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!
BUSINESS MATHEMATICS CHAPTER 4: Non-linear functions and applications Lecturer: Dr Trinh Thi Huong (Hường) Department of Mathematics and Statistics Email: trinhthihuong@tmu.edu.vn CONTENT 4.1 Quadratic, Cubic and Other Polynomial Functions 4.2 Exponential Functions 4.3 Logarithmic Functions 4.4 Hyperbolic (Rational) Functions of the Form 𝑎 𝑏𝑥 + 𝑐 4.5 Excel for Non-linear Functions 4.1 QUADRATIC, CUBIC AND OTHER POLYNOMIAL FUNCTIONS An example: A linear total revenue function 𝑇𝑅 = 3.5 Q o In case the firm is a monopolist, the total revenue is 𝑇𝑅 = 𝑃𝑄 The demand function is: 𝑃 = 50 − 2𝑄 𝑇𝑅 = 𝑃𝑄 = 50 − 2𝑄 𝑄 = 50𝑄 − 2𝑄2 This is a quadratic function 4.1.1 SOLVING A QUADRATIC EQUATION A quadratic equation has the general form 𝑎𝑥 + 𝑏𝑥 + 𝑐 = Δ = 𝑏 − 4𝑎𝑐 Solutions −𝑏 ± Δ x= 2𝑎 Δ < 0: Imaginary Solutions, recall: 𝑖 = −1 𝑎𝑛𝑑 𝑖 = −1 Δ = 0: A repeated real root Δ > 0: Two real roots 4.1.2 PROPERTIES AND GRAPHS OF QUADRATIC 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 DEFINITION AND GRAPHS OF EXPONENTIAL FUNCTIONS The exponential function has the general form 𝑦 = 𝑎 𝑥 or f x = 𝑎 𝑥 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 𝑥 f x = 𝑒 is often referred to as the natural exponential function to distinguish it from f x = 𝑎 𝑥 ,the general exponential function GRAPHS OF EXPONENTIAL FUNCTIONS 4.2.2 SOLVING EQUATIONS THAT CONTAIN EXPONENTIALS 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 = 𝑎𝑒 𝑟𝑡 , 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 𝑀 𝑦 𝑡 = + 𝑎𝑒 −𝑟𝑀𝑡 where M, a and r are constants 4.3 LOGARITHMIC FUNCTIONS o From the exponential function 𝑦 = 𝑎 𝑥 ⇔ 𝑥 = log 𝑎 𝑥 Log to base e 𝑦 = 𝑒 𝑥 ⇔ 𝑥 = ln 𝑥 RECALL: RULES OF LOGS 4.4 HYPERBOLIC (RATIONAL) FUNCTIONS OF THE FORM Form 𝑎 𝑦= 𝑏𝑥 + 𝑐 The simplest form: 𝑦 = 𝑥 EQUATIONS AND APPLICATIONS 𝑎 Functions of the form 𝑦 = model average 𝑏𝑥+𝑐 cost, supply, demand and other functions which grow or decay at increasing or decreasing rates ... Quadratic, Cubic and Other Polynomial Functions 4.2 Exponential Functions 4.3 Logarithmic Functions 4.4 Hyperbolic (Rational) Functions of the Form