Math 141: Business Mathematics I
Fall 2015
§1.3 Linear Functions and Mathematical Models
Instructor: Yeong-Chyuan Chung
Outline
• Mathematical Models
• Functions
• Applications of Linear Functions
– Linear Depreciation
– Linear Cost, Revenue, and Profit Functions
– Linear Supply and Demand Curves
Mathematical Models
Mathematical modelling is a process by which a real-world problem is analyzed and solved
using mathematical tools. It generally consists of four steps.
1. Formulate: Convert a real-world problem into a mathematical problem. In this and
the next chapter, we will see how some real-world problems can be converted into
mathematical problems involving linear equations or inequalities. Later on, we will
also see other mathematical models involving probability and statistics.
2. Solve: Use the appropriate mathematical techniques to solve the problem.
3. Interpret: Interpret the solution of the mathematical problem in the context of the
original real-world problem.
4. Test: Mathematical models may not describe the original real-world problem with
complete accuracy. In such cases, we test the accuracy of the model by observing how
well it describes the original real-world problem and how well it predicts past/future
behavior. If the results are unsatisfactory, we may have to make changes to the model.
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§1.3 Linear Functions and Mathematical Models
Functions
A function f is
We often write f (x), read “f of x”, for the value y. This indicates that y depends on x.
The variable x is called
The variable y is called
When we plot a function, we usually put the
variable on the horizontal
axis, and the
variable on the vertical axis. (Why?)
Note: Sometimes we use letters other than x and y. For instance, we may use t if the
variable represents time.
Linear Functions
In this and the next chapter, we are mostly interested in a simple but important class of
functions known as linear functions. Such functions are used to model many business and
economic problems.
A linear function is a function of the form
Some examples of linear functions are
Why are such functions called linear functions?
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Linear Depreciation
Linear depreciation is often used by businesses to describe assets that lose value over time.
The book value of an asset is
The scrap value of an asset is
Example (Exercise 16 in the text). An automobile purchased for use by the manager of a
firm at a price of $24,000 is to be depreciated using the linear depreciation method over 5
years. Assume that the scrap value is $0.
(a) Sketch the graph of the depreciation function.
(b) What is the rate of depreciation of the automobile?
(c) What will be the book value of the automobile at the end of 3 years?
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Linear Cost, Revenue, and Profit Functions
Operating costs, revenue from the sale of products/services, and the profits realized are three
quantities that the owner of any business will be very interested in. We can use functions to
express these quantities in mathematical terms.
Let x be the total number of units of a product manufactured or sold.
The total cost function is C(x) = Total cost of manufacturing x units of the product.
The revenue function is R(x) = Total revenue from the sale of x units of the product.
The profit function is P (x) = Total profit from manufacturing and selling x units of the
product =
In general, a company’s cost, revenue, and profit functions will probably be nonlinear. But
linear ones do arise, and we will only consider such functions in this course.
In fact, if the cost function and the revenue function are both linear, then the profit function
will automatically be linear. (Why?)
The cost function must take into account two types of costs.
Fixed costs are costs that are more or less constant and independent of production or sales.
Examples are rental fees and salaries.
Variable costs are costs that vary with production or sales.
If the fixed cost is F dollars and the production cost is c dollars per unit, then the cost
function is C(x) =
Example (Exercise 20 in the text). AutoTime, a manufacturer of electronic digital timers,
has a monthly fixed cost of $48,000 and a production cost of $8 for each timer manufactured.
The timers sell for $14 each.
(a) What is the cost function?
(b) What is the revenue function?
(c) What is the profit function?
(d) Compute the profit/loss corresponding to production levels of 4000, and 10,000 timers,
respectively.
§1.3 Linear Functions and Mathematical Models
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Linear Demand and Supply Curves
In a free-market economy, consumer demand for a commodity depends on the commodity’s
unit price. In general, the higher the unit price, the lower the demand.
A demand equation expresses this relationship, and the graph of the demand equation
is called a demand curve. Note that both variables in the demand equation take only
positive values. (Why?)
When we sketch a demand curve, we usually put
and we put
on the horizontal axis.
on the vertical axis,
In a competitive market, a relationship also exists between the unit price of a commodity
and its availability in the market (i.e. supply). In general, the higher the unit price, the
higher the supply.
A supply equation expresses this relationship, and the graph of the supply equation is
called a supply curve. Note that both variables in the supply equation take only positive
values.
When we sketch a supply curve, we usually put
and we put
on the horizontal axis.
on the vertical axis,
In this course, we will only consider demand equations and supply equations that are linear.
Example (Exercise 42 in the text). At a unit price of $25, the quantity demanded of a
certain sports watch is 1000 units. At a unit price of $35, the demand drops to 600 units.
(a) Find the demand equation (assuming it is linear).
(b) Sketch the demand curve.
§1.3 Linear Functions and Mathematical Models
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(c) What would the demand be if the watch were free?
(d) What is the highest price (theoretically) anyone would pay for the watch?
Example (Exercise 48 in the text). Producers of a certain refrigerator will make 2000
refrigerators available when the unit price is $330. At a unit price of $390, 6000 refrigerators
will be marketed.
(a) Find the supply equation (assuming it is linear).
(b) Sketch the supply curve.
(c) How many refrigerators will be marketed when the unit price is $450?
(d) What is the unit price when 1000 refrigerators are made available?
What do you think will happen if we sketch the demand and supply curves for a single
commodity on the same diagram? What information can we get from such a diagram? We
will discuss this in the next section.