Question 1

Find the average value of the function ​  ​
Over the plane region R.
​  and R is the triangle with vertices (0, 0), (0, 1) and (1, 1).
​

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 2

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)   ; relative minimum value:  
B)   ; relative maximum value:  
C)   ; saddle point:  
D)  there are no critical points

Question 3

Use a double integral to find the volume of the solid shown in the figure. ​  ​

Accepted Answer

A)  6 
B)  4 
C)  9 
D)  14

Question 4

Evaluate the first partial derivatives of the function at the given point. ​  ​

Accepted Answer

A)   ​ 
B)  ​  ​ 
C)  ​  ​ 
D)  ​  ​

Question 5

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  also gives rise to the unconstrained relative extremum of  .

Accepted Answer

The answer of Determine whether the statement is true or...

Question 6

The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by ​  ​
Where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold per week. Currently, the weekly output is 130 finished and 105 unfinished units. Determine the approximate change in the total weekly profit if the sole proprietor of the Country Workshop decides to increase the number of finished units to 140 per week and decrease the number of unfinished units to 100 per week.
​

Accepted Answer

A)  $63.33 per week 
B)  $950.00 per week 
C)  $190.00 per week 
D)  $95.00 per week 
E)  $9,500.00 per week

Question 7

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 $8/running foot and the steel fencing costs $4/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places.
​

Accepted Answer

The answer of The management of UNICO Department Store decides...

Question 8

Minimize the function
​  ​
subject to the constraint  .

Accepted Answer

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

Question 9

Minimize the function
​  ​
subject to the constraint  .

Accepted Answer

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

Question 10

Evaluate the double integral ​  ​
For the given function f(x, y) and the region R.
​  ; R is the rectangle defined by  and  .
​

Accepted Answer

A)  -10 
B)  10e 
C)  9 
D)  -9e

Question 11

Evaluate the double integral ​  ​
For the function  and the region R.
​  and R is the rectangle defined by  and  .
​

Accepted Answer

A)  8 
B)  6 
C)  2 
D)  3 
E)  5

Question 12

Find the first partial derivatives of the function. ​  ​

Accepted Answer

A)   ​ 
B)  ​  ​ 
C)  ​  ​ 
D)  ​  ​

Question 13

Find the domain of the function.
​

Accepted Answer

The answer of Find the domain of the function.
​ ...

Question 14

Find the average value of the given function  over the plane region R. ​  ; R is the region bounded by the graph of  and  from  to  .
​

Accepted Answer

A)   
B)  ​  
C)  ​  
D)  ​  
E)  ​

Question 15

Find the total differential of the function.
​

Accepted Answer

The answer of Find the total differential of the function.
​...

Question 16

Sketch the level curves of the function corresponding to the given values of z. ​  ;  ​

Accepted Answer

A)  ​  
B)  ​  
C)  ​  
D)  ​

Question 17

A building in the shape of a rectangular box is to have a volume of 6,000ft3 (see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $3/square foot for the front and back, and $2/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)? ​  ​

Accepted Answer

A)  x = 30; y = 20; z = 10; C = 3,500 
B)  x = 20; y = 30; z = 30; C = 3,500 
C)  x = 30; y = 20; z = 20; C = 3,600 
D)  x = 20; y = 30; z = 10; C = 3,600

Question 18

The total daily revenue (in dollars) that Weston Publishing realizes in publishing and selling its English-language dictionaries is given by
​  ​
where x denotes the number of deluxe copies and y denotes the number of standard copies published and sold daily. The total daily cost of publishing these dictionaries is given by
​  ​
dollars. Determine how many deluxe copies and how many standard copies Weston should publish each day to maximize its profits. Please round the answers to the nearest integer.
​
__________ deluxe copies
​
__________ standard copies
​
What is the maximum profit realizable? Please round the answer to the nearest dollar.
​
$__________

Accepted Answer

The answer of The total daily revenue (in dollars) that...

Question 19

Minimize the function ​  ​
Subject to the constraint  .
​

Accepted Answer

A)   
B)  ​  
C)  ​  
D)  ​

Question 20

Find the approximate change in z when the point  changes from  to  . ​  ; from  to  ​

Accepted Answer

A)  -0.004 
B)  ​0.002 
C)  ​-0.04 
D)  ​0.004 
E)  ​-0.008

Exam 8: Calculus of Several Variables

Find the average value of the function Over the plane region R. and R is the triangle with vertices (0, 0) , (0, 1) and (1, 1) .

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.

Use a double integral to find the volume of the solid shown in the figure.

Evaluate the first partial derivatives of the function at the given point.

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 also gives rise to the unconstrained relative extremum of .

Minimize the function subject to the constraint .

Minimize the function subject to the constraint .

Evaluate the double integral For the given function f(x, y) and the region R. ; R is the rectangle defined by and .

Evaluate the double integral For the function and the region R. and R is the rectangle defined by and .

Find the first partial derivatives of the function.

Find the domain of the function.

Find the average value of the given function over the plane region R. ; R is the region bounded by the graph of and from to .

Find the total differential of the function.

Sketch the level curves of the function corresponding to the given values of z. ;

Minimize the function Subject to the constraint .

Find the approximate change in z when the point changes from to . ; from to

Exam 8: Calculus of Several Variables

Find the average value of the function ​ ​ Over the plane region R. ​ and R is the triangle with vertices (0, 0) , (0, 1) and (1, 1) . ​

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. ​ ​

Use a double integral to find the volume of the solid shown in the figure. ​ ​

Evaluate the first partial derivatives of the function at the given point. ​ ​

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 also gives rise to the unconstrained relative extremum of .

Minimize the function ​ ​ subject to the constraint .

Minimize the function ​ ​ subject to the constraint .

Evaluate the double integral ​ ​ For the given function f(x, y) and the region R. ​ ; R is the rectangle defined by and . ​

Evaluate the double integral ​ ​ For the function and the region R. ​ and R is the rectangle defined by and . ​

Find the first partial derivatives of the function. ​ ​

Find the domain of the function. ​

Find the average value of the given function over the plane region R. ​ ; R is the region bounded by the graph of and from to . ​

Find the total differential of the function. ​

Sketch the level curves of the function corresponding to the given values of z. ​ ; ​

Minimize the function ​ ​ Subject to the constraint . ​

Find the approximate change in z when the point changes from to . ​ ; from to ​

Find the average value of the function Over the plane region R. and R is the triangle with vertices (0, 0) , (0, 1) and (1, 1) .

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.

Use a double integral to find the volume of the solid shown in the figure.

Evaluate the first partial derivatives of the function at the given point.

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 also gives rise to the unconstrained relative extremum of .

Minimize the function subject to the constraint .

Minimize the function subject to the constraint .

Evaluate the double integral For the given function f(x, y) and the region R. ; R is the rectangle defined by and .

Evaluate the double integral For the function and the region R. and R is the rectangle defined by and .

Find the first partial derivatives of the function.

Find the domain of the function.

Find the average value of the given function over the plane region R. ; R is the region bounded by the graph of and from to .

Find the total differential of the function.

Sketch the level curves of the function corresponding to the given values of z. ;

Minimize the function Subject to the constraint .

Find the approximate change in z when the point changes from to . ; from to