11DIFFERENTIATION - Department of Mathematics

712 11 DIFFERENTIATION
FIGURE 11.4
After a spectacular rise, the sales begin to
taper off.
Exploring with Technology
Two years from the date of release, the sales are changing at the rate of
5(1 4)
S(2) 3
2 (4 1) 5 0.6
That is, they are decreasing at the rate of $600,000 per year.
The graph of the function S is shown in Figure 11.4.
Millions of dollars
3
2
1
S (t)
2 4 6
Years
8 10
Refer to Example 6.
1. Use a graphing utility to plot the graph of the function S using the viewing rectangle [0, 10] [0, 3].
2. Use TRACE and ZOOM to determine the coordinates of the highest point on the graph of S in the interval
[0, 10]. Interpret your results.
EXAMPLE 7
Group Discussion
Suppose the revenue of a company is given by R(x) xp(x), where
x is the number of units of the product sold at a unit price of p(x) dollars.
1. Compute R(x) and explain, in words, the relationship between R(x)
and p(x) and/or its derivative.
2. What can you say about R(x) ifp(x) is constant? Is this expected?
t
When organic waste is dumped into a pond, the oxidation process that takes
place reduces the pond’s oxygen content. However, given time, nature will
restore the oxygen content to its natural level. Suppose the oxygen content t
days after organic waste has been dumped into the pond is given by
f(t) 100t2 10t 100
t2 (0 t )
20t 100
percent of its normal level.