Essential Mathematics for Economics and Business / Edition 4

by Teresa Bradley

Overview

Now 4 colour and includes an outstanding resources suite!

Essential Mathematics for Economics and Business is established as one of the leading introductory textbooks for non maths specialists taking economics and business degrees.

The fundamental mathematical concepts are explained as simply and briefly as possible, using a wide selection of worked examples, graphs and real-world applications.  It combines a non-rigorous approach to mathematics with applications in economics and business.

 

'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

Product Details

ISBN-13:	9781118358290
Publisher:	Wiley
Publication date:	05/06/2013
Edition description:	4th Revised ed.
Pages:	688
Product dimensions:	7.40(w) x 9.60(h) x 1.20(d)

About the Author

Until 2012, Teresa Bradley lectured in mathematics and statistics at Limerick Institute of Technology, and has been involved for many years with the University of London on the external Diploma in Economics as well as the BSc in Economics, Business and Management.
Teresa Bradley is also author of Essential Statistics for Economics, Business and Management, published by John Wiley & Sons, Ltd.

Read an Excerpt

Essential Mathematics for Economics and Business

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By Teresa Bradley Paul Patton

John Wiley & Sons

ISBN: 0-470-84466-3

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Chapter One

Mathematical Preliminaries

At the end of this chapter you should be able to:

Perform basic arithmetic operations and simplify algebraic expressions

Perform basic arithmetic operations with fractions

Solve equations in one unknown, including equations involving fractions

Understand the meaning of no solution and infinitely many solutions

Currency conversions

Solve simple inequalities

Calculate percentages

In addition, you will be introduced to the calculator and a spreadsheet.

Some mathematical preliminaries

Brackets in mathematics are used for grouping and clarity. Brackets may also be used to indicate multiplication. Brackets are used in functions to declare the independent variable (see later). Powers: positive whole numbers such as [2.sup.3], which means 2 x 2 x 2 = 8:

[(anything).sup.3] = (anything) x (anything) x (anything)

[(x).sup.3] = x x x x x

[(x + 4).sup.5] = (x + 4)(x + 4)(x + 4) (x + 4) (x + 4)

Note

Brackets: (A)(B) or A x B or AB all indicate A multiplied by B.

Variables and letters: When we don't know the value of a quantity, we give that quantity a symbol, such as x. We may then make general statements about the unknownquantity, x, for example 'For the next 15 weeks, if I save £x per week I shall have £4000 to spend on a holiday'. This statement may be expressed as a mathematical equation:

15 x weekly savings = 4000 15 x x = 4000 or 15x = 4000

Now that the statement has been reduced to a mathematical equation, we may solve the equation for the unknown, x:

15x = 4000 15x/15 = 4000/15 divide both sides of the equation by 15 x = 266.67

Square roots: the square root of a number is the reverse of squaring:

[(2).sup.2] = 4 [right arrow] [square root of (4)] = 2

[(2.5).sup.2] = 6:25 [right arrow] [square root of (6.25)] = 2:5

1.1 Arithmetic Operations

Addition and subtraction

Adding: If all the signs are the same, simply add all the terms and give the answer with the common overall sign.

Subtracting: When subtracting any two numbers or two similar terms, give the answer with the sign of the largest number or term.

If terms are identical, for example all x-terms, all xy-terms, all [x.sup.2]-terms, then they may be added or subtracted as shown in the following examples:

Add/subtract with numbers, mostly Add/subtract with variable terms

5 + 8 + 3 = 16 similarity [right arrow] 5x + 8x + 3x = 16x

5 + 8 + 3 + y = 16 + y similarity [right arrow] (i) 5x + 8x + 3x + y = 16x + y

The y-term is different, so it cannot be (ii) 5xy + 8xy + 3xy + added to the others y = 16xy + y

The y-term is different, so it cannot be added to the others

7 - 10 = -3 similarity [right arrow] (i) 7x - 10x = -3x

(ii) 7[x.sup.2] - 10[x.sup.2] = -3[x.sup.2]

7 - 10 - 10x = -3 - 10x similarity [right arrow] 7[x.sup.2] - 10[x.sup.2] - 10x = -3[x.sup.2] - 10x

The x-term is different, so it cannot be The x-term is different, so it cannot be subtracted from the others subtracted from the others

Worked example 1.1 Addition and subtraction

For each of the following, illustrate the rules for addition and subtraction:

(a) 2 + 3 + 2·5 = (2 + 3 + 2.5) = 7.5

(b) 2x + 3x + 2.5x = (2 + 3 + 2.5)x = 7.5x

(c) -3xy - 2.2xy - 6xy = (-3 -2.2 -6)xy = -11.2 xy

(d) 8x + 6xy -12x + 6 + 2xy = 8x 12x + 6xy + 2xy)6 = -4x + 8xy + 6

(e) 3[x.sup.2] + 4x + 7 - 2[x.sup.2] - 8x + 2 = 3[x.sup.2] - 2[x.sup.2] + 4x - 8[x + 7 + 2 = [x.sup.2] - 4x + 9

Multiplying and dividing

Multiplying or dividing two quantities with like signs gives an answer with a positive sign. Multiplying or dividing two quantities with different signs gives an answer with a negative sign.

Worked example 1.2 Multiplication and division

Each of the following examples illustrate the rules for multiplication.

(a) 5 x 7 = 35 (b) -5 x -7 = 35 (c) 5 x -7 = -35 (d) -5 x 7 = -35 (e) 7/5 = 1.4 (f) (-7)/(-5) = 1.4 (g) (-7)/5= -1.4 (h) 7/(-5) = -1.4 (i) 5(7) = 35 (j) (-5)(-7) = 35 (k) (-5)y = -5y (l) (-x)(-y) = xy (m) 2(x + 2) = 2x + 4 (n) (x + 4)(x + 2) = x(x + 2) + 4(x + 2) = [x.sup.2] + 2x + 4x + 8 = [x.sup.2] + 6x + 8

(o) [(x + y).sup.2]

= (x + y)(x + y) = x(x + y) + y(x + y) = xx + xy + yx + yy = [x.sup.2] + 2xy + [y.sup.2]

* Remember

It is very useful to remember that a minus sign is a -1, so -5 is the same as -1 x 5

* Remember

0 x (any real number) = 0 0 / (any real number) = 0 But you cannot divide by 0

multiply each term inside the bracket by the term outside the bracket multiply the second bracket by x, then multiply the second bracket by (+4) and add,

multiply each bracket by the term outside it add or subtract similar terms, such as 2x + 4x = 6x

multiply the second bracket by x and then by y add the similar terms: xy + yx = 2xy

The following identities are important:

1. (x + y).sup.2] = [x.sup.2] + 2xy + [y.sup.2] 2. (x - y).sup.2] = [x.sup.2] - 2xy + [y.sup.2] 3. (x + y) = (x - y) = [x.sup.2] -[y.sup.2]

* Remember: Brackets are used for grouping terms together in maths for:

(i) Clarity (ii) Indicating the order in which a series of operations should be carried out

1.2 Fractions

Terminology:

fraction = numerator/denominator

3/7

3 is called the numerator

7 is called the denominator

1.2.1 Add/subtract fractions: method

The method for adding or subtracting fractions is:

Step 1: Take a common denominator, that is, a number or term which is divisible by the denominator of each fraction to be added or subtracted. A safe bet is to use the product of all the individual denominators as the common denominator.

Step 2: For each fraction, divide each denominator into the common denominator, then multiply the answer by the numerator.

Step 3: Simplify your answer if possible.

Worked example 1.3 Add and subtract fractions

Each of the following illustrates the rules for addition and subtraction of fractions.

Numerical example

1/7 + 2/3 - 4/5

Step 1: The common denominator is (7)(3)(5)

Step 2: 1/7 + 2/3 - 4/5

=1(3)(5)+2(7)(5)-4(7)(3)/(7)(3)(5)

Step 3: = 15 + 70 - 84/105 =1/105

1/7 + 2/3

Step 1: The common denominator is (7)(3)

Step 2: 1/7 + 2/3 = 1(3) + 2(7)/(7)(3)

Step 3: = 3 + 14/21 = 17/21

Same example, but with variables

x/7 + 2x/3-4x/5

= x(3)(5) + 2x(7)(5)-4x(7)(3) (7)(3)(5)

= 15x + 70x - 84x/105

= x/105

1/x + 4 + 2/x = 1(x) + 2(x + 4)/(x + 4) (x)

= x + 2x + 8/[x.sup.2]+ 4x

= 3x + 8/[x.sup.2]+ 4x

1.2.2 Multiplying fractions

In multiplication, write out the fractions, multiply the numbers across the top lines and multiply the numbers across the bottom lines.

Note: Write whole numbers as fractions by putting them over 1.

Terminology: RHS means right-hand side and LHS means left-hand side.

Worked example 1.4

Multiplying fractions

(a) (2/3) (5/7) = (2)(5)/(3)(7) = 10/21

(b) (-2/3) (7/5) = (-2)(7)/(3)(5) = -14/15

(c) 3 x 2/5 (3/1)(2/5)=(3)(2)/(1)(5) = 6/5 = 1 1/5

The same rules apply for fractions involving variables, x, y, etc.

(d) (3/x) (x + 3)/(x - 5) = 3(x + 3)/x(x - 5) = 3x + 9/[x.sup.2]- 5x]

1.2.3 Dividing fractions

General rule:

Dividing by a fraction is the same as multiplying by the fraction inverted

Worked example 1.5

Division with fractions

The following examples illustrate how division with fractions operates.

(a) (2/3)/(5/11) = (2/3)(11/5) = 22/15

(b) 5/(3/4) = 5 x 4/3 = 5/1 x 4/3 = 20/3 = 6 2/3

(c) (7/3)/8 = (7/3)/8/1 = 7/3 x 1/8 = 7/24

(d) 2x/x + y/3x/2(x - y) = 2x/x + y 2(x - y)/ 3x

= 4x(x - y)/3x(x + y) = 4(x - y)/3(x + y)

Note: The same rules apply to all fractions, whether the fractions consist of numbers or variables.

Progress Exercises 1.1 Revision on Basics

Show, step by step, how the expression on the left-hand side simplifies to that on the right.

1. 2x + 3x + 5(2x - 3) = 15(x - 1)

2. 4[x.sup.2] + 7x + 2x (4x - 5) = 3x(4x - 1)

3. 2x (y + 2) - 2y(x + 2) = 4(x - y)

4. (x + 2)(x - 4) - 2(x - 4) = x(x - 4)

5. (x + 2)(y - 2) + (x - 3)(y + 2)

= 2xy - y - 10

6. [(x + 2).sup.2] + [(x - 2).sup.2] = 2[(x.sup.2] + 4)

7. [(x + 2).sup.2] - [(x - 2).sup.2] = 8x

8. (x + 2)2 - x(x + 2) = 2(x + 2)

9. 1/3 + 3/5 + 5/7 = 1 68/105

10. x/2 - x/3 = x/6

11. (2/3)/(1/5) = 10/3

12. (2/7)/3 = 2/21

13. 2(2/x - x/2) = 4-[x.sup.2]/x = 4/x - x

14. -12/P (3P/2 + P/2) = -24

15. (3/x)/x + 3 = 3/x(x +3)

16. (5Q/P + 2)/(1/(P + 2)) = 5Q

1.3 Solving Equations

The solution of an equation is simply the value or values of the unknown(s) for which the left-hand side (LHS) of the equation is equal to the right-hand side (RHS).

For example, the equation, x + 4 = 10, has the solution x = 6. We say x = 6 'satisfies' the equation. We say this equation has a unique solution.

Not all equations have solutions. In fact, equations may have no solutions at all or may have infinitely many solutions. Each of these situations is demonstrated in the following examples.

Case 1: Unique solutions An example of this is given above: x + 4 = 10 etc.

Case 2: Infinitely many solutions The equation, x + y = 10 has solutions (x = 5, y = 5); (x = 4, y = 6); (x = 3, y = 7), etc. In fact, this equation has infinitely many solutions or pairs of values (x, y) which satisfy the formula, x + y = 10.

Case 3: No solution The equation, 0(x) = 5 has no solution. There is simply no value of x which can be multiplied by 0 to give 5.

* Methods for solving equations

Solving equations can involve a variety of techniques, many of which will be covered later.

(Continues...)

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Excerpted from Essential Mathematics for Economics and Business by Teresa Bradley Paul Patton Excerpted by permission.
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Table of Contents

Introduction xiii

Chapter 1

Mathematical Preliminaries 1

1.1 Some Mathematical Preliminaries 2

1.2 Arithmetic Operations 3

1.3 Fractions 6

1.4 Solving Equations 11

1.5 Currency Conversions 14

1.6 Simple Inequalities 18

1.7 Calculating Percentages 21

1.8 The Calculator. Evaluation and Transposition of Formulae 24

1.9 Introducing Excel 28

Chapter 2

The Straight Line and Applications 37

2.1 The Straight Line 38

2.2 Mathematical Modelling 54

2.3 Applications: Demand, Supply, Cost, Revenue 59

2.4 More Mathematics on the Straight Line 76

2.5 Translations of Linear Functions 82

2.6 Elasticity of Demand, Supply and Income 83

2.7 Budget and Cost Constraints 91

2.8 Excel for Linear Functions 92

2.9 Summary 97

Chapter 3

Simultaneous Equations 101

3.1 Solving Simultaneous Linear Equations 102

3.2 Equilibrium and Break-even 111

3.3 Consumer and Producer Surplus 128

3.4 The National Income Model and the IS-LM Model 133

3.5 Excel for Simultaneous Linear Equations 137

3.6 Summary 142

Appendix 143

Chapter 4

Non-linear Functions and Applications 147

4.1 Quadratic, Cubic and Other Polynomial Functions 148

4.2 Exponential Functions 170

4.3 Logarithmic Functions 184

4.4 Hyperbolic (Rational) Functions of the Form a/(bx + c) 197

4.5 Excel for Non-linear Functions 202

4.6 Summary 205

Chapter 5

Financial Mathematics 209

5.1 Arithmetic and Geometric Sequences and Series 210

5.2 Simple Interest, Compound Interest and Annual Percentage Rates 218

5.3 Depreciation 228

5.4 Net Present Value and Internal Rate of Return 230

5.5 Annuities, Debt Repayments, Sinking Funds 236

5.6 The Relationship between Interest Rates and the Price of Bonds 248

5.7 Excel for Financial Mathematics 251

5.8 Summary 254

Appendix 256

Chapter 6

Differentiation and Applications 259

6.1 Slope of a Curve and Differentiation 260

6.2 Applications of Differentiation, Marginal Functions, Average Functions 270

6.3 Optimisation for Functions of One Variable 286

6.4 Economic Applications of Maximum and Minimum Points 304

6.5 Curvature and Other Applications 320

6.6 Further Differentiation and Applications 334

6.7 Elasticity and the Derivative 347

6.8 Summary 357

Chapter 7

Functions of Several Variables 361

7.1 Partial Differentiation 362

7.2 Applications of Partial Differentiation 380

7.3 Unconstrained Optimisation 400

7.4 Constrained Optimisation and Lagrange Multipliers 410

7.5 Summary 422

Chapter 8

Integration and Applications 427

8.1 Integration as the Reverse of Differentiation 428

8.2 The Power Rule for Integration 429

8.3 Integration of the Natural Exponential Function 435

8.4 Integration by Algebraic Substitution 436

8.5 The Definite Integral and the Area under a Curve 441

8.6 Consumer and Producer Surplus 448

8.7 First-order Differential Equations and Applications 456

8.8 Differential Equations for Limited and Unlimited Growth 468

8.9 Integration by Substitution and Integration by Parts website only

8.10 Summary 474

Chapter 9

Linear Algebra and Applications 477

9.1 Linear Programming 478

9.2 Matrices 488

9.3 Solution of Equations: Elimination Methods 498

9.4 Determinants 504

9.5 The Inverse Matrix and Input/Output Analysis 518

9.6 Excel for Linear Algebra 531

9.7 Summary 534

Chapter 10

Difference Equations 539

10.1 Introduction to Difference Equations 540

10.2 Solution of Difference Equations (First-order) 542

10.3 Applications of Difference Equations (First-order) 554

10.4 Summary 564

Solutions to Progress Exercises 567

Worked Examples 653

Index 659