Question 1

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?
​

Accepted Answer

A)  450 deluxe copies, 330 standard copies 
B)  450 deluxe copies, 240 standard copies 
C)  480 deluxe copies, 330 standard copies 
D)  400 deluxe copies, 300 standard copies 
E)  480 deluxe copies, 300 standard copies

Question 2

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).
​

Accepted Answer

A)   
B)   
C)   
D)

Question 3

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.

Accepted Answer

The answer of A closed rectangular box having a volume...

Question 4

Find the average value of the function
​  ​
over the plane region R. Enter your answer as an expression.
​  and R is the triangle with vertices  ,  and  .

Accepted Answer

The answer of Find the average value of the function
​...

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 h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then
​  where  .
​

Accepted Answer

A)  From the definition follows      , so the statement is false.​ 
B)  From the definition follows      , so the statement is true.​ 
C)  From the definition follows      , so the statement is false.​

Question 6

Let  . Compute the following: ​  ​

Accepted Answer

A)  ​  
B)  ​  
C)  ​  
D)  ​

Question 7

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. ​
X
1
2
3
4
Y
4
6
8
9
​
​

Accepted Answer

A)  ​  
B)  ​  
C)  ​  
D)  ​

Question 8

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 y 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 180 and 200 and the number of unfinished units varies betweenand 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
​

Accepted Answer

A)   per week 
B)   per week 
C)   per week 
D)   per week 
E)   per week

Question 9

In a survey it was determined that the demand equation for VCRs is given by
​  ​
The demand equation for blank VCR tapes is given by
​  ​
where p and q denote the unit prices, respectively, and x and y denote the number of VCRs and the number of blank VCR tapes demanded each week. Determine whether these two products are substitute, complementary, or neither.

Accepted Answer

The answer of In a survey it was determined that...

Question 10

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.
​

Accepted Answer

A)  Let  and  , then the point  gives rise to a (constrained) relative extremum(minimum),  and  , so the statement is false. 
B)  Let  and  , then the point  gives rise to a (constrained) relative extremum(minimum),  and  , so the statement is true.

Question 11

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  ,  , and  .
​

Accepted Answer

A)   
B)   
C)

Question 12

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

Accepted Answer

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

Question 13

Find the total differential of the function.
​

Accepted Answer

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

Question 14

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

Accepted Answer

A)   ​ 
B)  ​  ​ 
C)  ​  ​ 
D)  ​  ​

Question 15

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.

Accepted Answer

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

Question 16

Find the volume of the solid bounded above by the surfac  and below by the plane region R.  ; R is the triangle with vertices (0, 0), (2, 0) and (0, 2).

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 17

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  .

Accepted Answer

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

Question 18

Minimize the function
​  ​
subject to the constraint  .

Accepted Answer

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

Question 19

Find the domain of the function.
​

Accepted Answer

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

Question 20

Find the average value of the given function ​  ​
Over the plane region R.
​  ; R is the region bounded by the graph of y = 8x and y = 0 from x = 1 to x = 7.
​

Accepted Answer

A)   ​ 
B)   ​ 
C)   ​ 
D)   ​

Exam 8: Calculus of Several Variables

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) .

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.

Find the average value of the function over the plane region R. Enter your answer as an expression. and R is the triangle with vertices , and .

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 .

Let . Compute the following:

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. X 1 2 3 4 Y 4 6 8 9

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.

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 , , and .

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

Find the total differential of the function.

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

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.

Find the volume of the solid bounded above by the surfac and below by the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (0, 2).

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 .

Minimize the function subject to the constraint .

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 y = 8x and y = 0 from x = 1 to x = 7.

Exam 8: Calculus of Several Variables

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) . ​

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.

Find the average value of the function ​ ​ over the plane region R. Enter your answer as an expression. ​ and R is the triangle with vertices , and .

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 . ​

Let . Compute the following: ​ ​

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. ​ X 1 2 3 4 Y 4 6 8 9 ​ ​

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. ​

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 , , and . ​

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

Find the total differential of the function. ​

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

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.