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Deck 8: Calculus of Several Variables
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Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
A) 9
B) 4
C) 72
D) 63
E) 8
for the given function f(x, y) and the region R. A) 9
B) 4
C) 72
D) 63
E) 8
Question
Use a double integral to find the volume of the solid shown in the figure. 
A) 14
B) 6
C) 4
D) 9
A) 14
B) 6
C) 4
D) 9
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. f(x, y) = 6x + 6y; R is bounded by x = 0, 
, y = 0 and y = 4.
A)
B)
C)
D)
for the given function f(x, y) and the region R. f(x, y) = 6x + 6y; R is bounded by x = 0, , y = 0 and y = 4.A)
B)
C)
D)
Question
Use a double integral to find the volume of the solid shown in the figure. 
A)
B)
C)
D)
A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. f(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.
A)
B)
C)
D)
for the given function f(x, y) and the region R. f(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. f(x, y) = 5 
; R is bounded by the lines x = 1, y = 0 and y = x.
A)
B)
C)
D)
E)
for the given function f(x, y) and the region R. f(x, y) = 5 ; R is bounded by the lines x = 1, y = 0 and y = x.A)
B)
C)
D)
E)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is the rectangle defined by 
A)
B)
C)
D)
for the given function f(x, y) and the region R. ; R is the rectangle defined by A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is bounded by x = 0, x = 1, y = 0 and y = x.
A) 4
B) 2e
C) - 2
D) - 4
for the given function f(x, y) and the region R. ; R is bounded by x = 0, x = 1, y = 0 and y = x.A) 4
B) 2e
C) - 2
D) - 4
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is the rectangle defined by 
A) 5
B) 0
C) 3
D) - 5
for the given function f(x, y) and the region R. ; R is the rectangle defined by A) 5
B) 0
C) 3
D) - 5
Question
Use a double integral to find the volume of the solid shown in the figure. 
A) 5,996
B) 5,994
C) 6,000
D) 6,008
A) 5,996
B) 5,994
C) 6,000
D) 6,008
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. f(x, y) = 2y + x; R is the rectangle defined by 
A)
B)
C)
D) 26
for the given function f(x, y) and the region R. f(x, y) = 2y + x; R is the rectangle defined by A)
B)
C)
D) 26
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is bounded by 
and y = x.
A)
B)
C)
D)
for the given function f(x, y) and the region R. ; R is bounded by and y = x.A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
A)
B)
C)
D)
for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.A)
B)
C)
D)
Question
Find the volume of the solid bounded above by the surface z = f(x, y)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
A)
B)
C)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).A)
B)
C)
Question
Use a double integral to find the volume of the solid shown in the figure. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
R is the rectangle defined by 
A) 10e
B) 9
C) - 9e
D) - 10
for the given function f(x, y) and the region R. R is the rectangle defined by A) 10e
B) 9
C) - 9e
D) - 10
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
A) 14
B) 13
C) 11
D) 18
E) 2
for the given function f(x, y) and the region R. A) 14
B) 13
C) 11
D) 18
E) 2
Question
Find the volume of the solid bounded above by the surface z = f(x, y)
And below by the plane region R.
; R is the region bounded by 
.
A)
B)
C)
And below by the plane region R.
; R is the region bounded by .A)
B)
C)
Question
Evaluate the double integral 
for the function 
and the region 
. 
and 
is the rectangle defined by 
and 
.
A)
B)
C)
D)
E)
for the function and the region . and is the rectangle defined by and .A)
B)
C)
D)
E)
Question
Evaluate the double integral 
for the function 
and the region 
. 
and 
is bounded by 
, 
, 
and 
.
A)
B)
C)
D)
E)
for the function and the region . and is bounded by , , and .A)
B)
C)
D)
E)
Question
Find the average value of the given function f(x, y)
Over the plane region R.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 3.
A)
B)
C)
D)
Over the plane region R.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 3.A)
B)
C)
D)
Question
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function 
where 
stands for the number of finished units and 
stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 
and 
and the number of unfinished units varies between 
and 
per week. Please round your answer to the nearest dollar, if necessary. 
$__________ per week.
where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary. $__________ per week. Question
Use a double integral to find the volume of the solid shown in the figure. 
Question
Evaluate the double integral 
for the given function f(x, y) and the region R.
f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
for the given function f(x, y) and the region R.f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is the rectangle defined by 
for the given function f(x, y) and the region R. ; R is the rectangle defined by Question
Evaluate the double integral 
for the given function f(x, y) and the region R.
f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R.f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.
Question
Evaluate the double integral 
for the given function f(x, y) and the region R.
f(x, y) = 2x + 2y; R is bounded by x = 0,
, y = 0 and y = 4.
for the given function f(x, y) and the region R.f(x, y) = 2x + 2y; R is bounded by x = 0,
, y = 0 and y = 4. Question
Find the average value of the function 
over the plane region 
. 
and 
is the triangle with vertices (0, 0), (0, 1) and (1, 1).
A)
B)
C)
D) e
E)
over the plane region . and is the triangle with vertices (0, 0), (0, 1) and (1, 1).A)
B)
C)
D) e
E)
Question
Evaluate the double integral 
for the function 
and the region 
. 
and 
is bounded by 
, 
, 
and 
.
for the function and the region . and is bounded by , , and . Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R. ; R is bounded by x = 0, x = 1, y = 0 and y = x. Question
Find the average value of the given function f(x, y)
Over the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).
A)
B)
C)
D)
Over the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is the rectangle defined by 
for the given function f(x, y) and the region R. ; R is the rectangle defined by Question
Evaluate the double integral 
for the given function f(x, y) and the region R.
f(x, y) = 2y + 3x; R is the rectangle defined by
for the given function f(x, y) and the region R.f(x, y) = 2y + 3x; R is the rectangle defined by
Question
The population density of a certain city is given by the function 
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by 
A)
B)
C)
D)
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by A)
B)
C)
D)
Question
Use a double integral to find the volume of the solid shown in the figure. 
Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then ![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d973_bd69_b1b6_59b71036c616_TB6026_11.jpg)
where ![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d973_bd6a_b1b6_a3016793e416_TB6026_11.jpg)
A) From the definition follows![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d973_e47b_b1b6_5f1d86d3579e_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_0b8c_b1b6_7daad9a35ffe_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_0b8d_b1b6_e943e9ba6017_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_329e_b1b6_e14b4cf7232b_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_59af_b1b6_399bcbacf9f5_TB6026_11.jpg)
, so the statement is true.
B) From the definition follows![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_80c0_b1b6_b1d55a522883_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_80c1_b1b6_73bf039417a5_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_a7d2_b1b6_15c2ba687213_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_a7d3_b1b6_5d27e7597df5_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_cee4_b1b6_47cf4c44e522_TB6026_11.jpg)
, so the statement is false.
C) From the definition follows![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d974_f5f5_b1b6_95e64b3a9da5_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d975_1d06_b1b6_cb2ce721719e_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d975_1d07_b1b6_f91325f40d4a_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d975_4418_b1b6_75554aad7fa0_TB6026_11.jpg)
![<strong>Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where </strong> A) From the definition follows , so the statement is true. B) From the definition follows , so the statement is false. C) From the definition follows , so the statement is false. <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d975_4419_b1b6_01cc3848185a_TB6026_11.jpg)
, so the statement is false.
where A) From the definition follows
, so the statement is true.B) From the definition follows
, so the statement is false.C) From the definition follows
, so the statement is false. Question
Evaluate the double integral 
for the function 
and the region R. 
and R is the rectangle defined by 
and 
.
for the function and the region R. and R is the rectangle defined by and . Question
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function 
where 
stands for the number of finished units and 
stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between 
and 
and the number of unfinished units varies between 
and 
per week. Please round your answer to the nearest dollar, if necessary.
A)
per week
B)
per week
C)
per week
D)
per week
E)
per week
where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary.A)
per weekB)
per weekC)
per weekD)
per weekE)
per week Question
Find the volume of the solid bounded above by the surface 
and below by the plane region 
. 
and 
is the region bounded by the graphs of 
and 
.
A)
B)
C)
D)
E)
and below by the plane region . and is the region bounded by the graphs of and .A)
B)
C)
D)
E)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
R is the rectangle defined by 
for the given function f(x, y) and the region R. R is the rectangle defined by Question
Minimize the function 
subject to the constraint 
.
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Question
Evaluate the double integral 
for the given function f(x, y) and the region R. 
; R is bounded by 
and y = x.
for the given function f(x, y) and the region R. ; R is bounded by and y = x. Question
Minimize the function 
subject to the constraint 
.
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Question
Maximize the function 
subject to the constraints 
.
A) Max. of
at
B) Max. of
at
C) Max. of
at
D) Max. of
at
E) Max. of
at
subject to the constraints .A) Max. of
atB) Max. of
atC) Max. of
atD) Max. of
atE) Max. of
at Question
Maximize the function 
subject to the constraints 
.
A) Max. of
at
B) Max. of -2 at
C) Max. of
at
D) Max. of
at
E) Max. of 6 at
subject to the constraints .A) Max. of
atB) Max. of -2 at
C) Max. of
atD) Max. of
atE) Max. of 6 at
Question
Minimize the function 
subject to the constraint 
.
A)
B)
C)
D)
E)
subject to the constraint .A)
B)
C)
D)
E)
Question
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
z = f(x, y)
and below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5). Question
Find the average value of the given function
f(x, y)
over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
f(x, y)
over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2). Question
Maximize the function 
subject to the constraint 
.
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Question
Maximize the function 
subject to the constraints 
.
A) Max. of
at
B) Max. of
at
C) Max. of
at
D) Max. of
at
E) Max. of
at
subject to the constraints .A) Max. of
atB) Max. of
atC) Max. of
atD) Max. of
atE) Max. of
at Question
The population density of a certain city is given by the function 
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by 
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by Question
Find the average value of the function 
over the plane region 
. 
and 
is the triangle with vertices 
, 
and 
.
over the plane region . and is the triangle with vertices , and . Question
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R.
; R is the region bounded by 
.
z = f(x, y)
and below by the plane region R.
; R is the region bounded by . Question
Find the average value of the given function
f(x, y)
over the plane region R.
; R is the region bounded by the graph of y = 6x and y = 0 from x = 1 to x = 5.
f(x, y)
over the plane region R.
; R is the region bounded by the graph of y = 6x and y = 0 from x = 1 to x = 5. Question
Maximize the function 
subject to the constraints 
.
A) Max. of
at
B) Max. of
at
C) Max. of
at
D) Max. of
at
E) Max. of
at
subject to the constraints .A) Max. of
atB) Max. of
atC) Max. of
atD) Max. of
atE) Max. of
at Question
Minimize the function 
subject to the constraints 
A) Min. of
at
B) Min. of
at
C) Min. of
at
D) Min. of
at
E) Min. of
at
subject to the constraints A) Min. of
atB) Min. of
atC) Min. of
atD) Min. of
atE) Min. of
at Question
Use a double integral to find the volume of the solid shown in the figure. 
Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then![Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d981_786c_b1b6_f3f19210a87b_TB6026_11.jpg)
where ![Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where <div style=padding-top: 35px>](https://storage.examlex.com/TB6026/11eaa8b1_d981_786d_b1b6_296c145eb06d_TB6026_11.jpg)
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where Question
Find the volume of the solid bounded above by the surface 
and below by the plane region 
. 
and 
is the region bounded by the graphs of 
and 
.
and below by the plane region . and is the region bounded by the graphs of and . Question
Use a double integral to find the volume of the solid shown in the figure. 
Question
Minimize the function 
subject to the constraint 
.
subject to the constraint . Question
The total weekly profit (in dollars) realized by Country Workshop in manufacturing and selling its rolltop desks is given by the profit function 
where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The company's management decides to restrict the manufacture of these desks to a total of exactly 100 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit?
A) 80 finished and 20 unfinished units
B) 70 finished and 30 unfinished units
C) 60 finished and 40 unfinished units
D) 110 finished and -10 unfinished units
where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The company's management decides to restrict the manufacture of these desks to a total of exactly 100 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit?A) 80 finished and 20 unfinished units
B) 70 finished and 30 unfinished units
C) 60 finished and 40 unfinished units
D) 110 finished and -10 unfinished units
Question
The management of UNICO Department Store decides to enclose an 
area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls of the store. Two sides of the enclosure will be made of pine board, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places. 
A) 7.07 ft by 42.43 ft
B) 8.61 ft by 40.89 ft
C) 5.06 ft by 44.44 ft
D) 7.24 ft by 42.26 ft
area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls of the store. Two sides of the enclosure will be made of pine board, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places. A) 7.07 ft by 42.43 ft
B) 8.61 ft by 40.89 ft
C) 5.06 ft by 44.44 ft
D) 7.24 ft by 42.26 ft
Question
Maximize the function 
subject to the constraint 
.
subject to the constraint . Question
Minimize the function 
subject to the constraint 
.
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Question
Find the maximum and minimum values of the function 
subject to the constraint 
.
A)
B)
C)
D)
E)
subject to the constraint .A)
B)
C)
D)
E)
Question
Maximize the function 
subject to the constraint 
.
A)
B)
C)
D)
E)
subject to the constraint .A)
B)
C)
D)
E)
Question
Maximize the function 
subject to the constraint 
.
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Question
Find the maximum and minimum values of the function 
subject to the constraint 
.
A) Maxima :
Minima :
B) Maxima :
Minima :
C) Maxima :
Minima :
D) Maxima :
Minima :
subject to the constraint .A) Maxima :
Minima :B) Maxima :
Minima :C) Maxima :
Minima :D) Maxima :
Minima : Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If 
gives rise to a (constrained) relative extremum of 
subject to the constraint 
, then , 
and 
, simultaneously.
A) Let
and
, then the point
gives rise to a (constrained) relative extremum(minimum),
and
, so the statement is true.
B) Let
and
, then the point
gives rise to a (constrained) relative extremum(minimum),
and
, so the statement is false.
gives rise to a (constrained) relative extremum of subject to the constraint , then , and , simultaneously.A) Let
and , then the point gives rise to a (constrained) relative extremum(minimum), and , so the statement is true.B) Let
and , then the point gives rise to a (constrained) relative extremum(minimum), and , so the statement is false. Question
The total daily profit (in dollars) realized by Weston Publishing in publishing and selling its dictionaries is given by the profit function 
where x stands for the number of deluxe editions and y denotes the number of standard editions sold daily. Weston's management decides that publication of these dictionaries should be restricted to a total of exactly 700 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit?
A) 320 deluxe copies, 330 standard copies
B) 450 deluxe copies, 240 standard copies
C) 320 deluxe copies, 300 standard copies
D) 450 deluxe copies, 330 standard copies
E) 400 deluxe copies, 300 standard copies
where x stands for the number of deluxe editions and y denotes the number of standard editions sold daily. Weston's management decides that publication of these dictionaries should be restricted to a total of exactly 700 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit?A) 320 deluxe copies, 330 standard copies
B) 450 deluxe copies, 240 standard copies
C) 320 deluxe copies, 300 standard copies
D) 450 deluxe copies, 330 standard copies
E) 400 deluxe copies, 300 standard copies
Question
A building in the shape of a rectangular box is to have a volume of 
(see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $7/square foot for the front and back, and $6/square foot for the sides. Find the dimensions of the building that will result in a minimal annual heating and cooling cost. What is the minimal annual heating and cooling cost(C)? 
A)
B)
C)
D)
(see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $7/square foot for the front and back, and $6/square foot for the sides. Find the dimensions of the building that will result in a minimal annual heating and cooling cost. What is the minimal annual heating and cooling cost(C)? A)
B)
C)
D)
Question
An open rectangular box is to be constructed from material that costs 
for the bottom and 
for its sides. Find the dimensions of the box of greatest volume that can be constructed for 
.
A)
B)
C)
D)
for the bottom and for its sides. Find the dimensions of the box of greatest volume that can be constructed for .A)
B)
C)
D)
Question
The Ross-Simons Company has a monthly advertising budget of $40,000. Their marketing department estimates that if they spend x dollars on newspaper advertising and y dollars on television advertising, then the monthly sales will be given by 
dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads each month to maximize its monthly sales.
A) $31,000 on newspaper advertisements and $11,000 on television advertisements
B) $28,000 on newspaper advertisements and $8,000 on television advertisements
C) $33,000 on newspaper advertisements and $13,000 on television advertisements
D) $30,000 on newspaper advertisements and $10,000 on television advertisements
dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads each month to maximize its monthly sales.A) $31,000 on newspaper advertisements and $11,000 on television advertisements
B) $28,000 on newspaper advertisements and $8,000 on television advertisements
C) $33,000 on newspaper advertisements and $13,000 on television advertisements
D) $30,000 on newspaper advertisements and $10,000 on television advertisements
Question
A closed rectangular box having a volume of 
is to be constructed. If the material for the sides costs 
and the material for the top and bottom costs 
, find the dimensions of the box that can be constructed with minimum cost.
A)
B)
C)
D)
E)
is to be constructed. If the material for the sides costs and the material for the top and bottom costs , find the dimensions of the box that can be constructed with minimum cost.A)
B)
C)
D)
E)
Question
Minimize the function 
subject to the constraint 
.
subject to the constraint . Question
John Mills- the propietor of Mills Engine Company, a manufacturer of mdel airplane engine-finds that it takes 
units of labor and 
units of capital to produce 
units of the product. If a unit of labor costs 
, a unit of capital costs 
and 
is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.
A)
units of labour and
units of capital.
B)
units of labour and
units of capital.
C)
units of labour and
units of capital.
D)
units of labour and
units of capital.
E)
units of labour and
units of capital.
units of labor and units of capital to produce units of the product. If a unit of labor costs , a unit of capital costs and is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.A)
units of labour and units of capital.B)
units of labour and units of capital.C)
units of labour and units of capital.D)
units of labour and units of capital.E)
units of labour and units of capital. Question
Find the maximum and minimum values of the function 
subject to the constraint 
.
A) Maxima :
Minima :
B) Maxima :
Minima :
C) Maxima :
Minima :
D) Maxima :
Minima :
subject to the constraint .A) Maxima :
Minima :B) Maxima :
Minima :C) Maxima :
Minima :D) Maxima :
Minima : Question
The Company requires that its corned beef hash containers have a capacity of 
, be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs 
and the metal for the pull-off lid costs 
.
A)
B)
C)
D)
, be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs and the metal for the pull-off lid costs .A)
B)
C)
D)
Question
Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 105 in. Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail. What is the volume of such a package? Hint: The length plus the girth is 
, and the volume is 
. 
A)
B)
C)
D)
E)
, and the volume is . A)
B)
C)
D)
E)
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Deck 8: Calculus of Several Variables
1
Evaluate the double integral for the given function f(x, y) and the region R.
A) 9
B) 4
C) 72
D) 63
E) 8
for the given function f(x, y) and the region R. A) 9
B) 4
C) 72
D) 63
E) 8
72
2
Use a double integral to find the volume of the solid shown in the figure.
A) 14
B) 6
C) 4
D) 9
A) 14
B) 6
C) 4
D) 9
6
3
Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 6x + 6y; R is bounded by x = 0, , y = 0 and y = 4.
A)
B)
C)
D)
for the given function f(x, y) and the region R. f(x, y) = 6x + 6y; R is bounded by x = 0, , y = 0 and y = 4.A)
B)
C)
D)
4
Use a double integral to find the volume of the solid shown in the figure.
A)
B)
C)
D)
A)
B)
C)
D)
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5
Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.
A)
B)
C)
D)
for the given function f(x, y) and the region R. f(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.A)
B)
C)
D)
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6
Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 5 ; R is bounded by the lines x = 1, y = 0 and y = x.
A)
B)
C)
D)
E)
for the given function f(x, y) and the region R. f(x, y) = 5 ; R is bounded by the lines x = 1, y = 0 and y = x.A)
B)
C)
D)
E)
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7
Evaluate the double integral for the given function f(x, y) and the region R. ; R is the rectangle defined by
A)
B)
C)
D)
for the given function f(x, y) and the region R. ; R is the rectangle defined by A)
B)
C)
D)
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8
Evaluate the double integral for the given function f(x, y) and the region R. ; R is bounded by x = 0, x = 1, y = 0 and y = x.
A) 4
B) 2e
C) - 2
D) - 4
for the given function f(x, y) and the region R. ; R is bounded by x = 0, x = 1, y = 0 and y = x.A) 4
B) 2e
C) - 2
D) - 4
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9
Evaluate the double integral for the given function f(x, y) and the region R. ; R is the rectangle defined by
A) 5
B) 0
C) 3
D) - 5
for the given function f(x, y) and the region R. ; R is the rectangle defined by A) 5
B) 0
C) 3
D) - 5
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10
Use a double integral to find the volume of the solid shown in the figure.
A) 5,996
B) 5,994
C) 6,000
D) 6,008
A) 5,996
B) 5,994
C) 6,000
D) 6,008
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11
Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 2y + x; R is the rectangle defined by
A)
B)
C)
D) 26
for the given function f(x, y) and the region R. f(x, y) = 2y + x; R is the rectangle defined by A)
B)
C)
D) 26
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12
Evaluate the double integral for the given function f(x, y) and the region R. ; R is bounded by and y = x.
A)
B)
C)
D)
for the given function f(x, y) and the region R. ; R is bounded by and y = x.A)
B)
C)
D)
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13
Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
A)
B)
C)
D)
for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.A)
B)
C)
D)
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14
Find the volume of the solid bounded above by the surface z = f(x, y)
And below by the plane region R. ; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
A)
B)
C)
And below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).A)
B)
C)
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15
Use a double integral to find the volume of the solid shown in the figure.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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16
Evaluate the double integral for the given function f(x, y) and the region R. R is the rectangle defined by
A) 10e
B) 9
C) - 9e
D) - 10
for the given function f(x, y) and the region R. R is the rectangle defined by A) 10e
B) 9
C) - 9e
D) - 10
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17
Evaluate the double integral for the given function f(x, y) and the region R.
A) 14
B) 13
C) 11
D) 18
E) 2
for the given function f(x, y) and the region R. A) 14
B) 13
C) 11
D) 18
E) 2
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18
Find the volume of the solid bounded above by the surface z = f(x, y)
And below by the plane region R. ; R is the region bounded by .
A)
B)
C)
And below by the plane region R.
; R is the region bounded by .A)
B)
C)
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19
Evaluate the double integral for the function and the region . and is the rectangle defined by and .
A)
B)
C)
D)
E)
for the function and the region . and is the rectangle defined by and .A)
B)
C)
D)
E)
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20
Evaluate the double integral for the function and the region . and is bounded by , , and .
A)
B)
C)
D)
E)
for the function and the region . and is bounded by , , and .A)
B)
C)
D)
E)
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21
Find the average value of the given function f(x, y)
Over the plane region R. ; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 3.
A)
B)
C)
D)
Over the plane region R.
; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 3.A)
B)
C)
D)
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22
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary. $__________ per week.
where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary. $__________ per week. Unlock Deck
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23
Use a double integral to find the volume of the solid shown in the figure.
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24
Evaluate the double integral for the given function f(x, y) and the region R.
f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
for the given function f(x, y) and the region R.f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
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25
Evaluate the double integral for the given function f(x, y) and the region R. ; R is the rectangle defined by
for the given function f(x, y) and the region R. ; R is the rectangle defined by Unlock Deck
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26
Evaluate the double integral for the given function f(x, y) and the region R.
f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R.f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.
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27
Evaluate the double integral for the given function f(x, y) and the region R.
f(x, y) = 2x + 2y; R is bounded by x = 0, , y = 0 and y = 4.
for the given function f(x, y) and the region R.f(x, y) = 2x + 2y; R is bounded by x = 0,
, y = 0 and y = 4. Unlock Deck
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28
Find the average value of the function over the plane region . and is the triangle with vertices (0, 0), (0, 1) and (1, 1).
A)
B)
C)
D) e
E)
over the plane region . and is the triangle with vertices (0, 0), (0, 1) and (1, 1).A)
B)
C)
D) e
E)
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29
Evaluate the double integral for the function and the region . and is bounded by , , and .
for the function and the region . and is bounded by , , and . Unlock Deck
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30
Evaluate the double integral for the given function f(x, y) and the region R. ; R is bounded by x = 0, x = 1, y = 0 and y = x.
for the given function f(x, y) and the region R. ; R is bounded by x = 0, x = 1, y = 0 and y = x. Unlock Deck
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31
Find the average value of the given function f(x, y)
Over the plane region R. ; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).
A)
B)
C)
D)
Over the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).A)
B)
C)
D)
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32
Evaluate the double integral for the given function f(x, y) and the region R. ; R is the rectangle defined by
for the given function f(x, y) and the region R. ; R is the rectangle defined by Unlock Deck
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33
Evaluate the double integral for the given function f(x, y) and the region R.
f(x, y) = 2y + 3x; R is the rectangle defined by
for the given function f(x, y) and the region R.f(x, y) = 2y + 3x; R is the rectangle defined by
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34
The population density of a certain city is given by the function where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
A)
B)
C)
D)
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by A)
B)
C)
D)
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35
Use a double integral to find the volume of the solid shown in the figure.
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36
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where
A) From the definition follows
, so the statement is true.
B) From the definition follows
, so the statement is false.
C) From the definition follows
, so the statement is false.
where A) From the definition follows
, so the statement is true.B) From the definition follows
, so the statement is false.C) From the definition follows
, so the statement is false. Unlock Deck
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37
Evaluate the double integral for the function and the region R. and R is the rectangle defined by and .
for the function and the region R. and R is the rectangle defined by and . Unlock Deck
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38
The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary.
A) per week
B) per week
C) per week
D) per week
E) per week
where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary.A)
per weekB)
per weekC)
per weekD)
per weekE)
per week Unlock Deck
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39
Find the volume of the solid bounded above by the surface and below by the plane region . and is the region bounded by the graphs of and .
A)
B)
C)
D)
E)
and below by the plane region . and is the region bounded by the graphs of and .A)
B)
C)
D)
E)
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40
Evaluate the double integral for the given function f(x, y) and the region R. R is the rectangle defined by
for the given function f(x, y) and the region R. R is the rectangle defined by Unlock Deck
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41
Minimize the function subject to the constraint .
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
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42
Evaluate the double integral for the given function f(x, y) and the region R. ; R is bounded by and y = x.
for the given function f(x, y) and the region R. ; R is bounded by and y = x. Unlock Deck
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43
Minimize the function subject to the constraint .
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
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44
Maximize the function subject to the constraints .
A) Max. of at
B) Max. of at
C) Max. of at
D) Max. of at
E) Max. of at
subject to the constraints .A) Max. of
atB) Max. of
atC) Max. of
atD) Max. of
atE) Max. of
at Unlock Deck
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45
Maximize the function subject to the constraints .
A) Max. of at
B) Max. of -2 at
C) Max. of at
D) Max. of at
E) Max. of 6 at
subject to the constraints .A) Max. of
atB) Max. of -2 at
C) Max. of
atD) Max. of
atE) Max. of 6 at
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46
Minimize the function subject to the constraint .
A)
B)
C)
D)
E)
subject to the constraint .A)
B)
C)
D)
E)
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47
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R. ; R is the triangle with vertices (0, 0), (5, 0) and (0, 5).
z = f(x, y)
and below by the plane region R.
; R is the triangle with vertices (0, 0), (5, 0) and (0, 5). Unlock Deck
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48
Find the average value of the given function
f(x, y)
over the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2).
f(x, y)
over the plane region R.
; R is the triangle with vertices (0, 0), (2, 0) and (2, 2). Unlock Deck
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49
Maximize the function subject to the constraint .
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Unlock Deck
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50
Maximize the function subject to the constraints .
A) Max. of at
B) Max. of at
C) Max. of at
D) Max. of at
E) Max. of at
subject to the constraints .A) Max. of
atB) Max. of
atC) Max. of
atD) Max. of
atE) Max. of
at Unlock Deck
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51
The population density of a certain city is given by the function where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by Unlock Deck
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52
Find the average value of the function over the plane region . and is the triangle with vertices , and .
over the plane region . and is the triangle with vertices , and . Unlock Deck
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53
Find the volume of the solid bounded above by the surface
z = f(x, y)
and below by the plane region R. ; R is the region bounded by .
z = f(x, y)
and below by the plane region R.
; R is the region bounded by . Unlock Deck
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54
Find the average value of the given function
f(x, y)
over the plane region R. ; R is the region bounded by the graph of y = 6x and y = 0 from x = 1 to x = 5.
f(x, y)
over the plane region R.
; R is the region bounded by the graph of y = 6x and y = 0 from x = 1 to x = 5. Unlock Deck
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55
Maximize the function subject to the constraints .
A) Max. of at
B) Max. of at
C) Max. of at
D) Max. of at
E) Max. of at
subject to the constraints .A) Max. of
atB) Max. of
atC) Max. of
atD) Max. of
atE) Max. of
at Unlock Deck
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k this deck
56
Minimize the function subject to the constraints
A) Min. of at
B) Min. of at
C) Min. of at
D) Min. of at
E) Min. of at
subject to the constraints A) Min. of
atB) Min. of
atC) Min. of
atD) Min. of
atE) Min. of
at Unlock Deck
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k this deck
57
Use a double integral to find the volume of the solid shown in the figure.
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58
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where
If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
where Unlock Deck
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59
Find the volume of the solid bounded above by the surface and below by the plane region . and is the region bounded by the graphs of and .
and below by the plane region . and is the region bounded by the graphs of and . Unlock Deck
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k this deck
60
Use a double integral to find the volume of the solid shown in the figure.
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61
Minimize the function subject to the constraint .
subject to the constraint . Unlock Deck
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62
The total weekly profit (in dollars) realized by Country Workshop in manufacturing and selling its rolltop desks is given by the profit function where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The company's management decides to restrict the manufacture of these desks to a total of exactly 100 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit?
A) 80 finished and 20 unfinished units
B) 70 finished and 30 unfinished units
C) 60 finished and 40 unfinished units
D) 110 finished and -10 unfinished units
where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The company's management decides to restrict the manufacture of these desks to a total of exactly 100 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit?A) 80 finished and 20 unfinished units
B) 70 finished and 30 unfinished units
C) 60 finished and 40 unfinished units
D) 110 finished and -10 unfinished units
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63
The management of UNICO Department Store decides to enclose an area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls of the store. Two sides of the enclosure will be made of pine board, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places.
A) 7.07 ft by 42.43 ft
B) 8.61 ft by 40.89 ft
C) 5.06 ft by 44.44 ft
D) 7.24 ft by 42.26 ft
area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls of the store. Two sides of the enclosure will be made of pine board, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places. A) 7.07 ft by 42.43 ft
B) 8.61 ft by 40.89 ft
C) 5.06 ft by 44.44 ft
D) 7.24 ft by 42.26 ft
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64
Maximize the function subject to the constraint .
subject to the constraint . Unlock Deck
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65
Minimize the function subject to the constraint .
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
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k this deck
66
Find the maximum and minimum values of the function subject to the constraint .
A)
B)
C)
D)
E)
subject to the constraint .A)
B)
C)
D)
E)
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67
Maximize the function subject to the constraint .
A)
B)
C)
D)
E)
subject to the constraint .A)
B)
C)
D)
E)
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68
Maximize the function subject to the constraint .
A)
B)
C)
D)
subject to the constraint .A)
B)
C)
D)
Unlock Deck
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69
Find the maximum and minimum values of the function subject to the constraint .
A) Maxima : Minima :
B) Maxima : Minima :
C) Maxima : Minima :
D) Maxima : Minima :
subject to the constraint .A) Maxima :
Minima :B) Maxima :
Minima :C) Maxima :
Minima :D) Maxima :
Minima : Unlock Deck
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70
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If gives rise to a (constrained) relative extremum of subject to the constraint , then , and , simultaneously.
A) Let and
, then the point
gives rise to a (constrained) relative extremum(minimum),
and
, so the statement is true.
B) Let and
, then the point
gives rise to a (constrained) relative extremum(minimum),
and
, so the statement is false.
gives rise to a (constrained) relative extremum of subject to the constraint , then , and , simultaneously.A) Let
and , then the point gives rise to a (constrained) relative extremum(minimum), and , so the statement is true.B) Let
and , then the point gives rise to a (constrained) relative extremum(minimum), and , so the statement is false. Unlock Deck
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71
The total daily profit (in dollars) realized by Weston Publishing in publishing and selling its dictionaries is given by the profit function where x stands for the number of deluxe editions and y denotes the number of standard editions sold daily. Weston's management decides that publication of these dictionaries should be restricted to a total of exactly 700 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit?
A) 320 deluxe copies, 330 standard copies
B) 450 deluxe copies, 240 standard copies
C) 320 deluxe copies, 300 standard copies
D) 450 deluxe copies, 330 standard copies
E) 400 deluxe copies, 300 standard copies
where x stands for the number of deluxe editions and y denotes the number of standard editions sold daily. Weston's management decides that publication of these dictionaries should be restricted to a total of exactly 700 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit?A) 320 deluxe copies, 330 standard copies
B) 450 deluxe copies, 240 standard copies
C) 320 deluxe copies, 300 standard copies
D) 450 deluxe copies, 330 standard copies
E) 400 deluxe copies, 300 standard copies
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72
A building in the shape of a rectangular box is to have a volume of (see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $7/square foot for the front and back, and $6/square foot for the sides. Find the dimensions of the building that will result in a minimal annual heating and cooling cost. What is the minimal annual heating and cooling cost(C)?
A)
B)
C)
D)
(see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $7/square foot for the front and back, and $6/square foot for the sides. Find the dimensions of the building that will result in a minimal annual heating and cooling cost. What is the minimal annual heating and cooling cost(C)? A)
B)
C)
D)
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73
An open rectangular box is to be constructed from material that costs for the bottom and for its sides. Find the dimensions of the box of greatest volume that can be constructed for .
A)
B)
C)
D)
for the bottom and for its sides. Find the dimensions of the box of greatest volume that can be constructed for .A)
B)
C)
D)
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74
The Ross-Simons Company has a monthly advertising budget of $40,000. Their marketing department estimates that if they spend x dollars on newspaper advertising and y dollars on television advertising, then the monthly sales will be given by dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads each month to maximize its monthly sales.
A) $31,000 on newspaper advertisements and $11,000 on television advertisements
B) $28,000 on newspaper advertisements and $8,000 on television advertisements
C) $33,000 on newspaper advertisements and $13,000 on television advertisements
D) $30,000 on newspaper advertisements and $10,000 on television advertisements
dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads each month to maximize its monthly sales.A) $31,000 on newspaper advertisements and $11,000 on television advertisements
B) $28,000 on newspaper advertisements and $8,000 on television advertisements
C) $33,000 on newspaper advertisements and $13,000 on television advertisements
D) $30,000 on newspaper advertisements and $10,000 on television advertisements
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75
A closed rectangular box having a volume of is to be constructed. If the material for the sides costs and the material for the top and bottom costs , find the dimensions of the box that can be constructed with minimum cost.
A)
B)
C)
D)
E)
is to be constructed. If the material for the sides costs and the material for the top and bottom costs , find the dimensions of the box that can be constructed with minimum cost.A)
B)
C)
D)
E)
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76
Minimize the function subject to the constraint .
subject to the constraint . Unlock Deck
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77
John Mills- the propietor of Mills Engine Company, a manufacturer of mdel airplane engine-finds that it takes units of labor and units of capital to produce units of the product. If a unit of labor costs , a unit of capital costs and is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.
A) units of labour and
units of capital.
B) units of labour and
units of capital.
C) units of labour and
units of capital.
D) units of labour and
units of capital.
E) units of labour and
units of capital.
units of labor and units of capital to produce units of the product. If a unit of labor costs , a unit of capital costs and is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.A)
units of labour and units of capital.B)
units of labour and units of capital.C)
units of labour and units of capital.D)
units of labour and units of capital.E)
units of labour and units of capital. Unlock Deck
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78
Find the maximum and minimum values of the function subject to the constraint .
A) Maxima : Minima :
B) Maxima : Minima :
C) Maxima : Minima :
D) Maxima : Minima :
subject to the constraint .A) Maxima :
Minima :B) Maxima :
Minima :C) Maxima :
Minima :D) Maxima :
Minima : Unlock Deck
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79
The Company requires that its corned beef hash containers have a capacity of , be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs and the metal for the pull-off lid costs .
A)
B)
C)
D)
, be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs and the metal for the pull-off lid costs .A)
B)
C)
D)
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80
Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 105 in. Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail. What is the volume of such a package? Hint: The length plus the girth is , and the volume is .
A)
B)
C)
D)
E)
, and the volume is . A)
B)
C)
D)
E)
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Unlock Deck
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