Math 1320
Chapter 1: Linear Functions
1.2 Functions and Models
Modeling
• To mathematically model a situation means to
represent it in mathematical terms. The
particular representation is called a
mathematical model.
• Mathematical models do not always represent
a situation perfectly or completely. Some
represent a situation only approximately, or
only some of the time, whereas others
represent only some aspects of the situation.
Type of Models
• Analytic models are obtained by analyzing the
situation being modeled.
• Curve-fitting models are obtained by finding
mathematical formulas that approximate
observed data.
• Continuous models are defined by functions
whose domains ae intervals of the real line.
• Discrete models have only set points as the
domain. Discrete models are used extensively in
probability and statistics.
Example 1
The amount of free space left on your hard drive
is 50 GB and is decreasing by 5 GB per month.
Find a mathematical model for this situation.
Example 2
My square orchid garden is right off the side of
the house so that the house itself forms the
northern boundary. The fencing for the southern
boundary costs $4 per foot, and the fencing for
the east and west sides cost $2 per foot. Find
the total cost function as a function of the
length x of a side.
Definition
• Cost – A cost function specifies the cost C as a
function of the number of items x. Thus, C(x) is
the cost of x items and has the form: Cost =
variable cost + fixed cost, where the variable cost
is a function of x and the fixed cost is constant.
• A cost function of the form 𝐶 𝑥 = 𝑚𝑥 + 𝑏 is
called a linear cost function:
– The variable cost is mx,
– The fixed cost is b,
– The marginal cost, the cost for one more, is m.
Definition
• Revenue – The revenue resulting from one or
more business transactions is the total
payment received, sometimes call the gross
proceeds.
• If R(x) is the revenue from selling x items at a
price of m each, then revenue is the linear
function 𝑅 𝑥 = 𝑚𝑥.
• The selling price m can also be called the
marginal revenue.
Definition
• Profit – The profit is the net proceeds, or what
is left of the revenue when costs are
subtracted.
• If the profit depends linearly on the number of
items, the slope m is called the marginal
profit.
• Profit, revenue, and cost are related by: Profit
= Revenue – Cost, or 𝑃 𝑥 = 𝑅 𝑥 − 𝐶(𝑥)
Definition
• Break-even – The break-even point is the
number of items x at which profit is zero.
• Note that if 𝑃 𝑥 = 0, then 0 = 𝑅 𝑥 − 𝐶(𝑥)
and so revenue is equal to cost.
Example 3
Your college newspaper has fixed production costs of $70
per edition and marginal distribution costs of 38¢ per
copy. The paper sells for 48¢ per copy.
a. Write down the associated cost function C(x) in
dollars.
b. Write down the revenue function R(x) in dollars.
c. Write down the profit function P(x) in dollars.
d. What profit (or loss) results from the sale of 100
copies of the paper? What about 500 copies?
e. How many copies should be sold in order to
break even?
Example 4
The Metropolitan Company sells its latest
product at a unit price of $5. Variable costs are
estimated to be 30% of the total revenue, while
fixed costs amount to $7,000 per month. How
many units should the company sell per month
in order to break even?
Definitions
• Demand – A demand equation or demand
function expresses demand q (for quantity) as a
function of the unit price p (price per item).
• Supply – A supply equation or supply function
expresses supply q as a function of the unit price
p.
• Equilibrium – Occurs when supply = demand.
• It is usually the case that demand decreases and
supply increases as the unit price increases.
Example 5
The demand for your hand-made skateboards,
in weekly sales, is 𝑞 = −3𝑝 + 700 if the selling
price is $p. You are prepared to supply 𝑞 = 2𝑝 −
500 per week at the price $p. At what price
should you sell your skateboards so that there is
neither a shortage nor a surplus?
Example 6
Worldwide quarterly sales of a brand of cell phones
were approximately 𝑞 = −𝑝 + 156 million phones
when the wholesale price was $p.
a. If the cellphone company was prepared to
supply 𝑞 = 4𝑝 − 334 million phones per
quarter at the wholesale price of $p, what
would have been the equilibrium price?
b. The actual wholesale price was $93 in the
fourth quarter of 2004. Estimate the
projected shortage or surplus at that price.