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1.3 Linear Functions and Mathematical Models
•Mathematics can be used to solve real-world problems.
•Regardless of the field from which the real-world problem is
drawn, the problem is analyzed using a process called
mathematical modeling.
The relation between two quantities is conveniently described
in mathematics by using the concept of a function.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Functions
• A function f is a rule that assigns to each value of x one and
only one value of y.
• The value y is normally denoted by f(x), emphasizing the
dependency of y on x.
Ex. y  3x 2  2 is a function.
We write f (x) , read “f of x”, in place of y to show the dependency
of y on x .
So
f ( x)  3x 2  2
and
f (5)  3(5) 2  2  77
NOTE: It is not f times x
Function
Range
Domain
-1
1
3
-4
1
-6
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Domain and Range
Suppose we are given the function y = f(x).
• The variable x is referred to as the independent variable,
and the variable y is called the dependent variable.
• The set of all the possible values of x is called the domain
of the function f.
• The set of all the values of f(x) resulting from all the
possible values of x in its domain is called the range of f.
• The output f(x) associated with an input x is unique:
– Each x must correspond to one and only one value of
f(x).
Linear Function
A linear function can be expressed in the form
f ( x)  mx  b
m and b are
constants
It can be used for
• Simple Depreciation
• Linear Supply and Demand Functions
• Linear Cost, Revenue, and Profit Functions
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Simple Depreciation
Ex. A computer with original value $2000 is linearly
depreciated to a value of $200 after 4 years. Find an equation
for the value, V, of the computer at the end of year t. What will
be the computer value at the end of the third year? What is the
rate of depreciation of the computer?
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Solution: Since the depreciation is linear, V is a linear function of t.
Equivalently, The graph of the function is a straight line. Observe
that V=2000 when t=0, and V=200 when t=4. This tells us that the
line passes through the points
(t , V ) : (0, 2000) and (4, 200)
The slope of the line is given by
m
2000  200
 450;
04
Use the point-slope form of the equation of a line with the point
(0, 2000) and the slope -450, we have
V-2000= -450(t-0)
thus
V (t )  450t  2000
The computer value at the end of the third year is given by
V= -450(3)+2000=650.
The rate of depreciation of the computer is given by the negative of
the slope of the depreciation line. Since the slope is m= -450, the
rate of depreciation is $450 per year.
Example: A car purchased for use by the manager of a firm at a
price of $24,000 is to be depreciated using the straight line method
over 5 years. What will be the book value of the car at the end of 3
year? (Assume the scrap value is $0.)
Solution: Let V(t) be the value of the car at the end of tth year.
When t = 0, V = 24,000; when t = 5, V = 0
(t ,V ) : (0, 24000), (5, 0)
Then b = 24,000
24000  0
Slope m 
 4800
05
Then the equation is V=-4,800t+24,000
When t=3, V=-4,800(3)+24,000=9,600
So the book value of the car at the end of 3 year is $9,600.
Cost, Revenue, and Profit Functions
Let x denote the number of nits of a product manufactured or sold.
•
Then the total cost function is
C (x)=Total cost of manufacturing x units of the product
•
The revenue function is
R(x) = Total revenue realized from the sale of x units of the
product.
•
The profit function is
P(x)= Total profit realized from manufacturing and selling x
units of product.
Suppose a firm has a fixed cost of F dollars, a production cost of
c dollars per unit, and a selling price of s dollars per unit.
Then the cost function C(x), the revenue function R(x), and the
profit function P(x) for the firm are given by
C (x) = c x + F
R (x) = s x
P (x) = R (x) – C (x)
= (s - c) x- F
Ex. A shirt producer has a fixed monthly cost of $3600. If each
shirt costs $3 and sells for $12 find:
a.
The cost function
Cost: C(x) = 3x + 3600 where x is the number
of shirts produced.
b.
The revenue function
Revenue: R(x) = 12x where x is the number of
shirts sold.
c.
The profit from selling 900 shirts
Profit: P(x) = Revenue – Cost
= 12x – (3x + 3600) = 9x – 3600
P(900) = 9(900) – 3600 = 4500 or $4500
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Linear Demand and Supply Curves
A demand equation expresses the relation between the unit price
and the quantity demanded. The corresponding graph of the
demand equation is called a demand curve.
A demand function, defined by p=f(x), where p measures the unit
price and x measures the number of units of commodity, is
generally characterized as a decreasing function of x; that is, p=f(x)
decreases as x increases.
Linear Demand
Ex. The quantity demanded of a particular game is 5000 games
when the unit price is $6. At $10 per unit the quantity
demanded drops to 3400 games. Find a demand equation
relating the price p, and the quantity demanded, x (in units of
100). What is the unit price corresponding to a quantity demand
of 4000 games? What is the quantity demanded if the unit price
is $8?
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Solution: When p=6, x=50, and when p=12, x=32. Then the
following two points lie on the demand curve.
( x, p) : (50, 6) and (34, 10)
Since the demand equation is linear, its graph is a straight line.
The slope of the line is given by
m
10  6
1

34  50
4
Using the point-slope form of an equation of a line with the
Point (50, 6), we find that
1
p  6   ( x  50)
4
1
37
p  x
4
2
is the required fucntion.
If the quantity demanded is 4000 games (x=40), the demand
Equation yields
1
37
p   (40) 
 8.5
4
2
Next, if the unit price is $8 (p=8), the demand equation yields
1
37
8 x
4
2
1
21
x
4
2
x  42
Supply Function
An equation that expresses the relationship between the unit price
And the quantity supplied is called a supply equation, and the
Corresponding graph is called a supply curve.
A supply function, defined by p = f (x), is generally characterized by
Increasing function of x; that is, p = f (x) increases as x increases.