Question 1

Find the first partial derivatives of the function.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 2

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.

Accepted Answer

The answer of Evaluate the double integral  for the...

Question 3

Let  Compute the following.
g(1, 1, 1) = __________
g(- 1, 0, 1) = __________
g(1, - 1, - 1) = __________

Accepted Answer

The answer of Let  Compute the following.
g(1, 1, 1)...

Question 4

Maximize the function  subject to the constraint  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 5

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

Accepted Answer

A)   
B)   
C)   
D)

Question 6

Minimize the function  subject to the constraint  .

Accepted Answer

The answer of Minimize the function  subject to the...

Question 7

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

Accepted Answer

A)   
B)   
C)   
D) 26

Question 8

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 108 in.Find the dimensions of the rectangular package that would have the greatest possible volume under these regulations.(Hint: Let the dimensions of the box be  by  by  (see the figure below).Then,  , and the volume  .So that  .Maximize  .)

Accepted Answer

A) x = 20 inches, y = 37 inches, z = 15 inches, V = 11,100 cubic inches 
B) x = 18 inches, y = 18 inches, z = 15 inches, V = 4,860 cubic inches 
C) x = 20 inches, y = 36 inches, z = 20 inches, V = 11,664 cubic inches 
D) x = 20 inches, y = 36 inches, z = 15 inches, V = 11,100 cubic inches 
E) x = 18 inches, y = 36 inches, z = 18 inches, V = 11,664 cubic inches

Question 9

Find the first partial derivative of the following function.

Accepted Answer

A)   
 
B)   
 
C)   
 
D)   
 
E)

Question 10

The following table gives the total sales of drugs (in billions of dollars) in one country from 1999  through 2003:  Find an equation of the least-square line for these data.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 11

Evaluate the double integral  for the given function f(x, y) and the region R.  ; R is bounded by  and y = x.

Accepted Answer

The answer of Evaluate the double integral  for the...

Question 12

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.

Accepted Answer

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 13

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 the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.

Accepted Answer

A) True 
 B)False

Question 14

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  .

Accepted Answer

The answer of The Company requires that its corned beef...

Question 15

The total weekly revenue (in dollars) of Country Workshop associated with manufacturing and selling their rolltop desks is given by the function  where x denotes the number of finished units and y denotes the number of unfinished units manufactured and sold each week.Compute  and  when x = 200 and y = 180.Interpret your results.

Accepted Answer

A) $36/unit; $24/unit The weekly revenue increases by $36/unit for each additional finished desk produced (beyond 200) when the level of production of unfinished desks remains fixed at 180; the revenue increases by $24/unit when each additional unfinished desk (beyond 180) is produced and the level of production of finished desks remains fixed at 200. 
B) - $380/unit; - $20/unit The weekly revenue decreases by $380/unit for each additional finished desk produced (beyond 200) when the level of production of unfinished desks remains fixed at 180; the revenue decreases by $20/unit when each additional unfinished desk (beyond 180) is produced and the level of production of finished desks remains fixed at 200. 
C) $36/unit; - $24/unit The weekly revenue increases by $36/unit for each additional finished desk produced (beyond 200) when the level of production of unfinished desks remains fixed at 180; the revenue decreases by $24/unit when each additional unfinished desk (beyond 180) is produced and the level of production of finished desks remains fixed at 200. 
D) $24/unit; $36/unit The weekly revenue increases by $36/unit for each additional finished desk produced (beyond 200) when the level of production of unfinished desks remains fixed at 180; the revenue decreases by $24/unit when each additional unfinished desk (beyond 180) is produced and the level of production of finished desks remains fixed at 200.

Question 16

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

Accepted Answer

A) (0, 0); relative maximum value: f(0, 0) = 2 
B) (2, 4); relative maximum value: f(0, 0) = 3 
C) (3, 0); relative maximum value: f(0, 0) = 4

Question 17

Find the second order partial derivative of the following function.Show that the mixed partial derivative  and  are equal.

Accepted Answer

A)    
B)    
C)    
D)    
E)

Question 18

The volume of a cylindrical tank of radius r and height h is given by  Find the volume of a cylindrical tank of radius 1.5 ft and height 4 ft.

Accepted Answer

The answer of The volume of a cylindrical tank of...

Question 19

An open rectangular box having a volume of  is to be constructed from a tin sheet.Find the dimensions of such a box if the amount of material used in its construction is to be minimal.
Hint: Let the dimensions of the box be x by y by z.Then, xyz = 108 and the amount of material used is given by S = xy + 2yz + 2xz.Show that  Minimize f(x, y)
x = __________
y = __________
z = __________

Accepted Answer

The answer of An open rectangular box having a volume...

Question 20

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

Accepted Answer

A) (0, 0) and  ; saddle point: f(0, 0) = 5; relative minimum value:
 
B) (0, 0) and  ; saddle point: f(0, 0) = 5; relative maximum value:
 
C)   and (0, 0); relative minimum:  ; relative maximum value: f(0, 0) = - 5 
D) there are no critical points

Exam 8: Calculus of Several Variables

Find the first partial derivatives of the function.

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.

Let Compute the following. g(1, 1, 1) = g(- 1, 0, 1) = g(1, - 1, - 1) = __

Maximize the function subject to the constraint .

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

Minimize the function subject to the constraint .

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

Find the first partial derivative of the following function.

The following table gives the total sales of drugs (in billions of dollars) in one country from 1999 through 2003: Find an equation of the least-square line for these data.

Evaluate the double integral for the given function f(x, y) and the region R. ; R is bounded by and y = x.

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 the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.

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 .

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

Find the second order partial derivative of the following function.Show that the mixed partial derivative and are equal.

The volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 1.5 ft and height 4 ft.

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

Exam 8: Calculus of Several Variables

Find the first partial derivatives of the function.

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.

Let Compute the following. g(1, 1, 1) = __________ g(- 1, 0, 1) = __________ g(1, - 1, - 1) = __________

Maximize the function subject to the constraint .

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

Minimize the function subject to the constraint .

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

Find the first partial derivative of the following function.

The following table gives the total sales of drugs (in billions of dollars) in one country from 1999 through 2003: Find an equation of the least-square line for these data.

Evaluate the double integral for the given function f(x, y) and the region R. ; R is bounded by and y = x.

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 the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.

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 .

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

Find the second order partial derivative of the following function.Show that the mixed partial derivative and are equal.

The volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 1.5 ft and height 4 ft.

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

Let Compute the following. g(1, 1, 1) = g(- 1, 0, 1) = g(1, - 1, - 1) = __