James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

206 Chapter 3 Differentiation Rules
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79. The displacement of a particle on a vibrating string is given
by the equation sstd − 10 1 1 4 sins10td where s is measured
in centimeters and t in seconds. Find the velocity of the
particle after t seconds.
80. If the equation of motion of a particle is given by
s − A cosst 1 d, the particle is said to undergo simple
harmonic motion.
(a) Find the velocity of the particle at time t.
(b) When is the velocity 0?
81. A Cepheid variable star is a star whose brightness alternately
increases and decreases. The most easily visible such star is
Delta Cephei, for which the interval between times of maximum
brightness is 5.4 days. The average brightness of this
star is 4.0 and its brightness changes by 60.35. In view of
these data, the brightness of Delta Cephei at time t, where t
is mea sured in days, has been modeled by the function
Bstd − 4.0 1 0.35 sinS
5.4D
2t
(a) Find the rate of change of the brightness after t days.
(b) Find, correct to two decimal places, the rate of increase
after one day.
82. In Example 1.3.4 we arrived at a model for the length of daylight
(in hours) in Philadelphia on the tth day of the year:
Lstd − 12 1 2.8 sinF G
2
365 st 2 80d
Use this model to compare how the number of hours of
day light is increasing in Philadelphia on March 21 and
May 21.
83. The motion of a spring that is subject to a frictional force or
a damping force (such as a shock absorber in a car) is often
modeled by the product of an exponential function and a sine
or cosine function. Suppose the equation of motion of a
point on such a spring is
sstd − 2e 21.5t sin 2t
where s is measured in centimeters and t in seconds. Find
the velocity after t seconds and graph both the position and
velocity functions for 0 < t < 2.
84. Under certain circumstances a rumor spreads according to
the equation
1
pstd −
1 1 ae 2k t
where pstd is the proportion of the population that has heard
the rumor at time t and a and k are positive constants. [In
Sec tion 9.4 we will see that this is a reasonable equation
for pstd.]
(a) Find lim t l ` pstd.
(b) Find the rate of spread of the rumor.
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(c) Graph p for the case a − 10, k − 0.5 with t measured in
hours. Use the graph to estimate how long it will take for
80% of the population to hear the rumor.
85. The average blood alcohol concentration (BAC) of eight
male subjects was measured after consumption of 15 mL of
ethanol (corresponding to one alcoholic drink). The resulting
data were modeled by the concentration function
Cstd − 0.0225te 20.0467t
where t is measured in minutes after consumption and C is
measured in mgymL.
(a) How rapidly was the BAC increasing after 10 minutes?
(b) How rapidly was it decreasing half an hour later?
Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after
Oral Administration in the Fasting State,” Journal of Pharmacokinetics and
Biopharmaceutics 5 (1977): 207–24.
86. In Section 1.4 we modeled the world population from 1900
to 2010 with the exponential function
Pstd − s1436.53d ? s1.01395d t
where t − 0 corresponds to the year 1900 and Pstd is
measured in millions. According to this model, what was the
rate of increase of world population in 1920? In 1950? In
2000?
87. A particle moves along a straight line with displacement sstd,
velocity vstd, and acceleration astd. Show that
astd − vstd dv
ds
Explain the difference between the meanings of the
derivatives dvydt and dvyds.
88. Air is being pumped into a spherical weather balloon. At
any time t, the volume of the balloon is Vstd and its radius
is rstd.
(a) What do the derivatives dVydr and dVydt represent?
(b) Express dVydt in terms of drydt.
89. The flash unit on a camera operates by storing charge on a
capacitor and releasing it suddenly when the flash is set off.
The following data describe the charge Q remaining on the
capacitor (measured in microcoulombs, mC) at time t (measured
in seconds).
t 0.00 0.02 0.04 0.06 0.08 0.10
Q 100.00 81.87 67.03 54.88 44.93 36.76
(a) Use a graphing calculator or computer to find an exponential
model for the charge.
(b) The derivative Q9std represents the electric current (measured
in microamperes, mA) flowing from the capacitor
to the flash bulb. Use part (a) to estimate the current
when t − 0.04 s. Compare with the result of Example
2.1.2.
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