Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graphs of the first and second derivatives of a function y = f(x) are given. Select a possible graph of f that passes
through the point P. (NOTE: Vertical scales may vary from graph to graph.)
1)
f’ f”
1)
A)
B)
C)
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
2)
Sketch the graph of f(x) = x +3
2)
Sketch a graph of a single function that has these properties.
3)
a) Continuous and differentiable for all real numbers
b) f(x) < 0 on (– , –3 ) and ( 3 , )
c) f(x) > 0 on (–3 , 3)
d) f(x) > 0 on (– , 0 )
e) f(x) < 0 on ( 0 , )
f) f (–3) =f(3) = 0
g) An inflection point at (0,0)
3)
Provide an appropriate response.
4)
Identify the intervals where f(x) is decreasing.
4)
5)
Write the sign chart that corresponds to the following graph of f(x):
5)
Sketch a graph of a single function that has these properties.
6)
a) Continuous and differentiable for all real numbers
b) f(x) > 0 on (–3 , –1) and ( 2 , )
c) f(x) < 0 on (–, –3) and ( –1 , 2)
d) f(x) > 0 on (– , –2) and ( 1 , )
e) f(x) < 0 on (–2 , 1)
f) f (–3) =f (–1) =f(2) = 0
g) f(x) = 0 at (–2 , 0) and (1, 1)
6)
Solve the problem.
7)
The financial analysis department of a software design company determined that the cost
of producing x palm assistants is C(x) = 5000 + 3x. The department also determined the
associated price–demand equation to be p = 23 –x
500, where p is price in dollars.
a) Obtain the profit function.
b) Determine the maximum profit.
7)
4
Provide an appropriate response.
8)
Identify the intervals where f(x) > 0.
8)
9)
Sketch the graph of f(x) =3x2+ 2x + 5
6x2+ 2 . Include sketch of all asymptotes.
9)
10)
Write the sign chart that corresponds to the following graph of f(x)::
10)
Solve the problem.
11)
A backpack manufacturer is planning to expand its work force. They estimate that the
number of backpacks produced by hiring new workers is given by
T(x) = –0.25x4+4x3, 0 x
12. Determine when the rate of backpacks is increasing and
when it is decreasing. Determine the point of diminishing returns and the maximum rate
of change of backpack production.
11)
Provide an appropriate response.
12)
Sketch the graph of f(x) =2x2+ 5x – 3
x2– 9 . Include sketch of all asymptotes.
12)
13)
Find the absolute minimum value of f(x) = 4x ln x – 7x. Round your answer to three
decimal places.
13)
14)
Find the absolute maximum value of f(x) = 3x – ln x . Round your answer to four decimal
places.
14)
15)
Consider the function f(x) = –0.25x4–x3+ 2. Determine the intervals where f(x) is
increasing and decreasing, concave up and concave down and all local extrema. Use that
information to obtain a sketch of the function.
15)
Sketch a graph of a single function that has these properties.
16)
a) Continuous for all real numbers
b) Differentiable everywhere except x = 0
c) f(x) < 0 on (– , 0)
d) f(x) > 0 on ( 0 , )
e) f(x) < 0 on (– , 0) and (0, )
f) f(–2) = f (2) = 5
g) y–intercept and x–intercept at (0, 0)
16)
Solve the problem.
17)
A logo baseball cap manufacturer has a uniform annual demand of 25,000 caps. It costs $1
to store one baseball cap for 1 year and $500 to set up the plant for production of the logo
baseball caps. How many times a year should the company produce the caps in order to
minimize the total storage and set–up costs? (Assume that there are 250 working days per
year.)
17)
Provide an appropriate response.
18)
Suppose f is a continuous function. Describe the graph of f at (1, f(1)) if f’(1) = 0 and
f”(x) < 0.
18)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
19)
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?
19)
A)
24 min
B)
6 min
C)
12 min
D)
30 min
20)
Find the critical values and determine the intervals where f(x) is decreasing and the intervals where
f(x) is increasing for f(x) =3x4–6x2+ 7.
20)
A)
increasing on (–1, 0) and (1, ); decreasing on ( , –1) and (0, 1)
B)
decreasing on (–1, 0) and (1, ); increasing on ( , –1)
C)
decreasing on (–1, 0) and (1, ); increasing on ( , –1) and (0, 1)
D)
increasing on (–1, 0); decreasing on ( , –1) and (0, 1)
Find the intervals where f”(x) < 0 ir f”(x) > 0 as indicated.
21)
f”(x) < 0
21)
A)
(–2, )
B)
(–5, 2)
C)
(–, –2)
D)
(–5, 5)
Find the limit, if it exists.
22)
Find lim
x –2
x2+ x – 2
x2– 4 .
22)
A)
3
4
B)
0
C)
D)
2
Find the domain and intercepts.
23)
h(x) =5x
x + 3
23)
A)
Domain: All real numbers except -3; y intercept: 0; no x intercept
B)
Domain: All real numbers except -3; y intercept: 0; x intercept: 0
C)
Domain: All real numbers except 3; y intercept: 0; x intercept: 0
D)
Domain: ( , -3); y intercept: 0; x intercept: 0
Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
24)
Concave downward
24)
A)
(–1, 0) , (1, ); x = 0
B)
(–, –1); no inflection points
C)
(–1, 0); no inflection points
D)
(–1, 0) , (1, ); x = 0 and x = 1
Provide an appropriate response.
25)
Find y” for y =x4–8x1/2
25)
A)
12x2+2
x x
B)
4x3+4x
C)
12x2–4x
D)
4x3–4x
A
Sketch a graph of the function.
26)
f(x) = 3x4–4x3
26)
10
A
A)
B)
C)
D)
Solve the problem.
27)
A carpenter is building a rectangular room with a fixed perimeter of 440 ft. What are the
dimensions of the largest room that can be built? What is its area?
27)
A)
220 ft by 220 ft; 48,400 ft2
B)
44 ft by 396ft; 17,424 ft2
C)
110 ft by 330 ft; 36,300 ft2
D)
110 ft by 110 ft; 12,100 ft2
28)
Find two numbers whose difference is 14 and whose product is a minimum.
28)
A)
0 and 14
B)
1 and 15
C)
28 and 14
D)
7 and –7
29)
Find y” for y = – 1
3x + 4 .
29)
A)
y” = – 2
(3x + 4)3
B)
y” = – 18
(3x + 4)3
C)
y” = – 6
(3x + 4)3
D)
y” =18
(3x + 4)3
Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the
function is concave up or concave down, and all inflection points.
30)
f(x) =x4 ex, –< x <
30)
A)
f is increasing on (–, –4] and [0, )
and decreasing on [–4, 0]. f is concave
up on (–, –6) and (–2, ) and concave
down on (–6, –2). f has inflection points
at x = –6 and x = –2.
12
B)
f is increasing on (–, –4] and [0, )
and decreasing on [–4, 0]. f is concave
up on (– 2, ) and concave down on
(–, –2). f has an inflection point at x = –2.
C)
f is increasing on [–4, 0] and decreasing
on (–, –4] and [0, ). f is concave up on
(–6, –2) and concave down on (–, –6) and
(–2, ). f has inflection points at x = –6
and x = –2.
D)
f is increasing on [0, 4] and decreasing
on (–, 0] and [4, ). f is concave up on
(–, 2) and (6, ) and concave down on
(2, 6). f has inflection points at x = 2
and x = 6.
13
Provide an appropriate response.
31)
Find the inflection point(s) for f(x) =x3– 6x – 1.
31)
A)
(1, –1)
B)
(0, –6)
C)
(–1, 6)
D)
(0, –1)
Solve the problem.
32)
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.
32)
A)
6°C
B)
19°C
C)
5°C
D)
20°C
Find the limit, if it exists.
33)
lim
x
3x –2x2+4x3
7– 2x –x3
33)
A)
–4
B)
3
2
C)
4
D)
Provide an appropriate response.
34)
Find vertical asymptotes for f(x) =7x – 2
x2– 3x – 4.
34)
A)
x = 1, x = –4
B)
x = –1, x = 4
C)
x = –1, x = –4
D)
x = 1, x = 4
35)
f(x) =3x2+ 30x
35)
A)
Min: (-5, -75)
No inflection points
B)
Min: (10, -30)
No inflection points
C)
Min: (5, -75)
No inflection points
D)
Min: (-10, -30)
No inflection points
15
Find the domain and intercepts.
36)
f(x) =3x +6
36)
A)
Domain: ( , ); y intercept: –2; x intercept: 6
B)
Domain: (–2, ); y intercept: 6; x intercept: –2
C)
Domain: ( , ); y intercept: 6; x intercept: –2
D)
Domain: ( , ); y intercept: 6; x intercept: 2
Sketch the graph and show all local extrema and inflection points.
37)
f(x) =6x
x2+ 9
37)
A)
Local minimum: (-3, –1
2)
Local maximum: (3, 1
2)
Inflection point: (0, 0)
B)
Local minimum: (-3, -1)
Local maximum: (3, 1)
Inflection points: (0, 0), (-3 3, –3
23),
(3 3, 3
23)
16
C)
Local minimum: (3, -1)
Local maximum: (-3, 1)
Inflection point: (0, 0)
D)
Maximum: (0, 2
3)
No inflection point
Solve the problem.
38)
A local office supply store has an annual demand for 10,000 cases of photocopier paper per year. It
costs $3 per year to store a case of photocopier paper, and it costs $40 to place an order. Find the
optimum number of cases of photocopier paper per order.
38)
A)
516 cases
B)
266,667 cases
C)
163 cases
D)
365 cases
Provide an appropriate response.
39)
Find the absolute minimum value of f(x) = 5 + 4x +16
x for x > 0.
39)
A)
min f(x) = f(2) = 13
B)
min f(x) = f(1) = 25
C)
min f(x) = f(2) = 21
D)
min f(x) = f(0) = 5
Sketch a graph of the function.
17
40)
f(x) = 2x3+ 15x2+ 24x
40)
A)
B)
C)
D)
18
Solve the problem.
41)
Because of material shortages, it is increasingly expensive to produce 6.2L diesel engines. In fact,
the profit in millions of dollars from producing x hundred thousand engines is approximated by
P(x) = –x3+ 30x2+ 10x – 52, where 0
x
20. Find the inflection point of this function to determine
the point of diminishing returns.
41)
A)
(10, 1958)
B)
(10, 2048)
C)
(7.50, 2048)
D)
(10, 114.67)
Decide if the given value of x is a critical number for f, and if so, decide whether the point for x on f is a local minimum,
local maximum, or neither.
42)
f(x) =(x +4)4; x = – 4
42)
A)
Not a critical number.
B)
Critical number but not an extreme point.
C)
Critical number; minimum at (–4 , 0)
D)
Critical number; maximum at (–4 , 0)
C
Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
43)
Concave upward
43)
A)
(–, –2); no inflection points
B)
(–, ); x = –2
C)
(–, ); no inflection points
D)
(–2, ); no inflection points
C
B
Find the domain and intercepts.
44)
f(x) =x +36
44)
A)
Domain: [–36, ); y intercept: –36; x intercept: 6
B)
Domain: [–36, ); y intercept: 6; x intercept: –36
C)
Domain: (–36, ); y intercept: 6; x intercept: –36
D)
Domain: ( , ); y intercept: 6; x intercept: –36
Provide an appropriate response.
45)
Find f”(x) for f(x) = 5x4– 6x2+ 7.
45)
A)
f”(x) = 60x2– 12x
B)
f”(x) = 20x2– 12x
C)
f”(x) = 20x2– 12
D)
f”(x) = 60x2– 12
Sketch a graph of the function.
46)
f(x) =x4– 2x2+5
46)
A)
B)
20
C)
D)
Find the limit, if it exists.
47)
Find lim
x +
x2
ex .
47)
A)
ex
B)
–
C)
D)
0
Provide an appropriate response.
48)
Find two numbers whose sum is 500 and whose product is a maximum.
48)
A)
250 and 250
B)
10 and 490
C)
1 and 499
D)
249 and 251
Solve the problem.
49)
A company wishes to manufacture a box with a volume of 24 cubic feet that is open on top and is
twice as long as it is wide. Find the width of the box that can be produced using the minimum
amount of material. Round to the nearest tenth, if necessary.
49)
A)
2.6 ft
B)
6.4 ft
C)
5.2 ft
D)
3.2 ft
Provide an appropriate response.
50)
Find f”(x) for f(x) =(4x + 5)3 .
50)
A)
f”(x) = 12x + 15
B)
f”(x) = 384x + 480
C)
f”(x) = 24x + 30
D)
f”(x) = 4x + 5
Sketch a graph of the function.
51)
f(x) =4x + 1
x
51)
A)
B)
C)
D)
Provide an appropriate response.
52)
Find the critical values and determine the intervals where f(x) is increasing and f(x) is decreasing if
f(x) = 1 +3
x+2
x2.
52)
A)
increasing on (–4, 0); decreasing on ( , –4) and (0, )
B)
decreasing on –4
3, 0 ; increasing on , –4
3and (0, )
C)
decreasing on (–4, 0); increasing on ( , –4) and (0, )
D)
increasing on –4
3, 0 ; decreasing on , –4
3and (0, )
53)
Given f(x) = x +16
x, x < 0, find the values of x corresponding to local maxima and local minima.
53)
A)
local minimum at x = –4 (no local maximum)
B)
local maximum at x = –4 (no local minimum)
C)
local maximum at x = –4, local minimum at x = 4
D)
no local maximum or minimum
Sketch a graph of the function.
54)
f(x) =4x2+ 24x
54)
23
A)
B)
C)
D)
Answer:
C
Explanation:
A)
B)
C)
D)
Provide an appropriate response.
55)
Determine the intervals for which the function f(x) =x3+18x2+ 2, is decreasing.
55)
A)
(0, 12) and (12, )
B)
( , –12) and (0, )
C)
( , –12) and (–12, 0)
D)
(–12, 0)
56)
Use the first derivative test to determine the local extrema, if any, for the function:
f(x) =x3+3x2– 24x + 6
56)
A)
local max at x = 2 and local min at – 4
B)
local max at x = – 4
C)
local min at x = 2
D)
local max at x = – 4 and local min at x = 2
Answer:
D
Explanation:
A)
B)
C)
D)
Answer:
D
Explanation:
A)
B)
C)
D)
57)
Find the critical values and determine the intervals where f(x) is decreasing for f(x) =3(x – 4)2/3 + 6.
57)
A)
f(x) is decreasing on ( , 4); increasing on (4, )
B)
f(x) is decreasing on ( , 6); increasing on (6, )
C)
f(x) is increasing on ( , 4); decreasing on (4, )
D)
f(x) is decreasing on ( , –4); increasing on (–4, )
58)
Find the absolute maximum value of f(x) =x4
ex for x > 0. Round your answer to three decimal
places.
58)
A)
4 at x = 4.689
B)
4.689 at x = 0
C)
0.7439 at x = 4
D)
4.689 at x = 4
Solve the problem.
59)
Suppose that the total–cost function for a certain company to produce x units of a product is given
by C(x) =4x2+40. Graph the average cost function A(x) = C(x)/x.
59)
A)
B)
25
C)
D)
Provide an appropriate response.
60)
Find y” for y =5x2+ 4 .
60)
A)
y” =20
(5x2+ 4)3/2
B)
y” = – 25x2
(5x2+ 4)1/2
C)
y” = – 1
4(5x2+ 4)3/2
D)
y” = – 25x2
(5x2+ 4)3/2
61)
Find the absolute minimum value of f(x) = x +9
x on (0, ).
61)
A)
Absolute maximum is 3 at x = 6.
B)
Absolute minimum is 6 at x = 3.
C)
Absolute maximum is 6 at x = 3.
D)
Absolute minimum is 3 at x = 6.
26
62)
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.
62)
A)
x = 2
B)
x = 2 and x = –2
C)
x = –2
D)
neither x = 2 nor x = –2 correspond to a local minimum
63)
Find horizontal asymptotes, if any, for f(x) =2x2– 2
4x3– 3.
63)
A)
y =1
2
B)
y = 0
C)
y = – 2
3
D)
y =2
3
B
Solve the problem.
64)
The Olympic flame at the 1992 Summer Olympics was lit by a flaming arrow. As the arrow moved
d feet horizontally from the archer, assume that its height h(d), in feet, was approximated by the
function h(d) = –0.002d2+ 0.7d + 6.9. Find the relative maximum of the function.
64)
A)
(350, 129.4)
B)
(0, 6.9)
C)
(175, 68.15)
D)
(175, 61.25)
C
Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
65)
Concave downward
65)
A)
(–5, 5); no inflection points
B)
(–5, 2); x = 0
C)
(–, –2); x = –2
D)
(–2, ); x = 2
C
A
Provide an appropriate response.
66)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) =8
x2+ 1
66)
A)
No relative extrema
B)
Relative maximum: (0, –8)
C)
Relative maximum: (–1, 8)
D)
Relative maximum: (0, 8)
Use the given graph of f(x) to find the intervals on which f(x) > 0.
67)
67)
A)
f(x) > 0 on (–, 4], f(x) < 0 on [4, )
B)
f(x) > 0 on [-4, 4], f(x) < 0 on (–, -4]
[4, )
C)
f(x) > 0 on (–, -4] [4, ), f(x) < 0 on [-4, 4]
D)
f(x) > 0 on [-16, 16], f(x) < 0 on (–, -16] [16, )
Find the limit, if it exists.
68)
Find lim
x
3x + 4
4x2– 3.
68)
A)
–4
3
B)
4
3
C)
3
4
D)
0
69)
lim
x–
3x3+3x2
7x2– x
69)
A)
3
7
B)
–3
C)
0
D)
–
Sketch the graph and show all local extrema and inflection points.
70)
f(x) =1
16 – x2
70)
A)
Local max: (0, 1)
No inflection point
B)
Local min: (0, 1
4)
No inflection point
29
C)
Local max: (0, 1
4)
No inflection point
D)
Local min: (0, 1)
No inflection point
Solve the problem.
71)
A 60 room hotel is filled to capacity every night at a rate of $40 per room. The management wants
to determine if a rate increase would increase their profit. They are not interested in a rate decrease.
Suppose management determines that for each $2 increase in the nightly rate, five fewer rooms will
be rented. If each rented room costs $8 a day to service, how much should the management charge
per room to maximize profit?
71)
A)
$45
B)
$50
C)
$42
D)
The management should leave the rate as it is.
Sketch a graph of the function.
72)
f(x) =12x –x3
72)
30
A)
B)
C)
D)
Find the intervals where f”(x) < 0 ir f”(x) > 0 as indicated.
73)
f”(x) > 0
73)
A)
(0, 3)
B)
(0, )
C)
(–3, 3)
D)
(–3, )
Provide an appropriate response.
74)
Use the first derivative test to determine the local extrema, if any, for the function:
f(x) =3x4–6x2+ 7.
74)
A)
local max at x = 1 and local min at x = 0
B)
local max at x = 0 and local min at x = –1 and x = 1
C)
local min at x = 0 and local max at x = –1 and x = 1
D)
local max at x = –1 and local min at x = 0 and x = 1
75)
Use a graphing utility to approximate the intervals where f(x) is decreasing and intervals where
f(x) is increasing for the function f(x) =x4–3x3–2x2+ 5x. Round your answer to two decimal
places.
75)
A)
increasing on ( , –0.82) and (0.62, 2.45); decreasing on (–0.82, 0.62) and (2.45, )
B)
decreasing on ( , –0.82); increasing on (–0.82, 0.62)
C)
increasing on ( , –0.82); decreasing on (–0.82, 0.62)
D)
decreasing on ( , –0.82) and (0.62, 2.45); increasing on (–0.82, 0.62) and (2.45, )
Solve the problem.
76)
A company manufactures and sells x pocket calculators per week. If the weekly cost and demand
equations are given by:
C(x) = 8,000 + 5x
p = 14 –x
4,000,0
x
25,000
Find the production level that maximizes profit.
76)
A)
14,000 pocket calculators per week
B)
8000 pocket calculators per week
C)
18,000 pocket calculators per week
D)
2000 pocket calculators per week
Find the limit, if it exists.
77)
Find lim
x +
ln x
x .
77)
A)
1
B)
–
C)
0
D)
Provide an appropriate response.
78)
Use the first derivative test to determine the local extrema, if any, for the function:
f(x) =3(x – 4)2/3 + 6.
78)
A)
f(x) has a local maximum at x = 4.
B)
f(x) has a local minimum at 6
C)
f(x) has a local minimum at x = 4.
D)
f(x) has no local extrema
Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
79)
Concave upward
79)
A)
(–3, ); x = 0
B)
(0, ); x = 0
C)
(0, ); no inflection points
D)
(–3, 3); x = 0
B
D)
Provide an appropriate response.
80)
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 maximum.
80)
A)
x = 2 and x = – 2
B)
x = – 2
C)
x = 2
D)
x = 0 and x = 2
B
D)
33
C
D)
Solve the problem.
81)
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?
81)
A)
50 < t < 60
B)
t > 40
C)
40 < t < 50
D)
40 < t < 60
Provide an appropriate response.
82)
Find the critical values and determine the intervals where f(x) is increasing and the intervals where
f(x) is decreasing for the function f(x) =x3+3x2– 24x + 6.
82)
A)
decreasing on ( , –4) and (2, ); increasing on (–4, 2)
B)
increasing on ( , –4) and (2, ); decreasing on (–4, 2)
C)
increasing on ( , –4) and (2, ); decreasing on (–4, )
D)
increasing on ( , –4); decreasing on (–4, 2)
Solve the problem.
83)
Find the approximate number of batches (to the nearest whole number) of an item that should be
produced annually if 240,000 units are to be made. It costs $1 to store a unit for one year, and it
costs $580 to set up the factory to produce each batch.
83)
A)
16 batches
B)
12 batches
C)
14 batches
D)
10 batches
Provide an appropriate response.
84)
Determine the interval(s) over which f(x) = (x +3)3 is concave upward.
84)
A)
(–, )
B)
(–, –3)
C)
(–3, )
D)
(–, 3)
Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the
function is concave up or concave down, and all inflection points.
34
85)
f(x) =x2
x2+16, –< x <
85)
A)
f is increasing on [0, ) and decreasing
on (–, 0]. f is concave up on –43, 43
and concave down on –, –43 and
43, . f has inflection points at
x = – 43 and x =43.
B)
f is increasing on (–, 0] and decreasing
on [0, ). f is concave up on –, –43
and 43, and concave down on
–43, 43. f has inflection points
at x = – 43 and x =43.
35
C)
f is increasing on [0, ) and decreasing
on (–, 0]. f is concave up on (–, ).
f has no inflection points.
D)
f is increasing on (–, 0] and decreasing
on [0, ). f is concave up on –, –43
and 43, and concave down on
–43, 43. f has inflection points
at x = – 43 and x =43.
Provide an appropriate response.
86)
Find the inflection point(s) for f(x) =x + 7.
86)
A)
(–6, 1)
B)
(–3, 2)
C)
(–7, 0)
D)
There are no points of inflection.
D
87)
Find f”(x) for f(x) = –7x9+ 5x2.
87)
A)
f”(x) = 504x7– 10
B)
f”(x) = –63x8+ 10x
C)
f”(x) = –504x7+ 10
D)
f”(x) = 504x8+ 10
C
36
A
88)
Find the absolute minimum value of f(x) =ex
x3 for x > 0. Round your answer to three decimal
places.
88)
A)
1 at x = 2.718
B)
3 at x = 0.7439
C)
0.7439 at x = 3
D)
2.718 at x = 1
Sketch the graph and show all local extrema and inflection points.
89)
f(x) = 2x3– 12x2+ 18x
89)
A)
Local min: (2, 10)
No inflection point
B)
No extrema
Inflection point: (0, 0)
37
C)
Local maximum: (0, 0)
Local minimum: (1, -1)
Inflection point: (0.5, -0.5)
D)
Local max: (1, 8), min: (3, 0)
Inflection point: 2,4
Find the limit, if it exists.
90)
Find: lim
x
5x2+ 3x – 1
6x2– x + 7
90)
A)
–1
7
B)
–5
6
C)
1
7
D)
5
6
Provide an appropriate response.
91)
Find the absolute maximum and absolute minimum values of the function f(x) =x4–6x2 on the
interval [0, 3].
91)
A)
This function has no absolute maximum or minimum on the given interval.
B)
Absolute maximum: f(3) = –27; absolute minimum: f( 3) = –9
C)
Absolute maximum: f(0) =0; absolute minimum: f(2) = –8
D)
Absolute maximum: f(3) = 27; absolute minimum: f( 3) = –9
Find the limit, if it exists.
92)
Find lim
x
3
x4– 81
x – 3 .
92)
A)
81
B)
108
C)
3
D)
0
Provide an appropriate response.
93)
Find all inflection points for f(x) =x4–10x3+24x2+ 3x + 5.
93)
A)
Inflection points at x = 1, x = 4
B)
Inflection points at x = 0, x = 1, x = 4
C)
Inflection points at x –0.06, x
2.43, x
5.13
D)
This function does not have any inflection points.
Solve the problem.
94)
A computer software company sells 20,000 copies of a certain computer game each year. It costs the
company $1.00 to store each copy of the game for one year. Each time it must produce additional
copies, it costs the company $625 to set up production. How many copies of the game should the
company produce during each production run in order to minimize its total storage and set–up
costs?
94)
A)
5000 copies in 4 production runs
B)
20,000 copies in 1 production run
C)
4000 copies in 5 production runs
D)
10,000 copies in 2 production runs
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)
1 unit
B)
12 units
C)
652 units
D)
5 units
Use the given graph of f(x) to find the intervals on which f(x) > 0.
96)
96)
A)
f(x) > 0 on [-2, ), f(x) < 0 on decreasing on (–, -2]
B)
f(x) > 0 on [-5, ), f(x) < 0 on (–, -5]
C)
f(x) < 0 on (–,)
D)
f(x) > 0 on (–, –5] [1, ), f(x) < 0 on [–5, 1]
97)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) = 5 –x2
97)
A)
Relative minima: (–5, 0); ( 5, 0)
B)
Relative maximum: (5, 5)
C)
Relative minimum: (0, 5)
D)
Relative maximum: (0, 5)
98)
The percent of concentration of a certain drug in the bloodstream x hr after the drug is
administered is given by K(x) =4x
x2+ 49 . How long after the drug has been administered is the
concentration a maximum? Round answer to the nearest tenth, if necessary.
98)
A)
7 hr
B)
4 hr
C)
2.5 hr
D)
4.9 hr
Provide an appropriate response.
99)
Find f”(x) for f(x) = 4x – 6.
99)
A)
f”(x) = 0
B)
f”(x) = 4
C)
f”(x) =4
x
D)
f”(x) = 4x3– 6x2
100)
Determine the interval(s) where f(x) =x2
x – 3 is decreasing.
100)
A)
( , 0) and (6, )
B)
(0, 3) and (6, )
C)
(0, 3) and (3, 6)
D)
(0, 6)
C
Solve the problem.
101)
The cost of manufacturing x electric woks in one day is given by C(x) = 2x3–16x2+ 4x. Find the
average cost per electric wok and the interval where the average cost per electric wok is decreasing.
101)
A)
C(x) = 6x2– 32x + 4; x < 4
B)
C(x) = 2x2– 16x + 4; 0 < x < 4
C)
C(x) = 6x2– 32x + 4; 0 < x < 4
D)
C(x) = 2x2– 32x + 4; 0 < x < 4
B
Find the intervals where f”(x) < 0 ir f”(x) > 0 as indicated.
102)
f”(x) < 0
102)
A)
(1, )
B)
(–1, 0) , (1, )
C)
(–1, 0)
D)
(–, –1)
B
A
Provide an appropriate response.
103)
Determine the interval(s) over which f(x) = (x –4)3 is concave downward.
103)
A)
(–, –4)
B)
(4, )
C)
(–, 4)
D)
(–4, )
Solve the problem.
104)
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 120 in. What dimensions will give a box with a square
end the largest possible volume?
104)
A)
20 in. ×20 in. ×100 in.
B)
40 in. ×40 in. ×40 in.
C)
20 in. ×40 in. ×40 in.
D)
20 in. ×20 in. ×40 in.
Length + Girth = 108 ; L + 4s = <a>
Solve for L: L = <a> – 4s
The Volume can be expressed as a function of s by:
V(x) =s2(<a> – 4s)
V'(x) = –4s2+ 2s(<a> – 4s) = –12s2+ <2a>s
Solve V'(x) = 0: s = <a
6>
L = <a> – 4(<a
6>) = <a
3>
The dimensions that give the largest possible volume would be <a
6> x <a
6> x <a
3>
Provide an appropriate response.
105)
The percent of concentration of a acid absorbed in a new manufacturing process after x hr after the
acid has been mixed is given by A(x) =4x
x2+ 49 How long after the acid has been added is the
concentration a maximum? Round answer to the nearest tenth, if necessary.
105)
A)
4.9 hr
B)
2.5 hr
C)
4 hr
D)
7 hr
106)
Find the absolute minimum value of f(x) = 4x +x2+ 2 on [0, ).
106)
A)
Absolute minimum is 2 at x = 2.
B)
Absolute minimum is 4 at x = 2.
C)
Absolute minimum is 2 at x = 0.
D)
Absolute minimum is 2 at x = 6.
43
Solve the problem.
107)
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.
107)
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 $10 when x = 100 cell phones.
D)
The minimum average cost is $875 when x = 75 cell phones.
Provide an appropriate response.
108)
Find the absolute maximum and minimum values of the function f(x) =4x
x2+ 1 on the interval
[–3, 0].
108)
A)
Absolute maximum is 0 at x = 0. Absolute minimum 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 = 0. Absolute maximum is – 2 at x = –1.
D)
Absolute minimum is 0 at x = –1. Absolute maximum is – 2 at x = 0.
109)
Find the absolute maximum and minimum values of f(x) = 9x3– 54x2+ 81x + 13 on the interval
[–6, 2].
109)
A)
max f(x) = f(–6) = –4361
min f(x) = f(1) = 49
B)
max f(x) = f(1) = 49
min f(x) = f(–6) = –4361
C)
max f(x) = f(1) = 4361
min f(x) = f(–6) = 49
D)
max f(x) = f(1) = 4361
min f(x) = f(–6) = –49
44
Solve the problem.
110)
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?
110)
A)
$1,820,000
B)
$1,520,000
C)
$1,720,000
D)
$1,620,000
Provide an appropriate response.
111)
Find the inflection point(s) for f(x) =1
4x4–x3+ 6.
111)
A)
(0, 6) and (2, 2)
B)
(0, 0) and (2, 2)
C)
(0, 6) and (2, –4)
D)
(0, 0)
112)
Find y” for y = 2 x3/2 – 6x1/2 .
112)
A)
y” =3
2x– 1/2 +3
2x– 3/2
B)
y” = 3x– 1/2 + 3x– 3/2
C)
y” =3
2x1/2 +3
2x– 1/2
D)
y” = 3x1/2 – 3x– 1/2
113)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) = 20x3– 3x5
113)
A)
Relative minimum: (–2 , –64)
Relative maximum: (2 , 64)
B)
Relative minimum: (–2, –64)
Relative maximum: (0, 0)
C)
Relative minimum: (–2 , –64)
Relative minimum: (0, 0)
Relative maximum: (2 , 64)
D)
Relative maximum: (0, 0)
Relative minimum: (2, 64)
Solve the problem.
114)
The annual revenue and cost functions for a manufacturer of grandfather clocks are approximately
R(x) =500x – 0.01x2 and C(x) =160x + 100,000, where x denotes the number of clocks made. What is
the maximum annual profit?
114)
A)
$3,090,000
B)
$2,890,000
C)
$2,790,000
D)
$2,990,000
115)
A bookstore has an annual demand for 21,000 copies of a best–selling book. It costs $0.90 to store
one copy for one year, and it costs $100 to place an order. Find the optimum number of copies per
order.
115)
A)
2160 copies
B)
1944 copies
C)
2277 copies
D)
2219 copies
Provide an appropriate response.
116)
Use a graphing utility to approximate where the local extrema of the function
f(x) =x4–3x3–2x2+ 5x are to two decimal places.
116)
A)
local max at x
0.62; local min at x –0.82 and x
2.45
B)
local min at x –0.62 and x
2.45
C)
local min at x
0.62; local max at x –0.82 and x
2.45
D)
local max at x
0.82
Answer Key
Testname: C4
47
Answer Key
Testname: C4
Answer Key
Testname: C4
Answer Key
Testname: C4