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

284 Chapter 4 Applications of Differentiation
37. hstd − t 3y4 2 2t 1y4 38. tsxd − s 3 4 2 x 2
39. Fsxd − x 4y5 sx 2 4d 2 40. tsd − 4 2 tan
41. f sd − 2 cos 1 sin 2 42. hstd − 3t 2 arcsin t
43. fsxd − x 2 e 23x 44. fsxd − x 22 ln x
; 45–46 A formula for the derivative of a function f is given.
How many critical numbers does f have?
45. f 9sxd − 5e 20.1 | x | sinx 2 1 46. f 9sxd − 100 cos2 x
2 1
10 1 x 2
;
;
47–62 Find the absolute maximum and absolute minimum
values of f on the given interval.
47. f sxd − 12 1 4x 2 x 2 , f0, 5g
48. f sxd − 5 1 54x 2 2x 3 , f0, 4g
49. f sxd − 2x 3 2 3x 2 2 12x 1 1, f22, 3g
50. f sxd − x 3 2 6x 2 1 5, f23, 5g
51. f sxd − 3x 4 2 4x 3 2 12x 2 1 1, f22, 3g
52. f std − st 2 2 4d 3 , f22, 3g
53. f sxd − x 1 1 , f0.2, 4g
x
54.
x
f sxd −
, f0, 3g
x 2 2 x 1 1
55. f std − t 2 s 3 t , f21, 4g
56. f std − st , f0, 2g
2
1 1 t
57. f std − 2cos t 1 sin 2t, f0, y2g
58. f std − t 1 cotsty2d, fy4, 7y4g
59. f sxd − x 22 ln x , f 1 2 , 4g
60. f sxd − xe xy2 , f23, 1g
61. f sxd − lnsx 2 1 x 1 1d, f21, 1g
62. f sxd − x 2 2 tan 21 x, f0, 4g
63. If a and b are positive numbers, find the maximum value
of f sxd − x a s1 2 xd b , 0 < x < 1.
64. Use a graph to estimate the critical numbers of
f sxd − | 1 1 5x 2 x 3 | correct to one decimal place.
65–68
(a) Use a graph to estimate the absolute maximum and
minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum
values.
65. f sxd − x 5 2 x 3 1 2, 21 < x < 1
66. f sxd − e x 1 e 22x , 0 < x < 1
;
67. f sxd − xsx 2 x 2
68. f sxd − x 2 2 cos x, 22 < x < 0
69. After the consumption of an alcoholic beverage, the concentration
of alcohol in the bloodstream (blood alcohol concentration,
or BAC) surges as the alcohol is absorbed, followed by a
gradual decline as the alcohol is metabolized. The function
Cstd − 1.35te 22.802t
models the average BAC, measured in mgymL, of a group of
eight male subjects t hours after rapid consumption of 15 mL
of ethanol (corresponding to one alcoholic drink). What is the
maximum average BAC during the first 3 hours? When does
it occur?
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.
70. After an antibiotic tablet is taken, the concentration of the
antibiotic in the bloodstream is modeled by the function
Cstd − 8se 20.4t 2 e 20.6t d
where the time t is measured in hours and C is measured in
mgymL. What is the maximum concentration of the antibiotic
during the first 12 hours?
71. Between 0°C and 30°C, the volume V (in cubic centimeters)
of 1 kg of water at a temperature T is given approximately by
the formula
V − 999.87 2 0.06426T 1 0.0085043T 2 2 0.0000679T 3
Find the temperature at which water has its maximum density.
72. An object with weight W is dragged along a horizontal plane
by a force acting along a rope attached to the object. If the
rope makes an angle with the plane, then the magnitude of
the force is
W
F −
sin 1 cos
where is a positive constant called the coefficient of friction
and where 0 < < y2. Show that F is minimized when
tan − .
73. The water level, measured in feet above mean sea level, of
Lake Lanier in Georgia, USA, during 2012 can be modeled
by the function
Lstd − 0.01441t 3 2 0.4177t 2 1 2.703t 1 1060.1
where t is measured in months since January 1, 2012. Estimate
when the water level was highest during 2012.
74. On May 7, 1992, the space shuttle Endeavour was launched
on mission STS-49, the purpose of which was to install a new
perigee kick motor in an Intelsat communications satellite.
The table gives the velocity data for the shuttle between liftoff
and the jettisoning of the solid rocket boosters.
(a) Use a graphing calculator or computer to find the cubic
polynomial that best models the velocity of the shuttle for
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