Find the double integral over the rectangular region R with the given boundaries.
28)
(5xy) dx dy
R
0 x 3, 0 y 1
28)
A)
9
4
B)
45
C)
45
4
D)
45
2
Find the indicated relative minimum or maximum.
29)
Minimum of f(x,y) = x2+ y2,
subject to x + y = 1
29)
A)
f 1
2, 1
2= 1
B)
f(0, 1) =1
2
C)
f(0, 1) = 1
D)
f 1
2, 1
2=1
2
Find the average value of the function f over the region R.
30)
f(x, y) =e7x; 0 x 1
7, 0 y 1
7
30)
A)
e – 1
B)
e – 1
7
C)
2e – 1
49
D)
2e – 1
Solve the problem.
31)
A rectangular box with square base and no top is to have a volume of 32 ft3. What is the least
amount of material required?
31)
A)
42 ft2
B)
36 ft2
C)
40 ft2
D)
48 ft2
32)
Find two positive numbers x and y such that x + y =72 and xy2 is maximized.
32)
A)
x =24 and y =48
B)
x = 1 and y =71
C)
x =18 and y =54
D)
x =36 and y =36
Find the second–order partial derivative.
33)
Find fyy when f(x,y) = 8x3y – 7y2+ 2x.
33)
A)
–28
B)
48xy
C)
24x2
D)
–14
D)
Solve the problem.
34)
The material for the bottom of a rectangular box costs $3 per square foot while the material for the
sides and top costs $1 per square foot. Find the greatest capacity such a box can have if the total
amount available for material is $12.
34)
A)
4 ft3
B)
2 ft3
C)
3 ft3
D)
1 ft3
D)
Evaluate the iterated integral.
35)
2
0
2
–3
(x3+ 2x2y – y3+ xy) dy dx
35)
A)
211
6
B)
209
6
C)
203
6
D)
205
6
D)
22
D)
Find the indicated derivative.
36)
Find fxz for f(x, y, z) =7x6y6+ 10x9z3.
36)
A)
42x5y6+ 270x8z2
B)
30x9z2
C)
270x8z2
D)
3z2
Evaluate the iterated integral.
37)
2
0
1
0
y y2+ 7x dy dx
37)
A)
2
15 (155/2 –145/2 – 1)
B)
2
105(155/2 –145/2 )
C)
2
15 (155/2 – 1)
D)
2
105(155/2 –145/2 – 1)
D)
Evaluate dz.
38)
z =x2+9y
7x2– y ; x = – 3, y =4, dx = – 0.01, dy = – 0.02
38)
A)
0.00097
B)
–0.00386
C)
–0.00772
D)
–0.00493
D)
Solve the problem.
39)
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.
39)
A)
2 ft by 4 ft by 2 ft
B)
2 ft by 2 ft by 4 ft
C)
2 ft by 2 ft by 8 ft
D)
1 ft by 1 ft by 16 ft
D)
D)
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.
40)
z = x3; 0 x 2, 0 y 3
40)
A)
4
B)
18
C)
6
D)
12
Solve the problem.
41)
A computer firm markets two kinds of electronic calculator that compete with one another. The
total revenue function is Rp, 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?
41)
A)
$5 and $90
B)
$20 and $120
C)
$15 and $155
D)
$40 and $170
Evaluate the function.
42)
Find f(3, 0, 8) when f(x, y, z) =3x2+3y2–z2.
42)
A)
–55
B)
91
C)
–37
D)
–31
Find values of x and y such that both fx(x, y) = 0 and fy(x, y) = 0.
43)
f(x, y) = x2+ xy + y2– 3x + 2
43)
A)
x = 2, y = – 1
B)
x = 1, y = – 1
2
C)
x = 0, y = 0
D)
x = – 2, y = 1
Find the partial derivative.
44)
Find fx(4, –3) when f(x, y) =(6x +2y)2.
44)
A)
108
B)
288
C)
216
D)
16
24
Evaluate dz.
45)
z =7x2+ 5xy + 8y2;
x = – 4, y =7, dx = – 0.02, dy = – 0.01
45)
A)
1.63
B)
–1.63
C)
–0.50
D)
0.50
Use the total differential to approximate the quantity. Then use a calculator to approximate the quantity, and give the
absolute value of the difference of the two results to four decimal places.
46)
1.022+3.982
46)
A)
4; 4.1086; 0.1086
B)
3.150; 3.1590; 0.0090
C)
3.1086; 4.1086; 1.0000
D)
4.1086; 4.1086; 0
Evaluate the double integral.
47)
2
0
x2
0
(x2– y2) dy dx
47)
A)
32
115
B)
44
105
C)
44
115
D)
32
105
Find all points where the function has any relative extrema or saddle points and identify the type of relative extremum.
48)
f(x,y) = 4xy – x2y – xy2
48)
A)
Relative maximum at 4
3, 4
3 and saddle point at 2
3, 2
3
B)
Relative maximum at 4
3, 4
3 and saddle point at (0, 0)
C)
Relative minimum at 4
3, 4
3 and saddle point at (0, 0)
D)
Relative minimum at 2
3, 2
3 and saddle point at (0, 0)
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.
49)
z = (x + y)2; –1 x 1, –1 y 1
49)
A)
2
3
B)
8
3
C)
4
3
D)
1
3
Find the double integral over the rectangular region R with the given boundaries.
50)
(x +x2y4– 3) dy dx
R
0 x 1, 0 y 2
50)
A)
–43
15
B)
11
15
C)
14
15
D)
13
15
Find fx(x, y).
51)
f(x, y) =x2+y2
xy
51)
A)
fx(x, y) =2x2+ 2y2–x2y2
xy
B)
fx(x, y) =yx2–y3
x2y2
C)
fx(x, y) =x2–y2
xy
D)
fx(x, y) =x2–y2
x2
Solve the problem.
52)
If a, b, and c are all nonzero, then determine if the plane ax + by + cz = 1 always intersects all three
coordinate planes (i.e. the planes x = 0, y = 0, z = 0), sometimes intersects them, or never intersects
them.
52)
A)
Always
B)
Never
C)
Sometimes
26
Find fx(x, y).
53)
f(x, y) =x19 – 5x8y23 + 4y–7
53)
A)
19x18 –40x8y23
B)
19x18 – 5x8y23 + 4y–7
C)
19x18 –40x7y23
D)
19x18 –40x8y22 – 28y–8
Solve the problem.
54)
Approximate the amount of aluminum needed for a beverage can of radius 2.8 cm and height 16
cm. Assume the walls of the can are 0.1 cm thick.
54)
A)
4.93 cm3
B)
33.07 cm3
C)
28.15 cm3
D)
330.75 cm3
Find fx(x, y).
55)
f(x, y) =e–x
x2+y2
55)
A)
e–x(x2+y2+ 2x)
(x2+y2)2
B)
–e–x(x2+y2+ x)
(x2+y2)2
C)
–2xe–x
(x2+y2)2
D)
–e–x(x2+y2+ 2x)
(x2+y2)2
Answer the question.
56)
Given f(x,y) = x2+ y2, the purpose of finding the critical points is which of the following?
(i) find local minima, if they exist
(ii) find local maxima, if they exist
(iii) find saddle points, if they exist
(iv) find interior points, if they exist
56)
A)
(i), (ii), and (iii) are all correct.
B)
Only (iii) is correct.
C)
All are correct.
D)
Only (i) and (ii) are correct.
27
Find the second–order partial derivative.
57)
Find gyx when g(x, y) =x
y.
57)
A)
1
y
B)
0
C)
–1
y2
D)
x2
y2
Evaluate the iterated integral.
58)
2
0
4
0
(1 + x + y) dx dy
58)
A)
26
B)
11
C)
32
D)
18
Find the second–order partial derivative.
59)
Find fxy when f(x,y) = 8x3y – 7y2+ 2x.
59)
A)
–28
B)
48xy
C)
–14
D)
24x2
Evaluate the function.
60)
Find h(3, 1) when h(x, y) = (x + y)3.
60)
A)
12
B)
28
C)
64
D)
16
28
Evaluate the double integral.
61)
1
0
y
0
ex + y dx dy
61)
A)
1
e(e2– e)2
B)
1
2(e – 1)2
C)
1
2(e2– e)2
D)
1
3(e – 1)2
Solve the problem.
62)
Determine the type of curve given by vertical slices of the graph of the equation
f(x, y) =5x2+ 3y2+ 1.
62)
A)
Circle
B)
Ellipse
C)
Parabola
D)
None of these
63)
Determine the type of curve given by horizontal slices of the graph of the equation
f(x, y) =3x2+ 5y2+ 2.
63)
A)
Paraboloid
B)
Ellipse
C)
Circle
D)
None of these
Evaluate the integral.
64)
12
1
x x2+ 5y dx
64)
A)
1
3[(x2+12)3/2 –(x2+ 1)3/2 ]
B)
2
3[(144 + 5y)3/2 –(1 + 5y)3/2 ]
C)
1
3[(144 + 5y)3/2 –(1 + 5y)3/2 ]
D)
1
3[12(144 + 5y)3/2 –(1 + 5y)3/2 ]
29
Solve the problem.
65)
The resistance R of a circuit containing two resistors in parallel is 1
R=1
R1
+1
R2, where the two
resistors have resistances R1 and R2. For resistors with R1=180 ±0.5 ohm and R2=110 ±0.7 ohm,
approximate the error in R.
65)
A)
1.0
B)
–0.2
C)
0.8
D)
0.3
Use the total differential to approximate the quantity. Then use a calculator to approximate the quantity, and give the
absolute value of the difference of the two results to four decimal places.
66)
0.98ln(1.02)
66)
A)
1; .02; 0.9800
B)
0.02; 0.01; 0.0100
C)
0.02; 0.0194; 0.0006
D)
0; 0.0194; 0.0194
Solve the problem.
67)
Determine if the graph of the equation f(x, y) =5– x2– y2 is a sphere, a hemisphere, a
paraboloid, or none of these.
67)
A)
Sphere
B)
Paraboloid
C)
Hemisphere
D)
None of these
68)
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+y2–11x –3y. What is the maximum temperature on the plate?
68)
A)
126°
B)
18°
C)
75°
D)
No maximum
Use the region R to evaluate the double integral.
69)
x2+y2 dx dy
R
R bounded by x =3, y = 0, and y =10x
69)
A)
1545
2
B)
4635
2
C)
7290
D)
13,905
2
Evaluate the iterated integral.
70)
5
2
4
1
x
y+4x
3 dy dx
70)
A)
21
2ln 4 +168
3
B)
45 +15
2ln 5 –15
2ln 2
C)
45 +15
2ln 5
D)
21
2 ln 4 + 42
Find the indicated relative minimum or maximum.
71)
Minimum of f(x,y) = x2+ y2– xy,
subject to x – y = 10
71)
A)
f(5, –5) = 75
B)
f(1, 2) = 3
C)
f(5, 5) = 25
D)
f(2, –1) = 7
Solve the problem.
72)
The production function for a certain country is z = x0.5y0.4, where x stands for units of labor and y
for units of capital. At present, x is 33 and y is 51. Use differentials to estimate the change in z if x
becomes 39 and y becomes 53.
72)
A)
2.95
B)
3.82
C)
2.14
D)
1.27
Graph the level curves in the first quadrant of the xy–plane for the given function at heights of z = 0, z = 2, and z = 4.
31
73)
–y2
5+ x =4z
73)
A)
z = 4
z = 2
z = 0
B)
z = 4
z = 2
z = 0
C)
z = 4
z = 2
z = 0
D)
z = 4
z = 2
z = 0
Find the partial derivative.
74)
Let z = f(x,y) =9x2– 20xy + 7y3. Find z
x.
74)
A)
18x – 20y + 21y2
B)
18x – 20y
C)
18x2– 20x
D)
–20x + 21y2
Solve the problem.
75)
The profit (in thousands of dollars) that a company earns from producing x tons of
brass and y tons of steel can be approximated by P(x, y) =68xy – 8x3–y3. Find the amount of brass
and steel that maximize profit and find the value of the maximum profit.
75)
A)
17
3 tons of brass and 34
3 tons of steel; maximum profit is $1,455,704
B)
34
3 tons of brass and 17
3 tons of steel; maximum profit is $1,455,704
C)
34
5 tons of brass and 17
2 tons of steel; maximum profit is $800,858
D)
34
7 tons of brass and 68
5 tons of steel; maximum profit is $1,064,667
Evaluate the integral.
76)
11
2
3x + 4y dy
76)
A)
2
27 (4y + 33)3/2 –2
27 (4y + 6)3/2
B)
1
18 (3x + 44)3/2 –1
18 (3x + 8)3/2
C)
2
9(4y + 33)3/2 –2
9(4y + 6)3/2
D)
1
6(3x + 44)3/2 –1
6(3x + 8)3/2
33
Find fx(x, y).
77)
f(x, y) = y ln (4x +9y)
77)
A)
fx(x, y) =4xy
4x +9y
B)
fx(x, y) =4y
4x +9y
C)
fx(x, y) = y ln (4x +9y)
D)
fx(x, y) = ln (4x +9y) +4
4x +9y
Answer the question.
78)
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 saddle point at (a,b) if
(i) D > 0 and fxx(a,b) > 0.
(ii) D > 0 and fxx(a,b) < 0.
(iii) D < 0.
78)
A)
Only (i) is correct.
B)
Only (iii) is correct.
C)
Only (ii) is correct.
D)
None of the above is correct.
Find fx(x, y).
79)
f(x, y) =
5x7y4
79)
A)
7x64x7y4
B)
75x7y4
5x
C)
7x6y4
5x7y4
D)
75x7y4
5xy4
Solve the problem.
80)
The total cost to hand–produce x large dolls and y small dolls is given by
C(x,y) = 2x2+ 5y2+ 4xy + 20. If a total of 20 dolls must be made, how should production be
allocated so that the total cost is minimized?
80)
A)
Make 10 large dolls and 10 small ones
B)
Make 20 large dolls and 0 small ones
C)
Make 0 large dolls and 20 small ones
D)
Make 19 large dolls and 1 small one
Find the partial derivative.
81)
Let z = f(x, y) =4(x +4y –3)2. Find z
y.
81)
A)
32x +128y –96
B)
16x +64y –48
C)
16x +64y
D)
32x +128y +96
82)
f(x,y) =2x + 3x2y2– 4y2. Find fx(x, y).
82)
A)
6x2y – 8y
B)
2+ 6xy2
C)
6xy2– 8y
D)
2+ 6x2y
Answer the question.
83)
If (a,b) is a critical point in the interior of the domain of f(x,y) and if
H = fxx(a,b) fyy(a,b) –fxy(a,b) 2, then f has a local minimum at (a,b) if
(i) H > 0 and fxx(a,b) > 0.
(ii) H > 0 and fxx(a,b) < 0.
(iii) H < 0.
83)
A)
Only (i) is correct.
B)
Only (iii) is correct.
C)
Only (ii) is correct.
D)
(i), (ii), and (iii) are all incorrect.
Find fx(x, y).
84)
f(x, y) = ln y5
x10
84)
A)
–ln 10y5
x11
B)
–ln 10
x
C)
–10
x
D)
–10y5
x
Evaluate the double integral.
85)
6
0
x/2
0
(x + y) dy dx
85)
A)
36
B)
45
C)
63
D)
54
Find all points where the function has any relative extrema or saddle points and identify the type of relative extremum.
86)
f(x, y) =e–(x2+y2–8y)
86)
A)
Relative maximum at at (0, 4) and relative minimum at at (0, –4)
B)
Relative maximum at (0, 4)
C)
Saddle point at (0, 4)
D)
No relative extremum or saddle points.
87)
f(x, y) =3xy
87)
A)
Saddle point at (0, 0)
B)
Relative minimum at (–1, –1), saddle point at (0, 0)
C)
Relative maximum at (0, 0)
D)
No relative extrema or saddle points
Solve the problem.
88)
A company has the following production function for a certain product:
p(x, y) =28x0.2y0.8 .
Find the marginal productivity with fixed labor, py .
88)
A)
22.4 x
y
0.2
B)
22.4 y
x
0.8
C)
22.4yx0.2
D)
22.4 y
x
0.2
Use the total differential to approximate the quantity. Then use a calculator to approximate the quantity, and give the
absolute value of the difference of the two results to four decimal places.
89)
(5.012–0.932)1/3
89)
A)
1.8941; 2.8939; 1.0002
B)
2.8941; 2.8939; 0.0002
C)
2; 2.8939; 0.8939
D)
2.8941; 2; 0.8941
Find the indicated derivative.
90)
Find fx for f(x, y, z) =2x10y9z8 ln 6x7.
90)
A)
20x9y9z8 ln 42x6
B)
840x9y9z8 ln 42x6
C)
14x10y9z8+ 20x9y9z8 ln 42x6
D)
14x9y9z8+ 20x9y9z8 ln 6x7
Evaluate the function.
91)
Find g(–2, –8) when g(x, y) =3y2– 9xy.
91)
A)
–132
B)
–140
C)
50
D)
48
Answer the question.
92)
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 maximum at (a,b) if
(i) D > 0 and fxx(a,b) > 0.
(ii) D > 0 and fxx(a,b) = 0.
(iii) D < 0.
92)
A)
(i), (ii), and (iii) are all incorrect.
B)
Only (iii) is correct.
C)
Only (i) is correct.
D)
Only (ii) is correct.
Evaluate the double integral.
93)
8
0
8x
0
x2 dy dx
93)
A)
4096
15
B)
8192
7
C)
4096
7
D)
8192
15
Find the indicated relative minimum or maximum.
94)
Minimum of f(x,y) = x2– 14x + y2– 16y,
subject to 2x + 3y = 12
94)
A)
f(1, 5) = – 68
B)
f(2, 0) = – 24
C)
f(0, 1) = – 15
D)
f(3, 2) = – 61
Solve the problem.
95)
The number of liters of blood pumped through the lungs in one minute is given by
C =b
a – v
Suppose a = 150, b = 180, v = 120. Estimate the change in C if a becomes 140, b becomes 186, and v
changes to 128.
95)
A)
–0.2 liters
B)
3.8 liters
C)
3.4 liters
D)
0.6 liters
Find all points where the function has any relative extrema or saddle points and identify the type of relative extremum.
96)
f(x, y) =1
x+1
y
96)
A)
Relative maximum at (1, 1)
B)
Relative minimum at (100, 100)
C)
Saddle point at (0, 0)
D)
No relative extrema
Solve the problem.
97)
Assuming that a cylindrical container can be mailed only if the sum of its height and circumference
do not exceed 240 centimeters, what are the dimensions of the cylinder with the largest volume that
can be mailed?
97)
A)
Height 80 centimeters and radius 240/ centimeters
B)
Height 240 centimeters and radius 80/ centimeters
C)
Height 160 centimeters and radius 80 centimeters
D)
Height 80 centimeters and radius 80/ centimeters
Find the indicated derivative.
98)
Find fx for f(x, y, z) =6x6y9+ 4x10z5+ 4y4.
98)
A)
324x5y8+ 200x9z4
B)
36x5+ 40x9
C)
54x6y8+ 20x10z4
D)
36x5y9+ 40x9z5
Answer the question.
99)
True or false? Consider the double integral
2
0
6
3
x2+ y dy dx.
The first step in calculating this integral involves holding y constant.
99)
A)
True
B)
False
Evaluate the integral.
100)
3
0
xy(5x –1) dx
100)
A)
29
B)
3x(5x –1)
C)
81
2y
D)
42y
Evaluate the iterated integral.
101)
1
–3
4
0
x2+y2 dx dy
101)
A)
704
3
B)
92
3
C)
284
3
D)
368
3
Find fx(x, y).
102)
f(x, y) =e5x – 5y
102)
A)
fx(x, y) =e5x – 5
B)
fx(x, y) =5e5x
C)
fx(x, y) = – 5e5x – 5y
D)
fx(x, y) =5e5x – 5y
Answer the question.
103)
True or false? Consider the double integral
2
0
6
3
x2+ y dx dy.
The first step in calculating this integral involves holding y constant.
103)
A)
True
B)
False