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

chapter 3 Review 269
(b) Find the rate of change of the volume with respect to the
radius if the height is constant.
91. The mass of part of a wire is xs1 1 sx d kilograms, where
x is measured in meters from one end of the wire. Find the
linear density of the wire when x − 4 m.
92. The cost, in dollars, of producing x units of a certain commodity
is
Csxd − 920 1 2x 2 0.02x 2 1 0.00007x 3
(a) Find the marginal cost function.
(b) Find C9s100d and explain its meaning.
(c) Compare C9s100d with the cost of producing the
101st item.
93. A bacteria culture contains 200 cells initially and grows at a
rate proportional to its size. After half an hour the population
has increased to 360 cells.
(a) Find the number of bacteria after t hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 10,000?
94. Cobalt-60 has a half-life of 5.24 years.
(a) Find the mass that remains from a 100-mg sample after
20 years.
(b) How long would it take for the mass to decay to 1 mg?
95. Let Cstd be the concentration of a drug in the bloodstream. As
the body eliminates the drug, Cstd decreases at a rate that is
proportional to the amount of the drug that is present at the
time. Thus C9std − 2kCstd, where k is a positive number
called the elimination constant of the drug.
(a) If C 0 is the concentration at time t − 0, find the concentration
at time t.
(b) If the body eliminates half the drug in 30 hours, how long
does it take to eliminate 90% of the drug?
96. A cup of hot chocolate has temperature 80°C in a room kept
at 20°C. After half an hour the hot chocolate cools to 60°C.
(a) What is the temperature of the chocolate after another
half hour?
(b) When will the chocolate have cooled to 40°C?
97. The volume of a cube is increasing at a rate of 10 cm 3 ymin.
How fast is the surface area increasing when the length of an
edge is 30 cm?
98. A paper cup has the shape of a cone with height 10 cm and
radius 3 cm (at the top). If water is poured into the cup at a
rate of 2 cm 3 ys, how fast is the water level rising when the
water is 5 cm deep?
99. A balloon is rising at a constant speed of 5 ftys. A boy is
cycling along a straight road at a speed of 15 ftys. When he
passes under the balloon, it is 45 ft above him. How fast is the
distance between the boy and the balloon increasing 3 s later?
;
;
100. A waterskier skis over the ramp shown in the figure at a
speed of 30 ftys. How fast is she rising as she leaves the
ramp?
15 ft
4 ft
101. The angle of elevation of the sun is decreasing at a rate of
0.25 radyh. How fast is the shadow cast by a 400-ft-tall
building increasing when the angle of elevation of the sun
is y6?
102. (a) Find the linear approximation to f sxd − s25 2 x 2
near 3.
(b) Illustrate part (a) by graphing f and the linear
approximation.
(c) For what values of x is the linear approximation accurate
to within 0.1?
103. (a) Find the linearization of f sxd − s 3 1 1 3x at a − 0.
State the corresponding linear approximation and use it
to give an approximate value for s 3 1.03 .
(b) Determine the values of x for which the linear approximation
given in part (a) is accurate to within 0.1.
104. Evaluate dy if y − x 3 2 2x 2 1 1, x − 2, and dx − 0.2.
105. A window has the shape of a square surmounted by a semi -
circle. The base of the window is measured as having width
60 cm with a possible error in measurement of 0.1 cm. Use
differentials to estimate the maximum error possible in
computing the area of the window.
106–108 Express the limit as a derivative and evaluate.
x 17 2 1
106. lim
x l1 x 2 1
cos 2 0.5
108. lim
l y3 2 y3
s 4 16 1 h 2 2
107. lim
h l 0 h
109. Evaluate lim
x l 0
s1 1 tan x 2 s1 1 sin x
x 3 .
110. Suppose f is a differentiable function such that f stsxdd − x
and f 9sxd − 1 1 f f sxdg 2 . Show that t9sxd − 1ys1 1 x 2 d.
111. Find f 9sxd if it is known that
d
dx f f s2xdg − x 2
112. Show that the length of the portion of any tangent line to the
astroid x 2y3 1 y 2y3 − a 2y3 cut off by the coordinate axes is
constant.
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