11DIFFERENTIATION - Department of Mathematics

55. C ONSUMER P RICE I NDEX An economy’s consumer price index
(CPI) is described by the function
I(t) 0.2t 3 3t 2 100 (0 t 10)
where t 0 corresponds to 1990.
a. At what rate was the CPI changing in 1995? In 1997?
In 2000?
b. What was the average rate of increase in the CPI
over the period from 1995 to 2000?
56. E FFECT OF A DVERTISING ON S ALES The relationship between
the amount of money x that the Cannon Precision Instruments
Corporation spends on advertising and the company’s
total sales S(x) is given by the function
S(x) 0.002x 3 0.6x 2 x 500 (0 x 200)
where x is measured in thousands of dollars. Find the
rate of change of the sales with respect to the amount
of money spent on advertising. Are Cannon’s total sales
increasing at a faster rate when the amount of money
spent on advertising is (a) $100,000 or (b) $150,000?
57. P OPULATION G ROWTH Astudy prepared for a Sunbelt
town’s chamber of commerce projected that the town’s
population in the next 3 yr will grow according to the
rule
P(t) 50,000 30t 3/2 20t
where P(t) denotes the population t months from now.
How fast will the population be increasing 9 mo and 16
mo from now?
58. C URBING P OPULATION G ROWTH Five years ago, the government
of a Pacific island state launched an extensive propaganda
campaign toward curbing the country’s population
growth. According to the Census Department, the
population (measured in thousands of people) for the
following 4 yr was
P(t) 1
3 t3 64t 3000
where t is measured in years and t 0 at the start of
the campaign. Find the rate of change of the population
at the end of years 1, 2, 3, and 4. Was the plan working?
59. C ONSERVATION OF S PECIES 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.
11.1 B ASIC R ULES OF D IFFERENTIATION 705
Find the rate of growth of the turtle population when
t 2 and t 8. What will be the population 10 yr after
the conservation measures are implemented?
60. F LIGHT OF A R OCKET The altitude (in feet) of a rocket t
sec into flight is given by
s f(t) 2t3 114t2 480t 1 (t0) a. Find an expression v for the rocket’s velocity at any
time t.
b. Compute the rocket’s velocity when t 0, 20, 40, and
60. Interpret your results.
c. Using the results from the solution to part (b), find
the maximum altitude attained by the rocket.
Hint: At its highest point, the velocity of the rocket is zero.
61. S TOPPING D ISTANCE OF A R ACING C AR During a test by the
editors of an auto magazine, the stopping distance s (in
feet) of the MacPherson X-2 racing car conformed to
the rule
s f(t) 120t 15t2 (t 0)
where t was the time (in seconds) after the brakes were
applied.
a. Find an expression for the car’s velocity v at any
time t.
b. What was the car’s velocity when the brakes were
first applied?
c. What was the car’s stopping distance for that particular
test?
Hint: The stopping time is found by setting v 0.
62. D EMAND F UNCTIONS The demand function for the Luminar
desk lamp is given by
p f(x) 0.1x2 0.4x 35
where x is the quantity demanded (measured in thousands)
and p is the unit price in dollars.
a. Find f(x).
b. What is the rate of change of the unit price when the
quantity demanded is 10,000 units (x 10)? What is the
unit price at that level of demand?
63. I NCREASE IN T EMPORARY W ORKERS According to the Labor
Department, the number of temporary workers (in millions)
is estimated to be
N(t) 0.025t2 0.255t 1.505 (0 t 5)
where t is measured in years, with t 0 corresponding
to 1991.
a. How many temporary workers were there at the beginning
of 1994?
b. How fast was the number of temporary workers growing
at the beginning of 1994?
Source: Labor Department