A)
Local max: (–3, 0), min: (–1, –8)
Inflection point: – 2,– 4
B)
No extrema
Inflection point: (0, 0)
C)
Local maximum: (0, 0)
Local minimum: (–7, 343)
Inflection point: (–3.5, 171.5)
D)
Local min: (2, 10)
No inflection point
Answer:
A
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.
57)
f(x) =(x +3)4; x = – 3
A)
Critical number; minimum at (–3 , 0)
B)
Critical number but not an extreme point.
C)
Not a critical number.
D)
Critical number; maximum at (–3 , 0)
Answer:
A
18
Sketch the graph and show all local extrema and inflection points.
58)
f(x) =8x
x2+ 1
A)
Local minimum: (1, –4)
Local maximum: (–1, 4)
Inflection point: (0, 0)
B)
Local minimum: (–1, –4)
Local maximum: (1, 4)
Inflection points: (0, 0), (–1 3, –2 3),
(1 3, 2 3)
C)
Maximum: (0, 8)
No inflection point
D)
Local minimum: (–1, –2)
Local maximum: (1, 2)
Inflection point: (0, 0)
Answer:
B
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
59)
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.
Answer:
The rate of change of backpack production is increasing when hiring between 0 and 8 new workers and
decreasing when hiring between 8 and 12 new workers. The point of diminishing returns is 8 new
workers and the maximum rate of change is 1024 backpacks.
19
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
60)
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?
A)
B)
C)
40 < t < 50
D)
Answer:
C
61)
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.
A)
B)
C)
(10, 114.67)
D)
Answer:
B
Find the limit, if it exists.
62)
lim
x
3x –4x2+9x3
5– 2x –x3
A)
B)
C)
D)
Answer:
A
63)
lim
x–
2x3+3x2
7x2– x
A)
B)
C)
–2
D)
Answer:
D
64)
Find: lim
x
5x2+ 3x – 1
6x2– x + 7
A)
B)
C)
5
6
D)
Answer:
C
65)
Find lim
x
3
x4– 81
x – 3 .
A)
B)
C)
0
D)
Answer:
B
66)
Find lim
x –2
x2+ x – 2
x2– 4 .
A)
B)
C)
2
D)
Answer:
B
20
67)
Find lim
x
3x + 4
4x2– 3 .
A)
B)
C)
–4
3
D)
Answer:
D
68)
Find lim
x
+
x2
ex .
A)
B)
C)
–
D)
Answer:
B
69)
Find lim
x
+
ln x
x .
A)
B)
C)
D)
Answer:
D
Provide an appropriate response.
70)
Find horizontal asymptotes, if any, for f(x) =2x2– 2
4x3– 3 .
A)
B)
C)
y =1
2
D)
Answer:
D
71)
Find vertical asymptotes for f(x) =7x – 2
x2– 3x – 4 .
A)
B)
C)
x = 1, x = – 4
D)
Answer:
B
Sketch a graph of the function.
72)
f(x) =4x + 1
x
21
A)
B)
C)
D)
Answer:
D
22
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Sketch a graph of a single function that has these properties.
73)

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)
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the domain and intercepts.
74)
f(x) =x +9
A)
Domain: [–9,
); y intercept: –9; x intercept: 3
B)
Domain: ( ,
); y intercept: 3; x intercept: –9
C)
Domain: [–9,
); y intercept: 3; x intercept: –9
D)
Domain: (–9,
); y intercept: 3; x intercept: –9
Answer:
C
75)
f(x) =2x +4
A)
Domain: ( ,
); y intercept: –2; x intercept: 4
B)
Domain: ( ,
); y intercept: 4; x intercept: 2
C)
Domain: (–2,
); y intercept: 4; x intercept: –2
D)
Domain: ( ,
); y intercept: 4; x intercept: –2
Answer:
D
76)
f(x) =6x
x – 7
A)
Domain: ( , 7); y intercept: 0; x intercept: 0
B)
Domain: All real numbers except 7; y intercept: 0; x intercept: 0
C)
Domain: All real numbers except –7; y intercept: 0; x intercept: 0
D)
Domain: All real numbers except 7; y intercept: 0; no x intercept
Answer:
B
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.
23
77)
f(x) =x4 ex, –< x <
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.
B)
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.
C)
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.
D)
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.
Answer:
A
24
78)
f(x) =x2
x2+1, –< x <
A)
f is increasing on (–
, 0] and decreasing
on [0,
). f is concave up on –
, –1
3
and 1
3, and concave down on
–1
3, 1
3. f has inflection points
at x = – 1
3 and x =1
3.
B)
f is increasing on (–
, 0] and decreasing
on [0,
). f is concave up on –
, –1
3
and 1
3, and concave down on
–1
3, 1
3. f has inflection points
at x = – 1
3 and x =1
3.
25
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 –1
3, 1
3
and concave down on –
, –1
3 and
1
3, . f has inflection points at
x = – 1
3 and x =1
3.
Answer:
D
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
79)
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.
Answer:
f(x) is increasing on (–
, –3), decreasing on (–3, 0) (0,
); concave up on (–2, 0), concave down on
(–
, –2) (0,
).Local max is at 8.75 at x = – 3.
26
80)
Sketch the graph of f(x) =3x2+ 2x + 5
6x2+ 2 . Include sketch of all asymptotes.
Answer:
81)
Sketch the graph of f(x) =2x2+ 5x – 3
x2– 9 . Include sketch of all asymptotes.
Answer:
82)
Sketch the graph of f(x) = x +3
x2. Include sketch of all asymptotes.
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
83)
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.
A)
The minimum average cost is $875 when x = 75 cell phones.
B)
The minimum average cost is $74 when x = 20 cell phones.
C)
The minimum average cost is $75 when x = 875 cell phones.
D)
The minimum average cost is $10 when x = 100 cell phones.
Answer:
D
27
84)
Suppose that the total–cost function for a certain company to produce x units of a product is given by
C(x) =3x2+35. Graph the average cost function A(x) = C(x)/x.
A)
B)
C)
D)
Answer:
C
28
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.)
85)
f’ f”
A)
B)
C)
D)
Answer:
A
Provide an appropriate response.
86)
Find the absolute minimum value of f(x) =ex
x3 for x > 0. Round your answer to three decimal places.
A)
B)
C)
1 at x = 2.718
D)
Answer:
D
29
87)
Find the absolute maximum value of f(x) =x4
ex for x > 0. Round your answer to three decimal places.
A)
B)
C)
0.7439 at x = 4
D)
Answer:
A
88)
Find the absolute maximum and minimum values of f(x) = 9x3– 54x2+ 81x + 13 on the interval [–6, 2].
A)
max f(x) = f(1) = 4361
min f(x) = f(–6) = – 49
B)
max f(x) = f(–6) = – 4361
min f(x) = f(1) = 49
C)
max f(x) = f(1) = 4361
min f(x) = f(–6) = 49
D)
max f(x) = f(1) = 49
min f(x) = f(–6) = – 4361
Answer:
D
89)
Find the absolute maximum and minimum values of the function f(x) =4x
x2+ 1 on the interval [–3, 0].
A)
Absolute maximum is 0 at x = 0. Absolute minimum is –2 at x = – 1.
B)
Absolute minimum is 0 at x = 0. Absolute maximum is – 2 at x = – 1.
C)
Absolute maximum is 0 at x = 0. Absolute minimum is 2 at x = – 1.
D)
Absolute minimum is 0 at x = – 1. Absolute maximum is – 2 at x = 0.
Answer:
A
90)
Find the absolute minimum value of f(x) = 5 + 4x +16
x for x > 0.
A)
B)
C)
min f(x) = f(0) = 5
D)
Answer:
A
91)
Find the absolute maximum and absolute minimum values of the function f(x) =x4–6x2 on the interval [0, 3].
A)
Absolute maximum: f(3) = – 27; absolute minimum: f( 3) = – 9
B)
Absolute maximum: f(0) =0; absolute minimum: f(2) = – 8
C)
This function has no absolute maximum or minimum on the given interval.
D)
Absolute maximum: f(3) = 27; absolute minimum: f( 3) = – 9
Answer:
D
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
92)
Find the absolute minimum value of f(x) = 4x ln x – 7x. Round your answer to three decimal places.
Answer:
–8.468 at e3/4
93)
Find the absolute maximum value of f(x) = 3x – ln x . Round your answer to four decimal places.
Answer:
No absolute maximum
30
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
94)
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?
A)
B)
C)
24 min
D)
Answer:
C
95)
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.
A)
B)
C)
2.5 hr
D)
Answer:
D
96)
Find the absolute minimum value of f(x) = x +9
x on (0,
).
A)
Absolute minimum is 6 at x = 3.
B)
Absolute maximum is 3 at x = 6.
C)
Absolute minimum is 3 at x = 6.
D)
Absolute maximum is 6 at x = 3.
Answer:
A
97)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) =8
x2+ 1
A)
Relative maximum: (0, 8)
B)
No relative extrema
C)
Relative maximum: (–1, 8)
D)
Relative maximum: (0, –8)
Answer:
A
98)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) = 20x3– 3x5
A)
Relative minimum: (–2 , –64)
Relative minimum: (0, 0)
Relative maximum: (2 , 64)
B)
Relative minimum: (–2 , –64)
Relative maximum: (2 , 64)
C)
Relative minimum: (–2, –64)
Relative maximum: (0, 0)
D)
Relative maximum: (0, 0)
Relative minimum: (2, 64)
Answer:
B
99)
Find the absolute minimum value of f(x) = 4x +x2+ 2 on [0,
).
A)
Absolute minimum is 2 at x = 6.
B)
Absolute minimum is 2 at x = 0.
C)
Absolute minimum is 2 at x = 2.
D)
Absolute minimum is 4 at x = 2.
Answer:
B
100)
Find the relative extrema of the function. List your answer(s) in terms of ordered pair(s).
f(x) = 5 –x2
A)
Relative maximum: (0, 5)
B)
Relative minima: (–5, 0); ( 5, 0)
C)
Relative maximum: (5, 5)
D)
Relative minimum: (0, 5)
Answer:
A
31
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
101)
Suppose f is a continuous function. Describe the graph of f at (1, f(1)) if f'(1) = 0 and f”(x) < 0.
Answer:
The graph of f at (1, f(1)) is a local maximum.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
102)
Find two numbers whose sum is 310 and whose product is a maximum.
A)
B)
C)
10 and 300
D)
Answer:
D
103)
Find two numbers whose difference is 4 and whose product is a minimum.
A)
B)
C)
1 and 5
D)
Answer:
B
Solve the problem.
104)
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?
A)
44 ft by 396ft; 17,424 ft2
B)
220 ft by 220 ft; 48,400 ft2
C)
110 ft by 330 ft; 36,300 ft2
D)
110 ft by 110 ft; 12,100 ft2
Answer:
D
105)
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 90 in. What dimensions will give a box with a square end the largest possible
volume?
A)
15 in. ×30 in. ×30 in.
B)
30 in. ×30 in. ×30 in.
C)
15 in. ×15 in. ×30 in.
D)
15 in. ×15 in. ×75 in.
Answer:
C
32
106)
A company wishes to manufacture a box with a volume of 44 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.
A)
B)
C)
6.6 ft
D)
Answer:
B
107)
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?
A)
$50
B)
$45
C)
$42
D)
The management should leave the rate as it is.
Answer:
D
108)
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.
A)
2000 pocket calculators per week
B)
18,000 pocket calculators per week
C)
8000 pocket calculators per week
D)
14,000 pocket calculators per week
Answer:
B
109)
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?
A)
B)
C)
$1,720,000
D)
Answer:
B
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
110)
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.
Answer:
a) P(x) =x2
500 + 20x – 5000
b) $45,000
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
111)
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?
A)
B)
C)
652 units
D)
Answer:
B
33
112)
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?
A)
20,000 copies in 1 production run
B)
4000 copies in 5 production runs
C)
5000 copies in 4 production runs
D)
10,000 copies in 2 production runs
Answer:
C
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
113)
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.)
Answer:
The company will minimize its costs by making 5000 caps five times during the year.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
114)
Find the approximate number of batches (to the nearest whole number) of an item that should be produced
annually if 300,000 units are to be made. It costs $3 to store a unit for one year, and it costs $580 to set up the
factory to produce each batch.
A)
B)
C)
30 batches
D)
Answer:
D
115)
A bookstore has an annual demand for 79,000 copies of a best–selling book. It costs $0.20 to store one copy for
one year, and it costs $55 to place an order. Find the optimum number of copies per order.
A)
B)
C)
5933 copies
D)
Answer:
D
116)
A local office supply store has an annual demand for 40,000 cases of photocopier paper per year. It costs $4 per
year to store a case of photocopier paper, and it costs $50 to place an order. Find the optimum number of cases
of photocopier paper per order.
A)
B)
C)
1000 cases
D)
Answer:
C
34