38
Figure 2.1 Graph of quantity demanded (Q) at set prices (P) Page: 38
2.1 The Straight Line Page: 38
2.1.1 The straight line: slope, intercept and graph Page: 39
Introductory background on graphs Page: 39
Figure 2.2 Plotting points on a graph Page: 39
How to define a straight line Page: 39
Slope Page: 40
Figure 2.3 Lines with different slopes and different intercepts Page: 40
Measuring the slope of a line Page: 40
Figure 2.4 Measuring slope Page: 40
WORKED EXAMPLE 2.1 PLOTTING LINES GIVEN SLOPE AND INTERCEPT Page: 42
Figure 2.5 45° line through the origin Page: 42
Figure 2.6 Line with slope = 1, intercept = 2 Page: 43
PROGRESS EXERCISES 2.1 Page: 43
2.1.2 The equation of a line Page: 43
WORKED EXAMPLE 2.2 DETERMINE THE EQUATION OF A LINE GIVEN SLOPE AND INTERCEPT Page: 44
Figure 2.7 The formula y = 1 + 3x calculates the y-coordinate for any given value of x on the line; y = 1 + 3x is called the equation of the line Page: 44
Figure 2.8 Graph of the line y = x Page: 45
Figure 2.9 Comparing the lines y = x and y = x + 2 Page: 46
Horizontal intercepts Page: 46
WORKED EXAMPLE 2.3 CALCULATION OF HORIZONTAL AND VERTICAL INTERCEPTS Page: 47
Equation of horizontal and vertical lines Page: 47
Figure 2.10 Equation of horizontal and vertical lines Page: 47
For any straight line Page: 48
2.1.3 To graph a straight line from its equation Page: 48
To graph a straight line when its equation is given in the form y = mx + c Page: 48
WORKED EXAMPLE 2.4 TO GRAPH A STRAIGHT LINE FROM ITS EQUATION y = mx + c Page: 49
Table 2.2 Calculating the y-coordinate given the x-coordinate Page: 49
Figure 2.11 Plot the line y = 2x − 1 Page: 49
To graph a straight line given its equation in the form ax + by + d = 0 Page: 50
WORKED EXAMPLE 2.5 PLOT THE LINE ax + by + d = 0 Page: 51
Figure 2.12 Plotting the line y = −2x + 4 Page: 52
Some mathematical notations Page: 53
General notations Page: 53
PROGRESS EXERCISES 2.2 Page: 54
2.2 Mathematical Modelling Page: 54
2.2.1 Mathematical modelling Page: 55
Figure 2.13 The place of mathematical modelling in the scheme of modelling Page: 55
Figure 2.14 Steps in mathematical modelling Page: 56
Suggested steps in the construction of a mathematical model Page: 56
2.2.2 Economic models Page: 57
Circular flow of economic activity Page: 58
Figure 2.15 The circular flow model Page: 58
2.3 Applications: Demand, Supply, Cost, Revenue Page: 59
2.3.1 Demand and supply Page: 59
The demand function Page: 59
Figure 2.16 Demand functions (a) Q = f(P) and (b) P = g(Q) Page: 60
The equation of the demand function Page: 61
Figure 2.17 Demand function P = a − bQ Page: 61
WORKED EXAMPLE 2.6 LINEAR DEMAND FUNCTION Page: 62
Table 2.3 Demand schedule Page: 63
Figure 2.18 Demand function P = 100 − 0.5Q Page: 63
Figure 2.19 Demand function Q = 200 − 2P Page: 63
The supply function Page: 64
The equation of the supply function Page: 64
Figure 2.20 Supply function P = c + dQ Page: 65
WORKED EXAMPLE 2.7 ANALYSIS OF THE LINEAR SUPPLY FUNCTION Page: 65
Figure 2.21 Supply function Q = −10 + 2P Page: 66
Figure 2.22 Supply function P = 5 + 0.5Q Page: 67
WORKED EXAMPLE 2.8 LINEAR SUPPLY FUNCTION 2 Page: 67
Table 2.4 Supply schedule Page: 68
Figure 2.23 Supply function P = 10 + 0.5Q Page: 68
PROGRESS EXERCISES 2.3 Page: 68
2.3.2 Cost Page: 70
WORKED EXAMPLE 2.9 LINEAR TOTAL COST FUNCTION Page: 71
Table 2.5 Fixed, variable and total costs Page: 71
Figure 2.24 Linear total cost function Page: 72
2.3.3 Revenue Page: 72
WORKED EXAMPLE 2.10a LINEAR TOTAL REVENUE FUNCTION Page: 73
Table 2.6 Total revenue Page: 73
Figure 2.25 Linear total revenue function Page: 73
2.3.4 Profit Page: 74
WORKED EXAMPLE 2.10b LINEAR PROFIT FUNCTION Page: 74
Table 2.7 Page: 74
Figure 2.26 Linear profit function Page: 75
PROGRESS EXERCISES 2.4 Page: 75
2.4 More Mathematics on the Straight Line Page: 76
2.4.1 Calculating the slope of a line given two points on the line Page: 76
WORKED EXAMPLE 2.11 CALCULATING THE SLOPE GIVEN TWO POINTS ON THE LINE Page: 77
Figure 2.27 Measuring the slope given two points Page: 77
2.4.2 The equation of a line given the slope and any point on the line Page: 78
WORKED EXAMPLE 2.12 EQUATION OF A LINE GIVEN THE SLOPE AND A POINT ON THE LINE Page: 78
Table 2.8 Calculating points on the line, y = 1.6 + 1.7x Page: 78
Figure 2.28 Graph of line y = 1.6 + 1.7x Page: 79
2.4.3 The equation of a line given two points Page: 79
WORKED EXAMPLE 2.13 EQUATION OF A LINE GIVEN TWO POINTS ON THE LINE Page: 79
Table 2.9 Calculating points on the line y = 5.5 − 0.75x Page: 80
Figure 2.29 Graph of line y = 5.5 − 0.75x Page: 80
Summary Page: 81
PROGRESS EXERCISES 2.5 Page: 81
2.5 Translations of Linear Functions Page: 82
www.wiley.com/college/bradley Page: 82
2.6 Elasticity of Demand, Supply and Income Page: 83
2.6.1 Price elasticity of demand Page: 84
Remember Page: 84
Point elasticity of demand Page: 85
WORKED EXAMPLE 2.19 DETERMINING THE COEFFICIENT OF POINT ELASTICITY OF DEMAND Page: 85
Figure 2.35 Variation of price elasticity of demand with price along the demand function P = 2400 − 0.5Q Page: 88
Coefficient of price elasticity of demand Page: 88
Figure 2.36 Numerical scale for the coefficient of price elasticity of demand Page: 89
www.wiley.com/college/bradley Page: 89
Point elasticity of demand depends on price and vertical intercept only Page: 89
Arc price elasticity of demand Page: 90
Price elasticity of supply Page: 90
www.wiley.com/college/bradley Page: 90
PROGRESS EXERCISES 2.7 Page: 91
2.7 Budget and Cost Constraints Page: 91
www.wiley.com/college/bradley Page: 92
2.8 Excel for Linear Functions Page: 92
Figure 2.44 Excel sheet Page: 92
WORKED EXAMPLE 2.24 USE EXCEL TO SHOW THE EFFECT OF PRICE AND INCOME CHANGES ON A BUDGET CONSTRAINT Page: 93
Figure 2.45 Budget constraint: skating versus pool Page: 94
Figure 2.46 Budget constraint: skating versus pool Page: 94
Figure 2.47 Budget constraints and the decrease in the price of skating Page: 95
Figure 2.48 Budget constraints and the decrease in pocket money Page: 95
PROGRESS EXERCISES 2.9 Page: 96
2.9 Summary Page: 97
Mathematics Page: 97
Applications Page: 98
www.wiley.com/college/bradley Page: 99
TEST EXERCISES 2 Page: 99
3 Simultaneous Equations Page: 101
Chapter Objectives Page: 101
3.1 Solving Simultaneous Linear Equations Page: 102
Reminder Page: 102
3.1.1 Two equations in two unknowns Page: 102
WORKED EXAMPLE 3.1 SOLVING SIMULTANEOUS EQUATIONS 1 Page: 102
Figure 3.1 Unique solution Page: 103
WORKED EXAMPLE 3.2 SOLVING SIMULTANEOUS EQUATIONS 2 Page: 104
Figure 3.2 Unique solution Page: 105
3.1.2 Solve simultaneous equations by methods of elimination and substitution Page: 105
The method of elimination Page: 105
The method of substitution Page: 106
WORKED EXAMPLE 3.3 SOLVING SIMULTANEOUS EQUATIONS 3 Page: 106
3.1.3 Unique, infinitely many and no solutions of simultaneous equations Page: 107
Unique solution Page: 107
No solution Page: 107
WORKED EXAMPLE 3.4 SIMULTANEOUS EQUATIONS WITH NO SOLUTION Page: 107
Figure 3.3 No solution Page: 108
Infinitely many solutions Page: 108
WORKED EXAMPLE 3.5 SIMULTANEOUS EQUATIONS WITH INFINITELY MANY SOLUTIONS Page: 108
Figure 3.4 Infinitely many solutions Page: 109
3.1.4 Three simultaneous equations in three unknowns Page: 109
WORKED EXAMPLE 3.6 SOLVE THREE EQUATIONS IN THREE UNKNOWNS Page: 110
PROGRESS EXERCISES 3.1 Page: 111
3.2 Equilibrium and Break-even Page: 111
3.2.1 Equilibrium in the goods and labour markets Page: 112
Goods market equilibrium Page: 112
WORKED EXAMPLE 3.7 GOODS MARKET EQUILIBRIUM Page: 112
Figure 3.5 Goods market equilibrium Page: 113
Labour market equilibrium Page: 113
WORKED EXAMPLE 3.8 LABOUR MARKET EQUILIBRIUM Page: 113
Figure 3.6 Labour market equilibrium Page: 114
3.2.2 Price controls and government intervention in various markets Page: 114
Price ceilings Page: 114
WORKED EXAMPLE 3.9 GOODS MARKET EQUILIBRIUM AND PRICE CEILINGS Page: 115
Figure 3.7 Price ceiling and black market Page: 115
Price floors Page: 116
WORKED EXAMPLE 3.10 LABOUR MARKET EQUILIBRIUM AND PRICE FLOORS Page: 116
PROGRESS EXERCISES 3.2 Page: 117
3.2.3 Market equilibrium for substitute and complementary goods Page: 118
WORKED EXAMPLE 3.11 EQUILIBRIUM FOR TWO SUBSTITUTE GOODS Page: 119
3.2.4 Taxes, subsidies and their distribution Page: 120
Fixed tax per unit of output Page: 120
WORKED EXAMPLE 3.12 TAXES AND THEIR DISTRIBUTION Page: 121
Remember Page: 121
Figure 3.8 Goods market equilibrium and taxes Page: 121
Remember Page: 122
Subsidies and their distribution Page: 123
WORKED EXAMPLE 3.13 SUBSIDIES AND THEIR DISTRIBUTION Page: 123
Figure 3.9 Goods market equilibrium and subsidies Page: 123
Distribution of taxes/subsidies Page: 124
3.2.5 Break-even analysis Page: 125
WORKED EXAMPLE 3.14 CALCULATING THE BREAK-EVEN POINT Page: 125
Figure 3.10 Break-even point Page: 125
PROGRESS EXERCISES 3.3 Page: 126
3.3 Consumer and Producer Surplus Page: 128
3.3.1 Consumer and producer surplus Page: 128
Consumer surplus (CS) Page: 128
Figure 3.11 Consumer surplus Page: 128
Producer surplus (PS) Page: 129
Figure 3.12 Producer surplus Page: 129
Total surplus (TS) Page: 130
Figure 3.13 Area of triangle = 0.5 × area of rectangle = 0.5 × (b × h) Page: 130
WORKED EXAMPLE 3.15 CONSUMER AND PRODUCER SURPLUS AT MARKET EQUILIBRIUM Page: 130
Figure 3.14 Consumer and producer surplus Page: 131
PROGRESS EXERCISES 3.4 Page: 131
3.4 The National Income Model and the IS-LM Model Page: 133
3.4.1 National income model Page: 133
Steps for deriving the equilibrium level of national income Page: 133
Equilibrium level of national income when E = C + I Page: 134
WORKED EXAMPLE 3.16 EQUILIBRIUM NATIONAL INCOME WHEN E = C + I Page: 134
Figure 3.15 Equilibrium national income with consumption and investment Page: 135
PROGRESS EXERCISES 3.5 Page: 136
www.wiley.com/college/bradley Page: 136
3.5 Excel for Simultaneous Linear Equations Page: 137
WORKED EXAMPLE 3.21 COST, REVENUE, BREAK-EVEN, PER UNIT TAX WITH EXCEL Page: 137
Figure 3.18 Break-even with tax and no tax Page: 138
WORKED EXAMPLE 3.22 DISTRIBUTION OF TAX WITH EXCEL Page: 138
Figure 3.19 Market equilibrium Page: 139
Figure 3.20 Distribution of tax for equations (i) Page: 139
Figure 3.21 Distribution of tax for equations (ii) Page: 140
Remember Page: 141
PROGRESS EXERCISES 3.7 Page: 141
3.6 Summary Page: 142
Mathematics Page: 142
Applications Page: 142
www.wiley.com/college/bradley Page: 143
Appendix Page: 143
Figure 3.22 Distribution of tax Page: 144
TEST EXERCISES 3 Page: 144
4 Non-linear Functions and Applications Page: 147
Chapter Objectives Page: 147
4.1 Quadratic, Cubic and Other Polynomial Functions Page: 148
Figure 4.1 Non-linear total revenue function Page: 148
Figure 4.2 A cubic total cost function Page: 149
Warning Page: 149
4.1.1 Solving a quadratic equation Page: 149
Remember Page: 149
The roots of quadratic equations: an overview Page: 150
WORKED EXAMPLE 4.1 SOLVING LESS GENERAL QUADRATIC EQUATIONS Page: 150
Reasons for three different types of solutions (roots) Page: 151
Figure 4.3 Quadratics: (a) real roots, (b) repeated roots, (c) complex roots Page: 151
WORKED EXAMPLE 4.2 SOLVING QUADRATIC EQUATIONS Page: 151
PROGRESS EXERCISES 4.1 Page: 152
4.1.2 Properties and graphs of quadratic functions: f(x) = ax2 + bx + c Page: 153
Graphical representation of the roots of a quadratic Page: 153
WORKED EXAMPLE 4.3 SKETCHING A QUADRATIC FUNCTION f(x) = ±x2 Page: 153
Table 4.1 Calculation of points for y = x2 and y = −x2 Page: 153
Figure 4.4 y = x2 and y = −x2 Page: 153
www.wiley.com/college/bradley Page: 154
Graphs and equations of translated quadratics Page: 154
WORKED EXAMPLE 4.5 VERTICAL AND HORIZONTAL TRANSLATIONS OF QUADRATIC FUNCTIONS Page: 155
Figure 4.6a Vertical translations of y = x2 Page: 155
Figure 4.6b Horizontal translations of y = x2 Page: 155
To sketch any quadratic y = ax2 + bx + c Page: 156
WORKED EXAMPLE 4.6 SKETCHING ANY QUADRATIC EQUATION Page: 156
Table 4.3 Calculation of points for y = 2x2 − 7x − 9 Page: 156
Figure 4.7 Graph for Worked Example 4.5 Page: 156
Summary to date Page: 157
PROGRESS EXERCISES 4.2 Page: 158
4.1.3 Quadratic functions in economics Page: 158
Non-linear supply and demand functions Page: 158
WORKED EXAMPLE 4.7 NON-LINEAR DEMAND AND SUPPLY FUNCTIONS Page: 159
Table 4.4 Points for Ps = Q2 + 6Q + 9 and Pd = Q2 −10Q + 25 Page: 159
Figure 4.8 Market equilibrium with non-linear demand and supply functions Page: 159
Total revenue for a profit-maximising monopolist Page: 160
Remember Page: 160
WORKED EXAMPLE 4.8 NON-LINEAR TOTAL REVENUE FUNCTION Page: 160
Table 4.5 Points for TR = 50Q − 2Q2 Page: 160
Figure 4.9 Non-linear total revenue function Page: 160
WORKED EXAMPLE 4.9 CALCULATING BREAK-EVEN POINTS Page: 162
Table 4.6 Total revenue and total cost Page: 162
Figure 4.10 Total revenue and total cost: break-even points Page: 163
PROGRESS EXERCISES 4.3 Page: 163
4.1.4 Cubic functions Page: 165
WORKED EXAMPLE 4.10a PLOTTING CUBIC FUNCTIONS Page: 165
Figure 4.11 Graphs for Worked Example 4.10a Page: 165
Table 4.7 Calculation of points for graphs of (a) y = x3 and (b) y = −x3 Page: 165
WORKED EXAMPLE 4.10b GRAPHS OF MORE GENERAL CUBIC FUNCTIONS Page: 166
Table 4.8 Points for plotting graphs in Worked Example 4.10b Page: 167
Figure 4.12a Graph (a) for Worked Example 4.10b Page: 167
Figure 4.12b Graph (b) for Worked Example 4.10b Page: 167
General properties of cubic equations Page: 168
Polynomials Page: 168
WORKED EXAMPLE 4.11 TR, TC AND PROFIT FUNCTIONS Page: 168
Table 4.9 TR and TC for Worked Example 4.11 Page: 169
Figure 4.13 Quadratic TR and cubic TC functions Page: 169
PROGRESS EXERCISES 4.4 Page: 170
4.2 Exponential Functions Page: 170
4.2.1 Definition and graphs of exponential functions Page: 170
The number e Page: 170
Graphs of exponential functions Page: 171
WORKED EXAMPLE 4.12 GRAPHING EXPONENTIAL FUNCTIONS Page: 171
Table 4.10 Points for the functions y = 2x and y = 2−x Page: 171
Figure 4.14 Graph for Table 4.10 Page: 171
Table 4.11 Points for the functions y = (3.5)x and y = ex Page: 172
Figure 4.15 Graphs for Tables 4.10 and 4.11 Page: 172
Properties of exponential functions Page: 173
Figure 4.16 Various graphs of y = ax Page: 173
Remember Page: 173
Rules for using exponential functions Page: 173
Table 4.12 The rules for indices Page: 174
WORKED EXAMPLE 4.13 SIMPLIFYING EXPONENTIAL EXPRESSIONS Page: 174
PROGRESS EXERCISES 4.5 Page: 177
4.2.2 Solving equations that contain exponentials Page: 178
WORKED EXAMPLE 4.14 SOLVING EXPONENTIAL EQUATIONS Page: 178
Remember Page: 178
PROGRESS EXERCISES 4.6 Page: 179
4.2.3 Applications of exponential functions Page: 180
The laws of growth Page: 180
Unlimited growth Page: 180
WORKED EXAMPLE 4.15 UNLIMITED GROWTH: POPULATION GROWTH Page: 180
Table 4.13 Population values for different time periods Page: 180
Figure 4.17 Population growth Page: 181
Limited growth Page: 181
WORKED EXAMPLE 4.16 LIMITED GROWTH: CONSUMPTION AND CHANGES IN INCOME Page: 182
Table 4.14 Consumption values for different income levels Page: 182
Figure 4.18 Consumption with limited growth Page: 182
Logistic growth Page: 183
Figure 4.19 Logistic growth Page: 183
PROGRESS EXERCISES 4.7 Page: 183
PROGRESS EXERCISES 4.8 Page: 184
4.3 Logarithmic Functions Page: 184
4.3.1 How to find the log of a number Page: 184
What is the log of a number? Page: 185
Logs to base 10 and logs to base e Page: 186
PROGRESS EXERCISES 4.9 Page: 186
Solving equations containing exponentials ax, i.e., where a is any real number Page: 187
WORKED EXAMPLE 4.17 USE LOGS TO SOLVE CERTAIN EQUATIONS Page: 187
WORKED EXAMPLE 4.18 FINDING THE TIME FOR THE GIVEN POPULATION TO GROW TO 1750 Page: 188
PROGRESS EXERCISES 4.10 Page: 188
4.3.2 Graphs and properties of logarithmic functions Page: 189
WORKED EXAMPLE 4.19 GRAPHS OF LOGARITHMIC FUNCTIONS Page: 189
Table 4.15 Values of log(x) and ln(x) Page: 189
Figure 4.20 Graphs of log(x) and ln(x) Page: 189
4.3.3 Rules for logs Page: 190
Table 4.16 The rules for logs Page: 191
WORKED EXAMPLE 4.20 USING LOG RULES Page: 191
4.3.4 Solving equations using the log rules Page: 193
WORKED EXAMPLE 4.21 SOLVING CERTAIN EQUATIONS WITH RULE 3 FOR LOGS Page: 193
Solving equations that contain logs Page: 194
WORKED EXAMPLE 4.22 SOLVE EQUATIONS CONTAINING LOGS AND EXPONENTIALS Page: 195
PROGRESS EXERCISES 4.11 Page: 196
4.4 Hyperbolic (Rational) Functions of the Form a/(bx + c) Page: 197
4.4.1 Define and sketch rectangular hyperbolic functions Page: 197
Figure 4.21 Graph of y = 1/x Page: 197
Table 4.17 Calculation of points for Figure 4.21 Page: 198
Functions of the form y = a/(bx + c) Page: 198
WORKED EXAMPLE 4.23 SKETCHES OF HYPERBOLIC FUNCTIONS Page: 198
Figure 4.22 Hyperbolic functions Page: 199
The main features of y = a/(bx + c) Page: 199
PROGRESS EXERCISES 4.12 Page: 199
4.4.2 Equations and applications Page: 200
WORKED EXAMPLE 4.24 HYPERBOLIC DEMAND FUNCTION Page: 200
Figure 4.23 Market equilibrium Page: 201
PROGRESS EXERCISES 4.13 Page: 201
4.5 Excel for Non-linear Functions Page: 202
Remember Page: 203
WORKED EXAMPLE 4.25 TOTAL COST FUNCTIONS WITH EXCEL Page: 203
Figure 4.24 Linear TC and quadratic TR functions for Plane Soap Co. Page: 204
Figure 4.25 Cubic TC and quadratic TR functions for Round Soap Co. Page: 204
4.6 Summary Page: 205
Mathematics Page: 205
www.wiley.com/college/bradley Page: 206
Applications Page: 206
TEST EXERCISES 4 Page: 206
5 Financial Mathematics Page: 209
Chapter Objectives Page: 209
5.1 Arithmetic and Geometric Sequences and Series Page: 210
Definitions Page: 210
Arithmetic series (or arithmetic progression denoted by AP) Page: 210
Table 5.1 Arithmetic sequence Page: 211
WORKED EXAMPLE 5.1 SUM OF AN ARITHMETIC SERIES Page: 211
Geometric series (or geometric progression denoted by GP) Page: 211
Table 5.2 Geometric sequence Page: 211
WORKED EXAMPLE 5.2 SUM OF A GEOMETRIC SERIES Page: 212
The sum of an infinite number of terms of a GP Page: 212
PROGRESS EXERCISES 5.1 Page: 213
WORKED EXAMPLE 5.3 APPLICATION OF ARITHMETIC AND GEOMETRIC SERIES Page: 214
PROGRESS EXERCISES 5.2 Page: 216
5.2 Simple Interest, Compound Interest and Annual Percentage Rates Page: 218
Simple interest Page: 218
www.wiley.com/college/bradley Page: 218
Compound interest Page: 219
Deriving the compound interest formula Page: 219
WORKED EXAMPLE 5.5 COMPOUND INTEREST CALCULATIONS Page: 220
Present value at compound interest Page: 220
WORKED EXAMPLE 5.6 FUTURE AND PRESENT VALUES WITH COMPOUND INTEREST Page: 220
Other applications of the compound interest formula Page: 221
WORKED EXAMPLE 5.7 CALCULATING THE COMPOUND INTEREST RATE AND TIME PERIOD Page: 221
PROGRESS EXERCISES 5.3 Page: 222
When interest is compounded several times per year Page: 223
WORKED EXAMPLE 5.8 COMPOUNDING DAILY, MONTHLY AND SEMI-ANNUALLY Page: 224
Continuous compounding Page: 225
WORKED EXAMPLE 5.9 CONTINUOUS COMPOUNDING Page: 225
Annual percentage rate (APR)* Page: 225
WORKED EXAMPLE 5.10 ANNUAL PERCENTAGE RATES Page: 227
PROGRESS EXERCISES 5.4 Page: 228
5.3 Depreciation Page: 228
Straight-line depreciation Page: 229
Reducing-balance depreciation Page: 229
WORKED EXAMPLE 5.11 FUTURE VALUE OF ASSET AND REDUCING-BALANCE DEPRECIATION Page: 229
WORKED EXAMPLE 5.12 PRESENT VALUE OF ASSET AND REDUCING-BALANCE DEPRECIATION Page: 230
5.4 Net Present Value and Internal Rate of Return Page: 230
Net present value (NPV) Page: 230
Table 5.3 Cash flows of an investment project Page: 230
WORKED EXAMPLE 5.13 CALCULATING NET PRESENT VALUE Page: 231
Internal rate of return (IRR) Page: 232
Table 5.4 Calculation of NPVs for various interest rates Page: 232
To calculate the IRR for a given project: (a) graphically, (b) by calculation Page: 233
WORKED EXAMPLE 5.14 IRR DETERMINED GRAPHICALLY (EXCEL) AND BY CALCULATION Page: 233
Table 5.5 Excel sheet for calculating NPVs at different interest rates Page: 234
Figure 5.1 Graphical determination of IRR Page: 234
Comparison of appraisal techniques: NPV, IRR Page: 235
PROGRESS EXERCISES 5.5 Page: 236
5.5 Annuities, Debt Repayments, Sinking Funds Page: 236
5.5.1 Compound interest for fixed deposits at regular intervals of time Page: 236
WORKED EXAMPLE 5.15 COMPOUND INTEREST FOR FIXED PERIODIC DEPOSITS Page: 238
5.5.2 Annuities Page: 238
WORKED EXAMPLE 5.16 ANNUITIES Page: 239
The present value of an annuity Page: 240
WORKED EXAMPLE 5.17 PRESENT VALUE OF ANNUITIES Page: 241
5.5.3 Debt repayments Page: 242
WORKED EXAMPLE 5.18 MORTGAGE REPAYMENTS Page: 243
How much of the repayment is interest? Page: 244
WORKED EXAMPLE 5.19 HOW MUCH OF THE REPAYMENT IS INTEREST? Page: 244
Sinking funds Page: 245
WORKED EXAMPLE 5.20 SINKING FUNDS Page: 246
PROGRESS EXERCISES 5.6 Page: 247
5.6 The Relationship between Interest Rates and the Price of Bonds Page: 248
WORKED EXAMPLE 5.21 THE INTEREST RATE AND THE PRICE OF BONDS Page: 249
Table 5.6 The rates and the NPV for the cash flow for a £1000 bond Page: 249
PROGRESS EXERCISES 5.7 Page: 251
5.7 Excel for Financial Mathematics Page: 251
WORKED EXAMPLE 5.22 GROWTH OF AN INVESTMENT USING DIFFERENT METHODS OF COMPOUNDING (EXCEL) Page: 252
Table 5.7 Different methods of compounding Page: 252
Table 5.8 Data for plotting Figure 5.2 Page: 253
Figure 5.2 Growth of £1 at i = 50% using different compounding methods Page: 253
5.8 Summary Page: 254
Series Page: 254
Financial mathematics Page: 254
Excel Page: 255
www.wiley.com/college/bradley Page: 255
Appendix Page: 256
Figure 5.3 Estimating the internal rate of return Page: 257
TEXT EXERCISES 5 Page: 257
6 Differentiation and Applications Page: 259
Chapter Objectives Page: 259
6.1 Slope of a Curve and Differentiation Page: 260
6.1.1 The slope of a curve is variable Page: 260
Figure 6.1 The slope of a curve is variable Page: 260
6.1.2 Slope of a curve and turning points Page: 260
Figure 6.2 (a) Slope of chord approximates slope of curve. (b) Point C → C * as C moves towards B. Slope of tangent at B = slope of curve at B Page: 260
WORKED EXAMPLE 6.1 EQUATION FOR THE SLOPE OF y = x2 FROM FIRST PRINCIPLES Page: 261
Figure 6.3 Slope of curve y = x2 at x = −1 and x = 1.5 Page: 263
6.1.3 The derivative Page: 263
The power rule for differentiation Page: 263
6.1.4 How to use the power rule for differentiation Page: 264
WORKED EXAMPLE 6.2 USING THE POWER RULE Page: 264
Some important points to note before using the power rule Page: 265
The slope of a curve at a point Page: 265
Practical problems Page: 266
6.1.5 Working rules for differentiating sums and differences of several functions Page: 266
WORKED EXAMPLE 6.3 MORE DIFFERENTIATION USING THE POWER RULE Page: 267
6.1.6 Higher derivatives Page: 268
WORKED EXAMPLE 6.4 CALCULATING HIGHER DERIVATIVES Page: 268
PROGRESS EXERCISES 6.1 Page: 268
www.wiley.com/college/bradley Page: 270
6.2 Applications of Differentiation, Marginal Functions, Average Functions Page: 270
6.2.1 Marginal functions: an introduction Page: 270
Marginal revenue Page: 271
WORKED EXAMPLE 6.6 CALCULATING MARGINAL REVENUE GIVEN THE DEMAND FUNCTION Page: 271
Table 6.3a Total revenue and marginal revenue calculated by differentiation Page: 271
Figure 6.7 Marginal revenue measured along a chord Page: 272
WORKED EXAMPLE 6.7 CALCULATING MARGINAL REVENUE OVER AN INTERVAL Page: 272
Table 6.3b Total revenue and marginal revenue calculated over an interval, ΔQ = 1 Page: 273
Marginal cost Page: 273
WORKED EXAMPLE 6.8 DERIVE MARGINAL COST EQUATION FROM TOTAL COST FUNCTION Page: 274
6.2.2 Average functions: an introduction Page: 275
Average revenue (AR) Page: 275
The relationship between AR and price Page: 275
Average cost Page: 276
WORKED EXAMPLE 6.9 MR, AR FOR A PERFECTLY COMPETITIVE FIRM AND A MONOPOLIST Page: 276
Figure 6.8 A perfectly competitive firm’s AR and MR functions Page: 277
Figure 6.9 A monopolist’s AR and MR functions Page: 277
Table 6.4 Marginal revenue and average revenue for a monopolist Page: 277
Marginal and average revenue functions for a perfectly competitive firm and a monopolist: a summary Page: 278
Table 6.5 Average revenue and marginal revenue functions: a summary Page: 278
Total cost from average cost Page: 278
WORKED EXAMPLE 6.10 DERIVE MARGINAL COST FROM AVERAGE COST Page: 279
PROGRESS EXERCISES 6.3 Page: 280
6.2.3 Production functions and the marginal and average product of labour Page: 281
WORKED EXAMPLE 6.11 DEDUCE THE EQUATION FOR THE MARGINAL AND AVERAGE PRODUCT OF LABOUR FROM A GIVEN PRODUCTION FUNCTION Page: 282
www.wiley.com/college/bradley Page: 283
6.2.5 Marginal and average propensity to consume and save Page: 283
Marginal propensity to consume and save Page: 283
WORKED EXAMPLE 6.14 MPC, MPS, APC, APS Page: 284
www.wiley.com/college/bradley Page: 285
PROGRESS EXERCISES 6.4 Page: 285
6.3 Optimisation for Functions of One Variable Page: 286
6.3.1 Slope of a curve and turning points Page: 286
Remember Page: 286
Figure 6.12 Turning points Page: 287
Table 6.7 Some terminology used in optimisation Page: 287
WORKED EXAMPLE 6.16 FINDING TURNING POINTS Page: 288
Figure 6.13 Locating turning points Page: 288
Remember Page: 289
Figure 6.14 Graph of y = 1/x Page: 289
Remember Page: 289
PROGRESS EXERCISES 6.5 Page: 289
6.3.2 Determining maximum and minimum turning points Page: 290
Testing for minimum and maximum points Page: 290
Figure 6.15 (a) Maximum point. (b) Minimum point Page: 290
WORKED EXAMPLE 6.17 MAXIMUM AND MINIMUM TURNING POINTS Page: 292
Figure 6.16 Graph of y = −x3 + 9x2 − 24x + 26 Page: 292
PROGRESS EXERCISES 6.6 Page: 295
6.3.3 Intervals along which a function is increasing or decreasing Page: 295
WORKED EXAMPLE 6.18 INTERVALS ALONG WHICH A CURVE IS INCREASING OR DECREASING Page: 296
Remember Page: 296
Figure 6.17 Interval along which AC is decreasing or increasing Page: 297
6.3.4 Graphs of y, y′, y": derived curves Page: 297
Figure 6.18 Graphs of y, y′, y" Page: 297
WORKED EXAMPLE 6.19 DERIVED CURVES Page: 297
Table 6.8 Selected points for graphs in Figure 6.18 Page: 298
PROGRESS EXERCISES 6.7 Page: 300
6.3.5 Curve sketching and applications Page: 300
Figure 6.19 Incorrect curve of y = 1/(x − 0.23) Page: 300
Table 6.9 Points for Figure 6.19 Page: 300
Figure 6.20 Correct curve of y = 1/(x − 0.23) Page: 300
Remember Page: 301
Some key features to look for when sketching a curve Page: 301
WORKED EXAMPLE 6.20 SKETCHING FUNCTIONS Page: 302
Figure 6.21 Graph of Q = 100 − P2 Page: 302
Figure 6.22 Graph of AC = 5/Q Page: 303
Table 6.10 Points for Figure 6.22 Page: 303
PROGRESS EXERCISES 6.8 Page: 304
6.4 Economic Applications of Maximum and Minimum Points Page: 304
WORKED EXAMPLE 6.21 MAXIMUM TR AND A SKETCH OF THE TR FUNCTION Page: 304
Figure 6.23 TR is at a maximum when MR = 0 Page: 305
Remember Page: 306
WORKED EXAMPLE 6.22 BREAK-EVEN, PROFIT, LOSS AND GRAPHS Page: 307
Figure 6.24 Total revenue and total cost functions Page: 307
Figure 6.25 Profit function Page: 308
WORKED EXAMPLE 6.23 MAXIMUM AND MINIMUM OUTPUT FOR A FIRM OVER TIME Page: 309
Figure 6.26 A firm’s output function over time Page: 310
Table 6.11 Points for sketching the Q function in Figure 6.26 Page: 310
To show that MR = MC and (MR)’ < (MC)’ when profit is maximised Page: 310
Price discrimination Page: 311
WORKED EXAMPLE 6.24 PROFIT MAXIMISATION AND PRICE DISCRIMINATION Page: 311
Profit maximisation in perfect competition and monopoly (goods market) Page: 313
WORKED EXAMPLE 6.25 PROFIT MAXIMISATION FOR A PERFECTLY COMPETITIVE FIRM Page: 313
Figure 6.27 TR, TC, MR, MC and π functions for a perfectly competitive firm Page: 314
Table 6.12 Points for profit maximisation of a PC firm Page: 314
WORKED EXAMPLE 6.26 PROFIT MAXIMISATION FOR A MONOPOLIST Page: 315
Figure 6.28 TR, TC, MR, MC and π functions for a monopolist Page: 316
Table 6.13 Points for profit maximisation of a monopolist Page: 316
Summary Page: 317
Figure 6.29 Summary of turning points Page: 317
PROGRESS EXERCISES 6.9 Page: 318
6.5 Curvature and Other Applications Page: 320
6.5.1 Second derivative and curvature Page: 320
Table 6.14 Summary of relationship between y, y′ and y″ Page: 322
Curvature in economics Page: 322
Figure 6.30 Concave up Page: 322
Figure 6.31 Concave down Page: 322
WORKED EXAMPLE 6.27 CURVATURE OF CURVES: CONVEX OR CONCAVE TOWARDS THE ORIGIN Page: 323
Figure 6.32 Graph of y = 3x4 + 20 Page: 323
Table 6.15 Points for Figure 6.32 Page: 323
Figure 6.33 Graph of Q = 25/L Page: 324
Table 6.16 Points for Figure 6.33 Page: 324
6.5.2 Points of inflection Page: 324
Figure 6.34 Point of inflection Page: 324
Stationary points of inflection Page: 325
Figure 6.35 Stationary points of inflection Page: 325
Points of inflection in economics Page: 325
WORKED EXAMPLE 6.28 LOCATE THE POINT OF INFLECTION, POI = POINT AT WHICH MARGINAL RATE CHANGES Page: 326
PROGRESS EXERCISES 6.10 Page: 326
www.wiley.com/college/bradley Page: 327
Points of inflection and curvature for total cost functions Page: 327
WORKED EXAMPLE 6.31 RELATIONSHIP BETWEEN TC AND MC Page: 327
Figure 6.37 TC, TVC and MC functions Page: 328
WORKED EXAMPLE 6.32 RELATIONSHIP BETWEEN AC, AVC, AFC AND MC FUNCTIONS Page: 329
Table 6.18 Points for plotting the total, average and marginal cost functions Page: 330
Figure 6.38 Total, average and marginal cost functions Page: 331
PROGRESS EXERCISES 6.11 Page: 333
6.6 Further Differentiation and Applications Page: 334
6.6.1 Derivatives of other standard functions Page: 334
Table 6.19 Rules for finding derivatives Page: 335
WORKED EXAMPLE 6.33 DERIVATIVES OF EXPONENTIALS AND LOGS Page: 335
PROGRESS EXERCISES 6.12 Page: 335
6.6.2 Chain rule for differentiating a function of a function Page: 336
What is a function of a function? Page: 336
Stages of the chain rule Page: 336
WORKED EXAMPLE 6.34 USING THE CHAIN RULE FOR DIFFERENTIATION Page: 336
PROGRESS EXERCISES 6.13 Page: 338
6.6.3 Product rule for differentiation Page: 338
Product rule Page: 338
Stages of the product rule Page: 338
WORKED EXAMPLE 6.35 USING THE PRODUCT RULE FOR DIFFERENTIATION Page: 339
PROGRESS EXERCISES 6.14 Page: 340
6.6.4 Quotient rule for differentiation Page: 340
WORKED EXAMPLE 6.36 USING THE QUOTIENT RULE FOR DIFFERENTIATION Page: 341
PROGRESS EXERCISES 6.15 Page: 342
WORKED EXAMPLE 6.37 FIND MC GIVEN A LOGARITHMIC TC FUNCTION Page: 343
Figure 6.39 TC function Page: 343
Table 6.20 Points for plotting TC = 120 ln(Q + 10) Page: 343
WORKED EXAMPLE 6.38 DEMAND, TR, MR EXPRESSED IN TERMS OF EXPONENTIALS Page: 344
Table 6.21 Points for plotting P = 150e−0.02Q Page: 344
Figure 6.40 Demand function P = 150e−0.02Q Page: 344
PROGRESS EXERCISES 6.16 Page: 345
6.7 Elasticity and the Derivative Page: 347
6.7.1 Point elasticity of demand and the derivative Page: 348
WORKED EXAMPLE 6.39 EXPRESSIONS FOR POINT ELASTICITY OF DEMAND IN TERMS OF P, Q OR BOTH FOR LINEAR AND NON-LINEAR DEMAND FUNCTIONS Page: 349
WORKED EXAMPLE 6.40 POINT ELASTICITY OF DEMAND FOR NON-LINEAR DEMAND FUNCTIONS Page: 351
6.7.2 Constant elasticity demand function Page: 352
WORKED EXAMPLE 6.41 CONSTANT ELASTICITY DEMAND FUNCTION Page: 352
6.7.3 Price elasticity of demand, marginal revenue, total revenue and price changes Page: 353
Remember Page: 354
www.wiley.com/college/bradley Page: 354
PROGRESS EXERCISES 6.17 Page: 355
6.8 Summary Page: 357
Mathematics Page: 357
Applications Page: 358
www.wiley.com/college/bradley Page: 359
TEST EXERCISES 6 Page: 359
7 Functions of Several Variables Page: 361
Chapter Objectives Page: 361
7.1 Partial Differentiation Page: 362
7.1.1 Functions of two or more variables Page: 362
Figure 7.1 z = x + 2y + 4: a three-dimensional plane Page: 362
Table 7.1 Calculate points for the function z = x + 2y + 4 Page: 363
Graphical representation of functions of two variables in economics Page: 363
WORKED EXAMPLE 7.1 PLOT ISOQUANTS FOR A GIVEN PRODUCTION FUNCTION Page: 364
Table 7.2 Production functions and isoquants Page: 364
Table 7.3 Calculation of points of isoquants Page: 364
Figure 7.2 A three-dimensional view of isoquants Page: 365
Figure 7.3 A two-dimensional view of isoquants Page: 365
7.1.2 Partial differentiation: first-order partial derivatives Page: 366
WORKED EXAMPLE 7.2 PARTIAL DIFFERENTIATION: A FIRST EXAMPLE Page: 367
WORKED EXAMPLE 7.3 DETERMINING FIRST-ORDER PARTIAL DERIVATIVES Page: 368
PROGRESS EXERCISES 7.1 Page: 369
7.1.3 Second-order partial derivatives Page: 370
WORKED EXAMPLE 7.4 DETERMINING SECOND-ORDER PARTIAL DERIVATIVES Page: 371
PROGRESS EXERCISES 7.2 Page: 374
7.1.4 Differentials and small changes (incremental changes) Page: 374
Incremental changes Page: 374
Figure 7.4 Differentials and small changes Page: 375
WORKED EXAMPLE 7.5 DIFFERENTIALS FOR FUNCTIONS OF ONE VARIABLE Page: 375
Differentials for functions of two variables Page: 376
WORKED EXAMPLE 7.6a DIFFERENTIALS AND INCREMENTAL CHANGES Page: 376
WORKED EXAMPLE 7.6b INCREMENTAL CHANGES Page: 378
Summary Page: 378
PROGRESS EXERCISES 7.3 Page: 379
7.2 Applications of Partial Differentiation Page: 380
7.2.1 Production functions Page: 380
Marginal functions in general Page: 381
Table 7.4 Summary of marginal functions for Q = ALα Kβ: 0 < α < 1, 0 < β < 1 Page: 381
Remember Page: 382
WORKED EXAMPLE 7.7 MPL AND MPK: INCREASING OR DECREASING? Page: 382
The relationship between marginal and average functions Page: 382
Table 7.5 Average and marginal functions for the production function, Q = ALα Kβ Page: 382
Production conditions Page: 382
Graphical representation of production functions: isoquants Page: 383
The slope of an isoquant (MRTS) Page: 383
Figure 7.5 MRTS = slope of isoquant, slope is diminishing Page: 384
Remember Page: 384
The slope of an isoquant is the ratio of the marginal products Page: 384
WORKED EXAMPLE 7.8 SLOPE OF AN ISOQUANT IN TERMS OF MPL, MPK Page: 385
Table 7.6 Points for plotting the isoquants K = 25/L and K = 49/L Page: 385
Figure 7.6 Isoquants Page: 385
The MRTS in reduced form for a Cobb–Douglas production function Page: 387
PROGRESS EXERCISES 7.4 Page: 387
7.2.2 Returns to scale Page: 388
Homogeneous functions of degree r Page: 389
WORKED EXAMPLE 7.9 CONSTANT, INCREASING AND DECREASING RETURNS TO SCALE Page: 389
Incremental changes Page: 390
7.2.3 Utility functions Page: 390
Marginal utility Page: 390
Graphical representation of utility functions Page: 391
Slope of an indifference curve Page: 391
Figure 7.7 MRS = slope of an indifference curve, slope is diminishing Page: 392
Remember Page: 392
WORKED EXAMPLE 7.10 INDIFFERENCE CURVES AND SLOPE Page: 392
Table 7.7 Points for plotting indifference curves Page: 393
Figure 7.8 Indifference curves, U = f(x, y) Page: 393
PROGRESS EXERCISES 7.5 Page: 393
7.2.4 Partial elasticities Page: 395
Partial elasticities of demand Page: 395
Price elasticity of demand Page: 395
WORKED EXAMPLE 7.11 PARTIAL ELASTICITIES OF DEMAND Page: 396
Partial elasticity of labour and capital Page: 396
WORKED EXAMPLE 7.12 PARTIAL ELASTICITIES OF LABOUR AND CAPITAL Page: 397
7.2.5 The multipliers for the linear national income model Page: 397
WORKED EXAMPLE 7.13 USE PARTIAL DERIVATIVES TO DERIVE EXPRESSIONS FOR VARIOUS MULTIPLIERS Page: 398
PROGRESS EXERCISES 7.6 Page: 399
7.3 Unconstrained Optimisation Page: 400
7.3.1 Find the optimum points for functions of two variables Page: 400
Optimisation of functions of one variable revisited Page: 400
Optimisation of functions of two variables: method Page: 401
Figure 7.9 (a) z = − x2 − y2 + 40 has a maximum point at x = 0, y = 0, z = 40. (b) z = x2 + y2 + 20 has a minimum point at x = 0, y = 0, z = 20. (c) z = x2 − y2 −4x + 4y: a saddle point at x = 2, y = 2, z = 0. Page: 401
WORKED EXAMPLE 7.14 OPTIMUM POINTS FOR FUNCTIONS OF TWO VARIABLES Page: 402
PROGRESS EXERCISES 7.7 Page: 403
7.3.2 Total revenue maximisation and profit maximisation Page: 403
WORKED EXAMPLE 7.15 MONOPOLIST MAXIMISING TOTAL REVENUE FOR TWO GOODS Page: 404
WORKED EXAMPLE 7.16 MAXIMISE PROFIT FOR A MULTI-PRODUCT FIRM Page: 405
7.3.3 Price discrimination Page: 406
To find the prices which should be charged in each market to maximise profit Page: 406
WORKED EXAMPLE 7.17 MONOPOLIST: PRICE AND NON-PRICE DISCRIMINATION Page: 407
PROGRESS EXERCISES 7.8 Page: 408
7.4 Constrained Optimisation and Lagrange Multipliers Page: 410
7.4.1 What is a constrained maximum or minimum? Page: 410
7.4.2 Finding the constrained extrema with Lagrange multipliers Page: 411
WORKED EXAMPLE 7.18 MAXIMISING TOTAL REVENUE SUBJECT TO A BUDGET CONSTRAINT Page: 411
Figure 7.10 Constrained maximum: TR = 36x − 3x2 + 56y − 4y2, subject to 5x + 10y = 80 Page: 412
Table 7.8 TR evaluated at selected points on the constraint: 80 = 5x + 8y(y = 8 − 0.5x) Page: 412
7.4.3 Maximum utility subject to a budget constraint Page: 413
WORKED EXAMPLE 7.19 LAGRANGE MULTIPLIERS AND UTILITY MAXIMISATION Page: 413
Graphical analysis for locating maximum utility, −(Ux/Uy) = − (PX/PY) Page: 414
Figure 7.11 Utility subject to a budget constraint Page: 414
WORKED EXAMPLE 7.20 USE LAGRANGE MULTIPLIERS TO DERIVE THE IDENTITY Ux/Uy = PX/PY Page: 415
Interpretation of the Lagrange multiplier λ Page: 416
WORKED EXAMPLE 7.21 MEANING OF λ Page: 416
7.4.4 Production functions Page: 417
WORKED EXAMPLE 7.22 MAXIMISE OUTPUT SUBJECT TO A COST CONSTRAINT Page: 417
7.4.5 Minimising cost subject to a production constraint Page: 419
WORKED EXAMPLE 7.23 MINIMISE COSTS SUBJECT TO A PRODUCTION CONSTRAINT Page: 419
PROGRESS EXERCISES 7.9 Page: 420
7.5 Summary Page: 422
Function of one variable y = f(x) Page: 422
Function of two variables: z = f(x, y) Page: 423
Unconstrained optimisation Page: 423
Constrained optimisation: Lagrange multipliers Page: 424
Applications Page: 424
Partial elasticity Page: 424
National income model multipliers Page: 425
www.wiley.com/college/bradley Page: 425
TEST EXERCISES 7 Page: 425
8 Integration and Applications Page: 427
Chapter Objectives Page: 427
8.1 Integration as the Reverse of Differentiation Page: 428
Figure 8.1 Integration reverses differentiation Page: 428
Figure 8.2 Integration reverses differentiation Page: 428
8.2 The Power Rule for Integration Page: 429
Deduce the power rule for integration Page: 429
Figure 8.3 Integration reverses differentiation Page: 429
WORKED EXAMPLE 8.1 USING THE POWER RULE FOR INTEGRATION Page: 430
The minus one exception to the power rule Page: 431
Figure 8.4 ∫1xdx=ln⁡(x)+c since integration reverses differentiation Page: 431
The integral of a constant term Page: 432
Working rules for integration Page: 432
WORKED EXAMPLE 8.2 INTEGRATING SUMS AND DIFFERENCES, CONSTANT MULTIPLIED BY VARIABLE TERM Page: 433
WORKED EXAMPLE 8.3 INTEGRATING MORE GENERAL FUNCTIONS Page: 434
PROGRESS EXERCISES 8.1 Page: 435
8.3 Integration of the Natural Exponential Function Page: 435
Figure 8.5 Integration of ex Page: 435
WORKED EXAMPLE 8.4 INTEGRATING FUNCTIONS CONTAINING ex Page: 435
8.4 Integration by Algebraic Substitution Page: 436
8.4.1 Using substitution to integrate functions of linear functions Page: 436
Remember Page: 436
WORKED EXAMPLE 8.5 INTEGRATING FUNCTIONS OF LINEAR FUNCTIONS BY SUBSTITUTION Page: 437
8.4.2 General functions of linear functions Page: 438
WORKED EXAMPLE 8.6 INTEGRATING LINEAR FUNCTIONS RAISED TO A POWER Page: 439
WORKED EXAMPLE 8.7 MORE EXAMPLES ON INTEGRATING FUNCTIONS OF LINEAR FUNCTIONS Page: 440
PROGRESS EXERCISES 8.2 Page: 441
8.5 The Definite Integral and the Area under a Curve Page: 441
The approximate area under a curve Page: 441
Figure 8.6 Area under the curve ≃ sum of areas of rectangles Page: 441
Figure 8.7 Decreasing the size of Δx gives a better approximation to area Page: 442
Figure 8.8 Area under the curve is determined exactly by integration Page: 442
WORKED EXAMPLE 8.8 EVALUATING THE DEFINITE INTEGRAL Page: 443
Figure 8.9 Area under f(x) = x + 2 Page: 444
WORKED EXAMPLE 8.9 DEFINITE INTEGRAL AND ex Page: 444
Definite integration gives the net area contained between the curve and the x-axis Page: 445
WORKED EXAMPLE 8.10 DEFINITE INTEGRATION AND NET AREA BETWEEN CURVE AND x-AXIS Page: 445
Figure 8.10 Definite integration calculates the net enclosed area Page: 446
Evaluation of the definite integral when F(x) = ln |x| Page: 446
WORKED EXAMPLE 8.11 DEFINITE INTEGRATION AND LOGS Page: 447
PROGRESS EXERCISES 8.3 Page: 447
8.6 Consumer and Producer Surplus Page: 448
Consumer surplus (CS) Page: 448
Figure 8.11 Consumer surplus for Worked Example 8.12(a) Page: 449
Figure 8.12 Consumer surplus for Worked Example 8.12(b) Page: 449
WORKED EXAMPLE 8.12 USING THE DEFINITE INTEGRAL TO CALCULATE CONSUMER SURPLUS Page: 449
Producer surplus (PS) Page: 451
WORKED EXAMPLE 8.13 USING THE DEFINITE INTEGRAL TO CALCULATE PRODUCER SURPLUS Page: 451
Figure 8.13 Producer surplus for Worked Example 8.13(a) Page: 452
Figure 8.14 Producer surplus for Worked Example 8.13(b) Page: 452
WORKED EXAMPLE 8.14 CONSUMER AND PRODUCER SURPLUS: EXPONENTIAL FUNCTIONS Page: 453
Area between two curves and other applications of definite integration Page: 455
Figure 8.15 Shaded area = area under upper curve − area under lower curve Page: 455
PROGRESS EXERCISES 8.4 Page: 455
8.7 First-order Differential Equations and Applications Page: 456
General and particular solutions of differential equations Page: 457
Figure 8.16 The general solution represented by a ‘family’ of related curves and a particular solution Page: 457
Solution of differential equations of the form dy/dx = f(x) Page: 458
WORKED EXAMPLE 8.15 SOLUTION OF DIFFERENTIAL EQUATIONS: dy/dx = f(x) Page: 458
PROGRESS EXERCISES 8.5 Page: 459
Differential equations in economics Page: 459
WORKED EXAMPLE 8.16 FIND TOTAL COST FROM MARGINAL COST Page: 459
Differential equations and rates of change Page: 460
Figure 8.17 The definite integral of the rate = total accumulation Page: 461
WORKED EXAMPLE 8.17 DIFFERENTIAL EQUATIONS AND RATES OF CHANGE Page: 461
Figure 8.18 Consumption of oil for years 0 to 20 Page: 462
PROGRESS EXERCISES 8.6 Page: 463
Solution of differential equations of the form dy/dx = ky Page: 464
WORKED EXAMPLE 8.18 SOLVING DIFFERENTIAL EQUATIONS OF THE FORM dy/dx = ky Page: 465
Figure 8.19 General solution, with particular solution indicated Page: 466
Solution of differential equations of the form dy/dx = f(x)g(y) Page: 467
WORKED EXAMPLE 8.19 SOLVING DIFFERENTIAL EQUATIONS OF THE FORM dy/dx = f(x)g(y) Page: 467
PROGRESS EXERCISES 8.7 Page: 467
8.8 Differential Equations for Limited and Unlimited Growth Page: 468
Law of unlimited growth Page: 468
Figure 8.20 Unlimited growth y = Aert Page: 468
Law of limited growth Page: 468
Figure 8.21 The solution of the differential equation: dy/dt = r(A − y) models limited growth Page: 468
WORKED EXAMPLE 8.20 LIMITED GROWTH Page: 469
Figure 8.22 Limited growth, where the limiting value is 700 Page: 469
Constant proportional rates of growth Page: 470
WORKED EXAMPLE 8.21 DETERMINING THE PROPORTIONAL RATES OF GROWTH Page: 470
PROGRESS EXERCISES 8.8 Page: 471
www.wiley.com/college/bradley Page: 473
8.10 Summary Page: 474
Rules for integration Page: 474
First-order differential equations Page: 474
Consumer and producer surplus Page: 474
Integrate marginal functions to obtain total functions Page: 475
Integrate rates (w.r.t. time) to obtain the total amount accumulated over a given time interval Page: 475
Solution of certain first-order differential equations Page: 475
www.wiley.com/college/bradley Page: 475
TEST EXERCISES 8 Page: 475
9 Linear Algebra and Applications Page: 477
Chapter Objectives Page: 477
9.1 Linear Programming Page: 478
Remember Page: 478
WORKED EXAMPLE 9.1 FIND THE MINIMUM COST SUBJECT TO CONSTRAINTS Page: 478
Table 9.1 Vitamin content of X and Y Page: 478
Table 9.2 Vitamin content of x and y portions of mix X and Y, respectively Page: 479
Figure 9.1 Inequality constraints and the feasible region Page: 480
Figure 9.2 Cost decreases as isocost lines move towards the origin Page: 481
Figure 9.3 Minimum cost at point V, subject to constraints Page: 481
The minimum cost by mathematical methods Page: 482
To demonstrate the extreme point theorem Page: 482
Maximisation Page: 482
WORKED EXAMPLE 9.2 PROFIT MAXIMISATION SUBJECT TO CONSTRAINTS Page: 483
Table 9.3 Requirements for gates type I and II Page: 483
Table 9.4 Requirements for x type I gates and y type II gates Page: 483
Figure 9.4 The constraints defining the feasible region for gate manufacturing Page: 484
Figure 9.5 Profit and revenue increases as isoprofit and isorevenue lines move upwards from the origin Page: 484
Figure 9.6 (a) Revenue is a maximum at point B. (b) Profit is a maximum at point C Page: 485
PROGRESS EXERCISES 9.1 Page: 487
9.2 Matrices Page: 488
9.2.1 Matrices: definition Page: 488
Special matrices Page: 489
9.2.2 Matrix addition and subtraction Page: 489
WORKED EXAMPLE 9.3 ADDING AND SUBTRACTING MATRICES Page: 490
Scalar multiplication Page: 491
WORKED EXAMPLE 9.4 MULTIPLICATION OF A MATRIX BY A SCALAR Page: 491
9.2.3 Matrix multiplication Page: 491
WORKED EXAMPLE 9.5 MATRIX MULTIPLICATION Page: 492
9.2.4 Applications of matrix arithmetic Page: 494
WORKED EXAMPLE 9.6 APPLICATIONS OF MATRIX ARITHMETIC Page: 494
Table 9.5 Number of computers sold in each shop Page: 494
Table 9.6 Selling price of computers in each shop Page: 495
PROGRESS EXERCISES 9.2 Page: 497
9.3 Solution of Equations: Elimination Methods Page: 498
9.3.1 Gaussian elimination Page: 498
WORKED EXAMPLE 9.7 SOLUTION OF A SYSTEM OF EQUATIONS: GAUSSIAN ELIMINATION Page: 499
WORKED EXAMPLE 9.8 MORE GAUSSIAN ELIMINATION Page: 500
9.3.2 Gauss–Jordan elimination Page: 502
WORKED EXAMPLE 9.9 GAUSS–JORDAN ELIMINATION Page: 502
PROGRESS EXERCISES 9.3 Page: 503
9.4 Determinants Page: 504
9.4.1 Evaluate 2 × 2 determinants Page: 504
Determinants: definitions Page: 504
Warning Page: 505
9.4.2 Use determinants to solve equations: Cramer’s rule Page: 505
WORKED EXAMPLE 9.10 USING DETERMINANTS TO SOLVE SIMULTANEOUS EQUATIONS Page: 505
Cramer’s rule Page: 507
WORKED EXAMPLE 9.11 USING CRAMER’S RULE TO SOLVE SIMULTANEOUS EQUATIONS Page: 509
WORKED EXAMPLE 9.12 FIND THE MARKET EQUILIBRIUM USING CRAMER’S RULE Page: 510
General expressions for equilibrium in the income-determination model Page: 511
WORKED EXAMPLE 9.13 USE CRAMER’S RULE FOR THE INCOME-DETERMINATION MODEL Page: 511
PROGRESS EXERCISES 9.4 Page: 512
9.4.3 Evaluate 3 × 3 determinants Page: 513
WORKED EXAMPLE 9.14 EVALUATION OF A 3 × 3 DETERMINANT Page: 513
WORKED EXAMPLE 9.15 SOLVE THREE SIMULTANEOUS EQUATIONS BY CRAMER’S RULE Page: 513
Applications Page: 515
WORKED EXAMPLE 9.16 EQUILIBRIUM LEVELS IN THE NATIONAL INCOME MODEL Page: 515
PROGRESS EXERCISES 9.5 Page: 517
9.5 The Inverse Matrix and Input/Output Analysis Page: 518
The inverse matrix Page: 518
9.5.1 To find the inverse of a matrix: elimination method Page: 518
WORKED EXAMPLE 9.17 THE INVERSE OF A MATRIX: ELIMINATION METHOD Page: 518
9.5.2 To find the inverse of a matrix: cofactor method Page: 520
The inverse of a 2 × 2 matrix Page: 521
WORKED EXAMPLE 9.18 THE INVERSE OF A 3 × 3 MATRIX Page: 521
To write a system of equations in matrix form Page: 523
To solve a set of equations using the inverse matrix Page: 524
WORKED EXAMPLE 9.19 SOLVE A SYSTEM OF EQUATIONS BY THE INVERSE MATRIX Page: 524
Input/output analysis Page: 525
WORKED EXAMPLE 9.20 INPUT/OUTPUT ANALYSIS Page: 527
PROGRESS EXERCISES 9.6 Page: 529
9.6 Excel for Linear Algebra Page: 531
WORKED EXAMPLE 9.21 USE EXCEL TO SOLVE SYSTEMS OF LINEAR EQUATIONS Page: 531
9.7 Summary Page: 534
Linear programming Page: 534
Matrix algebra Page: 535
Determinants Page: 535
The inverse of a square matrix Page: 536
Applications of inverse matrices Page: 536
Input/output analysis Page: 536
www.wiley.com/college/bradley Page: 537
TEST EXERCISES 9 Page: 537
10 Difference Equations Page: 539
Chapter Objectives Page: 539
10.1 Introduction to Difference Equations Page: 540
Table 10.1 Terminology associated with difference equations, (a), (b), (c) and (d) Page: 541
10.2 Solution of Difference Equations (First-order) Page: 542
WORKED EXAMPLE 10.1 SOLVING DIFFERENCE EQUATIONS BY ITERATION Page: 542
WORKED EXAMPLE 10.2 GENERAL SOLUTION OF A HOMOGENEOUS FIRST-ORDER DIFFERENCE EQUATION Page: 542
General and particular solutions of difference equations Page: 543
WORKED EXAMPLE 10.3 GENERAL AND PARTICULAR SOLUTIONS OF FIRST-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS Page: 544
Stability and the time path to stability Page: 545
WORKED EXAMPLE 10.4 STABILITY OF SOLUTIONS OF FIRST-ORDER DIFFERENCE EQUATIONS Page: 546
Figure 10.1 Yt = 450(2)t: Yt increases without bound Page: 547
Table 10.2 Points for Yt = 450(2)t Page: 547
Table 10.3 Points for Yt = −(−0.7)t Page: 547
Figure 10.2 Solution oscillates to stability at Y = 0 Page: 547
Table 10.4 Points for Yt = −10(−1)t Page: 548
Figure 10.3 Unstable time path: solution oscillates between +10 and −10 Page: 548
Non-homogeneous difference equations (RHS ≠ 0) Page: 548
WORKED EXAMPLE 10.5 SOLVE NON-HOMOGENEOUS FIRST-ORDER DIFFERENCE EQUATIONS 1 Page: 549
Table 10.5 Points for Yt = 1228(0.95)t + 20 000 Page: 550
Figure 10.4 Stable time path: solution decreases steadily to 20 000 Page: 550
WORKED EXAMPLE 10.6 SOLVE NON-HOMOGENEOUS FIRST-ORDER DIFFERENCE EQUATIONS 2 Page: 551
Table 10.6 Points for Yt = 890(−2/3)t + 10(0.8)t Page: 552
Figure 10.5 Solution oscillates to stability Page: 552
PROGRESS EXERCISES 10.1 Page: 553
10.3 Applications of Difference Equations (First-order) Page: 554
The lagged income model Page: 554
WORKED EXAMPLE 10.7 THE LAGGED INCOME MODEL Page: 555
Table 10.7 Points for Yt = 50(0.8)t + 50 (£000s) Page: 557
Figure 10.6 Solution decreases steadily to 50 (£000s) Page: 557
The cobweb model Page: 557
WORKED EXAMPLE 10.8 THE COBWEB MODEL Page: 558
Table 10.8 Pt = 10(−0.75)t + 50 Page: 560
Figure 10.7 The cobweb model: solution oscillates to a stable value of 50 Page: 560
The Harrod–Domar growth model Page: 560
WORKED EXAMPLE 10.9 THE HARROD–DOMAR GROWTH MODEL Page: 561
Table 10.9 Points for Yt = 8(1.04)t Page: 562
Figure 10.8 The Harrod–Domar growth model Page: 562
PROGRESS EXERCISES 10.2 Page: 563
10.4 Summary Page: 564
Applications Page: 564
www.wiley.com/college/bradley Page: 565
TEST EXERCISES 10 Page: 565
Back Matter Page: 567
Solutions to Progress Exercises Page: 567
Chapter 1 Page: 567
PE 1.2 Page: 567
PE 1.3 Page: 568
PE 1.4 Page: 569
Chapter 2 Page: 570
PE 2.1 Page: 570
Figure PE 2.1 Q3: cuts the horizontal at x = 0 Page: 571
Figure PE 2.1 Q4 Page: 571
Figure PE 2.1 Q5: cuts the horizontal at x = 0 Page: 571
Figure PE 2.1 Q6: cuts the horizontal at x = 2 Page: 571
PE 2.2 Page: 571
Figures PE 2.2 Q2(iii) Page: 571
Figure PE 2.2 Q8 Page: 572
PE 2.3 Page: 572
Figure PE 2.3 Q2 Page: 572
Figure PE 2.3 Q3 Page: 573
Figure PE 2.3 Q4 Page: 573
Figure PE 2.3 Q6 Page: 573
Figure PE 2.3 Q7 Page: 573
PE 2.4 Page: 574
Figure PE 2.4 Q1 Page: 574
Figure PE 2.4 Q2 Page: 574
Figure PE 2.4 Q3 Page: 574
PE 2.5 Page: 575
Figure PE 2.5 Q1(a)(i) Page: 575
Figure PE 2.5 Q1(a)(ii) Page: 575
Figure PE 2.5 Q3 Page: 575
Figure PE 2.5 Q4 Page: 575
PE 2.7 Page: 576
PE 2.9 Page: 577
Figure PE 2.9 Q1(b) Page: 577
Figure PE 2.9 Q2(c) Page: 577
Figure PE 2.9 Q3(a) Page: 577
Figure PE 2.9 Q4(b) Page: 578
Figure PE 2.9 Q4(d) Page: 578
Figure PE 2.9 Q5(b) Page: 578
Figure PE 2.9 Q5(d) Page: 578
Figure PE 2.9 Q7(a) Page: 579
Figure PE 2.9 Q7(b)) Page: 580
Figure PE 2.9 Q7(c) Page: 580
Figure PE 2.9 Q7(d) Page: 580
Table PE 2.9 Q8 Page: 580
Chapter 3 Page: 581
PE 3.1 Page: 581
PE 3.2 Page: 581
Figure PE 3.2 Q5 Page: 581
Figure PE 3.2 Q6 Page: 581
Figure PE 3.2 Q7(b) Page: 581
Figure PE 3.2 Q7(c) Page: 581
PE 3.3 Page: 582
Figure PE 3.3 Q1 Page: 582
Figure PE 3.3 Q5(a) Page: 582
Figure PE 3.3 Q7 Page: 582
Figure PE 3.3 Q10 Page: 583
PE 3.4 Page: 583
Figure PE 3.4 Q2(a) Page: 583
Figure PE 3.4 Q3(a) Page: 583
Figure PE 3.4 Q4(b) Page: 584
Figure PE 3.4 Q5(b) Page: 584
PE 3.5 Page: 584
Figure PE 3.5 Q2(b) Page: 584
Figure PE 3.5 Q3(b) Page: 584
PE 3.7 Page: 585
Figure PE 3.7 Q1(a) Page: 585
Figure PE 3.7 Q1(b) Page: 585
Figure PE 3.7 Q2 Page: 585
Figure PE 3.7 Q3(a) Page: 586
Figure PE 3.7 Q3(b) Page: 586
Figure PE 3.7 Q4(a) Page: 586
Figure PE 3.7 Q4(b) Page: 586
Chapter 4 Page: 587
PE 4.1 Page: 587
PE 4.2 Page: 587
Figure PE 4.2 Q1 Page: 587
Figure PE 4.2 Q2 Page: 587
Figure PE 4.2 Q3(a) Page: 587
Figure PE 4.2 Q3(b) Page: 587
Figure PE 4.2 Q3(c) Page: 588
Figure PE 4.2 Q4 Page: 588
Figure PE 4.2 Q5 Page: 588
Figure PE 4.2 Q6 Page: 588
Figure PE 4.2 Q7 Page: 589
PE 4.3 Page: 589
Figure PE 4.3 Q1 Page: 589
Figure PE 4.3 Q2 Page: 589
Figure PE 4.3 Q3 Page: 589
Figure PE 4.3 Q5 Page: 590
Figure PE 4.3 Q6 Page: 590
Figure PE 4.3 Q7 Page: 590
Figure PE 4.3 Q8 Page: 591
Figure PE 4.3 Q9 Page: 591
Figure PE 4.3 Q10 Page: 591
Figure PE 4.3 Q11 Page: 592
Figure PE 4.3 Q12 Page: 592
PE 4.4 Page: 592
Figure PE 4.4 Q1 Page: 592
Figure PE 4.4 Q2 Page: 593
Figure PE 4.4 Q3 Page: 593
Figure PE 4.4 Q4 Page: 593
PE 4.5 Page: 593
PE 4.6 Page: 594
PE 4.7 Page: 595
Figure PE 4.7 Q1 Page: 595
Figure PE 4.7 Q2 Page: 595
Figure PE 4.7 Q3 Page: 595
Figure PE 4.7 Q4 Page: 595
Figure PE 4.7 Q5 Page: 596
Figure PE 4.7 Q6 Page: 596
PE 4.8 Page: 596
Figure PE 4.8 Q1 Page: 596
Figure PE 4.8 Q2 Page: 597
Figure PE 4.8 Q3 Page: 597
PE 4.9 Page: 597
PE 4.10 Page: 598
PE 4.11 Page: 598
PE 4.12 Page: 599
Figure PE 4.12 Q1 Page: 599
Figure PE 4.12 Q2 Page: 599
Figure PE 4.12 Q3 Page: 599
Figure PE 4.12 Q4 Page: 599
Figure PE 4.12 Q5 Page: 599
Figure PE 4.12 Q6 Page: 599
Figure PE 4.12 Q7 Page: 600
Figure PE 4.12 Q8 Page: 600
Figure PE 4.12 Q9 Page: 600
PE 4.13 Page: 600
Figure PE 4.13 Q7 Page: 600
Figure PE 4.13 Q8(b) Page: 600
Figure PE 4.13 Q9 Page: 601
Figure Pe 4.13 Q10 Page: 601
Chapter 5 Page: 601
PE 5.1 Page: 601
PE 5.2 Page: 602
Figure PE 5.2 Q4 Page: 602
PE 5.3 Page: 602
PE 5.4 Page: 603
PE 5.5 Page: 603
PE 5.6 Page: 603
PE 5.7 Page: 604
Chapter 6 Page: 605
PE 6.1 Page: 605
Figure PE 6.1 Q1 Page: 605
PE 6.3 Page: 607
Figure PE 6.3 Q1 Page: 607
Figure PE 6.3 Q5(c) Page: 608
Figure PE 6.3 Q6d(i) Page: 608
Figure PE 6.3 Q6d(ii) Page: 609
PE 6.4 Page: 610
Figure PE 6.4 Q1(i) Page: 610
Figure PE 6.4 Q1(ii) Page: 610
Figure PE 6.4 Q2(i) Page: 610
Figure PE 6.4 Q2(ii) Page: 610
Figure PE 6.4 Q4(i) Page: 611
Figure PE 6.4 Q4(ii) Page: 611
PE 6.5 Page: 612
PE 6.6 Page: 612
PE 6.7 Page: 613
PE 6.8 Page: 614
Figure PE 6.8 Q1 Page: 614
Figure PE 6.8 Q2 Page: 614
Figure PE 6.8 Q3 Page: 614
Figure PE 6.8 Q4 Page: 614
Figure PE 6.8 Q5 Page: 615
Figure PE 6.8 Q6 Page: 615
Figure PE 6.8 Q7 Page: 615
Figure PE 6.8 Q8 Page: 615
Figure PE 6.8 Q9 Page: 615
Figure PE 6.8 Q10 Page: 615
PE 6.9 Page: 616
Figure PE 6.9 Q1 Page: 616
Figure PE 6.9 Q3(d)(i) Page: 616
Figure PE 6.9 Q3(d)(ii) Page: 616
Figure PE 6.9 Q4(d), TR, TC Page: 617
Figure PE 6.9 Q4(d), MR, MC Page: 617
Figure PE 6.9 Q5(c) Page: 617
Figure PE 6.9 Q5(d) Page: 617
PE 6.10 Page: 618
PE 6.11 Page: 619
PE 6.12 Page: 620
PE 6.13 Page: 620
PE 6.14 Page: 621
PE 6.15 Page: 621
PE 6.16 Page: 622
Figure PE 6.16 Q2 Page: 622
Figure PE 6.16 Q3 Page: 622
Figure PE 6.16 Q5 Page: 623
Figure PE 6.16 Q6 Page: 623
PE 6.17 Page: 624
Chapter 7 Page: 625
PE 7.1 Page: 625
Figure PE 7.1 Q14 Page: 626
Figure PE 7.1 Q15 Page: 626
PE 7.2 Page: 626
PE 7.3 Page: 627
PE 7.4 Page: 627
Figure PE 7.4 Q4(a) Page: 627
PE 7.5 Page: 628
Figure PE 7.5 Q2(b) Page: 628
PE 7.6 Page: 629
PE 7.7 Page: 629
PE 7.8 Page: 630
PE 7.9 Page: 630
Chapter 8 Page: 631
PE 8.1 Page: 631
PE 8.2 Page: 632
PE 8.3 Page: 633
Figure PE 8.3 Q20 Page: 633
Figure PE 8.3 Q21 Page: 634
Figure PE 8.3 Q22 Page: 634
Figure PE 8.3 Q23 Page: 634
Figure PE 8.3 Q24 Page: 634
Figure PE 8.3 Q25 Page: 634
PE 8.4 Page: 635
Figure PE 8.4 Q1 Page: 635
Figure PE 8.4 Q2 Page: 635
Figure PE 8.4 Q3 Page: 635
Figure PE 8.4 Q4 Page: 635
Figure PE 8.4 Q5 Page: 635
Figure PE 8.4 Q6 Page: 635
Figure PE 8.4 Q7 Page: 636
Figure PE 8.4 Q8 Page: 636
Figure PE 8.4 Q9 Page: 636
Figure PE 8.4 Q10 Page: 636
Figure PE 8.4 Q11 Page: 636
Figure PE 8.4 Q12 Page: 636
Figure PE 8.4 Q14 Page: 637
Figure PE 8.4 Q15 Page: 637
Figure PE 8.4 Q16 Page: 637
Figure PE 8.4 Q17 Page: 637
Figure PE 8.4 Q18 Page: 637
Figure PE 8.4 Q19 Page: 638
Figure PE 8.4 Q20 Page: 638
PE 8.5 Page: 638
PE 8.6 Page: 638
Figure PE 8.6 Q7(a) Page: 639
Figure PE 8.6 Q8(a) Page: 639
Figure PE 8.6 Q9 Page: 639
Figure PE 8.6 Q10 Page: 639
PE 8.7 Page: 639
PE 8.8 Page: 639
Figure PE 8.8 Q6 Page: 640
Figure PE 8.8 Q7 Page: 640
Chapter 9 Page: 641
PE 9.1 Page: 641
Figure PE 9.1 Q1(i) Page: 641
Figure PE 9.1 Q1(ii) Page: 641
Figure PE 9.1 Q2 Page: 642
Figure PE 9.1 Q3 Page: 642
Figure PE 9.1 Q4 Page: 642
Figure PE 9.1 Q5(a) Page: 643
Figure PE 9.1 Q5(b) Page: 643
Figure PE 9.1 Q5(c) Page: 643
Figure PE 9.1 Q6(b) Page: 644
Figure PE 9.1 Q7 Page: 644
Figure PE 9.1 Q8 Page: 644
Figure PE 9.1 Q9 Page: 644
Figure PE 9.1 Q10 Page: 645
Figure PE 9.1 Q11 Page: 645
PE 9.2 Page: 646
PE 9.3 Page: 647
PE 9.4 Page: 647
PE 9.5 Page: 648
PE 9.6 Page: 648
Chapter 10 Page: 649
PE 10.1 Page: 649
PE 10.2 Page: 650
Figure PE 10.2 Q5(c)(i) Page: 651
Figure PE 10.2 Q5(c)(ii) Page: 651
Figure PE 10.2 Q6(b) Page: 651
Figure PE 10.2 Q6(c) Page: 652
Worked Examples Page: 653
Index Page: 659