Lecture Business mathematics - Chapter 6: Differentiation and applications. The main topics covered in this chapter include: slope of a curve and differentiation; applications of differentiation, marginal functions, average functions; optimisation for functions of one variable; economic applications of maximum and minimum points; curvature and other applications;... Please refer to this chapter for details!
BUSINESS MATHEMATICS CHAPTER 6: DIFFERENTIATION AND APPLICATIONS Lecturer: Dr Trinh Thi Huong (Hường) Department of Mathematics and Statistics Email: trinhthihuong@tmu.edu.vn CONTENT 6.1 Slope of a Curve and Differentiation 6.2 Applications of Differentiation, Marginal Functions, Average Functions 6.3 Optimisation for Functions of One Variable 6.4 Economic Applications of Maximum and Minimum Points 6.5 Curvature and Other Applications 6.6 Further Differentiation and Applications 6.7 Elasticity and the Derivative 6.8 Summary 6.1 SLOPE OF A CURVE AND DIFFERENTIATION THE DERIVATIVE (ĐẠO HÀM) 𝑑𝑦 𝑑𝑥 is called the derivative of y with respect to x The process of finding 𝑑𝑦 𝑑𝑥 is called differentiation 𝑑𝑦 𝑑𝑥 is the equation for the slope of the curve at any point (x, y) on the curve 𝑦 𝑥 is the slope of the chord over a small interval x For very small intervals x, the slope of the curve is approximately equal to the slope of the chord, 𝑑𝑦 𝑦 i.e., ≈ 𝑑𝑥 𝑥 THE DERIVATIVE (ĐẠO HÀM), DIFFERENTIATION (VI PHÂN) 𝑦 = 𝑥 𝑛 , 𝑦 ′ = 𝑛𝑥 𝑛−1 : The power rule 𝑦 = 𝑒𝑥, 𝑦′ = 𝑒 𝑥 HIGH DERIVATIVES 6.2 APPLICATIONS OF DIFFERENTIATION, MARGINAL FUNCTIONS, AVERAGE FUNCTIONS 6.2.1 MARGINAL FUNCTIONS: AN INTRODUCTION The marginal revenue is the rate of change in total revenue per unit increase in output, Q Marginal revenue: 𝑀𝑅 = 𝑑(𝑇𝑅) 𝑑𝑄 = Δ𝑇𝑅 Δ𝑄 The marginal cost is the rate of change in total cost per unit increase in output, Q Marginal cost: 𝑀𝐶 = 𝑑(𝑇𝐶) 𝑑𝑄 = Δ𝑇𝐶 Δ𝑄 MARGINAL COST Marginal cost 𝑑(𝑇𝐶) 𝑀𝐶 = 𝑑𝑄 Total cost: 𝑇𝐶 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡 + 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑐𝑜𝑠𝑡 = 𝐹𝐶 + 𝑉𝐶 Where FC is a constant, then 𝑑(𝑇𝐶) 𝑑(𝐹𝐶 + 𝑉𝐶) 𝑑𝑉𝐶 𝑀𝐶 = = = = 𝑀𝑉𝐶 𝑑𝑄 𝑑𝑄 𝑑𝑄 MVC: Marginal variable cost o 6.2.3 PRODUCTION FUNCTIONS AND THE MARGINAL AND AVERAGE PRODUCT OF LABOUR Firms transform inputs (or factors of production) into units of output, including: labour, L; physical capital (buildings, machinery), K; raw materials, R; technology, 𝑇𝑒 ; land, S; and enterprise, E A general form of aproduction function: Q = f(L, K, R, 𝑇𝑒 , S, E ) o In a simple casse: Q = f(L) The marginal product of labour (MPL ) 𝑑(𝑄) 𝑀𝑃𝐿 = 𝑑𝐿 o The average product of labour (APL) 𝑄 𝐴𝑃𝐿 = 𝐿 6.2.5 MARGINAL AND AVERAGE PROPENSITY TO CONSUME AND SAVE Marginal propensity to consume (MPC) and marginal propensity save (MPS) Where: Y: income; C: Consumption, S: Saving and Y = C+S 6.3 OPTIMISATION FOR FUNCTIONS OF ONE VARIABLE 6.3.1 SLOPE OF A CURVE AND TURNING POINTS The slope of the curve at a point is the same as the slope of the tangent at that point Figure shows four turning points, two minima and two maxima, with the tangents drawn at these points Note these are called ‘local’ minimum and maximum points Find the x-coordinates of the turning points on a curve y = f(x), the following method is used: 𝑑𝑦 Step 1: Find for the given curve y = f(x) 𝑑𝑥 𝑑𝑦 𝑑𝑥 Step 2: Solve the equation = The solution of this equation gives the x-coordinates of the turning points 6.3.2 DETERMINING MAXIMUM AND MINIMUM TURNING POINTS THE MIN/MAX METHOD Step 1: Find 𝑑𝑦 𝑑𝑥 and 𝑑2 𝑦 𝑑𝑥 𝑑𝑦 Step 2: Solve the equation = 0; the solution of this 𝑑𝑥 equation gives the x-coordinates of the possible turning points Step 2a: Calculate the y-coordinate of each turning point Step 3: Determine whether the turning point is a maximum or minimum by substituting the xcoordinate of the turning point (from step 2) into the equation of the second derivative (method B): The point is a maximum if the value of d2y/dx2 is negative at the point The point is a minimum if the value of d2y/dx2 is positive at the point (Method A) 6.4 ECONOMIC APPLICATIONS OF MAXIMUM AND MINIMUM POINTS The first derivatives of economic functions were called marginal functions Therefore, the optimum value of functions, such as revenue, profit, cost, etc., will all occur when the corresponding marginal function (first derivative) is zero ... Minimum Points 6.5 Curvature and Other Applications 6.6 Further Differentiation and Applications 6.7 Elasticity and the Derivative 6.8 Summary 6.1 SLOPE OF A CURVE AND DIFFERENTIATION THE DERIVATIVE... Curve and Differentiation 6.2 Applications of Differentiation, Marginal Functions, Average Functions 6.3 Optimisation for Functions of One Variable 6.4 Economic Applications of Maximum and Minimum... 6.2.5 MARGINAL AND AVERAGE PROPENSITY TO CONSUME AND SAVE Marginal propensity to consume (MPC) and marginal propensity save (MPS) Where: Y: income; C: Consumption, S: Saving and Y = C+S 6.3