Lecture Business mathematics - Chapter 6: Differentiation and applications

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!

B USINESS M ATHEMATICS

CHAPTER 6:

DIFFERENTIATION AND

APPLICATIONS

Lecturer: Dr Trinh Thi Huong (Hường)

Department of Mathematics and

Statistics

Email: trinhthihuong@tmu.edu.vn

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 S LOPE OF A C URVE AND

D IFFERENTIATION

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, i.e., 𝑑𝑦

𝑑𝑥 ≈ 𝑦

𝑥

THE DERIVATIVE (ĐẠO HÀM), DIFFERENTIATION

(VI PHÂN)

𝑦 = 𝑥𝑛, 𝑦′ = 𝑛𝑥𝑛−1: The power rule

𝑦 = 𝑒𝑥, 𝑦′ = 𝑒𝑥

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

6.2.2 AVERAGE FUNCTIONS: AN INTRODUCTION

 AR is defined as average revenue per unit for the first

Q successive units sold AR is determined by dividing

total revenue by the quantity sold, Q

6.2.2 AVERAGE FUNCTIONS: AN INTRODUCTION

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:

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:

𝑑𝑥 for the given curve y = f(x).

𝑑𝑥 = 0.

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 𝑑2𝑦

𝑑𝑥2

 Step 2: Solve the equation 𝑑𝑦

𝑑𝑥 = 0; the solution of this

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

x-coordinate of the turning point (from step 2) into theequation 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.