Find the equation of the tangent line at the given value of x on the curve.
74)
2xy – 2x + y = – 14, x = 2
74)
A)
y =6
5x –22
5
B)
y = – 5
6x +22
6
C)
y =5
6x –22
6
D)
y = – 6
5x +22
5
Solve the problem.
75)
A hotel has 280 units. All rooms are occupied when the hotel charges $100 per day for a room. For
every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room
costs $24 per day to service and maintain. What should the hotel charge per day in order to
maximize daily profit?
75)
A)
$190
B)
$202
C)
$102
D)
$192
Find the equation of the tangent line at the given value of x on the curve.
76)
y3+ 2xy2+ 3 = 4y2+ x, x = 2
76)
A)
y = – 1
3x +5
3
B)
y =1
5x –7
5
C)
y = – 1
3x –1
3
D)
y =1
3x –5
3
Solve the problem.
77)
The demand equation for a certain product is 4p2+ q2=1000, where p is the price per unit in
dollars and q is the number of units demanded. Find dq/dp.
77)
A)
dq/dp = – 4p/q
B)
dq/dp = – p/4q
C)
dq/dp = – 4q/p
D)
dq/dp = – q/4p
21
Find dy/dx by implicit differentiation.
78)
xy + x = 2
78)
A)
1 + y
x
B)
–1 + x
y
C)
1 + x
y
D)
–1 + y
x
Solve the problem.
79)
The demand equation for a certain product is 4p2+ q2=1100, where p is the price per unit in
dollars and q is the number of units demanded. Find dp/dq.
79)
A)
dp/dq = – p/4q
B)
dp/dq = – q/4p
C)
dp/dq = – 4q/p
D)
dp/dq = – 4p/q
80)
Electrical systems are governed by Ohm’s law, which states that V = IR, where V = voltage,
I = current, and R = resistance. If the current in an electrical system is decreasing at a rate of 3 A/s
while the voltage remains constant at 30 V, at what rate is the resistance increasing when the
current is 52 A?
80)
A)
26
45 ohms/s
B)
135
26 ohms/s
C)
45
26 ohms/s
D)
45
1352 ohms/s
Find the equation of the tangent line at the given point on the curve.
81)
xy + x = 2; (1, 1)
81)
A)
y = 2x – 1
B)
y = – 1
2x +3
2
C)
y =1
2x +1
2
D)
y = – 2x + 3
22
Solve the problem.
82)
P(x) = – x3+ 12x2– 36x + 400, x
3 is an approximation to the total profit (in thousands of dollars)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must
be sold to maximize profit.
82)
A)
2 hundred thousand
B)
3 hundred thousand
C)
7 hundred thousand
D)
6 hundred thousand
Find the equation of the tangent line at the given value of x on the curve.
83)
2xy – y2= 1, x = 1
83)
A)
y = 1
B)
y = – x + 1
C)
y = x – 1
D)
x = 1
Explanation:
Solve the problem.
84)
Recent research has shown that the population f(S) of cod in the North Sea next year as a function
of this year’s population S (measured in thousands of tons) can be described by the Shepherd
model,
f(S) =aS
1 +(S/b)c
where a, b, and c are constants. The values of a, b, and c are 3.039, 247, and 3.25, respectively. Find
the approximate value of this year’s population that maximizes next year‘s population using this
model.
84)
A)
192 tons
B)
158,000 tons
C)
4000 tons
D)
192,000 tons
Explanation:
Find dy/dx by implicit differentiation.
85)
2xy – y2= 1
85)
A)
y
y – x
B)
x
y – x
C)
y
x – y
D)
x
x – y
Explanation:
Explanation:
Solve the problem.
86)
P(x) = – x3+ 15x2– 48x + 450, x
3 is an approximation to the total profit (in thousands of dollars)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must
be sold to maximize profit.
86)
A)
5 hundred thousand
B)
3 hundred thousand
C)
8 hundred thousand
D)
10 hundred thousand
87)
From a thin piece of cardboard 10 in. by 10 in., square corners are cut out so that the sides can be
folded up to make a box. What dimensions will yield a box of maximum volume? What is the
maximum volume? Round to the nearest tenth, if necessary.
87)
A)
3.3 in. by 3.3 in. by 3.3 in.; 37 in.3
B)
6.7 in. by 6.7 in. by 3.3 in.; 148.1 in.3
C)
6.7 in. by 6.7 in. by 1.7 in.; 74.1 in.3
D)
5 in. by 5 in. by 2.5 in.; 62.5 in.3
Explanation:
Find the location of the indicated absolute extremum for the function.
88)
Maximum
88)
A)
No maximum
B)
x = – 4
C)
x =11
4
D)
x = 0
Explanation:
Explanation:
Solve the problem.
89)
Find the approximate number of batches (to the nearest whole number) of an item that should be
produced annually if 100,000 units are to be made. It costs $4 to store a unit for one year, and it
costs $420 to set up the factory to produce each batch.
89)
A)
22 batches
B)
24 batches
C)
16 batches
D)
18 batches
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
90)
f(x) =1
x + 2 ; [–4, 1]
Minimum
90)
A)
1
3 at x = 1
B)
–1
2 at x = – 4
C)
No absolute minimum
D)
1
2 at x = 0
91)
f(x) =x + 3
x – 3 ; [–4, 4]
Maximum
91)
A)
–1 at x = 0
B)
7 at x = 4
C)
No absolute maximum
D)
1
7 at x = – 4
Graph the function on the indicated domain, and use the capabilities of your calculator to find the location and value of
the indicated absolute extremum.
92)
f(x) = x(x – 5)2/3; ( , )
Minimum
92)
A)
0 at x =5.0
B)
2.1 at x =4.7
C)
9 at x =6.6
D)
No absolute minimum
Solve the problem.
93)
Suppose c(x) =x3–24x2+30,000x is the cost of manufacturing x items. Find a production level that
will minimize the average cost per item of making x items.
93)
A)
13 items
B)
11 items
C)
12 items
D)
14 items
94)
Find the elasticity of demand E for the demand function q =12 – ln p
94)
A)
E =
–p
12 – ln p
B)
E =
–p
12p – ln p
C)
E =12 – ln p
p2
D)
E =1
12 – ln p
95)
If the price charged for a bolt is p cents, then x thousand bolts will be sold in a certain hardware
store, where p =37 –x
14 . How many bolts must be sold to maximize revenue?
95)
A)
518 bolts
B)
518 thousand bolts
C)
259 thousand bolts
D)
259 bolts
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
96)
f(x) =x4/3 –x2/3; [0, 2]
Minimum
96)
A)
0 at x = 1
B)
No absolute minimum
C)
0.9324 at x = 2
D)
–1
4 at x =2
4
Solve the problem.
97)
The graph gives the profit P(x) as a function of production level. Use graphical optimization to
estimate the production level that gives the maximum profit per item produced.
97)
A)
3 units
B)
6 units
C)
5 units
D)
4 units
Find dy for the given values of x and x.
98)
y =3x – 7
x – 1 ; x = 2, x = 0.1
98)
A)
6
B)
0.6
C)
0.4
D)
4
Assume x and y are functions of t. Evaluate dy/dt.
99)
x3ey–y3 ln x =7; dx/dt =2, x = 1, y =3
99)
A)
54 +e3
e3
B)
7
C)
54 –6e3
e3
D)
54 –6e3
3
Solve the problem.
100)
Researchers have discovered that by controlling both the temperature and the relative humidity in
a building, the growth of a certain fungus can be limited. The relationship between temperature
and relative humidity, which limits growth, can be described by
R(T) = – 0.00005T3+ 0.315T2–1.5572T + 97.086,
0 T
46,
where R(T) is the relative humidity (in %) and T is the temperature (in °C). Find the temperature at
which the relative humidity is minimized.
100)
A)
1.47°C
B)
–0.53°C
C)
4197.53°C
D)
2.47°C
101)
A bookstore has an annual demand for 67,000 copies of a best–selling book. It costs $0.60 to store
one copy for one year, and it costs $65 to place an order. Find the optimum number of copies per
order.
101)
A)
4365 copies
B)
3429 copies
C)
3810 copies
D)
4919 copies
102)
If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a
certain city, where p(x) =127 –x
34 . How many candy bars must be sold to maximize revenue?
102)
A)
2159 thousand candy bars
B)
4318 thousand candy bars
C)
4318 candy bars
D)
2159 candy bars
103)
Given the revenue and cost functions R =26x – 0.3x2 and C =5x + 11, where x is the daily
production, find the rate of change of profit with respect to time when 20 units are produced and
the rate of change of production is 4 units per day.
103)
A)
$72.00 per day
B)
$56.00 per day
C)
$81.60 per day
D)
$36.00 per day
28
Find the location of the indicated absolute extremum for the function.
104)
Maximum
104)
A)
x = 1
B)
x = 4
C)
No maximum
D)
x = – 1
Solve the problem.
105)
S(x) = – x3– 9x2+ 165x + 1300, 5 x
20 is an approximation to the number of salmon swimming
upstream to spawn, where x represents the water temperature in degrees Celsius. Find the
temperature that produces the maximum number of salmon.
105)
A)
5°C
B)
19°C
C)
20°C
D)
6°C
106)
Find the dimensions that produce the maximum floor area for a one–story house that is rectangular
in shape and has a perimeter of 167 ft.
106)
A)
41.75 ft ×167 ft
B)
13.92 ft ×41.75 ft
C)
41.75 ft ×41.75 ft
D)
83.5 ft ×83.5 ft
107)
The volume of a sphere is increasing at a rate of 10 cm3/s. Find the rate of change of its surface area
when its volume is 256
3 cm3
107)
A)
64
3 cm2/s
B)
5
3 cm2/s
C)
5 cm2/s
D)
10
3 cm2/s
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
108)
f(x) = 3x4+ 16x3+ 24x2+ 32; [–3, 1]
Maximum
108)
A)
59 at x = – 3
B)
32 at x = 0
C)
75 at x = 1
D)
48 at x = – 2
Find the absolute extrema if they exist as well as where they occur.
109)
f(x) =10x ln x
109)
A)
Absolute minimum of 0 at x = – 10; no absolute maximum
B)
Absolute minimum of 3.6788 at x =e–1; no absolute maximum
C)
Absolute maximum of 2,202,646.58 at x =e–10; no absolute minimum
D)
No absolute minimum or maximum
Solve the problem.
110)
A company estimates that the revenue (in dollars) from the sale of x units of dog houses is given by
C(x) =1425 +0.02x + 0.0006x2. Use the differential to approximate the change in revenue from the
sale of one more dog house when 900 dog houses are sold.
110)
A)
$110.00
B)
$0.92
C)
$0.110
D)
$92.00
Assume x and y are functions of t. Evaluate dy/dt.
111)
xy + x = 12; dx/dt = – 3, x = 2, y = 5
111)
A)
–3
B)
–9
C)
3
D)
9
Solve the problem.
112)
In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year
during the years 1990 – 2000 are given in the figure. Consider the closed interval [1990, 2000].
A(1990, 168) D (1993, 280) G (1996, 188) L(1999, 236)
B(1991, 204) E(1994, 211) H(1997, 255) M (2000, 270)
C(1992, 255) F(1995, 142) K (1998, 247)
Give all relative maxima and minima on the interval and the years when they occur.
112)
A)
Relative maxima of 280 in 1993, 255 in 1997, 270 in 2000
Relative minima of 168 in 1990, 142 in 1995, 236 in 1999
B)
Relative maxima of 280 in 1993 and 255 in 1997
Relative minima of 142 in 1995 and 236 in 1999
C)
Relative maxima of 280 in 1993, 255 in 1997, 270 in 2000
Relative minima of 142 in 1995 and 236 in 1999
D)
Relative maxima of 280 in 1993 and 255 in 1997
Relative minima of 168 in 1990, 142 in 1995, 236 in 1999
113)
The average daily metabolic rate for a hippopotamus living in the wild can be expressed as a
function of weight by m = 132.9w0.75, where w is the weight of the hippopotamus (in kg) and m is
the metabolic rate (in kcal/day). Determine dm/dt for a 2900–kg hippopotamus that is gaining
weight at a rate of 21.75 kg/day.
113)
A)
14 kcal/day2
B)
394 kcal/day2
C)
15,909 kcal/day2
D)
295 kcal/day2
114)
Maximize Q = xy2, where x and y are positive numbers, such that x +y2=10.
114)
A)
x = 1, y =3
B)
x = 0, y =10
C)
x =5, y =5
D)
x =5, y =5
115)
S(x) = – x3– 3x2+ 72x + 900, x
2 is an approximation to the number of salmon swimming
upstream to spawn, where x represents the water temperature in degrees Celsius. Find the
temperature that produces the maximum number of salmon.
115)
A)
6°C
B)
2°C
C)
4°C
D)
8°C
116)
The position of a particle at time t is given by s, where s3+ 9st + 4t3– 8t = 0. Find the velocity ds/dt.
116)
A)
ds/dt =8+ 9s – 12t2
3s2+ 9t
B)
ds/dt =8+ 9s – 12t2
3s2– 9t
C)
ds/dt =8– 9s – 12t2
3s2– 9t
D)
ds/dt =8– 9s – 12t2
3s2+ 9t
32