Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
The function f(x) = 4x3– 10x2– 8x+ 3 is decreasing on
1)
A)
(2, )
B)
(–, 2).
C)
–, 1
3.
D)
–1
3, .
E)
–1
3, 2 .
2)
If f(x) =x3– 7x2+ 2x– 5, then f is concave down on the interval
2)
A)
7
3, .
B)
(–, ).
C)
2
3, .
D)
–, 2
3.
E)
–, 7
3.
3)
If f(x) =x3+ 3x2– 24x+ 8, then f is
3)
A)
decreasing on (–, 4), concave up on (–1, ), and has no relative minimum point.
B)
increasing on (–4, 2), concave down on (–, ), and has a relative maximum when x= 2.
C)
decreasing on (–4, 2), concave down on (–, –1), and has a relative maximum when x= – 4.
D)
increasing on (2, ), concave up on (–, –1), and has a relative minimum when x= 2.
E)
decreasing on (1, 2), concave up (0, ), and has a relative minimum when x= – 4.
1
4)
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
4)
A)
100 units.
B)
200 units.
C)
400 units.
D)
800 units.
E)
none of the above
5)
The function y=x4– 8x2 has a relative maximum when x=
5)
A)
2.
B)
1.
C)
–1.
D)
–2.
E)
0.
6)
If f(x) =x3+ 2x2– 4x+ 10, then f is
6)
A)
increasing on –2, –2
3, concave down on 2
3, , and has a relative minimum when x= – 2.
B)
decreasing on (–2, ), concave down on –, 2
3, and has a relative maximum when x= – 2.
C)
increasing on –2, 2
3, concave up on (–, ), 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)
increasing on 2
3, , concave down on –, –2
3, and has a relative maximum when x=2
3.
7)
The function y=x3+ 15x2– 33x has a relative maximum when x=
7)
A)
1.
B)
0.
C)
11
D)
–11.
E)
–1.
2
8)
On the interval 0, 2 , the function y=x3+ 3x2– 9x+ 27 has an absolute maximum when x=
8)
A)
0.
B)
2.
C)
1
2.
D)
1.
E)
none of the above
9)
The function f(x) =x2– 1
x is concave up on the interval(s)
9)
A)
(1, ).
B)
(–, 0) and (0, 1).
C)
(–, –1).
D)
(0, ).
E)
(–, 0).
10)
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
10)
A)
2 ×2 × 3.
B)
3 × 3 ×4
3.
C)
4 × 4 ×3
4.
D)
2×2× 6.
E)
1 ×1 × 12.
B
11)
The function f(x) =x– 2
x+ 3 is concave down on the interval(s)
11)
A)
(–3, ).
B)
(–, –3) and (2, ).
C)
(–, –3).
D)
(–3, 0).
E)
(–, ).
12)
The function f(x) =x2– 6x+ 8 is decreasing on
12)
A)
(3, ).
B)
(–3, 3).
C)
(–, 3).
D)
(–, 2) and (4, ).
E)
(2, 4).
13)
How many critical values does the function f(x) = 3x4+ 4x3 have?
13)
A)
none
B)
one
C)
two
D)
three
E)
four
14)
If f(x) =x4– 6x2+ 3, then f has an inflection point when x=
14)
A)
3.
B)
0.
C)
2.
D)
3.
E)
1.
4
A
15)
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
15)
A)
50 units.
B)
100 units.
C)
750 units.
D)
1235 units.
E)
2500 units.
16)
An equation of a horizontal asymptote for the graph of y=2x
9x2– 1 is
16)
A)
y=2
9.
B)
x=2
9.
C)
y=1
3.
D)
y= 0.
E)
x=1
3.
17)
On the interval –1, 1 , the function y= 4 +x2–x3 has an absolute maximum when x=
17)
A)
1.
B)
0.
C)
1
2.
D)
–1.
E)
–1
2.
18)
An equation of a vertical asymptote for the graph of y=2x
4x– 1 is
18)
A)
y= 0.
B)
x=1
2.
C)
y=1
2.
D)
x=1
4.
E)
y=1
4.
19)
On the interval 0, 2 , the function y=3x4–4x3 has
19)
A)
an absolute min. at x= 1 and no absolute max.
B)
an absolute max. at x= 2 and an absolute min. at x= 1.
C)
an absolute max. at x= 2 and an absolute min. at x= 0.
D)
no absolute max. and no absolute min.
E)
an absolute max. at x= 0 and an absolute min. at x= 1.
20)
An equation of a horizontal asymptote for the graph of y=3x2– 2
x2– 4 is
20)
A)
x= 3.
B)
y= 2.
C)
x= 2.
D)
y= 0.
E)
y= 3.
Explanation:
21)
The function y=x5
20 +x4
12 +x– 3 has how may inflection points?
21)
A)
none
B)
one
C)
two
D)
three
E)
four
Explanation:
22)
An equation of a vertical asymptote for the graph of y=7x2– 4
2x+ 3 is
22)
A)
y=7
2.
B)
y= – 4
3.
C)
x= – 3
2.
D)
x= 0.
E)
x= 4.
Explanation:
Explanation:
23)
If f(x) =ex (x+ 4), then f has an inflection point when x=
23)
A)
–1.
B)
–5.
C)
0.
D)
–3.
E)
–6.
24)
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.
24)
A)
400,000
B)
500,000
C)
600,000
D)
700,000
E)
800,000
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
25)
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) = 16x5– 5x.
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)
27)
Find all the critical values of f(x) =2x– 1.
27)
28)
Use your graphing calculator to graph f(x) =1
20 x5+1
6x4–1
4x. Looking at the graph,
estimate the intervals where the function is concave up and concave down. Now find these
intervals using the second derivative.
28)
29)
Let f(x) =x
x– 2 .
(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.
29)
30)
Find all the critical values of f(x) =x8– 2x4.
30)
31)
Find all the critical values of f(x) =x ln x.
31)
32)
Sketch the graph of y=ex– 7x with the aid of intercepts, symmetry, and the
first–derivative test. Use your graphing calculator to verify your solution.
32)
33)
The revenue equation for a company is given by R(x) = 173.28x– 0.16x3. Determine when
relative extrema occur on the interval (0, ).
33)
34)
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.
34)
35)
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.
35)
Explanation:
36)
Use your graphing calculator to estimate the relative extrema of the function
y=x5+x4–x3+x2+x(estimate to 3 decimal places).
36)
Explanation:
37)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=2x– 3
x+ 5 .
37)
Explanation:
38)
Let f(x) = 3x5– 10x4+ 7x. Determine the intervals on which f is (a) concave up and (b)
concave down.
(c) Find the x–values of all inflection points.
38)
Explanation:
39)
Use your graphing calculator to graph y=x3– 6x2+ 9x– 2. Inspect the graph to estimate
where the relative extrema occur. Then use the first–derivative test to find the relative
extrema.
39)
Explanation:
10
Explanation:
40)
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.
40)
41)
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.
41)
42)
Find all the critical values of f(x) =x2+ 9
x.
42)
43)
If f(x) =x2ex, determine the intervals on which f is increasing and the intervals on which f
is decreasing.
43)
44)
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.
44)
45)
Let f(x) =x(x–3)5.
(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.
45)
46)
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?
46)
47)
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.
47)
48)
Find all the critical values of f(x) =x+ 2
x2– 3 .
48)
49)
Find the vertical asymptotes and non–vertical asymptotes of the following function. Do not
sketch its graph.
y=1 – 3x–x2
2x+ 1
49)
50)
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.
50)
51)
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.
51)
Explanation:
52)
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.
52)
Explanation:
53)
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) =x3+x2– 3x + 7.
53)
Explanation:
54)
Find all the critical values of f(x) =x4– 8x2+ 3.
54)
Explanation:
15
Explanation:
55)
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.
55)
56)
If f(x) = 7x2+ 3x– 2, determine the intervals on which f is increasing and the intervals on
which f is decreasing.
56)
57)
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) =x4
2–x3+ 5.
57)
58)
Let y=x2
x– 1
(a) Determine all x– and y– intercepts.
(b) Determine equations of all vertical 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.
58)
59)
If C= 4x– 4x2+x3 is a cost function, sketch the graph of this function with the aid of
intercepts, symmetry, and the first–derivative test.
59)
60)
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.
60)
61)
The demand equation for a monopolist’s product is p=10,000
q2+ 25 , where p is the price per
unit (in dollars) when q units are demanded.
(a) Determine the value of q for which revenue is maximum.
(b) What is the maximum revenue?
61)
62)
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) = 22t+ 15t2–t3. Find where the function is
concave up and where the function is concave down.
62)
63)
If f(x) =x4
4–2x3
3, determine the intervals on which f is increasing and the intervals on
which f is decreasing.
63)
64)
Find the vertical and non–vertical asymptotes of the following function. Do not sketch its
graph.
y=3x3
x2– 5
64)
65)
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.
65)
66)
A manufacturer has to produce annually 360 units of a product that are sold at a uniform
rate during the year. The production cost for each unit is $200, and carrying costs
(insurance, interest, storage, and so on) are estimated to be 10% of the value of average
inventory. Set–up costs per production run are $100. Find the economic lot size.
66)
67)
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.
67)
68)
The cost equation for a company is C(x) =x3+ 5x2– 8x+ 250. Use the second–derivative
test, if applicable, to find the relative maxima and the relative minima.
68)
69)
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.
69)
70)
A cable TV company has 500 customers paying $20 each month. For each $1 reduction in
price, the company attracts 50 more customers. Find the price that yields maximum
revenue.
70)
71)
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) = 20t+ 15t2–5
6t3. Find where the function
is concave up and where the function is concave down.
71)
72)
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.
72)
73)
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.
73)
74)
Find all the critical values of f(x) = 12x5– 65x3+ 45x+ 60.
74)
75)
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
ex
75)
76)
Let f(x) =x
x– 1 . Find the x values of (a) all relative maxima, (b) all relative minima, and (c)
all inflection points.
76)
77)
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.
77)
20