Lecture Note Function
Example 5
Since the beginning of the year, the price of whole-wheat bread at a local discount
supermarket has been rising at a constant rate of 2 cents per month. By November 1,
the price had reached $1.06 per loaf. Express the price of the bread as a function of
time and determine the price at the beginning of the year.
(^01210)
Solution
Let x: denote the number of months that have elapsed since January 1
y: denote the price of a loaf of bread (in cents).
Since y changes at a constant rate with respect to x, the function relating y to x must
be linear and its graph is a straight line. Because the price y increases by 2 each time x
increase by 1, the slope of the line must be 2. Then, we have to write the equation of
the line with slope 2 and passes throught the point(10,106). By the fomular, we
obtain
( )
()
00
106 2 10
yy mxx
yx
−= −
−=−
or
yx= 28 + 6
At the beginning of the year, we have x=^0 , then y= 86. Hence, the price of tbread
at the beginning of the year was 86 cents per loaf.
Example 6
The average scores of incoming students at an eastern liberal arts college in the SAT
mathematics examination have been declining at a constant rate in recent years. In
1986, the average SAT score was 575, while in 1991 it was 545.
a. Express the average SAT score as a function of time.
b. If the trend continues, what will the average SAT score of incoming students
be in 1996?
c. If the trend continues, when will the average SAT score be 527?
(Answer: a.yx=−+ 6 575 , b. 515, c. 8)
Example 7
A manufacturer’s total cost consists of a fixed overhead of $ 200 plus production
costs of $ 50 per unit. Express the total cost as a function of the number of units
produced and draw the graph.
Solution
Let x denote the number of units produced and C(x) the corresponding total cost.
Then, Total cost = (cost per unit) (number of units) + overhead.
Where Cost per unit = 50
Number of units = x
Overhead = 200
Hence,
C(x) = 50x + 200
Jan. 1 Nov. 1