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Essential Mathematics for Economics and Business PDF

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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

Description:

Essential Mathematics for Economics and Business is established as one of the leading introductory textbooks on mathematics for students of business and economics. Combining a user–friendly approach to mathematics with practical applications to the subjects, the text provides students with a clear and comprehensible guide to mathematics. The fundamental mathematical concepts are explained in a simple and accessible style, using a wide selection of worked examples, progress exercises and real–world applications.

New to this Edition

  • Fully updated text with revised worked examples and updated material on Excel and Powerpoint
  • New exercises in mathematics and its applications to give further clarity and practice opportunities
  • Fully updated online material including animations and a new test bank
  • The fourth edition is supported by a companion website at www.wiley.com/college/bradley, which contains: Animations of selected worked examples providing students with a new way of understanding the problems Access to the Maple T.A. test bank, which features over 500 algorithmic questions Further learning material, applications, exercises and solutions.
  • Problems in context studies, which present the mathematics in a business or economics framework.
  • Updated PowerPoint slides, Excel problems and solutions.

"The text is aimed at providing an introductory-level exposition of mathematical methods for economics and business students. In terms of level, pace, complexity of examples and user-friendly style the text is excellent - it genuinely recognises and meets the needs of students with minimal maths background."
—Colin Glass, Emeritus Professor, University of Ulster

"One of the major strengths of this book is the range of exercises in both drill and applications. Also the 'worked examples' are excellent; they provide examples of the use of mathematics to realistic problems and are easy to follow."
—Donal Hurley, formerly of University College Cork

"The most comprehensive reader in this topic yet, this book is an essential aid to the avid economist who loathes mathematics!"
—Amazon.co.uk

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