Lecture 01 linear and non linear equations w1

Lecture 01 linear and non linear equations w1 for Math for Business subject in International university in HCM national university

Lecture notes on Mathematics for business

VU TUAN ANH, MSA, CMA

I NTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

Lecture 1: Linear equations

Lecture outlines:

• Introduction to algebra

• Further algebra

• Graphs of linear equations

• Algebraic solution of simultaneous linear equations

• Supply and demand analysis

• Transposition of formulae

• National income determination (Not covered)

Introduction to algebra

Example:

Write down a formula for each situation:

(a) A plumber has a fixed call-out charge of $80 and has an hourly rate of $60 Workout

the total charge, C, for a job that takes L hours in which the cost of materials and parts is

$K.

(b) An airport currency exchange booth charges a fixed fee of $10 on all transactions

and offers an exchange rate of 1 dollar to 0.8 euros Work out the total charge, C, (in $) for buying x euros.

(c) A firm provides 5 hours of in-house training for each of its semi-skilled workers and

10 hours of training for each of its skilled workers Work out the total number of hours,

H, if the firm employs a semi-skilled and b skilled workers.

(b) The advertising budget is between $10 800 and $12 500 Write down and

solve an inequality to work out the minimum and maximum area that could be used

Graphs of linear equations

Example:

The number of people, N, employed in a chain of cafes is related to the number of cafes,

n, by the equation:

N = 10n + 120

(a) Illustrate this relation by plotting a graph of N against n for 0 ≤ n ≤ 20.

(b) Hence, or otherwise, calculate the number of

◦ (i) employees when the company has 14 cafes;

◦ (ii) cafes when the company employs 190 people.

Algebraic solution of simultaneous linear equation – DO IT !

Example:

If the following system of linear equations has

infinitely many solutions, find the value of k.

6x − 4y = 2

−3x + 2y = k

Supply and demand analysis – Market

equilibrium

 Demand Function : Qd = f(P)

 Supply Function: Qs = f(P)

supply curves intersect Qd = Qs

Supply and demand analysis – Market

equilibrium

are endogenously determined with other relevant factors held fixed For example:

demand curve or the whole supply curve or may change both

simultaneously

Supply and demand analysis – Market

equilibrium

Supply and demand analysis

Transposition of Formulae

Example:

Make x the subject of the formula

Transposition of Formulae

Solution:

Lecture 1: Non-Linear equations

Lecture outlines:

functions

Quadratic Function

Y = f(x) F(x) = ax 2 + bx + c

• (For linear model: y = a + bx , now the constant/intercept

is given the label of c When using general forms it is

important to be able to use labels interchangeably)

Quadratic Function - Solving

• Graphical approach: plot function and see where it cuts the x-asis (inefficient)

• Algebraic approach: use the general form and appropriate formula:

y = ax 2 + bx + c

• Solution is where y = 0:

ax 2 + bx+ c = 0

Quadratic Function

Example:

Solve the equation f(x) = 0 for each of

the following quadratic functions:

(d) f(x) = x^2 − 18x + 81

(e) f(x) = 2x^2 + 4x + 3

Quadratic Function

Example:

Sketch the graphs of the quadratic functions

given in (d) & (e)

Quadratic Function

Example:

Revenue, cost and profit

Cost functions

Cost is the total cost of producing output.

The cost function consists of two different types of cost:

- Variable costs

- Fixed costs.

Variable cost varies with output (the number of units produced) The

total variable cost can be expressed as the product of variable cost per unt and number of units produced If more items are produced cost is more.

Revenue, cost and profit

Cost function

C(x) = F +Vx

C = Total cost

F = Fixed cost

V = Variable cost Per unit

𝒙 = No of units produced and sold

It is called a linear cost function

Revenue, cost and profit

Revenue is the total payment received from

selling a good or performing a service The

revenue function, R(𝒙), reflects the revenue

from selling “𝒙” amount of output items at a

price of “p” per item.

Revenue, cost and profit

Revenue, cost and profit

Example:

Given that fixed costs are 500 and that

variable costs are 10 per unit, express TC and AC as functions of Q Hence sketch

their graphs

Revenue, cost and profit

Example:

The total cost of producing 500 items a day

in a factory is $40 000, which includes a

fixed cost of $2000.

(a) Work out the variable cost per item.

(b) Work out the total cost of producing 600 items a day

Revenue, cost and profit

Example:

Indices and Logarithms

Rules of indices

Indices and Logarithms

Examples:

Indices and Logarithms

Application: Production Functions

Indices and Logarithms

• Scale Effects:

– Constant Returns to Scale

– Increasing Returns to Scale

– Decreasing Returns to Scale

• Scale effect determined by degree of homogeneity

STEP 1: Determine whether function is homogenous:

A function Q = f(K,L) is said to be homogenous if

f(λK, λ L) = λ^n f(K,L)

Indices and Logarithms

STEP 2: Determine degree of homogeneity:

The power n will determine the degree of homogeneity:

If n<1 : decreasing returns to scale (Lợi nhuận giảm dần theo quy mô)

If n=1 : constant returns to scale (Lợi nhuận không thay đổi theo quy mô)

If n>1 : increasing returns to scale (Lợi nhuận giảm dần theo quy mô)

(PROVE)

Example:

Indices and Logarithms

Indices and Logarithms

Example:

Natural Logarithms and the Exponential

Function

Natural Logarithms and the Exponential

Function

Examples:

Natural Logarithms and the Exponential

Function

Examples:

Please check on blackboard for suggested exercises !