Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
The demand equation for a monopolist’s product is p=500
q, where p is the price per unit (in
dollars) for q units. If the total cost c (in dollars) of producing q units is given by c = 5q+ 2000, then
the level of production at which profit is maximized is
1)
A)
50 units.
B)
100 units.
C)
750 units.
D)
1235 units.
E)
2500 units.
2)
The function f(x) =x2– 1
x is concave up on the interval(s)
2)
A)
(1, ).
B)
(–, 0).
C)
(0, ).
D)
(–, –1).
E)
(–, 0) and (0, 1).
3)
On the interval 0, 2 , the function y=3x4–4x3 has
3)
A)
an absolute max. at x= 0 and an absolute min. at x= 1.
B)
an absolute min. at x= 1 and no absolute max.
C)
an absolute max. at x= 2 and an absolute min. at x= 1.
D)
an absolute max. at x= 2 and an absolute min. at x= 0.
E)
no absolute max. and no absolute min.
E
4)
You have been asked to design a rectangular box with a square base and lid. The volume of the box
must be 12 ft3. The cost per square foot of material for the base is $0.50, for the sides $0.20, and for
the lid $0.10. If the total cost of materials is a minimum, then the dimensions (in feet) of the box are
4)
A)
1 ×1 × 12.
B)
2×2× 6.
C)
4 × 4 ×3
4.
D)
2 ×2 × 3.
E)
3 × 3 ×4
3.
5)
If f(x) =x3+ 2x2– 4x+ 10, then f is
5)
A)
increasing on 2
3, , concave down on –, –2
3, and has a relative maximum when x=2
3.
B)
increasing on –2, 2
3, concave up on (–, ), and has a relative minimum when x= – 2.
C)
increasing on –2, –2
3, concave down on 2
3, , and has a relative minimum when x= – 2.
D)
decreasing on –2, 2
3, concave up on –2
3, , and has a relative minimum when x= – 2
3.
E)
decreasing on (–2, ), concave down on –, 2
3, and has a relative maximum when x= – 2.
6)
On the interval 0, 2 , the function y=x3+ 3x2– 9x+ 27 has an absolute maximum when x=
6)
A)
0.
B)
2.
C)
1
2.
D)
1.
E)
none of the above
2
7)
The function y=x5
20 +x4
12 +x– 3 has how may inflection points?
7)
A)
none
B)
one
C)
two
D)
three
E)
four
8)
How many critical values does the function f(x) = 3x4+ 4x3 have?
8)
A)
none
B)
one
C)
two
D)
three
E)
four
9)
The function f(x) =x– 2
x+ 3 is concave down on the interval(s)
9)
A)
(–3, ).
B)
(–3, 0).
C)
(–, –3) and (2, ).
D)
(–, –3).
E)
(–, ).
10)
If f(x) =x3– 7x2+ 2x– 5, then f is concave down on the interval
10)
A)
–, 2
3.
B)
(–, ).
C)
–, 7
3.
D)
7
3, .
E)
2
3, .
3
B
11)
If f(x) =x4– 6x2+ 3, then f has an inflection point when x=
11)
A)
1.
B)
0.
C)
2.
D)
3.
E)
3.
12)
An equation of a horizontal asymptote for the graph of y=3x2– 2
x2– 4 is
12)
A)
y= 0.
B)
y= 2.
C)
y= 3.
D)
x= 3.
E)
x= 2.
13)
The function y=x4– 8x2 has a relative maximum when x=
13)
A)
0.
B)
–2.
C)
1.
D)
–1.
E)
2.
14)
A manufacturer has determined that the total cost c of producing q units of a product is given by
c= 0.04q2+ 4q+ 6400. Average cost will be a minimum at a production level of
14)
A)
100 units.
B)
200 units.
C)
400 units.
D)
800 units.
E)
none of the above
15)
The function f(x) =x2– 6x+ 8 is decreasing on
15)
A)
(–, 3).
B)
(2, 4).
C)
(–, 2) and (4, ).
D)
(3, ).
E)
(–3, 3).
16)
On the interval –1, 1 , the function y= 4 +x2–x3 has an absolute maximum when x=
16)
A)
0.
B)
–1.
C)
1
2.
D)
1.
E)
–1
2.
17)
A rectangular plot adjacent to a stream is to be fenced in by using the stream as one side of the
enclosed area. If 2000 ft of fencing are to be used, find the maximum area that can be enclosed.
Assume the answer is in square feet.
17)
A)
400,000
B)
500,000
C)
600,000
D)
700,000
E)
800,000
18)
If f(x) =x3+ 3x2– 24x+ 8, then f is
18)
A)
decreasing on (–, 4), concave up on (–1, ), and has no relative minimum point.
B)
decreasing on (–4, 2), concave down on (–, –1), and has a relative maximum when x= – 4.
C)
increasing on (–4, 2), concave down on (–, ), and has a relative maximum when x= 2.
D)
decreasing on (1, 2), concave up (0, ), and has a relative minimum when x= – 4.
E)
increasing on (2, ), concave up on (–, –1), and has a relative minimum when x= 2.
5
A
19)
The function f(x) = 4x3– 10x2– 8x+ 3 is decreasing on
19)
A)
(–, 2).
B)
(2, )
C)
–, 1
3.
D)
–1
3, .
E)
–1
3, 2 .
20)
If f(x) =ex (x+ 4), then f has an inflection point when x=
20)
A)
–5.
B)
0.
C)
–1.
D)
–6.
E)
–3.
21)
An equation of a vertical asymptote for the graph of y=7x2– 4
2x+ 3 is
21)
A)
x= 0.
B)
x= 4.
C)
x= – 3
2.
D)
y=7
2.
E)
y= – 4
3.
22)
An equation of a vertical asymptote for the graph of y=2x
4x– 1 is
22)
A)
x=1
4.
B)
y=1
4.
C)
y=1
2.
D)
y= 0.
E)
x=1
2.
23)
The function y=x3+ 15x2– 33x has a relative maximum when x=
23)
A)
1.
B)
11
C)
–11.
D)
–1.
E)
0.
6
24)
An equation of a horizontal asymptote for the graph of y=2x
9x2– 1 is
24)
A)
x=2
9.
B)
x=1
3.
C)
y= 0.
D)
y=1
3.
E)
y=2
9.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
25)
Use the second derivative test to find the relative extremas of f(x) = 4x3– 6x and where
they occur. If the relative extremas can not be determined by the second derivative test,
state so.
25)
26)
Let y=x3– 3x2– 9x+ 10.
(a) Determine y’ and y”.
(b) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(c) Determine the coordinates of all relative maximum and relative minimum points
(d) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down;
(e) Determine the coordinates of all inflection points.
(f) With the aid of the information obtained in parts (a)–(e), give a reasonable sketch of
the curve.
26)
7
27)
A new book has its monthly revenue given by R(x) =40x
x2+ 25 where x is the number of
months after its release and R is in thousands of dollars. Find the relative extrema and use
this information to determine which month will bring the greatest revenue.
27)
28)
If C= – 4x+x3 is a cost function, sketch the graph of this function with the aid of intercepts,
symmetry, and the first–derivative test.
28)
29)
The graph of a function is given. Find
(a) the open intervals on which the function is increasing or decreasing;
(b) the coordinates of all relative extrema.
29)
30)
Find two positive numbers whose product is 25 and whose sum is a minimum.
30)
31)
The cost equation for a hot dog stand is given by C(x) = 2x3– 21x2+ 60x+ 500, where x is
the number of hot dogs sold, and C(x) is the cost in dollars. Determine where the graph of
the equation is concave up and where it is concave down and find any inflection points.
31)
32)
Let f(x) =x5+ 5x3. Determine the intervals of which f is (a) concave up and (b) concave
down. (c) Find the x–values of all inflection points.
32)
33)
A drug is injected into a patient’s bloodstream. The concentration of the drug in the
bloodstream t hours after the injection is approximated by C(t) =200t
t2+ 6t+ 9 . Find the
relative extrema and use this to determine when the drug is at its greatest concentration.
33)
34)
The cost equation for a bakery is given by C(x) = 10(x–2)2– 120(x– 2) + 450, where x is
the number of doughnuts made (in dozens), and C(x) is the cost in dollars. Use the
first–derivative test to find when relative extrema occur.
34)
35)
Suppose that the total number of units produced by a worker in t hours of an 8–hour shift
can be modeled by the production function P(t) = 4t+ 24t2– 2t3. Find where the function is
concave up and where the function is concave down.
35)
36)
Find all the critical values of f(x) =x4– 8x2+ 3.
36)
37)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=2x– 3
x+ 5 .
37)
9
38)
A released film has a revenue given by R(t) =20t
t+ 3 , where R(t) is in millions of dollars and
t is the number of weeks after its release. Sketch the graph of this function.
38)
39)
If C= 4x– 5x2+x3 is a cost function, sketch the graph of this function with the aid of
intercepts, symmetry, and the first–derivative test.
39)
40)
The cost equation for a company is C(x) = 2x3– 15x2– 84x+ 3100. Use the
second–derivative test, if applicable, to find the relative maxima and the relative minima.
40)
41)
It is estimated that under a short–term energy–assistance plan, n million households will
receive direct aid in meeting home–fuel costs after t years, where n= 2t3– 15t2+ 36t, 0 t
3. For what value of t will n be a maximum?
41)
42)
Let y=x3– 3x2
(a) Find y’ and y”.
(b) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(c) Determine the coordinates of all relative maximum and relative minimum points.
(d) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down.
(e) Determine the coordinates of all inflection points.
(f) With the aid of the information obtained in parts (a) – (e), give a reasonable sketch of
the curve.
42)
43)
An open box is to have a surface area of 300 in2. The base of the box is to be a square. Find
the dimensions of the box that maximize volume.
43)
44)
Here is a function and its first and second derivatives:
f(x)=x2
x– 2
f’(x)=x2– 4x
(x –2)2
f”(x)=8
(x –2)3
(a) Determine all x– and y–intercepts.
(b) Determine equations of all vertical asymptotes and non–vertical asymptotes.
(c) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(d) Determine the coordinates of all relative maximum and relative minimum points.
(e) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down.
(f) Determine the coordinates of all inflection points.
(g) With the aid of the information obtained in parts (a) – (g), give a reasonable sketch of
the curve.
44)
45)
Find all the critical values of f(x) =x2+ 9
x.
45)
12
46)
A released film has a revenue given by R(t) =50t
t3+ 8 , where R(t) is in millions of dollars
and t is the number of weeks after its release. Sketch the graph of this function.
46)
47)
The graph of a function is given. Find
(a) the open intervals on which the function is increasing or decreasing;
(b) the coordinates of all relative extrema.
47)
48)
The revenue equation for a company is given by R(x) = 173.28x– 0.16x3. Determine when
relative extrema occur on the interval (0, ).
48)
49)
Let y=2x– 4
x– 1
(a) Determine all x– and y–intercepts.
(b) Determine equations of all vertical asymptotes and non–vertical asymptotes.
(c) Determine y’ and y”.
(d) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(e) Determine the coordinates of all relative maximum and relative minimum points.
(f) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down.
(g) Determine the coordinates of all inflection points.
(h) With the aid of the information obtained in parts (a) – (g), give a reasonable sketch of
the curve.
49)
50)
Sensitivity to a drug depends on the dosage size x according to the equation s= 100x–x2.
Find the dosage that maximizes sensitivity.
50)
51)
If f(x) = 2x3+ 3x2– 36x+ 1, determine the intervals on which f is increasing and the
intervals on which f is decreasing.
51)
52)
The profit equation for a taco stand is given by P(x) = – 0.4x2+ 100x– 100, where x is the
number of tacos sold, and P(x) is the profit in dollars. Use the first–derivative test to find
when relative extrema occur.
52)
14
53)
Find all the critical values of f(x) =x8– 2x4.
53)
54)
The cost equation for a bakery is given by C(x) = 10(x–2)2– 120(x– 2) + 450, where x is
the number of doughnuts made (in dozens), and C(x) is the cost in dollars. Determine
where the graph of the equation is concave up and where it is concave down and find any
inflection points.
54)
55)
Let y= 3x2e2x
(a) Find y’ and y”.
(b) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(c) Determine the coordinates of all relative maximum and relative minimum points.
(d) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down.
(e) Determine the coordinates of all inflection points.
(f) With the aid of the information obtained in parts (a) – (e), give a reasonable sketch of
the curve.
55)
15
56)
Find the vertical asymptotes and non–vertical asymptotes of the following function. Do not
sketch its graph.
y=1 – 3x–x2
2x+ 1
56)
57)
Sketch the graph of y=x3– 3x2+x+ 1 with the aid of intercepts, symmetry, and the
first–derivative test. Use your graphing calculator to verify your solution.
57)
58)
Determine the intervals on which the function is increasing and on which it is decreasing.
Also determine the points of relative maxima and relative minima.
f(x) =8x
x2+ 6
58)
59)
Find the intervals where the graph of y=f(x) is concave up and where it is concave down.
Do not sketch its graph. Also determine its points of inflection. f(x) =x2
8–1
x.
59)
60)
If y=x4+ 4x3– 18x2– 3x+ 4, find the x–values of all inflection points.
60)
16
61)
An open box is to be made by cutting equal squares from each corner of a 20–in. square
piece of cardboard and then folding up the sides. Find the length of the side of the square
that must be cut out if the volume of the box is to be maximized.
61)
62)
If f(x) = 7x2+ 3x– 2, determine the intervals on which f is increasing and the intervals on
which f is decreasing.
62)
63)
The revenue equation for a company is given by R(x) = 68.04x– 0.07x3. Determine when
relative extrema occur on the interval (0, ).
63)
64)
Find the vertical and non–vertical asymptotes of the following function. Do not sketch its
graph.
y=2x2+ 7x– 22
2x– 3
64)
65)
Given y=f(x) =x3– 3x2+ 10 on the interval 0, 4 , find the values of x at which absolute
maxima and absolute minima occur.
65)
66)
If f(x) =x2ex, determine the intervals on which f is increasing and the intervals on which f
is decreasing.
66)
67)
Find all the critical values of f(x) = 12x5– 65x3+ 45x+ 60.
67)
17
68)
Let y=x+ 2
x2
(a) Determine all x– and y–intercepts.
(b) Determine equations of all vertical asymptotes and non–vertical asymptotes.
(c) Determine y’ and y”.
(d) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(e) Determine the coordinates of all relative maximum and relative minimum points.
(f) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down.
(g) Determine the coordinates of all inflection points.
(h) With the aid of the information obtained in parts (a) – (g), give a reasonable sketch of
the curve.
68)
69)
A real estate firm owns 70 apartments. At $200 per month each apartment can be rented.
However, for each $10 per month increase, there will be two vacancies with no possibility
of filling them. What rent per apartment will maximize monthly revenue?
69)
70)
Use the second derivative test to find the relative extremas of f(x) = 10xe5x and where they
occur. If the relative extremas can not be determined by the second derivative test, state so.
70)
71)
The demand equation for a monopolist’s product is p= 200 – 0.98q, where p is the price per
unit (in dollars) of producing q units. If the total cost c (in dollars) of producing 8 units is
given by c= 0.02q2+ 2q+ 8000, find the level of production at which profit is maximized.
71)
72)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=4x2– 2x+ 3
9 –x2.
72)
73)
The graph of a function is given. Find
(a) the open intervals on which the function is increasing or decreasing;
(b) the coordinates of all relative extrema.
73)
74)
A released film has a revenue given by R(t) =40t
t2+ 36 , where R(t) is in millions of dollars
and t is the number of weeks after its release. Sketch the graph of this function.
74)
75)
The demand function for a monopolist’s product is p= 100 – 3q, where p is the price per
unit (in dollars) for q units. If the average cost c (in dollars) per unit for q units is c= 4 +
100
q, find the output q at which profit is maximized.
75)
76)
Let f(x) =x4
4+x3
3–x2. Determine the intervals on which f is
(a) Determine the intervals on which f is increasing.
(b) Determine the intervals on which f is decreasing.
(c) Based on your answers to parts (a) and (b), find the values of x for which f has relative
maxima.
(d) Based on your answers to parts (a) and (b), find the values of x for which f has relative
minima.
76)
77)
A new movie has its monthly revenue given by R(x) =70x
x2+ 100 where x is the number of
days after its release and R is in millions of dollars. Find the relative extrema and use this
information to determine which day will bring the greatest revenue.
77)
78)
Determine the intervals on which the function is increasing and on which it is decreasing.
Also determine the points of relative maxima and relative minima.
f(x) = 16x5– 5x
78)
79)
Let f(x) =(x4+1)3.
(a) Determine the intervals on which f is increasing.
(b) Determine the intervals on which f is decreasing.
(c) Based on your answers to parts (a) and (b), find the values of x for which f has relative
maxima.
(d) Based on your answers to parts (a) and (b), find the values of x for which f has relative
minima.
79)
20