Download Math 141 Lecture Notes

Math 141 Lecture Notes
Section 1.3 Linear Functions and Mathematical Models
A function is a rule that assigns one and only one y-value for each x value. It is often
in the form of an equation and denoted “f(x)”.
The graph of a linear function is a straight line. The equation of a linear function can
be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept.
The domain is the set of all x-values for the function. The range is the set of all yvalues for the function. The dependent variable is y because it depends upon (or is
determined by) the value of the independent variable, x.
Example 1:
The number of U.S. broadband Internet households stood at 20 million at the beginning
of 2002 and is projected to grow at the rate of 7.5 million households per year for the
next 6 years. [p. 37 #29]
(a) Find a linear function f(t) giving the projected U.S. broadband Internet
households (in millions) in year t, where t = 0 corresponds to the beginning of
2002.
(b) What is the projected size of U.S. broadband Internet households at the
beginning of 2008?
Linear Cost Revenue, and Profit Functions
C(x) = cx + F, the Total Cost function, gives the total cost for manufacturing x units at
a unit cost of c and fixed costs F.
R(x) = px, the Revenue function, gives the total revenue realized from manufacturing
and selling x units at price p.
P(x) = R(x) – C(x) = px – (cx + F) = px – cx – F, the Profit function, gives the total profit
realized from manufacturing and selling x units.
Example 2: Rain-tec, a manufacturer of sprinkler parts, has determined that it costs
$20,400 to produce 20 sprinkler heads. The company has production costs of $20 per
unit and a selling price of $30 per unit. Find the cost, revenue, and profit functions for
Rain-Tec.
Linear Supply and Demand Curves
A demand equation expresses the relationship between the unit selling price and the
quantity demanded.
Example 3: The quantity demanded of the Sentinel alarm clock is 48,000 units when
the unit price is $8. At $12 per unit, the quantity demanded drops to 32,000 units. [p.
33 ex. 4]
(a) Find the demand equation, assuming that it is linear.
(b) What is the unit price corresponding to a quantity demanded of 40,000 units?
(c) What is the quantity demanded if the unit price is $14?
The supply equation expresses the relationship between the unit selling price and the
quantity supplied.
Example 4: Producers will make 2000 refrigerators available when the unit price is
$330. At a unit price of $390, 6000 refrigerators will be marketed. [p.38 #48]
(a) Find the equation relating the unit price of a refrigerator to the quantity supplied if
the equation is known to be linear.
(b) How many refrigerators will be marketed when the unit price is $450?
(c) What is the lowest price at which a refrigerator will be marketed?