Provide an appropriate response.
58)
Find the absolute maximum and minimum values of the function f(x) =4x
x2+ 1 on the interval
[–3, 0].
58)
A)
Absolute minimum is 0 at x = 0. Absolute maximum is – 2 at x = – 1.
B)
Absolute maximum is 0 at x = 0. Absolute minimum is 2 at x = – 1.
C)
Absolute minimum is 0 at x = – 1. Absolute maximum is – 2 at x = 0.
D)
Absolute maximum is 0 at x = 0. Absolute minimum is –2 at x = – 1.
Sketch the graph and show all local extrema and inflection points.
59)
f(x) = 2x3– 12x2+ 18x
59)
A)
Local maximum: (0, 0)
Local minimum: (1, -1)
Inflection point: (0.5, -0.5)
B)
No extrema
Inflection point: (0, 0)
21
C)
Local min: (2, 10)
No inflection point
D)
Local max: (1, 8), min: (3, 0)
Inflection point: 2,4
Provide an appropriate response.
60)
Use the first derivative test to determine the local extrema, if any, for the function:
f(x) =x3+3x2– 24x + 6
60)
A)
local max at x = 2 and local min at – 4
B)
local max at x = – 4 and local min at x = 2
C)
local min at x = 2
D)
local max at x = – 4
B
D)
61)
Find f”(x) for f(x) =(4x + 5)3 .
61)
A)
f”(x) = 384x + 480
B)
f”(x) = 4x + 5
C)
f”(x) = 12x + 15
D)
f”(x) = 24x + 30
A
D)
62)
Determine the intervals for which the function f(x) =x3+18x2+ 2, is decreasing.
62)
A)
( , –12) and (–12, 0)
B)
( , –12) and (0, )
C)
(0, 12) and (12, )
D)
(–12, 0)
D
D)
Solve the problem.
22
D
D)
63)
A private shipping company will accept a box for domestic shipment only if the sum of its length
and girth (distance around) does not exceed 102 in. What dimensions will give a box with a square
end the largest possible volume?
63)
A)
17 in. ×34 in. ×34 in.
B)
17 in. ×17 in. ×85 in.
C)
17 in. ×17 in. ×34 in.
D)
34 in. ×34 in. ×34 in.
Find the limit, if it exists.
64)
lim
x–
6x3+3x2
7x2– x
64)
A)
3
7
B)
–6
C)
–
D)
0
Provide an appropriate response.
65)
Find y” for y = 2 x3/2 – 6x1/2 .
65)
A)
y” = 3x1/2 – 3x– 1/2
B)
y” =3
2x1/2 +3
2x– 1/2
C)
y” =3
2x– 1/2 +3
2x– 3/2
D)
y” = 3x– 1/2 + 3x– 3/2
Solve the problem.
66)
The total cost, in dollars, of producing x cell phones is approximated by the function
C(x) = 2000 – 30x +x2
5. Find the minimum average cost.
66)
A)
The minimum average cost is $74 when x = 20 cell phones.
B)
The minimum average cost is $75 when x = 875 cell phones.
C)
The minimum average cost is $875 when x = 75 cell phones.
D)
The minimum average cost is $10 when x = 100 cell phones.
Use the given graph of f(x) to find the intervals on which f(x) > 0.
67)
67)
A)
f(x) > 0 on [-9, 9], f(x) < 0 on (–, -9]
[9, )
B)
f(x) > 0 on (–, -3] [3, ), f(x) < 0 on [-3, 3]
C)
f(x) > 0 on (–, 3], f(x) < 0 on [3, )
D)
f(x) > 0 on [-3, 3], f(x) < 0 on (–, -3]
[3, )
Solve the problem.
68)
With x representing the water temperature in degrees Celsius,
S(x) = – x3– 9x2+ 165x + 1300, 5
x
20 is an approximation to the number of salmon swimming
upstream to spawn. Find the temperature that produces the maximum number of salmon.
68)
A)
19°C
B)
20°C
C)
5°C
D)
6°C
Provide an appropriate response.
69)
The critical values of f(x) = 4x3– 48x + 24 are x = – 2 and x = 2. Use the first derivative test to
determine which of the critical values correspond to a local minimum.
69)
A)
x = 2
B)
x = – 2
C)
neither x = 2 nor x = – 2 correspond to a local minimum
D)
x = 2 and x = – 2
25
70)
Find y” for y =5x2+ 4 .
70)
A)
y” = – 1
4(5x2+ 4)3/2
B)
y” =20
(5x2+ 4)3/2
C)
y” = – 25x2
(5x2+ 4)3/2
D)
y” = – 25x2
(5x2+ 4)1/2
71)
Find f”(x) for f(x) = 4x – 6.
71)
A)
f”(x) = 4x3– 6x2
B)
f”(x) =4
x
C)
f”(x) = 4
D)
f”(x) = 0
Find the limit, if it exists.
72)
Find lim
x
3x + 4
4x2– 3 .
72)
A)
3
4
B)
4
3
C)
0
D)
–4
3
Solve the problem.
73)
Suppose that the total–cost function for a certain company to produce x units of a product is given
by C(x) =6x2+45. Graph the average cost function A(x) = C(x)/x.
73)
26
A)
B)
C)
D)
Find the limit, if it exists.
74)
Find lim
x +
x2
ex .
74)
A)
–
B)
0
C)
ex
D)
Solve the problem.
75)
The annual revenue and cost functions for a manufacturer of zip drives are approximately
R(x) = 520x – 0.02x2 and C(x) = 160x + 100,000, where x denotes the number of drives made. What
is the maximum annual profit?
75)
A)
$1,820,000
B)
$1,520,000
C)
$1,720,000
D)
$1,620,000
Provide an appropriate response.
76)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) =8
x2+ 1
76)
A)
No relative extrema
B)
Relative maximum: (0, –8)
C)
Relative maximum: (0, 8)
D)
Relative maximum: (–1, 8)
77)
A drug that stimulates reproduction is introduced into a colony of bacteria. After t minutes, the
number of bacteria is given approximately by:
N(t) = 1,000 + 36t2–t3,0
t 30
At what value of t is the number of bacteria a maximum?
77)
A)
30 min
B)
12 min
C)
24 min
D)
6 min
Sketch the graph and show all local extrema and inflection points.
78)
f(x) =16x
x2+ 16
78)
A)
Local minimum: (-4, -2)
Local maximum: (4, 2)
Inflection points: (0, 0), (-4 3, –4 3),
(4 3, 4 3)
B)
Maximum: (0, 1)
No inflection point
28
C)
Local minimum: (-4, –1)
Local maximum: (4, 1)
Inflection point: (0, 0)
D)
Local minimum: (4, -2)
Local maximum: (-4, 2)
Inflection point: (0, 0)
Solve the problem.
79)
The annual revenue and cost functions for a manufacturer of grandfather clocks are approximately
R(x) =520x – 0.03x2 and C(x) =200x + 100,000, where x denotes the number of clocks made. What is
the maximum annual profit?
79)
A)
$853,333
B)
$753,333
C)
$953,333
D)
$1,053,333
B
Provide an appropriate response.
80)
Find the inflection point(s) for f(x) =x + 7.
80)
A)
(–7, 0)
B)
(–6, 1)
C)
(–3, 2)
D)
There are no points of inflection.
D
29
A
81)
Find f”(x) for f(x) = – 7x9+ 5x2.
81)
A)
f”(x) = 504x7– 10
B)
f”(x) = 504x8+ 10
C)
f”(x) = – 63x8+ 10x
D)
f”(x) = – 504x7+ 10
82)
Find two numbers whose sum is 430 and whose product is a maximum.
82)
A)
10 and 420
B)
215 and 215
C)
1 and 429
D)
214 and 216
Find the limit, if it exists.
83)
lim
x
3x –6x2+7x3
5– 2x –x3
83)
A)
B)
3
2
C)
–7
D)
7
Provide an appropriate response.
84)
Find the absolute minimum value of f(x) =ex
x3 for x > 0. Round your answer to three decimal
places.
84)
A)
2.718 at x = 1
B)
1 at x = 2.718
C)
0.7439 at x = 3
D)
3 at x = 0.7439
Sketch a graph of the function.
30
85)
f(x) = 2x3– 12x2+ 18x
85)
A)
B)
C)
D)
31
Provide an appropriate response.
86)
Find y” for y = – 1
3x + 4 .
86)
A)
y” =18
(3x + 4)3
B)
y” = – 18
(3x + 4)3
C)
y” = – 2
(3x + 4)3
D)
y” = – 6
(3x + 4)3
Solve the problem.
87)
A company estimates that it will sell N(t) hair dryers after spending $t thousands on advertising as
given by:
N(t) = – 3t3+ 450t2– 21,600t + 1,100, 40 t
60
For which values of t is the rate of sales N'(t) increasing?
87)
A)
40 < t < 60
B)
50 < t < 60
C)
t > 40
D)
40 < t < 50
Provide an appropriate response.
88)
Find vertical asymptotes for f(x) =7x – 2
x2– 3x – 4 .
88)
A)
x = – 1, x = 4
B)
x = 1, x = 4
C)
x = – 1, x = – 4
D)
x = 1, x = – 4
Solve the problem.
89)
A carpenter is building a rectangular room with a fixed perimeter of 400 ft. What are the
dimensions of the largest room that can be built? What is its area?
89)
A)
100 ft by 300 ft; 30,000 ft2
B)
40 ft by 360ft; 14,400 ft2
C)
200 ft by 200 ft; 40,000 ft2
D)
100 ft by 100 ft; 10,000 ft2
Provide an appropriate response.
90)
Use the first derivative test to determine the local extrema, if any, for the function:
f(x) =3(x – 4)2/3 + 6.
90)
A)
f(x) has a local minimum at 6
B)
f(x) has a local minimum at x = 4.
C)
f(x) has no local extrema
D)
f(x) has a local maximum at x = 4.
Solve the problem.
91)
Find the approximate number of batches (to the nearest whole number) of an item that should be
produced annually if 250,000 units are to be made. It costs $3 to store a unit for one year, and it
costs $340 to set up the factory to produce each batch.
91)
A)
35 batches
B)
25 batches
C)
33 batches
D)
23 batches
Find the limit, if it exists.
92)
Find: lim
x
5x2+ 3x – 1
6x2– x + 7
92)
A)
1
7
B)
–5
6
C)
–1
7
D)
5
6
Sketch a graph of the function.
93)
f(x) = 3x4–4x3
93)
33
A)
B)
C)
D)
Sketch the graph and show all local extrema and inflection points.
94)
f(x) =4x2+ 24x
94)
34
A)
Min: (6, -24)
No inflection points
B)
Min: (3, -36)
No inflection points
C)
Min: (-6, -24)
No inflection points
D)
Min: (-3, -36)
No inflection points
Solve the problem.
95)
The average manufacturing cost per unit (in hundreds of dollars) for producing x units of a product
is given by:
C(x) = 2x3– 42x2+ 288x + 12, 1
x
5
At what production level will the average cost per unit be maximum?
95)
A)
652 units
B)
5 units
C)
12 units
D)
1 unit