Lecture Business mathematics - Chapter 4: Non-linear functions and applications

Lecture Business mathematics - Chapter 4: Non-linear functions and applications. The main topics covered in this chapter include: quadratic, cubic and other polynomial functions; exponential functions; logarithmic functions; hyperbolic (rational) functions of the form;... Please refer to this chapter for details!

B USINESS M ATHEMATICS

CHAPTER 4: Non-linear functions and applications

Lecturer: Dr Trinh Thi Huong (Hường)

Department of Mathematics and

Statistics

Email: trinhthihuong@tmu.edu.vn

4.1 QUADRATIC, CUBIC AND OTHER

o In case the firm is a

monopolist, the total

revenue is

𝑇𝑅 = 𝑃𝑄The demand function is:

4.1.1 SOLVING A QUADRATIC EQUATION

 A quadratic equation has the general form

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

Δ = 𝑏2 − 4𝑎𝑐Solutions

x = −𝑏 ± Δ

2𝑎

Δ < 0: Imaginary Solutions, recall: 𝑖2 = −1 𝑎𝑛𝑑 𝑖 =

−1

Δ = 0: A repeated real root

Δ > 0: Two real roots

4.1.2 PROPERTIES AND GRAPHS OFQUADRATIC FUNCTIONS

4.1.3 QUADRATIC FUNCTIONS IN ECONOMICS

Total revenue for a profit-maximising monopolist: Total revenue functions are

frequently represented by maximum type quadratics which pass through the origin.

4.1.4 CUBIC FUNCTIONS

 A cubic function is expressed by a cubic equation which has the general form

where a, b, c and d are constants.

In this section, in particular, calculating points by graph plotting is much easier and less time

consuming

4.2 EXPONENTIAL FUNCTIONS

4.2.1 D EFINITION AND GRAPHS OF EXPONENTIAL FUNCTIONS

 The exponential function has the general form

𝑦 = 𝑎𝑥 or f x = 𝑎𝑥

where:

• a is a constant and is referred to as the base of the

exponential function

• x is called the index or power of the exponential

function; this is the variable part of the function

 The number e, e = 2.718 2818

 f x = 𝑒𝑥 is often referred to as the natural exponential function to distinguish it from f x = 𝑎𝑥,the general

exponential function

GRAPHS OF EXPONENTIAL FUNCTIONS

4.2.2 SOLVING EQUATIONS THAT CONTAINEXPONENTIALS

4.2.3 APPLICATIONS OF EXPONENTIAL

FUNCTIONS

 The laws of growth: Exponential functions to

base e describe growth and decay in a wide range

of systems, as mentioned above There are three main laws of growth:

 Unlimited growth is modelled by the equation

𝑦 𝑡 = 𝑎𝑒𝑟𝑡, where a and r are constants.

 Limited growth is modelled by the equation

𝑦 𝑡 = 𝑀(1 − 𝑒−𝑟𝑡), where M and r are constants.

 Logistic growth is modelled by the equation

𝑦 𝑡 = 𝑀

1 + 𝑎𝑒−𝑟𝑀𝑡where M, a and r are constants

RECALL:

RULES

OF LOGS

𝑥

EQUATIONS AND APPLICATIONS

 Functions of the form 𝑦 = 𝑎

𝑏𝑥+𝑐 model average cost, supply, demand and other functions which grow or decay at increasing or decreasing rates