Greenwood press

100
RATES
The slope of the dotted line that is
tangent to the curve is the auto-
mobile’s instantaneous speed of
70 miles per hour at 1:34 
PM
.

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on behalf of the safety of the employees, as well as to support the economic ben-
efits of the company.
In consumer sales, the profit from a business is often dependent on the num-
ber of products sold. An ideal production level would be to determine the
moment when the change of profit, or the rate at which profit is changed, begins
to level off to zero. According to supply-and-demand principles, the company
would like to produce the appropriate amount of products in order to meet con-
sumer demand, but not end up with a surplus in inventory. If too many goods are
produced the rate of profit declines, because the company would lose money on
excess inventory. For example, suppose the price per cup of lemonade, 
l, de-
pended on the number of cups, 
n, purchased according to the equation
l = 2.00 − 0.01n. This equation suggests that the price of a cup of lemonade
would be $2.00 if none were sold, but the price will decline by one penny for
every cup sold. The revenue, 
r, obtained for selling lemonade would be the prod-
uct of the price per cup and the number of cups purchased. Therefore the total
revenue would be equivalent to 
r = lc = (2.00 − 0.01n)n = 2.00n − 0.01n
2
.
The cost to make the lemonade depends on start-up expenses and the quantity of
lemonade sold. If the lemonade stand costs $12.00 to set up and each cup costs
$0.14 to produce, then the cost, 
c, for the company to make lemonade can be rep-
resented by the equation 
c = 0.14n + 12.00. The profit, p, obtained by selling
lemonade is the difference between the revenue and costs, which is 
p =
r − c = (2.00n − 0.01n
2
) − (0.14n + 12.00) = –0.01n
2
+ 1.86n − 12.00. The
graph of the profit function is a parabola, illustrating that the rate of profit
changes, because the graph is nonlinear. Notice that the maximum profit of the
function occurs when ninety-three cups are sold—the moment when the rate of
profit is equal to zero or where the slope of tangent line equals zero, as shown in
the figure below.
online sources for further exploration
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