11DIFFERENTIATION - Department of Mathematics

706 11 DIFFERENTIATION
64. S ALES OF D IGITAL S IGNAL P ROCESSORS The sales of digital
signal processors (DSPs) in billions of dollars is projected
to be
S(t) 0.14t 2 0.68t 3.1 (0 t 6)
where t is measured in years, with t 0 corresponding
to the beginning of 1997.
a. What were the sales of DSPs at the beginning of
1997? What will be the sales at the beginning of 2002?
b. How fast was the level of sales increasing at the beginning
of 1997? How fast will the level of sales be increasing
at the beginning of 2002?
Source: World Semiconductor Trade Statistics
65. S UPPLY F UNCTIONS The supply function for a certain
make of transistor radio is given by
p f(x) 0.0001x 5/4 10
where x is the quantity supplied and p is the unit price
in dollars.
a. Find f(x).
b. What is the rate of change of the unit price if the
quantity supplied is 10,000 transistor radios?
66. P ORTABLE P HONES The percentage of the U.S. population
with portable phones is projected to be
P(t) 24.4t 0.34 (1 t 10)
where t is measured in years, with t 1 corresponding
to the beginning of 1998.
a. What percentage of the U.S. population is expected
to have portable phones by the beginning of 2006?
b. How fast is the percentage of the U.S. population
with portable phones expected to be changing at the
beginning of 2006?
Source: BancAmerica Robertson Stephens
67. A VERAGE S PEED OF A V EHICLE ON A H IGHWAY The average
speed of a vehicle on a stretch of Route 134 between
6 A.M. and 10 A.M. on a typical weekday is approximated
by the function
f(t) 20t 40t 50 (0 t 4)
where f(t) is measured in miles per hour and t is measured
in hours, t 0 corresponding to 6 A.M.
a. Compute f(t).
b. Compute f(0), f(1), and f(2) and interpret your results.
c. Compute f(), f(1), and f(2) and interpret your results.
68. H EALTH-CARE S PENDING Despite efforts at cost containment,
the cost of the Medicare program is increasing at
a high rate. Two major reasons for this increase are
an aging population and the constant development and
extensive use by physicians of new technologies. Based
on data from the Health Care Financing Administration
and the U.S. Census Bureau, health-care spending
through the year 2000 may be approximated by the function
S(t) 0.02836t3 0.05167t2 9.60881t
41.9 (0 t 35)
where S(t) is the spending in billions of dollars and t
is measured in years, with t 0 corresponding to the
beginning of 1965.
a. Find an expression for the rate of change of healthcare
spending at any time t.
b. How fast was health-care spending changing at the
beginning of 1980? How fast was health-care spending
changing at the beginning of 2000?
c. What was the amount of health-care spending at the
beginning of 1980? What was the amount of health-care
spending at the beginning of 2000?
Source: Health Care Financing Administration
69. O N-LINE S HOPPING Retail revenue per year from Internet
shopping is approximated by the function
f(t) 0.075t 3 0.025t 2 2.45t 2.4 (0 t 4)
where f(t) is measured in billions of dollars and t is
measured in years with t 0 corresponding to the beginning
of 1997.
a. Find an expression giving the rate of change of the
retail revenue per year from Internet shopping at any
time t.
b. How fast was the retail revenue per year from Internet
shopping changing at the beginning of the year
2000?
c. What was the retail revenue per year from Internet
shopping at the beginning of the year 2000?
Source: Forrester Research, Inc.
In Exercises 70 and 71, determine whether the
statement is true or false. If it is true, explain
why it is true. If it is false, give an example to
show why it is false.
70. If f and g are differentiable, then
d
[2f(x) 5g(x)] 2f(x) 5g(x)
dx
71. If f(x) x , then f(x) x x1
72. Prove the power rule (Rule 2) for the special case n 3.
Hint: Compute lim
h0
(x h)3 x3 h
.