11DIFFERENTIATION - Department of Mathematics

754 11 DIFFERENTIATION
EXAMPLE 6
SOLUTION ✔
FIGURE 11.15
The CPI of an economy is given by I(t).
200
180
160
140
120
100
I(t)
y = I(t)
2 4 6 8 9
Years
S ELF-CHECK E XERCISES 11.5
t
An economy’s CPI is described by the function
I(t) 0.2t3 3t2 100 (0 t 9)
where t 0 corresponds to the year 1992. Compute I(6) and I(6) and use
these results to show that even though the CPI was rising at the beginning of
1998, ‘‘inflation was moderating’’ at that time.
We find
I(t) 0.6t2 6t and I(t) 1.2t 6
Thus,
I(6) 0.6(6) 2 6(6) 14.4
I(6) 1.2(6) 6 1.2
Our computations reveal that at the beginning of 1998 (t 6), the CPI was
increasing at the rate of 14.4 points per year, whereas the rate of the inflation
rate was decreasing by 1.2 points per year. Thus, inflation was moderating at
that time (Figure 11.15). In Section 12.2, we will see that relief actually began
in early 1997.
1. Find the third derivative of
2. Let
f(x) 2x 5 3x 3 x 2 6x 10
f(x) 1
1 x
Find f(x), f (x), and f (x).
3. Acertain species of turtle faces extinction because dealers collect truckloads of
turtle eggs to be sold as aphrodisiacs. After severe conservation measures are
implemented, it is hoped that the turtle population will grow according to the rule
N(t) 2t 3 3t 2 4t 1000 (0 t 10)
where N(t) denotes the population at the end of year t. Compute N(2) and N(8)
and interpret your results.
Solutions to Self-CheckExercises 11.5 can be found on page 758.