Question 1

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

Accepted Answer

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

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

Question 3

Find the second-order partial derivatives of the function. ​   ​

Accepted Answer

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

Question 4

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

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

Question 5

The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function
​   ​
where x 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 between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary.
​
P = $__________ per week.

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Minimize the function ​
​   ​
Subject to the constraint   ​

Accepted Answer

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

Question 8

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   .

Accepted Answer

The answer of An open rectangular box is to be...

Question 9

Find the volume of the solid bounded above by the surface ​   ​
And below by the plane region R.
​
​   ​
R is the triangle with vertices   and   .
​

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Question 12

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, $5/square foot for the front and back, and $3/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

The answer of A building in the shape of a...

Question 13

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. ​   ​

Accepted Answer

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

Question 14

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

Question 15

Let   . Compute the following. ​   ,   ​

Accepted Answer

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

Question 16

The flow of blood through an arteriole in cubic centimeters per second is given by ​   ​
Where l is the length (in cm) of the arteriole, r is its radius (in cm), p is the difference in pressure between the two ends of the arteriole (in dyne/cm2), and k is the viscosity of blood (in dyne-sec/cm2). Find the approximate percentage change in the flow of blood if an error of 4% is made in measuring the length of the arteriole and an error of 3% is made in measuring its radius. Assume that p and k are constant.
​

Accepted Answer

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

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. ​
The least-squares line must pass through at least one data point.

Accepted Answer

A)  It is false. ​ Example. Find the least-squares line for the data ​   ​Solution. Here, we have n=5 and   The least-squares line for the data is given by linear equation y= f(x)= mx+b where the constants m and b satisfy the normal equations       Then, we obtain the normal equations   Solving them, we found m=1, b=0.4 Therefore, the required least-squares line is y= x+0.4 The scatter diagram and the least-squares line are shown in the figure. We can see that the line does not pass through any data point.
  
B)  It is true, Suppose that we are given two data points   If we try to fit a straight line to these data points, the line will miss the first and the second data points by the amounts d1, d2, respectively. 
 
The principle of least squares states that the straight line L that fits the data points best is the one chosen by requiring that the sum of the squares   be made as small as possible. But it is possible only in case when least-squares line passes through at least one data point. 
A)  It is false. ​ Example. Find the least-squares line for the data ​   ​Solution. Here, we have n=5 and   The least-squares line for the data is given by linear equation y= f(x)= mx+b where the constants m and b satisfy the normal equations       Then, we obtain the normal equations   Solving them, we found m=1, b=0.4 Therefore, the required least-squares line is y= x+0.4 The scatter diagram and the least-squares line are shown in the figure. We can see that the line does not pass through any data point.
  
B)  It is true, Suppose that we are given two data points   If we try to fit a straight line to these data points, the line will miss the first and the second data points by the amounts d1, d2, respectively. 
 
The principle of least squares states that the straight line L that fits the data points best is the one chosen by requiring that the sum of the squares   be made as small as possible. But it is possible only in case when least-squares line passes through at least one data point.

Question 18

Let   . Compute the following. ​   ,   ,   ​

Accepted Answer

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

Question 19

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

Accepted Answer

A)  12 
B)  96 
C)  64 
D)  256 
E)  384 
A)  12 
B)  96 
C)  64 
D)  256 
E)  384

Question 20

Maximize the function
​   ​
subject to the constraint   .

Accepted Answer

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

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

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 derivatives of the function.

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.

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

Minimize the function Subject to the constraint

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 .

Find the volume of the solid bounded above by the surface And below by the plane region R. R is the triangle with vertices and .

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

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

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.

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. ,

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. The least-squares line must pass through at least one data point.

Let . Compute the following. , ,

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

Maximize the function subject to the constraint .

Preliminaries

Functions, Limits, and the Derivative

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Differential Equations

Probability and Calculus

Taylor Polynomials and Infinite Series

Trigonometric Functions

Filters

Exam 8: Calculus of Several Variables

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

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 derivatives of the function. ​ ​

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. ​

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

Minimize the function ​ ​ ​ Subject to the constraint ​

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 .

Find the volume of the solid bounded above by the surface ​ ​ And below by the plane region R. ​ ​ ​ R is the triangle with vertices and . ​

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

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

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. ​ ​

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. ​ , ​

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. ​ The least-squares line must pass through at least one data point.

Let . Compute the following. ​ , , ​

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

Maximize the function ​ ​ subject to the constraint .

Preliminaries

Functions, Limits, and the Derivative

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Differential Equations

Probability and Calculus

Taylor Polynomials and Infinite Series

Trigonometric Functions

Filters

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

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 derivatives of the function.

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.

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

Minimize the function Subject to the constraint

Find the volume of the solid bounded above by the surface And below by the plane region R. R is the triangle with vertices and .

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

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

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.

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. ,

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. The least-squares line must pass through at least one data point.

Let . Compute the following. , ,

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

Maximize the function subject to the constraint .