MATHEMATICS FOR BUSINESS ECONOMICS Prof. dr. Herbert ... · MATHEMATICS FOR BUSINESS ECONOMICS HAMERS, KLEPPE AND KAPER HERBERT HAMERS JOHN KLEPPE THIRD EDITION BOB KAPER MATHEMATICS

www.boomhogeronderwijs.nl

MA

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BUSIN

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ERS, KLEPPE A

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KA

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HERBERT HAMERS

JOHN KLEPPE

BOB KAPERTHIRD EDITION

MATHEMATICS FOR BUSINESS ECONOMICS

Mathematics for Business Economics is intended for starting (business) economics students. The book repeats high school mathematics, but also extends to new domains. The focus is on economic applications: throughout the book many examples are given from microeconomics, modern portfolio theory, inventory management and statistics. A recurrent theme in these examples is the mathematical technique of optimization.

For a better understanding this book contains a large number of exercises. Furthermore, there is a freely accessible e-learning environment, where all mathematical topics are explained using text, multiple choice questions and films. Together, the book and e-learning environment provide a solid mathematical foundation for your study (business) economics.

Prof. dr. Herbert Hamers is professor in Game Theory and Operations Research at Tilburg University. Dr. Bob Kaper was professor in Mathematics at Tilburg University. Dr. John Kleppe was a�liated as lecturer mathematics at Tilburg University.

A Dutch language edition of this book is also available: Wiskunde voor bedrijfseconomen (ISBN: 9789024408481).

Mathemati s for Business E onomi s

Mathemati s for Business

E onomi s

Third edition

Herbert Hamers

Bob Kaper

John Kleppe

Basi design over: Dog & Pony, Amsterdam

Cover design: Co o Bookmedia, Amersfoort

2020 H. Hamers, B. Kaper, J. Kleppe & Boom uitgevers Amsterdam

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ISBN 978-90-2442-842-7

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NUR 123/782


Prefa e

This mathemati s book is intended for starting (business) e onomi s students. It is

based on many years of experien e in mathemati s le tures to students of Tilburg Uni-

versity and on tea hing high s hool students. The book repeats ontent of high s hool

(elementary al ulus, fun tions, derivatives), but also extends to new ontents (partial

derivatives, ( onstrained) optimization). The fo us is on e onomi appli ations. There

are many examples from mi roe onomi s, modern portfolio theory, inventory manage-

ment and statisti s. In this new edition, a hapter is added that ontains a large set of

diagnosti exer ises.

In addition to the book there is an e-learning environment where all mathemati al

topi s are explained using text, multiple hoi e questions and �lms. The address of this

e-learning environment is www.wiskundeoptiu.nl/business-e onomi s.

The solutions of all exer ises are available in the appendix.

This book would not have been written without the help of a large number of ol-

leagues that supported us with tips and tri ks. In parti ular, we want to mention Ruud

Brekelmans, Bart Husslage. Elleke Janssen and Marieke Quant, who ontributed to a

great extend to the development of the e-learning environment. Moreover, additional

thanks to Elleke for het support with L

A

T

E

X, in parti ular with respe t to the lay-out and

�gures. Further, we thank Marieke Quant and Jop S houten for their ontribution to

Chapter 7.

Herbert Hamers

Bob Kaper

John Kleppe

Tilburg, de ember 2019

v

Table of Contents

Prefa e v

Introdu tion xi

1 Fun tions of one variable 1

1.1 Introdu tion to fun tions of one variable . . . . . . . . . . . . . . . . . . 1

1.2 Overview of fun tions of one variable . . . . . . . . . . . . . . . . . . . 5

1.2.1 Polynomial fun tions . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Power fun tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.3 Exponential fun tions . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.4 Logarithmi fun tions . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.1 Break-even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.2 Market equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4 Diagnosti exer ises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Di�erentiation of fun tions of one variable 27

2.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Rules of di�erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1 Marginality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.2 Elasti ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.3 Rate produ tion pro ess . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Inverse fun tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.6 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6.1 Elasti ity and inverse . . . . . . . . . . . . . . . . . . . . . . . . 52


3 Fun tions of two variables 59

3.1 Introdu tion to fun tions of two variables . . . . . . . . . . . . . . . . . 59

3.2 Overview of fun tions of two variables . . . . . . . . . . . . . . . . . . . 61

3.3 Level urves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 Utility fun tions and indi�eren e urves . . . . . . . . . . . . . . 65

3.4.2 Modern portfolio theory . . . . . . . . . . . . . . . . . . . . . . 67


vii

viii Table of Contents

4 Di�erentiation of fun tions of two variables 71

4.1 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Partial marginality . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2 Partial elasti ity . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Tangent line to a level urve . . . . . . . . . . . . . . . . . . . . . . . . 86

4.5 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5.1 Marginal rate of substitution . . . . . . . . . . . . . . . . . . . . 89


5 Optimization 97

5.1 Optimization of fun tions of one variable . . . . . . . . . . . . . . . . . 97

5.2 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.1 Marginal output rule and produ tion rule . . . . . . . . . . . . . 109

5.2.2 Supply fun tion produ er . . . . . . . . . . . . . . . . . . . . . . 112

5.2.3 Inventory management . . . . . . . . . . . . . . . . . . . . . . . 115

5.3 Optimization of fun tions of two variables . . . . . . . . . . . . . . . . . 117

5.4 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.4.1 Pro�t maximization produ er . . . . . . . . . . . . . . . . . . . . 123

5.4.2 Regression line . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5 Optimization of onstrained extremum problems . . . . . . . . . . . . . 129

5.5.1 Substitution method . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5.2 First-order ondition for a onstrained extremum . . . . . . . . . 135

5.5.3 First-order ondition of Lagrange . . . . . . . . . . . . . . . . . 139

5.6 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.6.1 Utility maximization onsumer . . . . . . . . . . . . . . . . . . . 141

5.6.2 Cost minimization produ er . . . . . . . . . . . . . . . . . . . . 143

5.6.3 Optimal portfolio sele tion . . . . . . . . . . . . . . . . . . . . . 146

5.7 Optimization of onvex and on ave fun tions . . . . . . . . . . . . . . . 151

5.7.1 Convex and on ave fun tions of one variable . . . . . . . . . . . 151

5.7.2 Convex and on ave fun tions of two variables . . . . . . . . . . 154


6 Areas and integrals 161

6.1 Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.2 Antidi�erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.3 Area and integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.4 Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.4.1 Consumer surplus . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.4.2 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . 177


7 Extra diagnosti exer ises 181

Table of Contents ix

A Solutions to Exer ises 185

A.1 Solutions Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185







A.8 Figures a ompanying the solutions . . . . . . . . . . . . . . . . . . . . 203

Index 205

Introdu tion

This introdu tion �rst presents a short overview of the ontents of this book. Next,

we dis uss the stru ture of the book. Finally, some re ommendations for studying are

provided.

In Chapter 1 we introdu e fun tions of one variable. Besides an overview of elemen-

tary fun tions, also the determination of zeros, solving inequalities and al ulations using

powers are dis ussed.

Chapter 2 is devoted to the derivative of a fun tion of one variable. Besides a

number of di�erentiation rules, also the inverse fun tion and the e onomi notions of

marginality and elasti ity are presented.

In Chapter 3 we introdu e fun tions of two variables and the notion of a level urve.

Utility of a onsumer and a �rst introdu tion to modern portfolio theory are examples of

e onomi appli ations that will be addressed.

Partial derivatives and partial di�erentiation are the key elements of Chapter 4.

Moreover, the e onomi notion of marginal rate of substitution is related to the tangent

line to a level urve.

Chapter 5 is ompletely dedi ated to the determination of extrema, either with or

without onstraints. As appli ations from e onomi s we dis uss, among other things,

pro�t maximization of a produ er, the E onomi Order Quantity model, the regression

line, utility maximization of a onsumer, ost minimizing of a produ er and optimal port-

folio sele tion. Furthermore, also onvex and on ave fun tions are dis ussed, be ause

these fun tions frequently appear in e onomi models.

Finally, in Chapter 6 the notion of an integral is introdu ed and used as tool to

determine the area of a region. We apply integrals for the determination of onsumer

surplus and for the al ulation of probabilities.

In this book mathemati al and e onomi topi s are dis ussed in separate se tions. In the

mathemati al se tions a mathemati al notion or method is followed by a (mathemati al)

example and an exer ise. The example is usually a omplete solution or an illustration

of the orresponding notion or method; the exer ise is a way to fully apture a notion or

a method. A similar remark an be made on erning the e onomi -oriented se tions.

Ea h hapter is on luded with a se tion of diagnosti exer ises. The answers to

the exer ises are in luded in the appendix.

The �rst step in learning mathemati s is to visit the le tures that are o�ered at your

university. Understanding mathemati s an only be attained by trying to do the exer ises

yourselves. You will not always manage to solve an exer ise. Then it is suggested that

you study again the example dire tly before the exer ise. If this provides insu� ient

information, you have to onsult the theory in the book again. Another option is to

use the e-learning environment that an be found at www.wiskundeoptiu.nl/business-

xi

xii Introdu tion

e onomi s. Here the most important notions and methods are explained in an example

by a tea her. One an also pra ti e more with many multiple- hoi e questions. If you

have �nished all the exer ises in the text, then you an start with the diagnosti exer ises

of that hapter. These exer ises reveal whether you ompletely understand the hapter.

1

Fun tions of one variable

Fun tions are frequently used to des ribe relations between (e onomi ) variables. In this

hapter we introdu e the notion of a fun tion of one variable and provide an overview of

the lasses of polynomial fun tions, power fun tions, exponential fun tions and logarith-

mi fun tions. Moreover, elementary algebrai operations su h as solving equations and

inequalities are dis ussed.

List of notions

• Fun tion of one variable

• Zero of a fun tion

• Point of interse tion of graphs of fun tions

• Linear fun tion

• Quadrati fun tion

• Polynomial fun tion

• Power fun tion

• Exponential fun tion

• Logarithmi fun tion

1.1 Introdu tion to fun tions of one variable

To des ribe the relation between variables we an use the mathemati al notion of a fun -

tion. In this se tion we dis uss the notion of a fun tion of one variable.

We start this se tion with an example in whi h we show how a fun tion an be used

to des ribe the relation between the ost and the quantity of a produ ed good.

Example 1.1: ost fun tion

A manager has investigated a produ tion pro ess to analyze its osts. The result of this

investigation is a relation between the produ tion level q and the orresponding osts

C, whi h an be displayed graphi ally in a oordinate system (see Figure 1.1). In this

oordinate system the horizontal axis represents the produ tion level and the verti al

1

2 1. Fun tions of one variable

axis represents the orresponding osts. In other words, the independent variable (the

produ tion level) is displayed on the horizontal axis and the dependent variable (the

osts) is displayed on the verti al axis.

q

C

Figure 1.1 The ost fun tion of a produ tion pro ess.

A graphi al relation between the produ tion level and the osts is usually not su� ient.

A manager wants to know for ea h produ tion level q the exa t orresponding osts

C(q). Therefore, it is more useful to des ribe the relation between the produ tion

level and osts by a fun tion: that is a pres ription that al ulates for ea h feasible

produ tion level q the orresponding osts C. The pres ription that �ts the graph of

Figure 1.1 is

√q. This pres ription means that if the produ tion level q is equal to

9, the osts are

√9 = 3, and if the produ tion level q is equal to 16, the osts are√

16 = 4, et etera.

Ea h time the osts are obtained by applying the pres ription

√q to a value of

the produ tion level q. Sin e we assign no interpretation to produ tion levels with a

negative value, we restri t the pres ription to values q greater than or equal to 0. We

on lude that the variables C and q satisfy the equation C =√

q. We may say that

`the osts C is a fun tion of the produ tion level q'.

If the osts C is a fun tion of the produ tion level q and we do not spe ify the a tual

pres ription, then we denote the pres ription by C(q). The relation between C and q is

then given by the equation C = C(q). Hen e, in the above example C(q) represents√

q.

1.1. Introdu tion to fun tions of one variable 3

Fun tions of one variable

A fun tion of the variable x is a pres ription y(x) whi h al ulates a number,

the fun tion value, for any feasible value of the variable x.The set of all feasible values D of x is alled the domain of the fun tion.

The set of all possible fun tion values is alled the range of the fun tion.

If the domain D of a fun tion y(x) is not expli itly given, then the domain onsists of all

x for whi h the pres ription y(x) an be exe uted. If additional restri tions are provided,

then this will be expli itly stated. Su h a restri tion, like non-negativity of x, an for

example arise from the e onomi interpretation of the model.

The fun tion values y(x) an be interpreted as the values of a variable. If we all

this variable y, then y and x satisfy the equation

y = y(x).

The variable x in y(x) is alled the independent or input variable and the variable y is

alled the dependent or output variable.

Fun tions and variables are denoted by letters or ombinations of letters. Maybe you

are used to denoting variables by the letters x, y or z and a fun tion by the letter f(hen e f (x) instead of y(x), so that y = f (x)). In e onomi theory, however, one often

hooses a letter or a ombination of letters that is lose to the meaning of the variable

(p for pri e, L for Labour, K (from German `Kapital') for apital, w for wage, et etera),

or of the pres ription (MC for Marginal Cost fun tion, AC for Average Cost, et etera).

The graph of a fun tion y(x) is a graphi al representation of the fun tion in a oordinatesystem with two axes, the x-axis and the y-axis, onsisting of points with oordinates

(x, y(x)). It is ommon to use the horizontal axis for the independent variable x and the

verti al axis for the dependent variable y.

Example 1.2: graph of a fun tion

In Figure 1.2 the graph of the fun tion y(x) = 1 +√

x + 2 is shown. The domain of

y(x) is x ≥ −2 and the range is y ≥ 1. In interval notation the domain is [−2, ∞) andthe range [1, ∞).


x

y

y(x) = 1 +√

x + 2

−2

1

Figure 1.2 The graph of the fun tion y(x) = 1 +√

x + 2 in an (x, y)- oordinate system.

A point of interse tion of the graph of a fun tion y(x) with the x-axis an be determined

by al ulating a zero of the fun tion y(x).

Zero of a fun tion of one variable

A zero of a fun tion y(x) is a solution of the equation y(x) = 0.

A zero a of the fun tion y(x) gives the point of interse tion (a, 0) of the orresponding

graph and the x-axis.

Point of interse tion of two graphs

A point of interse tion of the graph y(x) with the graph of another fun tion

z(x) is a point (a, b) where a is a solution of the equation y(x) = z(x) and

b = y(a)(= z(a)).

For the determination of a point of interse tion of the graphs of the fun tions y(x) andz(x) we �rst al ulate the x- oordinate by solving the equation

y(x) = z(x).

Subsequently, the y- oordinate is obtained by substituting that x- oordinate into one of

the two fun tions.

A point of interse tion of the graph of a fun tion y(x) with the y-axis is obtainedby al ulating the fun tion value at x = 0. Hen e, a point of interse tion with the y-axisis (0, y(0)).

1.2. Overview of fun tions of one variable 5

Example 1.3: zero and point of interse tion

Consider the fun tions y(x) = −2x + 2 and z(x) = x − 4. The zero of y(x) is the

solution of y(x) = 0,

y(x) = 0 ⇔ −2x + 2 = 0

⇔ −2x = −2

⇔ x = 1.

Hen e, the point (1, 0) is the point of interse tion of the graph of y(x) with the x-axis.A point of interse tion of the graph of y(x) with the graph of z(x) is obtained by

solving the equation y(x) = z(x),

y(x) = z(x) ⇔ −2x + 2 = x − 4

⇔ −3x = −6

⇔ x = 2.

The x- oordinate of the point of interse tion is x = 2. Substituting x = 2 into the

fun tion y(x) we get the y- oordinate, whi h equals y(2) = −2. Obviously, we also

have z(2) = −2. The point of interse tion is (2,−2). Sin e y(0) = 2, the graph of

y(x) interse ts the y-axis at the point (0, 2).

In the up oming se tions of this hapter we dis uss several elementary fun tions of one

variable, show the orresponding graphs, provide some properties and al ulate zeros and

points of interse tion.

1.2 Overview of fun tions of one variable

1.2.1 Polynomial fun tions

In this subse tion we start with an overview of onstant, linear and quadrati fun tions.

Subsequently, we provide the general expression of a polynomial fun tion. In this overview

the solving of equations and inequalities is also dis ussed.

Constant fun tions

A fun tion of the form

y(x) = c,

where c is a number, is alled a onstant fun tion. A onstant fun tion has the same

value for ea h x. Hen eforth, the graph of a onstant fun tion is a horizontal line.


Example 1.4: graph of a onstant fun tion

An example of a onstant fun tion is the fun tion y(x) = 3 and its graph is shown in

Figure 1.3.

x

y

y(x) = 3

3

Figure 1.3 The graph of the onstant fun tion y(x) = 3.

Note that a onstant fun tion y(x) = c has no zeros if c 6= 0 and that every x is a zero

if c = 0.

Linear fun tions


y(x) = ax + b,

where a and b are numbers (a 6= 0), is alled a linear fun tion. Note that if a = 0 then

the fun tion y(x) is a onstant fun tion. The graph of a linear fun tion is a straight

line. The slope of the line is equal to the number a. The slope of the graph of a linear

fun tion indi ates the hange of the fun tion value if the input x in reases by one unit.

Sin e for a linear fun tion this hange is always equal to a, it holds that for any x

y(x + 1)− y(x) = a.

A linear fun tion has a positive slope if a > 0 and a negative slope if a < 0. Note that

the slope of a onstant fun tion is equal to 0.

Example 1.5: graph of a linear fun tion

The graphs of the linear fun tions y(x) = 3x + 2 and z(x) = −2x + 1 are shown in

Figure 1.4 in an (x, y)- oordinate system. The slope of the graph of y(x) is equal to3 and the slope of the graph of z(x) is equal to −2.


x

y

y(x) = 3x + 2

− 23

2

x

y

z(x) = −2x + 1

1

12

Figure 1.4 The graph of the fun tion y(x) = 3x + 2 (left).

The graph of the fun tion z(x) = −2x + 1 (right).

In the �nal example of Se tion 1.1 we have al ulated zeros and points of interse tion

for linear fun tions. It is straightforward to determine a general expression for the point

of interse tion of the graph of a linear fun tion and the x-axis or y-axis, respe tively. Todetermine the point of interse tion of the graph of a linear fun tion y(x) = ax + b with

the x-axis, we al ulate the zero of the fun tion y(x),

y(x) = 0 ⇔ ax + b = 0

⇔ ax = −b

⇔ x = − b

a.

Sin e a linear fun tion y(x) = ax + b has pre isely one zero, the graph has pre isely

one point of interse tion with the x-axis. Hen e, the point of interse tion of the graph

of a linear fun tion and the x-axis is (− ba , 0). The point of interse tion with the y-

axis is obtained by substituting x = 0 into the fun tion: y(0) = b. Hen e, the point

of interse tion with the y-axis is (0, b). We an on lude that for a linear fun tion

y(x) = ax + b the slope of the line is represented by a, the zero equals − ba and b is the

y- oordinate of the point of interse tion of the graph with the y-axis.

Exer ise 1.1 (zero and point of interse tion)

Consider the fun tions y(x) = 3x + 2 and z(x) = −5x + 4.a) Draw the graphs of the fun tions y(x) and z(x).b) Determine the zero of ea h of the two fun tions.

) Determine for the graph of ea h fun tion the point of interse tion with the x-axis.d) Determine for the graph of ea h fun tion the point of interse tion with the y-axis.e) Determine the point of interse tion of the graphs of these two fun tions.


Exer ise 1.2 (slope of a line)

The points (2, 4) and (3, 9) are on the graph of a linear fun tion y(x) = ax + b.

Determine a and b.

Quadrati fun tions


y(x) = ax2 + bx + c,

where a, b and c are numbers (a 6= 0) is alled a quadrati fun tion. Note that if a = 0the fun tion is either linear (if b 6= 0) or onstant (if b = 0). The graph of the fun tion

y(x) = ax2 + bx + c is a parabola. This parabola opens upward if a > 0 and opens

downward if a < 0.

Example 1.6: graphs of quadrati fun tions

The graphs of the quadrati fun tions y(x) = −x2 + 2x + 3 and z(x) = 2x2 + 1 are

shown in Figure 1.5. The graph of the fun tion y(x) is a parabola opened downward

be ause the oe� ient of x2is equal to −1 (negative), and the graph of the fun tion

z(x) is a parabola opened upward be ause the oe� ient of x2is equal to 2 (positive).

x

yy(x) = −x2 + 2x + 3

−1 3 x

y

z(x) = 2x2 + 1

1

Figure 1.5 The graph of y(x) = −x2 + 2x + 3 is a parabola opened downward (left).

The graph of z(x) = 2x2 + 1 is a parabola opened upward (right).

The points of interse tion of the graph of a quadrati fun tion y(x) = ax2 + bx + c andthe x-axis are determined by solving the quadrati equation

ax2 + bx + c = 0.

A quadrati equation an be solved using the quadrati formula. In this formula the

dis riminant is a key omponent.


The dis riminant of a quadrati equation ax2 + bx + c = 0 is equal to b2 − 4ac and is

denoted by D,

D = b2 − 4ac.

A quadrati equation has either two, one or zero solutions, depending on the value of

the dis riminant. The solutions of a quadrati equation are displayed in the following

dis riminant riterion.

Dis riminant riterion and quadrati formula of a quadrati equation

For a quadrati equation ax2 + bx + c = 0 with a 6= 0, the following holds for

D = b2 − 4ac:

(i) if D > 0, then the solutions of the quadrati equation are

x =−b +

√b2 − 4ac

2aand x =

−b −√

b2 − 4ac

2a.

(ii) if D = 0, then the solution of the quadrati equation is

x = − b

2a.

(iii) if D < 0, then the quadrati equation has no solutions.

Example 1.7: zeros of a quadrati fun tion

Consider the fun tion y(x) = −x2 + 2x+ 3. A zero of y(x) is a solution of the equation

−x2 + 2x + 3 = 0.

The dis riminant is D = 22 − 4 · −1 · 3 = 16. Be ause D > 0 we obtain a ording to

the quadrati formula the following two solutions:

x =−2 +

√22 − 4 · −1 · 3

2 · −1= −1 and x =

−2 −√

22 − 4 · −1 · 3

2 · −1= 3.

The points of interse tion of the graph with the x-axis are (−1, 0) and (3, 0), as an

be seen in Figure 1.5.

Exer ise 1.3 (zeros of a quadrati fun tion)

Determine the zeros of the following quadrati fun tions:

a) y(x) = x2 + 7x + 6. b) y(x) = 4x2 + 2x + 1.

Exer ise 1.4 (graph of a quadrati fun tion)

Consider the two fun tions y(x) = x2 + 4x + 3 and z(x) = −x2 + 6.a) Sket h the graphs of the fun tions y(x) and z(x).b) Determine the points of interse tion of the graphs of these fun tions.


When onsidering two fun tions the points of interse tion are not all that is interesting:

also the values of x where the value of one fun tion is greater than or less than the value

of the other fun tion ould be relevant. To solve this kind of problem we have to solve

inequalities. To solve the inequality f (x) ≥ g(x) we use the following step plan.

Solving inequality f (x) ≥ g(x)

Step 1. De�ne the fun tion h(x) = f (x)− g(x).Step 2. Determine the zeros of h(x).Step 3. Make a sign hart of h(x).Step 4. Observe in the sign hart where h(x) ≥ 0.

Con lusion: The values where h(x) ≥ 0 are identi al to the

values where f (x) ≥ g(x).

Obviously, we an solve the inequalities f (x) > g(x), f (x) ≤ g(x) and f (x) < g(x) in a

similar way. In the following example we illustrate the step plan for solving an inequality.

Example 1.8: solving inequalities

Consider the fun tions f (x) = x2 + 2 and g(x) = −3x. We want to �nd the values of

x where f (x) ≥ g(x).

Step 1. De�ne the fun tion h(x) = f (x)− g(x).We get h(x) = f (x)− g(x) = x2 + 2 − (−3x) = x2 + 3x + 2.

Step 2. Determine the zeros of h(x).

h(x) = 0 ⇔ x2 + 3x + 2 = 0

⇔ x = −1 or x = −2,

where the solutions are obtained by using the quadrati formula.

Step 3. Make a sign hart of h(x).The sign hart of h(x) is obtained by putting the zeros on a straight line. This divides

the line into three intervals: (−∞,−2), (−2,−1) and (−1, ∞). The fun tion values of

h(x) in one interval all have the same sign. This implies that either the fun tion values

are all positive or all negative in one interval. Hen e, by hoosing an arbitrary value

in (−∞,−2) we an determine the sign of h(x) in this interval. For instan e, hoose

x = −3. Then h(−3) = (−3)2 + 3 · (−3) + 2 = 2 > 0. Hen e, we an on lude

that h(x) > 0 for all x in the interval (−∞,−2). Sin e h(−1.5) = −0.25 it holds that

h(x) < 0 for all x in the interval (−2,−1). Similarly, we �nd that h(x) > 0 for all x in

the interval (−1, ∞), be ause h(0) = 2. Figure 1.6 shows the sign hart of h(x).


MA

THEM

ATIC

S FOR

BUSIN

ESS ECO

NO

MIC

SH

AM

ERS, KLEPPE A

ND

KA

PER

HERBERT HAMERS

JOHN KLEPPE

BOB KAPERTHIRD EDITION

MATHEMATICS FOR BUSINESS ECONOMICS

Mathematics for Business Economics is intended for starting (business) economics students. The book repeats high school mathematics, but also extends to new domains. The focus is on economic applications: throughout the book many examples are given from microeconomics, modern portfolio theory, inventory management and statistics. A recurrent theme in these examples is the mathematical technique of optimization.

For a better understanding this book contains a large number of exercises. Furthermore, there is a freely accessible e-learning environment, where all mathematical topics are explained using text, multiple choice questions and films. Together, the book and e-learning environment provide a solid mathematical foundation for your study (business) economics.

Prof. dr. Herbert Hamers is professor in Game Theory and Operations Research at Tilburg University. Dr. Bob Kaper was professor in Mathematics at Tilburg University. Dr. John Kleppe was a�liated as lecturer mathematics at Tilburg University.

A Dutch language edition of this book is also available: Wiskunde voor bedrijfseconomen (ISBN: 9789024408481).

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MATHEMATICS FOR BUSINESS ECONOMICS Prof. dr. Herbert ... · MATHEMATICS FOR BUSINESS ECONOMICS HAMERS, KLEPPE AND KAPER HERBERT HAMERS JOHN KLEPPE THIRD EDITION BOB KAPER MATHEMATICS