Chapter2 Advanced mathematics

Chapter 2 Non linear equations Chapter 2 Non linear equations Nguyen Thi Minh Tam ntmtam vnuagmail com October 24, 2020 1 2 1 Quadratic functions 2 2 2 Revenue, cost and profit 3 2 3 Indices and logarithms 4 2 4 Exponential and Natural logarithm functions 2 1 Quadratic functions A quadratic function is a function of the form f (x) = ax2 + bx + c, where a,b,c are constants, and a 6= 0 The graph of a quadratic function is a parabola The strategy for sketching the graph of a quadratic function f (.

Chapter 2: Non-linear equations

Nguyen Thi Minh Tam

ntmtam.vnua@gmail.com

October 24, 2020

1 2.1 Quadratic functions

2 2.2 Revenue, cost and profit

3 2.3 Indices and logarithms

4 2.4 Exponential and Natural logarithm functions

2.1 Quadratic functions

A quadratic functionis a function of the form

f (x ) = ax2+ bx + c,where a, b, c are constants, and a 6= 0

The graph of a quadratic function is a parabola

The strategy for sketching the graph of a quadratic function

f (x ) = ax2+ bx + c

1 Determine the basic shape The graph has a U shape if

a > 0, and an inverted U shape if a < 0

2 Determine the y intercept This is obtained by substituting

x = 0 into the function, which gives y = c

3 Determine the x intercepts (if any) These are obtained bysolving the quadratic equation

ax2+ bx + c = 0

Example 1 Given the supply and demand functions

P = 2QS2+ 10QS+ 10

P = −QD2 − 5QD + 52calculate the equilibrium price and quantity

2.2 Revenue, cost and profit

The total revenue, denoted by TR, is the amount of moneyreceived by the firm from the sale of its goods

If Q goods are sold and the price per good is P, then the totalrevenue is

TR = PQThe total cost, denoted by TC, is the amount of money thatthe firm has to spend to produce these goods

The profit functionis denoted by π and is defined by

π = TR − TC

Example 2 Given the demand function

Fixed costs, FC, are total costs that are independent of outputand include the cost of land, equipment, rent and possiblyskilled labour.

Variable costs, vary with output and include the cost of rawmaterials, components, energy and unskilled labour

If VC denotes the variable cost per unit of output, then thetotal variable cost, TVC, in producing Q goods is given by

TVC = (VC)QThe total cost is

TC = FC + (VC)QThe average cost function, AC, is defined by

AC = TC

FC

Q + VC

Example 3 Given that fixed costs are 100 and that variable costsare 2 per unit, express TC and AC as functions of Q Hence sketchtheir graphs.

Note When the variable cost, VC, is a constant, the graph of theaverage cost function is arectangular hyperbola(L-shaped curve)

Figure: The graph of an average cost function.

Figure: Typical TR and TC graphs sketched on the same diagram when the demand function is linear and the variable costs are constant.

The two curves intersect at two points, A and B,

corresponding to output levels QA and QB At these pointsthe cost and revenue are equal and the firmbreaks even

If Q < QA or Q > QB, then cost exceeds revenue and thefirm makes a loss

If QA < Q < QB, then revenue exceeds cost and the firmmakes a profit

Example 4 If fixed costs are 4, variable costs per unit are 1 andthe demand function is

P = 10 − 2Qobtain an expression for π in terms of Q and hence sketch a graph

of π against Q

a) For what values of Q does the firm break even?

b) What is the maximum profit?

2.3 Indices and logarithms

Index notation

If M = bn we say that bn is the exponential form of M tobase b

n is referred to as the index, poweror exponent

If n is a positive integer, then

= an

bn

The output Q, of any production process depends onfactors

of production such as capitalK andlabour L

The dependence of Q on K and L may be written as

If the degree of homogeneity, n, satisfies:

n < 1, the function is said to display decreasing returns toscale

n = 1, the function is said to display constant returns to scale

n > 1, the function is said to display increasing returns toscale

Example 5 Show that the following production functions arehomogeneous and comment on their returns to scale:

a) Q = 7KL2

b) Q = 50K1/4L3/4

All of the production functions considered so far are of theform

Q = AKαLβ,where A, α, β are positive constants

Such functions are called Cobb-Douglas production functions.They are homogeneous of degree α + β

2.4 Exponential and Natural logarithm functions

Natural exponential function

The function f (x ) = ex, where e = 2.718281828459 , is calledthenatural exponential function

Example 6 The percentage, y , of households possessing

microwave ovens t years after they have been launched is modelledby

1 + 800e−0.3t1) Find the percentage of households that have microwaves

a) at their launch;

b) after 10 years;

c) after 20 years.

2) What is the market saturation level?

3) Sketch a graph of y against t and hence give a qualitativedescription of the growth of microwave ownership over time

Example 7 An economy is forecast to grow continuously so thatthe gross national product (GNP), measured in billions of dollars,after t years is given by

GNP = 80e0.02tAfter how many years is GNP forecast to be $88 billion? Whatdoes the model predict about the value of GNP in the long run?

An economic variable which increases without bound is said todisplayunlimited growth

Exercise 2.3, 2.3*: 5 (page 168-169), 16, 17 (page 171)

Exercise 2.4, 2.4*: 1 (page 182), 5, 7, 1 (page 183), 3 (page 184),

9 (page 185)