Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1)
Write an “area” word problem for which finding the solution would involve evaluating the
double integral
5
0
x2
0
dy dx.
1)
2)
Write a “volume” word problem for which finding the solution would involve evaluating
the double integral
2
1
6
4(y + x) dy dx.
2)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find fx.
3)
f(x, y) =e5xy
3)
A)
fx(x, y) =5xye5xy
B)
fx(x, y) =5e5xy
C)
fx(x, y) =5ye5xy
D)
fx(x, y) =5(x + y)e5xy
4)
f(x, y) =1xy
4)
A)
Relative minimum = –25, saddle point = (0, 0)
B)
Saddle point = (0, 0)
C)
No relative extrema or saddle points
D)
Relative maximum =2, relative minimum = –1
5)
z =x3+3x2y;
z
x(2, 1)
5)
A)
24
B)
36
C)
30
D)
18
Evaluate the function.
6)
Find f(4, 7) when f(x, y) =e5yln x.
6)
A)
944,087,874.53
B)
2.20e+15
C)
1.14e+16
D)
2.60e+15
7)
f(x, y) =x2–y2
x2+y2
7)
A)
4xy2
(x2+y2)2
B)
4xy2
(x2+y2)4
C)
2y2
(x2+y2)2
D)
2x2y2
(x2+y2)2
8)
What is the greatest area that a rectangle can have if the length of its diagonal is 2 m?
8)
A)
5 m2
B)
1 m2
C)
2 2 m2
D)
2 m2
9)
Consider the data showing the average life expectancy of human beings in various years.
year life
expectancy
1900 61.2
1910 62.9
1920 63.75
1930 65.0
1940 66.5
1950 68.1
1960 69.9
1970 70.95
1980 73.7
Find the regression line. Let the year 1900 represent x = 0.
9)
A)
y = –0.11.2x + 23.1
B)
y = 8.3x + 22.1
C)
y = 0.15x + 60.9
D)
y = 0.37x – 21.8
10)
7
2
4
0
1–y2
0
dz dy dx
10)
A)
–86.67
B)
9
C)
5
D)
14
11)
f(x, y) = – x2+y2
11)
A)
B)
3
C)
D)
Use a 3D grapher to graph the function. Then estimate any relative extrema.
12)
f(x, y) =(x + 2y + 1) 2
12)
A)
Relative maximum = 4
B)
Relative minimum = 1
C)
No relative extrema
D)
Relative minimum = 0
Find the relative maximum and minimum values and the saddle points if they exist.
13)
f(x, y) =e–(x2+y2–8y)
13)
A)
Relative maximum =e16
B)
Relative maximum =e16, relative minimum =e–16
C)
Saddle point =0, 4
D)
No relative extremum or saddle points
4
Solve the problem.
14)
Of all points (x, y, z) that satisfy x – y – z = 1, find the one that minimizes x2+ 2y2+ 3z2.
14)
A)
–6
11, 3
11, –2
11
B)
–6
11, –3
11, –2
11
C)
–6
11, –3
11, 2
11
D)
6
11, –3
11, –2
11
Solve the problem. Assume that relative maximum and minimum values are absolute maximum and minimum values.
15)
A flat plate is located on a coordinate plane. The temperature of the plate, in degrees Fahrenheit, at
point (x, y) is given by T(x, y) =x2+y2–3x –7y. Find the coordinates of the point on the plate
where the temperature is minimal.
15)
A)
3
2, 7
2
B)
–3
2,–7
2
C)
(6, 14)
D)
(3, 14)
16)
f(x, y) =1
y –8x2
16)
A)
{(x, y)| y –8x2}
B)
{(x, y)| y 8x2}
C)
{(x, y)| y –8x2}
D)
{(x, y)| y 8x2}
17)
f(x, y) =x2
17)
A)
B)
C)
D)
18)
If f(x, y) is a function of two variables, then what does fx measure?
18)
A)
fx does not measure any real quantity
B)
The change in the function in the y direction
C)
The change in the function
D)
The change in the function in the x direction
Solve the problem.
19)
The population density of a city is given by p(x, y) =6x2+4y, where x and y are in miles, and p is
the number of people per square mile, in hundreds. The city limits are as shown in the graph
below. Determine the city’s population.
19)
A)
29,250 people
B)
26,850 people
C)
27,450 people
D)
53,100 people
Find the domain of the function of two variables.
20)
f(x, y) =7–x2–y2
20)
A)
{(x, y)|x2–y27}
B)
{(x, y)| x2+y27}
C)
{(x, y)| x2–y27}
D)
{(x, y)| x2+y27}
Solve the problem.
21)
Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its
center can be approximated by the function S =aV
0.51d4, where a is an atmospheric constant, and V
is the approximate volume of the tornado, in cubic feet. Assume that a is 0.78, and find Sd.
21)
A)
1.5V
d
B)
1.5V
d2
C)
–6.1V
d5
D)
1.5
d4
22)
Find g(–7, 3) when g(x, y) =4y2– 9xy.
22)
A)
388
B)
225
C)
232
D)
385
23)
2
0
x2
0
(x2– y2) dy dx
23)
A)
44
105
B)
32
115
C)
44
115
D)
32
105
Find fx.
24)
f(x, y) = x – 1 + 2y
1 + x
24)
A)
2 y
1 + x
B)
1
x + 1 (1 + x) 2
C)
1 + x
2 x – 1 – x – 1 – 2y (1 + x) –2
D)
x – 1 + 2y
(1 + x) 2
Solve the problem. Assume that relative maximum and minimum values are absolute maximum and minimum values.
25)
A company uses TV and magazines for advertising. They know that profit P is related to the
amounts T spent on TV and M spent on magazines by the equation P =24MT – 4M – 3T + 1, where
P, M, and T are in hundreds of thousands. Find the maximum profit.
25)
A)
$120,000
B)
$100,000
C)
$50,000
D)
$60,000
Find the requested partial derivative.
26)
f(x, y) =x2+y2; fx(2, –5)
26)
A)
229
29
B)
–229
29
C)
429
29
D)
–29
29
27)
2
–1
x+1
x–1
2y/x
0
3xyz dz dy dx
27)
A)
6
B)
5
C)
24
D)
9
28)
Find f(8, –5) when f(x, y) =7x + 3y – 3.
28)
A)
41
B)
31
C)
38
D)
35
Find fx, fy, and f.
29)
f(x, y, ) =
x +y2
29)
A)
fx=
2(x +y2)3/2; fy= –
(x +y2)3/2; f=1
x +y2
B)
fx= –
2(x +y2)3/2; fy=
(x +y2)3/2; f= – 1
x +y2
C)
fx=
2(x +y2)3/2; fy=
(x +y2)3/2; f=1
x +y2
D)
fx= –
2(x +y2)3/2; fy= –
(x +y2)3/2; f=1
x +y2
Solve the problem. Assume that relative maximum and minimum values are absolute maximum and minimum values.
30)
A closed rectangular box with a volume of 16 cubic feet is made from two kinds of materials. The
top and bottom are made of a material costing $0.10 per square foot, and the sides are made of a
material costing $0.05 per square foot. Find the dimensions of the box so that the cost of materials is
minimal.
30)
A)
2 ft by 2 ft by 8 ft
B)
1 ft by 1 ft by 16 ft
C)
2 ft by 4 ft by 4 ft
D)
2 ft by 2 ft by 4 ft
31)
z =3x3– 5xy – y;
z
x(1, 2)
31)
A)
–10
B)
–1
C)
–11
D)
–2
32)
Find fxy when f(x,y) = 8x3y – 7y2+ 2x.
32)
A)
–14
B)
24x2
C)
48xy
D)
–28
33)
Let f(x, y) =(8x5y5+ 7)2. Find fy.
33)
A)
80x5y4(8x5y5+ 7)
B)
80x4y5(8x5y5+ 7)
C)
40x5y4
D)
2(8x5y5+ 7)
10
Find the requested partial derivative.
34)
f(x, y) =x2+y2; fy(2, –3)
34)
A)
13
13
B)
–313
13
C)
–613
13
D)
–13
13
Use a 3D grapher to generate the graph of the function.
35)
f(x, y) = 1 – x – 2y
35)
A)
B)
C)
D)
Find the minimum or maximum value of f (as indicated) subject to the given constraint.
36)
Maximum of f(x, y) = xy,
subject to x + y = 100
36)
A)
Maximum = 0 at (10, 0)
B)
Maximum = 2500 at (50, 50)
C)
Maximum = 100 at (0, 100)
D)
Maximum = 100 at (50, 50)
37)
Let z = f(x, y) =(x + y)8. Find z
x.
37)
A)
8x(x + y)7
B)
8y(x + y)7
C)
8(x + y)
D)
8(x + y)7
Find fx.
38)
f(x, y) = (x + y) ln (xy)
38)
A)
fx(x, y) = ln (xy) +(x + y)
x
B)
fx(x, y) = x ln (xy) +(x + y)
y
C)
fx(x, y) = y ln (xy) +(x + y)
x
D)
fx(x, y) = x ln (xy) +(x + y)
x
Find the second–order partial derivative.
39)
Find gyx when g(x, y) =x
y.
39)
A)
1
y
B)
0
C)
–1
y2
D)
x2
y2
Find the requested partial derivative.
40)
z =4x3+ 5xy – y;
z
y(4, 2)
40)
A)
20
B)
212
C)
19
D)
211
Find the partial derivative.
41)
Let z = f(x, y) =3(x +8y –2)2. Find z
x.
41)
A)
3x +24y –6
B)
6x +48y +12
C)
24x +192y
D)
6x +48y –12
Solve the problem.
42)
Of all numbers whose difference is 20, find the two that have the minimum product.
42)
A)
1 and 21
B)
0 and 20
C)
10 and –10
D)
40 and 20
43)
The sum of all the forces acting on an object can be described mathematically by F(m, a) = ma,
where m is the mass of the object (in kg) and a is the acceleration felt by the object (in m/s2) as a
result of the forces (in newtons, N) acting on it. What is the force required to accelerate a 5 kg object
at 9 m/s2?
43)
A)
5 N
B)
45 N
C)
81 N
D)
14 N
Use the regression feature on your calculator to solve the problem.
44)
Consider the data showing the average life expectancy of human beings in various years.
year life
expectancy
1900 61.2
1910 62.9
1920 63.75
1930 65.0
1940 66.5
1950 68.1
1960 69.9
1970 70.95
1980 73.7
Use the regression line to predict the average life expectancy in the year 2005. Let the year 1900
represent x = 0.
44)
A)
77.18
B)
78.25
C)
79.11
D)
76.65
Use a 3D grapher to graph the function. Then estimate any relative extrema.
45)
f(x,y) =x2+ 2y2– xy2
45)
A)
Relative minimum = 0, relative maximum = 4
B)
Relative minima = 0, 4
C)
No relative extrema
D)
Relative maxima = 0, 4
Find the regression line without using a calculator.
46)
x 2 4 5 6
y 7 11 13 20
46)
A)
y = 2.8x + 0.15
B)
y = 3.0x + 0.15
C)
y = 3.0x
D)
y = 2.8x
Use a 3D grapher to graph the function. Then estimate any relative extrema.
47)
f(x, y) =x2+y2–4x –6y
47)
A)
Relative minimum = 0
B)
Relative maximum =13
C)
No relative extrema
D)
Relative minimum = – 13
Solve the problem.
48)
Managers rate employees according to job performance and attitude. The results for several
randomly selected employees are given below.
Performance, x Attitude, y
59
63
65
69
58
77
76
69
70
64
72
67
78
82
75
87
92
83
87
78
Find the regression line y = mx + b.
48)
A)
y = 2.02x – 47.3
B)
y = – 0.669x + 92.3
C)
y = 1.02x + 11.7
D)
y = 1.35x + 2.81
49)
The number of cows that can graze on a ranch is approximated by C(x,y) = 9x + 5y – 2, where x is
the number of acres of grass and y the number of acres of alfalfa. If the ranch has 75 acres of alfalfa
and 25 acres of grass, how many cows may graze?
49)
A)
600 cows
B)
800 cows
C)
798 cows
D)
598 cows
15
50)
Find two numbers whose sum is 64 and whose product is a maximum.
50)
A)
48 and 48
B)
16 and 48
C)
32 and 48
D)
32 and 32
Find the minimum or maximum value of f (as indicated) subject to the given constraint.
51)
Minimum of f(x, y) = x2– 14x + y2– 16y,
subject to 2x + 3y = 12
51)
A)
Minimum = –68 at (1, 5)
B)
Minimum = –15 at (0, 1)
C)
Minimum = –24 at (2, 0)
D)
Minimum = –61 at (3, 2)
Find the relative maximum and minimum values and the saddle points if they exist.
52)
f(x, y) = 2xy + 2x – 3y
52)
A)
Relative maximum = – 9
B)
Relative minimum =3
C)
Saddle point =3
2, 1 , saddle point = – 3
2, – 1
D)
Saddle point =3
2, – 1
Solve the problem.
53)
Poiseuille‘s law states that the resistance, R, for blood in a blood vessel varies directly as the length
of the vessel, L, and inversely as the fourth power of its diameter, d. This can be written as an
equation R(L, d) = k L
d4 where k is a constant. Find R(5, 0.3) and give the answer as the product of
k and a whole number.
53)
A)
617k
B)
62k
C)
630k
D)
17k
Evaluate the triple integral.
54)
5
1
2z
z–1
y+2z
0dx dy dz
54)
A)
536
3
B)
5
4
C)
55
2
D)
5
Find the volume of the solid capped by the surface z = f(x, y) over the region with the given boundaries on the xy–plane.
55)
z = x2+ y2; 0 x 1, 0 y 1
55)
A)
4
3
B)
1
3
C)
2
3
D)
8
3
56)
z = x3; 0 x 2, 0 y 3
56)
A)
6
B)
12
C)
18
D)
4
57)
f(x, y) = x3– 12x + y2
57)
A)
f(2, 0) = –16, relative minimum
B)
f(0, 2) = 4, relative maximum
C)
f(0, 0) = 0, relative minimum
D)
f(0, 0) = 0, relative maximum
58)
Find fx(0, y) by evaluating the limit of f(h, y) – f(0, y)
h as h approaches 0.
58)
A)
Undefined
B)
0
C)
y
D)
1
y
59)
The graph of a function of two variables, f(x, y), is a surface. In general, what geometrical
description can be given to the intersection of two surfaces?
59)
A)
Surface
B)
Curve
C)
Plane
D)
There is no general description.
60)
Minimum of f(x, y, z) = x + 2y – 2z,
subject to x2+y2+z2= 9
60)
A)
Minimum: –9 at (–1, –2, 2)
B)
Minimum: –8 at (–2, –1, 2)
C)
Minimum: 9 at (1, 2, –2)
D)
Minimum: –1 at (–1, 2, 2)
Solve the problem.
61)
A farmer has 620 m of fencing. Find the area of the largest rectangular field that he can enclose with
his fencing. Assume that no fencing is needed along one edge of the field.
61)
A)
148,050 m2
B)
48,050 m2
C)
96,100 m2
D)
69,000 m2
62)
9
0
3
0(9x2y + 10xy) dy dx
62)
A)
3888
B)
11,664
C)
1296
D)
432
Solve the problem.
63)
Suppose a continuous random variable has a joint probability density function given by
f(x, y) =2
5(2x + 3y), 0 x 1, 0 y 1.
Find 1
0
1
0f(x, y) dx dy.
63)
A)
1
B)
0.25
C)
0.75
D)
0.50
64)
Find f(100, 3) when f(x, y) = y log x.
64)
A)
60
B)
6
C)
30
D)
3
65)
Find fyx when f(x,y) = 8x3y – 7y2+ 2x.
65)
A)
24x2
B)
–14
C)
48xy
D)
–28
Solve the problem.
66)
Suppose that a continuous random variable has a joint probability density function given by
f(x, y) =y2– 2y +1
3xy –1
3x +17
12, 1 x 5, 0 y 1
Find the probability that a point is in the region bounded by 2 x 5 and 0 y 1.
66)
A)
2
3
B)
3
8
C)
1
2
D)
2
5
67)
Newton’s Universal Law of Gravitation states that the attractive force exerted on one spherically
symmetric object of mass M by a second spherically symmetric object of mass m is given by
F = GmM
r2, where r is the distance between the centers of the two objects and G is a constant equal
to 6.67 ×10–11 Nm2/kg2 when F is in newtons (N), M and m are in kilograms (kg), and r is in
meters (m). What is the magnitude of the gravitational force exerted on an object of mass 60 kg by
an object of mass 1 ×1012 kg if the objects are 20 m apart?
67)
A)
40,020.0 N
B)
10.0 N
C)
100.1 N
D)
200.1 N
68)
Minimum of f(x, y) = x2+ 2y2– xy,
subject to x + y = 8
68)
A)
Minimum = 25 at (6, 2)
B)
Minimum = 25 at (2, 6)
C)
Minimum = 28 at (5, 3)
D)
Minimum = 28 at (3, 5)
69)
f(x, y) =ex+ey
69)
A)
No relative extrema
B)
Relative minimum = 0
C)
Relative minimum = 2
D)
Relative minimum = 2e
70)
Find fxx when f(x, y) = ln( 2x + 9y ).
70)
A)
–4
(2x + 9y) 2
B)
–81
(2x + 9y) 2
C)
18
2x + 9y
D)
–18
(2x + 9y) 2
20
71)
Minimum of f(x, y) = xy,
subject to 9x2+ 4y2= 36
71)
A)
Minimum: 3 at 2, 3
22 and –2, –3
22 ;
B)
Minimum: –3 at 2, –3
22 and –2, 3
22
C)
Minimum: –3 at 2, 3
22
D)
Minimum: 3 at 2, –3
22
72)
f(x, y) =3x + 2y; fy(2, –1)
72)
A)
3
B)
4
C)
8
D)
2
Solve the problem.
73)
The production level P of a factory during one time period is modeled by P(x, y) = Kx1/2y1/2
where K is a positive integer, x is the number of units of labor scheduled and y is the number of
units of capital invested. If labor costs $3900/unit, capital costs $800/unit and the owner has
$1,700,000 available for one time period, what amount of labor and capital would maximize
production?
73)
A)
1062.5 units of labor and 217.9 units of capital
B)
212.5 units of labor and 944.4 units of capital
C)
435.9 units of labor and 2125.0 units of capital
D)
217.9 units of labor and 1062.5 units of capital
74)
Minimum of f(x, y) =x2y,
subject to x2+ 2y2= 6
74)
A)
Minimum: –4 at (±2, –1)
B)
Minimum: –18 at (±3, –2)
C)
Minimum: –18 at (–3, –2)
D)
Minimum: 4 at (2, 1)
75)
A flat plate is located on a coordinate plane. The temperature of the plate, in degrees Fahrenheit, at
point (x, y) is given by T(x, y) =x2+y2–7x –7y. What is the maximum temperature on the plate?
75)
A)
126°
B)
No maximum
C)
75°
D)
18°
76)
f(x, y) = x2+ y2
76)
A)
f(1, 1) = 2, relative maximum
B)
No relative extrema
C)
f(0, 0) = 0, relative maximum
D)
f(0, 0) = 0, relative minimum
Solve the problem.
77)
Suppose a continuous random variable has a joint probability density function given by
f(x, y) =2
5(2x + 3y), 0 x 1, 0 y 1.
Find the probability that a point (x, y) is in the region bounded by 0 x 1
2 and 1
2 y 1 by
evaluating the integral 1
1/2
1/2
0f(x, y) dx dy.
77)
A)
0.125
B)
0.275
C)
0.1
D)
0.4
Evaluate the integral.
78)
6
–2
5
–1xy2 dx dy
78)
A)
– 896
B)
–2912
3
C)
2912
3
D)
896
79)
Consider the following data on the growth of cactus grafts under controlled conditions.
weeks after
grafting
x
height
(inches)
y
1
2
4
5
2
2.4
5.1
7.3
Find the regression line y = mx + b.
79)
A)
y =1.50 x + .7
B)
y = 13x + 8.19
C)
y = –2.10x + .2
D)
y = 1.33x + .21
80)
The following data pertain to the residual chlorine in a swimming pool at various times after it has
been treated with chemicals.
number of hours, x residual chlorine
(parts per million), y
2
4
6
8
1.8
1.5
1.4
1.1
Find the regression line y = mx + b.
80)
A)
y = 0.22x + 3
B)
y = 0.025x + 4
C)
y = –0.37x + 8
D)
y = –0.11x + 2
81)
f(x, y, ) = xx + y
81)
A)
fx= x + y –x
2x + y ; fy= – xy
2x + y ; f= x x + y
B)
fx= x + y –x
x + y ; fy= – xy
x + y ; f= x x + y
C)
fx= x + y +x
2x + y ; fy=xy
2x + y ; f= x x + y
D)
fx= x + y +x
x + y ; fy=xy
x + y ; f= x x + y
Evaluate the integral.
82)
1
0
1
10ydx dy
82)
A)
6
B)
–9
2
C)
11
2
D)
– 4
83)
Find fyy when f(x, y) = 8xexy.
83)
A)
16x2ex
B)
16x2exy
C)
8x3ex
D)
8x3exy
Evaluate the integral.
84)
9
0
3
0(x + y) dx dy
84)
A)
6
B)
36
C)
162
D)
972
85)
0
–7
0
–3(3x + 8y) dy dx
85)
A)
–45
2
B)
–135
2
C)
–315
2
D)
–945
2
86)
f(x, y) = x3+ y3– 4xy
86)
A)
f(4
3, 4
3) = 2.37, relative maximum
B)
f(0, 0) = 0, relative minimum
C)
f(0, 0) = 0, relative maximum
D)
f(4
3, 4
3) = –2.37, relative minimum
87)
Maximum of f(x, y) = 4x + 6y,
subject to x2+y2= 13
87)
A)
Maximum: 36 at (3, 4)
B)
Maximum: –26 at (–2, –3)
C)
Maximum: 26 at (2, 3)
D)
Maximum: 0 at (0, 0)
25
Use a 3D grapher to generate the graph of the function.
88)
f(x, y) = 1 – x
88)
A)
B)
C)
D)
89)
Is it the case that every absolute maximum is always a relative maximum?
89)
A)
Yes
B)
No
90)
z = (x + y)2; –1 x 1, –1 y 1
90)
A)
8
3
B)
4
3
C)
1
3
D)
2
3
26
Evaluate the integral.
91)
1
0
y9
0
x dx dy
91)
A)
1
19
B)
1
38
C)
1
37
D)
1
39
92)
f(x, y) = x + y
92)
A)
B)
C)
D)
27
Find the volume of the solid capped by the surface z = f(x, y) over the region with the given boundaries on the xy–plane.
93)
z =8
(x + y)3; 0 x 1, 1 y 2
93)
A)
8
3
B)
4
3
C)
2
3
D)
1
3
Solve the problem.
94)
The intelligence quotient in psychology is given by Q(m, c) = 100m
c, where m is an individual’s
mental age and c is the individual’s chronological, or actual, age. Find Q
c .
94)
A)
–100m
c
B)
–100m
c2
C)
100m
c
D)
100m
95)
Find fyy when f(x, y) = x ln (y – x).
95)
A)
x
y – x
B)
–x
(y – x)2
C)
x
(y – x)2
D)
–1
(y – x)2
96)
f(x, y) = yex – 9
96)
A)
{(x, y)| x 9 and y 0}
B)
{(x, y)| x –9}
C)
{(x, y)| x 9}
D)
{(x, y)| x 9}
Find fx.
97)
f(x, y) = y ln (6x +8y)
97)
A)
fx(x, y) = ln (6x +8y) +6
6x +8y
B)
fx(x, y) =6y
6x +8y
C)
fx(x, y) =6xy
6x +8y
D)
fx(x, y) = y ln (6x +8y)
98)
x 1 3 5 7 9
y 143 116 100 98 90
98)
A)
y = 6.8x – 150.7
B)
y = 6.2x – 140.4
C)
y = –6.2x + 140.4
D)
y = –6.8x + 150.7
99)
The regression line relating dexterity scores (x) and productivity scores (y) for the employees of a
company is y = 1.91x + 5.50. Use the regression line to predict the productivity score for a person
whose dexterity score is 38.
99)
A)
78.1
B)
210.9
C)
56.3
D)
58.2
Find the minimum or maximum value of f (as indicated) subject to the given constraint.
100)
Minimum of f(x, y, z) = x2+ y2+ z2,
subject to x + 2y – z = 3
100)
A)
Minimum =3
2 at –1
2, 2, 1
2
B)
Minimum =3
2 at 1
2, 1, –1
2
C)
Minimum = 1 at (0, 1, –1)
D)
Minimum = 1 at (1, 1, 0)
29
Find the volume of the solid capped by the surface z = f(x, y) over the region with the given boundaries on the xy–plane.
101)
z = 6x2y; 0 x 4, 0 y 3
101)
A)
1256
B)
576
C)
2256
D)
676
Use a 3D grapher to generate the graph of the function.
102)
f(x, y) = 4x2+ 4y2+ 2
102)
A)
B)
C)
D)
30
Find the regression line without using a calculator.
103)
x 6 8 20 28 36
y 2 4 13 20 30
103)
A)
y = 0.9x + 3.79
B)
y = 0.9x – 2.79
C)
y = 0.8x – 3.79
D)
y = 0.9x – 3.79
104)
Find fy(x, 0) by evaluating the limit of f(x, h) – f(x, 0)
h as h approaches 0.
104)
A)
Undefined
B)
0
C)
1
x
D)
x
105)
Find fxx when f(x, y) =x2+ y –ex+y.
105)
A)
1 –ex+y
B)
2 +ex+y
C)
2 –ex+y
D)
2 –y2ex+y
106)
z = 4x2+ 9y2; 0 x 1, 0 y 1
106)
A)
26
3
B)
5
3
C)
17
3
D)
13
3
107)
6
0
x/2
0(x + y) dy dx
107)
A)
36
B)
45
C)
63
D)
54
108)
f(x, y) =1
x+ xy –8
y
108)
A)
f(–1
2, 4) = –6, relative maximum
B)
f(1
2, –4) = 2, relative minimum
C)
f(1
2, –4) = 2, relative maximum
D)
f(–1
2, 4) = –6, relative minimum
109)
f(x, y) = –x2–y2
109)
A)
B)
C)
D)
110)
z =x+y; 0 x 1, 0 y 1
110)
A)
1
3
B)
8
3
C)
2
3
D)
4
3
111)
f(x, y) =x2–y2; fx(5, 2)
111)
A)
10 21
21
B)
–221
21
C)
521
21
D)
–521
21
112)
The regression line relating attitude rating (x) and job performance rating (y) for the employees of a
company is y = 1.02x + 11.7. Use the regression line to predict the job performance rating for a
person whose attitude rating is 76.
112)
A)
12.6
B)
87.9
C)
80.1
D)
89.2
113)
z =(8x +5y)2;
z
x(–1, 2)
113)
A)
26
B)
–128
C)
16
D)
32
33
Use a 3D grapher to generate the graph of the function.
114)
f(x, y) =4 –x2–y2
114)
A)
B)
C)
D)
115)
Find the dimensions of the right circular cylinder with maximum volume if its surface area is
24in.2.
115)
A)
r = 3 in., h = 8 in.
B)
r = 2 in., h = 4 in.
C)
r = 3 in., h = 3 in.
D)
r = 2 in., h = 6 in.
116)
Is it true that fxy(a, b) =fyx(a, b) for every function?
116)
A)
Yes
B)
No
117)
Find h(9, 1) when h(x, y) = (x + y)3.
117)
A)
1000
B)
30
C)
730
D)
100
Solve the problem.
118)
The surface area of a certain mammal, in square meters, is approximated by
A(W, H) =0.33W0.49H0.62, where W is the weight of the animal in kilograms and H is the height
in meters. Find A
H.
118)
A)
0.16W0.49H–0.38
B)
0.16W–0.51H0.62
C)
0.2W0.49H–0.38
D)
0.2W–0.51H0.62
119)
Minimum of f(x, y, z) = xy + z,
subject to x + y + z = 1
119)
A)
Minimum = 0 at (1, 1, –1)
B)
Minimum =1
2 at (1, 1, –1)
C)
Minimum = 1 at (0, 0, 1)
D)
Minimum = 2 at (1, 1, 1)
35
Solve the problem.
120)
The population density of ants in a field is given by p(x, y) =1
100xy2 , where 0 x 20 and
0 y 80, x and y are in feet, and p is the number of ants per square foot. Determine the total
population of ants in the field.
120)
A)
85,333 ants
B)
341,333 ants
C)
682,667 ants
D)
1,024,000 ants
121)
Find gyy when g(x,y) =x
y.
121)
A)
2x
y3
B)
3x
y2
C)
–3x
y2
D)
–2x
y3
122)
Suppose that the labor cost for a building is approximated by C(x, y) =10x2+ 2y2– 400x – 120y
+ 26,000, where x is the number of days of skilled labor and y is the number of days of semiskilled
labor required. Find the x and y that minimize cost C.
122)
A)
x =40, y =60
B)
x =12, y =90
C)
x =20, y =30
D)
x =30, y =12
123)
Let z = g(x,y) =2x + 3x2y2– 4y2. Find z
x.
123)
A)
6xy2– 8y
B)
2+ 6xy2
C)
6x2y – 8y
D)
2+ 6x2y
36
Evaluate the integral.
124)
1
0
1
0(10x + 6y) dy dx
124)
A)
– 2
B)
2
C)
68
D)
8
125)
f(x, y) =ex–y
125)
A)
fx(x, y) = xex–y
B)
fx(x, y) =ex–y
C)
fx(x, y) = –ex–y
D)
fx(x, y) = –yex–y
126)
f(x, y) =y +2x
126)
A)
{(x, y)| y 2x}
B)
{(x, y)| y 2x}
C)
{(x, y)| y –2x }
D)
{(x, y)| y –2x}
127)
f(x, y) = x2+ xy + y2– 3x + 2
127)
A)
f(2, –1) = –1, relative maximum
B)
f(–2, 1) = 11, relative minimum
C)
f(2, –1) = –1, relative minimum
D)
f(–2, 1) = 11, relative maximum
37
128)
Find g(3, 4) when g(x, y) =x – 6y
x2+ y2.
128)
A)
–25
21
B)
–21
25
C)
– 5
21
D)
–21
5
129)
f(x, y) =x
y+1
y – 5
129)
A)
{(x, y)| y 0 and y –5}
B)
{(x, y)| x 0 and y 5}
C)
{(x, y)| x 0 and y 0 and y 5}
D)
{(x, y)| y 0 and y 5}
130)
Production of television sets is given by P(x,y) = 100 2
3x–2/3 +2
5y–1/3 –4, where x is work hours
and y is the amount of capital. If 64 work hours and 125 units of capital are used, what is the
production output?
130)
A)
456,366 television sets
B)
4564 television sets
C)
4631 television sets
D)
463,120 television sets
131)
f(x, y) =1
x+1
y
131)
A)
Relative minimum = 0
B)
Relative maximum = 100, relative minimum = 0.01
C)
Relative maximum = 2
D)
No relative extrema
132)
f(x, y) =5x7y4
132)
A)
7x64x7y4
B)
75x7y4
5xy4
C)
75x7y4
5x
D)
7x6y4
5x7y4
133)
z = 8x + 4y + 7; 0 x 1, 1 y 3
133)
A)
38
B)
36
C)
26
D)
28
134)
f(x, y) = x3+ y3– 9xy
134)
A)
No relative extrema
B)
f(2, 2) = –20, relative minimum
C)
f(1, 1) = –7, relative minimum
D)
f(3, 3) = –27, relative minimum
135)
f(x, y) =x2+4xy +y2
135)
A)
Relative maximum =96
B)
Relative minimum =16
C)
Saddle point =(0, 0), relative maximum =96
D)
Saddle point =(0, 0)
39
136)
z =x
y; 0 x 1, 1 y e
136)
A)
1
3
B)
1
4
C)
1
6
D)
1
2
137)
z = e2x + 3y; 0 x 1, 0 y 1
137)
A)
1
6(e5– e3– e2– 1)
B)
1
4(e5– e3– e2+ 1)
C)
1
4(e5– e3– e2– 1)
D)
1
6(e5– e3– e2+ 1)
138)
4
0
4x
0x2 dy dx
138)
A)
512
7
B)
256
7
C)
256
15
D)
512
15
139)
A company has the following production function for a certain product:
p(x, y) =30x0.8y0.2 .
Find the marginal productivity with fixed capital, py .
139)
A)
6yx0.8
B)
6y
x0.8
C)
6y
x0.2
D)
6x
y0.8
140)
Two different tests are designed to measure employee productivity and dexterity. Several
employees are randomly selected and tested with these results.
Productivity, x Dexterity, y
23
25
28
21
21
25
26
30
34
36
49
53
59
42
47
53
55
63
67
75
Find the regression line y = mx + b.
140)
A)
y = – 0.329x + 75.3
B)
y = 1.53x + 10.7
C)
y = 1.91x + 5.05
D)
y = 2.03x + 2.36
141)
f(x, y) =e5x + 3y
141)
A)
fx(x, y) =5e5x + 3y
B)
fx(x, y) =e5x + 3
C)
fx(x, y) =3e5x + 3y
D)
fx(x, y) =5e5x
142)
1
0
5
5x y dy dx
142)
A)
125
2
B)
25
3
C)
25
2
D)
125
3
Solve the problem.
143)
Newton’s Universal Law of Gravitation states that the attractive force exerted on one spherically
symmetric object of mass M by a second spherically symmetric object of mass m is given by
F = GmM
r2, where r is the distance between the centers of the two objects and G is a constant equal
to 6.67 ×10–11 Nm2/kg2 when F is in newtons (N), M and m are in kilograms (kg), and r is in
meters (m). What is the magnitude of the gravitational force exerted on an object of mass 1 kg by an
object of mass 1000 kg if the objects are 1 m apart?
143)
A)
6.67 ×10–8 N
B)
1 ×10–8 N
C)
6.67 ×10–14 N
D)
1 ×10–5 N
144)
f(x, y) =x3+y3– 48x – 147y – 5
144)
A)
Relative maximum =809
B)
Saddle point =(4, –7), saddle point =(–4, 7)
C)
Relative minimum = –819, saddle point =(4, –7), saddle point =(–4, 7), relative maximum =
809
D)
Relative maximum =809, relative minimum = –819
145)
Minimum of f(x, y) = x2+ y2,
subject to x + y = 1
145)
A)
Minimum =1
2 at 1
2, 1
2
B)
Minimum =1
2 at (0, 1)
C)
Minimum = 1 at (0, 1)
D)
Minimum = 1 at 1
2, 1
2
146)
Find fxy when f(x, y) = xy2+ yex2+ 5.
146)
A)
2yex2
B)
2xex2
C)
2y + 2xex2
D)
y + xex2
42
147)
Find fxx when f(x,y) = 8x3y – 7y2+ 2x.
147)
A)
48xy
B)
–14
C)
–28
D)
24x2
148)
Find fxy when f(x, y) = 8xexy.
148)
A)
8(2xyexy +exy)
B)
8(2xexy +x2yexy)
C)
8(2xexy + xy2exy)
D)
8x2y2exy
Solve the problem.
149)
A company has the following production function for a certain product:
p(x, y) =31x0.3y0.7 .
Find the marginal productivity with fixed capital, px .
149)
A)
9.3 y
x0.7
B)
9.3 y
x1.3
C)
9.3 x
y0.7
D)
9.3xy0.7
150)
f(x, y) =x3–y4
x2+y2
3
150)
A)
3x3–y4
x2+y2
2 x4+ 3x2y2+ 2xy4
(x2+y2)2
B)
x4–y4
x4+y4
C)
3 x3–y4
x2+y2
2 x4+ 3x2y2+ 2xy4
(x2+y2)4
D)
x3–y4
x2+y2
3(x4+ 3x2y2+ 2y4)(x2+y2)
Solve the problem.
151)
The total cost to hand–produce x large dolls and y small dolls is given by
C(x, y) = 2x2+ 9y2+ 4xy + 60. If a total of 60 dolls must be made, how should production be
allocated so that the total cost is minimized?
151)
A)
Make 0 large dolls and 60 small ones.
B)
Make 59 large dolls and 1 small one.
C)
Make 30 large dolls and 30 small ones.
D)
Make 60 large dolls and 0 small ones.
152)
Find two numbers x and y such that x + y =48 and xy2 is maximized.
152)
A)
x =24 and y =24
B)
x = 1 and y =47
C)
x =12 and y =36
D)
x =16 and y =32
Evaluate the function.
153)
Find f(–3, 1, 5) when f(x, y, z) =1
3x2– 8y5+ z – 4.
153)
A)
–5
B)
10
C)
1
D)
–4
Solve the problem.
154)
Under certain conditions the wind speed S, in miles per hour, of a tornado at a distance d from its
center can be approximated by the function S =aV
0.51d5, where a is an atmospheric constant, and V
is the approximate volume of the tornado, in cubic feet. Assume that a is 0.78, and find SV.
154)
A)
0.78
d5
B)
1.5
d5
C)
1
d5
D)
a
1.02d5
Evaluate the integral.
155)
1
0
x9
0
x dy dx
155)
A)
1
11
B)
1
5
C)
2
11
D)
1
10
156)
If (a, b) is a critical point in the interior of the domain of f(x, y) and if
D = fxx(a, b) fyy(a, b) –[fxy(a, b)]2, then f has a local minimum at (a, b) if
I. D > 0 and fxx(a, b) > 0.
II. D > 0 and fxx(a, b) < 0.
III. D < 0.
156)
A)
Only III is correct.
B)
Only II is correct.
C)
I, II, and III are all incorrect.
D)
Only I is correct.
157)
Suppose a continuous random variable has a joint probability density function given by
f(x, y) =2
5(2x + 3y), 0 x 1, 0 y 1.
Find the probability that a point (x, y) is in the region bounded by 1
2 x 1 and 0 y 1
2 by
evaluating the integral 1/2
0
1
1/2 f(x, y) dx dy.
157)
A)
0.275
B)
0.375
C)
0.225
D)
0.250
158)
2
1
ln x
0ey dy dx
158)
A)
1
2
B)
1
4
C)
9
2
D)
9
4
159)
1
0
1
0
1
0(2x +9y +10z) dz dy dx
159)
A)
64
B)
21
2
C)
7
2
D)
7
160)
Maximum of f(x, y) =9x2+3y2,
subject to x2+y2= 1
160)
A)
Maximum: 9 at (±1, 0)
B)
Maximum: 8 at (0, ±1)
C)
Maximum: 3 at (±1, 0)
D)
Maximum: 3 at (0, ±1)
161)
1
0
3z
0
4–x–2z
0(3x2–y2+ 2z2) dy dx dz
161)
A)
14.5
B)
2.3
C)
1
D)
4.7
Solve the problem. Assume that relative maximum and minimum values are absolute maximum and minimum values.
162)
A rectangular metal tank with an open top is to hold 256 cubic feet of liquid. What are the
dimensions of the tank that require the least material to build?
162)
A)
2 ft by 16 ft by 4 ft
B)
8 ft by 8 ft by 4 ft
C)
2 ft by 2 ft by 64 ft
D)
16 ft by 4 ft by 4 ft
163)
Find fxy when f(x,y) = 10x2y4– 7x3y5.
163)
A)
80xy3– 105x2y4
B)
160xy3– 21x2y4
C)
160xy3– 105x2y4
D)
80xy3– 21x2y4
164)
x 0 3 4 5 12
y 8 2 6 9 12
164)
A)
y = 0.63x + 4.88
B)
y = 0.43x + 4.98
C)
y = 0.53x + 4.88
D)
y = 0.73x + 4.98
Solve the problem.
165)
Ten students in a graduate program were randomly selected. Their grade point averages (GPAs)
when they entered the program were between 3.5 and 4.0. The following data were obtained
regarding their GPAs on entering the program versus their current GPAs.
Entering GPA, x Current GPA, y
3.5
3.8
3.6
3.6
3.5
3.9
4.0
3.9
3.5
3.7
3.6
3.7
3.9
3.6
3.9
3.8
3.7
3.9
3.8
4.0
Find the regression line y = mx + b.
165)
A)
y = 0.0313x + 3.67
B)
y = 0.329x + 2.51
C)
y = 0.497x + 5.81
D)
y = 0.0212x + 4.91
166)
The production function z for an industrial country was estimated as z = x3y5, where x is the
amount of labor and y the amount of capital. Find the marginal productivity of labor.
166)
A)
6x2y5
B)
3x2y5
C)
10x3y4
D)
5x3y4
167)
f(x, y) = 3 –x2
167)
A)
B)
C)
D)
Solve the problem.
168)
The material for the bottom of a rectangular box costs three times as much per square foot as the
material for the sides and top. Find the greatest capacity such a box can have if the total amount of
money available for material is $12.
168)
A)
3 ft3
B)
4 ft3
C)
2 ft3
D)
1 ft3
Find the indicated extreme value of f subject to the given constraint.
169)
Maximum of f(x, y, z) = x + y + z,
subject to 1
x+1
y+1
z= 1
169)
A)
Maximum: 1 at (1, 1, –1), (1, –1, 1), (–1, 1, 1)
B)
Maximum: 9 at (3, 3, 3)
C)
Maximum: 6 at (3, 3, 3)
D)
Maximum: 6 at (2, 2, 2)
Solve the problem.
170)
Find three numbers whose sum is 72 and whose product is a maximum.
170)
A)
36, 18, and 18
B)
24, 18, and 18
C)
36, 36, and 36
D)
24, 24, and 24
171)
Find the dimensions of the right circular cylinder with maximum surface area, if its volume is
64 ft3.
171)
A)
r = 8 ft, h =
16 ft
B)
r = 3 ft, h = 2 ft
C)
r = 6 ft, h = 1 ft
D)
r =2 ft, h =32
ft
172)
5
0
z
1
x/z
0
2xyz dy dx dz
172)
A)
24
B)
626
C)
156
D)
605
12
173)
A company that manufactures tennis rackets has determined that the demand functions for the two
types of rackets they produce are given by q1= 54 – 2x + y and q2= 398 + x – 9y where q1 is the
price of the standard tennis racket, q2 is the price of the competition tennis racket, x is the weekly
demand for standard rackets, and y is the weekly demand for competition rackets. How many of
each type of racket must be produced to maximize revenue?
173)
A)
26 standard rackets and 25 competitive rackets
B)
25 standard rackets and 25 competitive rackets
C)
25 standard rackets and 26 competitive rackets
D)
26 standard rackets and 26 competitive rackets
174)
Let z = f(x,y) =5x2– 15xy + 2y3. Find z
x.
174)
A)
10x – 15y
B)
–15x – 6y
C)
–15x + 6y2
D)
10x + 15y2
175)
f(x, y) = x3– 12xy + 8y3
175)
A)
f(1, 2) = 9, relative minimum
B)
f(2, 1) = –8, relative minimum
C)
f(2, 1) = –8, relative maximum
D)
f(1, 2) = 9, relative maximum
176)
z =6xy –5y;
z
y(1, 3)
176)
A)
1
B)
6
C)
18
D)
0
Solve the problem.
177)
A company’s monthly sales, in thousands, is given by S(x, y) =7x0.8y0.3, where x is the amount
spent on newspaper advertising per month in thousands of dollars and y is the amount spent on
radio advertising per month in thousands of dollars. Suppose the company currently spends $4000
on newspaper advertising per month and $5000 on radio advertising per month. What would be
the effect on sales if the company increases the amount spent on newspaper advertising to $5000,
while the amount spent on radio advertising remains constant?
177)
A)
Sales would increase by $2063.42.
B)
Sales would increase by $6577.87.
C)
Sales would decrease by $657.79.
D)
Sales would increase by $13,410.71.
178)
A farmer has 780 m of fencing. Find the dimensions of the rectangular field of maximum area that
can be enclosed by this amount of fencing.
178)
A)
78 m by 312 m
B)
195 m by 195 m
C)
195 m by 585 m
D)
185 m by 205 m
179)
f(x, y) =x2+y2–9x –11y
179)
A)
No relative extrema or saddle points
B)
Saddle point =9
2, 11
2
C)
Relative minimum = –50.5
D)
Relative maximum =202, relative minimum = –40, saddle point =9
2, 11
2
Solve the problem.
180)
The price–earnings ratio of a stock is given by R(P, E) =P
E, where P is the price per share of a stock,
and E is the earnings per share of the same stock. If the price per share of a given stock is $240, and
the earnings per share of the same stock are $27.65, what is the price–earnings ratio of the stock?
Give decimal notation to the nearest tenth.
180)
A)
267.7
B)
0.1
C)
6636.0
D)
8.7
Find the partial derivative.
181)
Let z = f(x, y) =6(x +7y –8)2. Find z
y.
181)
A)
42x +294y –336
B)
84x +588y +672
C)
42x +294y
D)
84x +588y –672
Solve the problem. Assume that relative maximum and minimum values are absolute maximum and minimum values.
182)
A firm produces two kinds of tennis balls, one for recreational players which sells for $2.50 per can,
and one for serious players which sells for $4.00 per can. The total revenue from the sale of x
thousand cans of the first ball and y thousand cans of the second ball is given by R(x, y) = 2.5x + 4y.
The company determines that the total cost, in thousands of dollars, of producing x thousand cans
of the first ball and y thousand cans of the second ball is given by C(x, y) =x2– 2xy + 2y2.Find the
number of each type of ball which must be produced and sold in order to maximize the profit.
182)
A)
4000 of the $2.50 cans and 3000 of the $4.00 cans
B)
3000 of the $2.50 cans and 4000 of the $4.00 cans
C)
5000 of the $2.50 cans and 3000 of the $4.00 cans
D)
2000 of the $2.50 cans and 5000 of the $4.00 cans
183)
5
–10
3
24x dy dx
183)
A)
– 450
B)
150
C)
– 30
D)
– 150
184)
Maximum of f(x, y) =x2+ 4y3,
subject to x2+ 2y2= 2
184)
A)
Maximum: –31 at (1, –2)
B)
Maximum: –4 at (0, –1)
C)
Maximum: 4 at (0, 1)
D)
Maximum: 8 at (2, 1)
53
Solve the problem.
185)
The intelligence quotient in psychology is given by Q(m, c) = 100 ·m
c, where m is a person’s mental
age, and c is his or her chronological , or actual, age. Find Q(19, 17) and round the answer to the
nearest whole number.
185)
A)
112
B)
11
C)
89
D)
100
186)
f(x, y) = 4xy – x2y – xy2
186)
A)
No relative extrema
B)
f(4
3, 4
3) = 2.37, relative maximum
C)
f(2
3, 2
3) = 1.185, relative maximum
D)
f(0, 0) = 0, relative minimum
Find the minimum or maximum value of f (as indicated) subject to the given constraint.
187)
Minimum of f(x, y) = x2+ 4y2+ 6,
subject to 2x – 8y = 20
187)
A)
Minimum = 26 at (2, –2)
B)
Minimum = 26 at (2, 2)
C)
Minimum = 26 at (–2, 2)
D)
Minimum = –26 at (–2, –2)
Evaluate the integral.
188)
1
0
y
0ex + y dx dy
188)
A)
1
2(e2– e)2
B)
1
2(e – 1)2
C)
1
e(e2– e)2
D)
1
3(e – 1)2
189)
8
0
3
0(9x – 3y) dx dy
189)
A)
12
B)
9
2
C)
36
D)
3
2
190)
What are the dimensions of a rectangular box, open at the top, which has maximum volume when
the surface area is 48 in.2?
190)
A)
x = 8 in., y = 2 in., z = 6 in.
B)
x = 4 in., y = 2 in., z = 2 in.
C)
x = 4 in., y = 4 in., z = 2 in.
D)
x = 6 in., y = 6 in., z = 3 in.
Find fx.
191)
f(x, y) = x ln (8x +5y)
191)
A)
fx(x, y) = ln (8x +5y) +8
8x +5y
B)
fx(x, y) =8x
8x +5y
C)
fx(x, y) = ln (8x +5y)
D)
fx(x, y) =8x
8x +5y + ln (8x +5y)
Find the second–order partial derivative.
192)
Find fyx when f(x, y) = ln(2x + 9y).
192)
A)
9
(2x + 9y)2
B)
–18
(2x + 9y)2
C)
18
(2x + 9y)2
D)
–9
(2x + 9y)2
55
Find the relative maximum and minimum values and the saddle points if they exist.
193)
f(x, y) = –5xy(x + y) + 6
193)
A)
Relative minimum =6
B)
Saddle point = (0, 0), relative maximum =1256
C)
Relative maximum =1256
D)
Saddle point = (0, 0)
194)
Let f(x, y) =x3– 5x2y + 7xy3. Find fx.
194)
A)
x2– 5xy + 7y3
B)
3x2+ 2xy + 7y3
C)
3x2
D)
3x2– 10xy + 7y3
195)
Let z = g(x,y) =2x + 9x2y2– 5y2. Find z
y.
195)
A)
2+ 18x2y
B)
18x2y – 10y
C)
2+ 18xy2
D)
18xy2– 10y
Find the second–order partial derivative.
196)
Find fyy when f(x,y) = 8x3y – 7y2+ 2x.
196)
A)
–28
B)
48xy
C)
24x2
D)
–14
197)
Find fxy when f(x, y) =x
x + y .
197)
A)
2y
(x + y)3(x + y)3
B)
x
(x + y)3
C)
y – x
(x + y)3
D)
x – y
(x + y)3
Find the indicated extreme value of f subject to the given constraint.
198)
Maximum of f(x, y, z) =x3+y3+z3,
subject to x2+y2+z2= 4
198)
A)
Maximum: –8 at (–2, 0, 0), (0, –2, 0), (0, 0, –2)
B)
Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2)
C)
Maximum: 8 at (2, 0, 0)
D)
Maximum: –8 at (2, 0, 0)
Solve the problem.
199)
Consider these midterm and final exam scores for three students in a calculus class.
midterm, x final, y
71
80
73
83
76
77
Find the regression line y = mx + b.
199)
A)
y = –x + 3.1
B)
y = –0.61x + 124.2
C)
y = –0.65x + 100.4
D)
y = 0.256x – 12.3
Find fx.
200)
f(x, y) =x2+y2
xy
200)
A)
fx(x, y) =2x2+ 2y2–x2y2
xy
B)
fx(x, y) =x2–y2
xy
C)
fx(x, y) =x2–y2
x2
D)
fx(x, y) =yx2–y3
x2y2
57
Solve the problem.
201)
Of all points (x, y, z) that satisfy x – y + 3z = 7, find the one that minimizes (x – 2)2+(y + 1)2+
(z – 4)2.
201)
A)
–14
11, 3
11, 20
11
B)
14
11, 3
11, –20
11
C)
14
11, –3
11, 20
11
D)
14
11, 3
11, 20
11
Find the minimum or maximum value of f (as indicated) subject to the given constraint.
202)
Minimum of f(x, y) = x2+ y2– xy,
subject to x – y = 10
202)
A)
Maximum = 3 at (1, 2)
B)
Maximum = 25 at (5, 5)
C)
Maximum = 75 at (5, –5)
D)
Maximum = 7 at (2, –1)
C
203)
Minimum of f(x, y) = xy,
subject to x2+y2=200
203)
A)
Minimum: 100 at (10, –10) and (–10, 10)
B)
Minimum: 100 at (10, 10)
C)
Minimum: –100 at (10, –10) and (–10, 10)
D)
Minimum: 0 at (0, 0)
C
204)
f(x, y) =e(x2+y2)
204)
A)
f(0, 1) = 0, relative minimum
B)
f(0, 0) = 1, relative minimum
C)
f(0, 1) = 0, relative maximum
D)
f(0, 0) = 1, relative maximum
B
58
C
Solve the problem.
205)
The yield of a stock is given by Y(D, P) =D
P, where D is the dividends per share of a stock, and P is
the price per share of the stock. If the price per share of a stock is $120, and the dividends per share
of the same stock are $4.35, then what is the yield of the stock to the nearest tenth of a percent? Give
percent notation to the nearest tenth of a percent.
205)
A)
522.0%
B)
3.6%
C)
27.6%
D)
1.8%
206)
–8
–10
9
54y dx dy
206)
A)
– 32
B)
– 64
C)
– 288
D)
224
207)
Minimum of f(x, y, z) = 4x – 3y + 2z,
subject to x2+y2= 6z
207)
A)
Minimum: 225
4 at 6, –9
2, 225
24
B)
Minimum: –75
4 at –6, 9
2, 225
24
C)
Minimum: 117
4 at 6, 9
2, 225
24
D)
Minimum: –33
4 at 6, 9
2, –225
24
208)
A computer firm markets two kinds of electronic calculator that compete with one another. The
total revenue function is R(p, q) = 80p – 6p2– 4pq + 68q – 2q2, where p is the price of the first
calculator (in multiples of $10), and q is the price of the second calculator (in multiples of $10).
What prices should be charged in order to maximize the total revenue?
208)
A)
$5 and $90
B)
$20 and $120
C)
$6.70 and $180
D)
$40 and $170
Solve the problem.
209)
Suppose that the manufacturing cost of a precision instrument is approximated by
M(x,y) =15x2+ 10y2– 8xy, where x is the cost of materials and y is the cost of labor. Find Mx(5, 2).
209)
A)
295
B)
134
C)
–40
D)
0
210)
The intelligence quotient in psychology is given by Q(m, c) = 100m
c, where m is an individual’s
mental age and c is the individual’s chronological, or actual, age. Find Q
m .
210)
A)
100
c
B)
m
c
C)
–100m
c2
D)
100m
211)
ln 5
0
5
eyey dx dy
211)
A)
9
B)
8
C)
18
D)
4
Solve the problem.
212)
The surface area of a human body (in square meters) is approximated by A = 0.202W0.425H0.725,
where W is the weight of the person in kilograms and H is the height in meters. Find A if W =65
and H =1.65.
212)
A)
1.62 m2
B)
1.96 m2
C)
1.72 m2
D)
1.71 m2
213)
f(x, y) =4x
y–y
4x
213)
A)
fx(x, y) =4x2
y+y
4x2
B)
fx(x, y) = – 4
y2–4
y
C)
fx(x, y) =4
y+y
4x2
D)
fx(x, y) =4
y–y
4x2
214)
f(x, y, ) =x2y +y2 + x2
214)
A)
fx= 2xy +2; fy=x2+ y; f=y2+ x
B)
fx= 2xy; fy=x2+ 2y; f=y2+ 2x
C)
fx= 2xy +2; fy=x2+ 2y; f=y2+ 2x
D)
fx= 2y +2; fy=x2+ 2; f=y2+ 2x
215)
1
–1
4
0
5
0(x2+y2+z2) dx dy dz
215)
A)
189
B)
–61
C)
23.2
D)
560
216)
f(x, y) = y ln 1
x
216)
A)
fx(x, y) = – 1
x
B)
fx(x, y) = – y
x2
C)
fx(x, y) =y
x2
D)
fx(x, y) = – y
x
Evaluate the function.
217)
Find f(4, 0, 9) when f(x, y, z) =4x2+4y2–z2.
217)
A)
–65
B)
–14
C)
145
D)
–17
218)
2
0
9–z2
0
x
0xy dy dx dz
218)
A)
37
5
B)
602
5
C)
20
D)
129.6
Use a 3D grapher to graph the function. Then estimate any relative extrema.
219)
f(x, y) =ex+y
219)
A)
No relative extrema
B)
Relative minimum = 1
C)
Relative minimum = e
D)
Relative minimum = 0
220)
f(x, y) = x2– y2
220)
A)
f(0, 0) = 0, relative minimum
B)
No relative extrema
C)
f(0, 0) = 0, relative maximum
D)
f(1, 1) = 0, relative maximum
221)
Let f(x, y) =3x –6y2–9. Find fx.
221)
A)
–6
B)
3x
C)
–12y
D)
3
Find the requested partial derivative.
222)
f(x, y) =6x + 4y; fx(6, –2)
222)
A)
6
B)
–8
C)
4
D)
36
223)
Let z = g(x,y) =6x + 9x2y2– 8y2. Find z
y.
223)
A)
6+ 18yx2
B)
18yx – 18y
C)
18yx – 16y
D)
18yx2– 16y
224)
–2
–8
10
3dy dx
224)
A)
42
B)
1
C)
–44
D)
4
225)
f(x, y) =x19 – 5x3y23 + 4y–7
225)
A)
19x18 –15x2y23
B)
19x18 –15x3y23
C)
19x18 –15x3y22 – 28y–8
D)
19x18 – 5x3y23 + 4y–7
63
Use a 3D grapher to generate the graph of the function.
226)
f(x, y) = 2 –x2–y2
226)
A)
B)
C)
D)
227)
Find f(0, 1, –1) when f(x, y, z) =8x–4yz +5x.
227)
A)
–3
B)
5
C)
4
D)
–4
64
Find the relative maximum and minimum values and the saddle points if they exist.
228)
f(x, y) =exy
228)
A)
No relative extrema or saddle points
B)
Saddle point = (0, 0)
C)
Relative minimum = 1
D)
Relative minimum = 0
229)
2
0
3
0
5
0xyz dx dy dz
229)
A)
150
B)
225
C)
225
2
D)
75
230)
Find h(3, 6) when h(x, y) =3x +y2.
230)
A)
10
B)
9
C)
5 3
D)
3 5
Find the minimum or maximum value of f (as indicated) subject to the given constraint.
231)
Maximum of f(x, y) = 4xy,
subject to x + y = 8
231)
A)
Maximum = 64 at (3, 5)
B)
Maximum = 72 at (0, 8)
C)
Maximum = 72 at (2, 6)
D)
Maximum = 64 at (4, 4)
232)
Do more data points always make a regression line a better predictor?
232)
A)
Yes
B)
No
Evaluate the triple integral.
233)
2
0
y2
2
z
5yz dx dz dy
233)
A)
68
3
B)
23
3
C)
–124
3
D)
–128
3
Answer Key
Testname: C6
Answer Key
Testname: C6
Answer Key
Testname: C6
Answer Key
Testname: C6
Answer Key
Testname: C6