MTH108 Business Math I

Lecture 10

Chapter 5

Linear functions; Applications

Objectives

• Present a discussion of the characteristics of linear functions

• Present a wide variety of applications of linear functions

Today’s Topics

• Linear functions

Review

Functions

Definition: A linear function f involving one independent variable x and a dependent variable y has the general form

where and are constants,

• The given form is a slope-intercept form with slope and y-intercept .

• This equation tells that a change in the value of y is directly proportional to a change in the value of x

• The rate of change is constant and is given by the slope .

Example

In the previous chapter we have seen the weekly salary function

where y is defined as the weekly salary and x in the no. of units sold per week.

Clearly, this is a weekly function in one independent variable x.

• 25 represents the base salary, i.e. when no units are sold per week and 3 is the commission of each unit sold.

• The change in weekly salary is directly proportional to the change in the no. of units sold.

• Slope of 3 indicates the increase in weekly salary associated with each additional unit sold.• The graph is in the first quadrant, allowing x and y to be positive values only.

Definition: A linear function f involving two independent variables, x1 and x2 and a dependent variable y has the general form

where and are non-zero constants and is a constant.

• This equation tells us that the variable y depends jointly on the values of x1 and x2.

• The value of y varies in direct proportion to changes in the values of x1 and x2.

Increase in x1 by 1 unit change in y by a1 units,Increase in x2 by 1 unit change in y by a2 units.

Example weekly Salary FunctionAssume that a salesperson salary depends on the no. of

units sold of each of two products, i.e. the salary function is given as

Where y= weekly salary,X1 = no. of units sold of product 1X2 = no. of units sold of product 2This salary function gives a base salary of 25 and

commission 5 on each unit sold of product 1 and 3 on each unit sold of product 2.

Definition: A linear function f invvolving n independent variables x1, x2, …, xn and a dependent variable y has the general form

where are non-zero constants and is a constant.

Organizations are mostly concerned with Costs which are often define in terms of two components:

total variable cost and total fixed cost.

Examples1) Linear Cost functions A firm which produces a single product is interested in

determining the functions that expresses annual total cost y as a function of the no. of units produced x.

Accountants indicate that the fixed expenditure each year are 50,000. They also have estimated that raw material costs for each unit produced are 5.50 and

labour costs per unit are 1.50 in the assembly department, 0.75 in the finishing room, and 1.25 in the packaging and shipping department.

Solution:The total cost function will be in the form:

=total variable cost + total fixed costTotal fixed cost = 50,000Total variable cost = total raw material cost + total labour cost

if x is the no. of units, then = 5.5x + 1.5x + 0.75x + 1.25x

Thus,

The 9 represents the combined variable cost per unit of 9 (dollars). That is, for each additional unit produced, total cost will increase by 9 (dollars).

Examples (contd.)

2) Linear Revenue functionThe money which flows into an organization from

either selling products or providing is often referred to as revenue.

The most fundamental way of computing total revenue from selling a product (or service) is

Total revenue = (price)(quantity sold)We assume that selling price is the same for all units

sold.

Suppose a firm sells n products. If xi equals the no. of units sold of products j and pj equals the price of product pj. Then, the function of revenue from the sale of the n products is

Examples (contd.)

3) Linear Profit FunctionsProfit for an organization is the difference between

total revenue and total cost.

If

where

• When total revenue exceeds total cost, profit is positive, usually called as net gain or net profit.

• When total cost exceeds total revenue, profit is negative, usually called as net loss or deficit.

• When the revenue and cost are linear functions of the same variable(s), the profit function is a linear function of the same variable(s).

A firm sells a single product for 65 dollars per unit. Variable costs per unit are 20 dollars for materials and 27.50 dollars for labour. Annual fixed costs are 100,000 dollars. Construct the profit function stated in terms of x, the no. of units produced and sold. What profit is earned if annual sales are 20,000 units.

Solution:

Note that P(x) is a linear function. The slope of 17.50 shows that for each additional unit produced and sold, total profit increases by 17.50 dollars. In business and economics, this is called as marginal profit.

If the firm sells 20,000 units during the year, then

Section 5.1 Follow-up ExercisesQ.1-5, Q.8-9

Review

• Linear function in one variable• Linear function in two variables• Linear function in n variables• Applications o Linear cost functiono Linear revenue functiono Linear profit function