Question 1

Minimize the function   subject to the constraint   .

Accepted Answer

The answer of Minimize the function   subject to...

Question 2

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = xy; z = - 2, - 1, 1, 2

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 3

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.

Accepted Answer

The answer of The Country Workshop's total weekly profit (in...

Question 4

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = 2x + 6y; z = - 2, - 1, 0, 1, 2

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 5

A level curves of a function   of two variables,   ,

Accepted Answer

A) Exists for all values of   
B) Exists for all values of   
C) Exists for all values of   
D) Exists for all values of   
E) Exists for all values of   
A) Exists for all values of   
B) Exists for all values of   
C) Exists for all values of   
D) Exists for all values of   
E) Exists for all values of

Question 6

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

Accepted Answer

A)      
B)      
C)      
D)      
E)      
A)      
B)      
C)      
D)      
E)

Question 7

The productivity of a South American country is given by the function   when x units of labor and y units of capital are used.What is the marginal productivity of labor and what is the marginal productivity of capital when the amounts expended on labor and capital are 288 units and 18 units, respectively? Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity? Please answer yes or no.

Accepted Answer

A) The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on capital rather than on labor. 
B) The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should not have encouraged increased spending on capital rather than on labor. 
C) The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should have encouraged increased spending on capital rather than on labor. 
D) The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on labor rather than on capital. 
A) The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on capital rather than on labor. 
B) The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should not have encouraged increased spending on capital rather than on labor. 
C) The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should have encouraged increased spending on capital rather than on labor. 
D) The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on labor rather than on capital.

Question 8

Minimize the function   subject to the constraints

Accepted Answer

A) Min.of   at
  
B) Min.of   at
  
C) Min.of   at
  
D) Min.of   at
  
E) Min.of   at
  
A) Min.of   at
  
B) Min.of   at
  
C) Min.of   at
  
D) Min.of   at
  
E) Min.of   at

Question 9

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, - 1); saddle point: f(0, - 1) = - 1 
B) (0, 0); relative minimum value: f(0, 0) = 0 
C) (1, 0); relative maximum value: f(1, 0) = 3 
D) there are no critical points 
A) (0, - 1); saddle point: f(0, - 1) = - 1 
B) (0, 0); relative minimum value: f(0, 0) = 0 
C) (1, 0); relative maximum value: f(1, 0) = 3 
D) there are no critical points

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.   then f must have a critical point at (a, b).

Accepted Answer

A) True 
 B)False

Question 11

Find the first partial derivatives of the function.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

Let f(x, y) = 2x + 3y - 4.Compute the following. f(1, 0), f(0, 1), f(1, 2)

Accepted Answer

A) f(1, 0) = - 2, f(0, 1) = - 1, f(1, 2) = 4 
B) f(1, 0) = 2, f(0, 1) = 1, f(1, 2) = - 4 
C) f(1, 0) = - 3, f(0, 1) = - 2, f(1, 2) = 5 
D) f(1, 0) = 3, f(0, 1) = 2, f(1, 2) = - 5 
A) f(1, 0) = - 2, f(0, 1) = - 1, f(1, 2) = 4 
B) f(1, 0) = 2, f(0, 1) = 1, f(1, 2) = - 4 
C) f(1, 0) = - 3, f(0, 1) = - 2, f(1, 2) = 5 
D) f(1, 0) = 3, f(0, 1) = 2, f(1, 2) = - 5

Question 13

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives   and   are equal.

Accepted Answer

A)    ;   ;
  ;
  
B)    ;

C)    ;

D)

A)    ;   ;
  ;
  
B)    ;

C)    ;

D)

Question 14

Find an equation of the least-squares line for the data.Please round the coefficients to the nearest hundredth.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

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

The answer of Find the average value of the given...

Question 16

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)

Accepted Answer

A) x = 1; y = 1; z = 2 
B) x = 2; y = 2; z = 1 
C) x = 2; y = 1; z = 1 
D) x = 2; y = 1; z = 2 
A) x = 1; y = 1; z = 2 
B) x = 2; y = 2; z = 1 
C) x = 2; y = 1; z = 1 
D) x = 2; y = 1; z = 2

Question 17

Find the first partial derivatives of the function.

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 18

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) (1, 5) and (2, 5); saddle point: f(1, 5) = - 47; relative minimum value: f(2, 5) = - 48 
B) (1, 5) and (2, 5); relative maximum value: f(1, 5) = - 48; relative minimum value: f(2, 5) = 47 
C) (1, 5); saddle point: f(1, 5) = - 47 
D) (2, 5); relative maximum value: f(2, 5) = 48 
E) there are no critical points 
A) (1, 5) and (2, 5); saddle point: f(1, 5) = - 47; relative minimum value: f(2, 5) = - 48 
B) (1, 5) and (2, 5); relative maximum value: f(1, 5) = - 48; relative minimum value: f(2, 5) = 47 
C) (1, 5); saddle point: f(1, 5) = - 47 
D) (2, 5); relative maximum value: f(2, 5) = 48 
E) there are no critical points

Question 19

The management of Kaldor, a manufacturer of electric motors, submitted the following data in the annual report to its stockholders.The table shows the net sales (in millions of dollars) during the 5 years that have elapsed since the new management team took over.(The first year the firm operated under the new management corresponds to the time period   , and the four subsequent years correspond to   .)   Determine the equation of the least-squares line for these data and use it to predict the net sales for the upcoming year.

Accepted Answer

A) 485.8 millions of dollars 
B) 529.8 millions of dollars 
C) 519.8 millions of dollars 
D) 512.8 millions of dollars 
E) 552.8 millions of dollars 
A) 485.8 millions of dollars 
B) 529.8 millions of dollars 
C) 519.8 millions of dollars 
D) 512.8 millions of dollars 
E) 552.8 millions of dollars

Question 20

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

Accepted Answer

The answer of Evaluate the double integral   for...

Minimize the function subject to the constraint .

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = xy; z = - 2, - 1, 1, 2

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = 2x + 6y; z = - 2, - 1, 0, 1, 2

A level curves of a function of two variables, ,

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

Minimize the function subject to the constraints

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.

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. then f must have a critical point at (a, b).

Find the first partial derivatives of the function.

Let f(x, y) = 2x + 3y - 4.Compute the following. f(1, 0), f(0, 1), f(1, 2)

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives and are equal.

Find an equation of the least-squares line for the data.Please round the coefficients to the nearest hundredth.

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).

Find the first partial derivatives of the function.

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.

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

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Exam 8: Calculus of Several Variables

Minimize the function subject to the constraint .

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = xy; z = - 2, - 1, 1, 2

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = 2x + 6y; z = - 2, - 1, 0, 1, 2

A level curves of a function of two variables, ,

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

Minimize the function subject to the constraints

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.

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. then f must have a critical point at (a, b).

Find the first partial derivatives of the function.

Let f(x, y) = 2x + 3y - 4.Compute the following. f(1, 0), f(0, 1), f(1, 2)

Find the second-order partial derivatives of the given function.Show that the mixed partial derivatives and are equal.

Find an equation of the least-squares line for the data.Please round the coefficients to the nearest hundredth.

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).

Find the first partial derivatives of the function.

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.

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

Preliminaries

Functions, Limits and the Derivative

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Filters