Question 1

Find the first partial derivatives of the function. ​   ​

Accepted Answer

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

Question 2

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

Accepted Answer

The answer of ​Find the approximate change in z when...

Question 3

Minimize the function ​   ​
Subject to the constraint   .
​

Accepted Answer

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

Question 4

The volume V (in liters) of a certain mass of gas is related to its pressure P (in millimeters of mercury) and its temperature T (in degrees Kelvin) by the law ​   ​
Compute   and   when T = 260 and P = 700.
​

Accepted Answer

A)    The volume increases by 0.044 L when the temperature decreases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume decreases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
B)    The volume increases by 0.044 L when the temperature increases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume decreases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
C)    The volume increases by 0.044 L when the temperature increases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume increases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
D)  ​   The volume increases by 0.044 L when the temperature increases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume increases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
A)    The volume increases by 0.044 L when the temperature decreases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume decreases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
B)    The volume increases by 0.044 L when the temperature increases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume decreases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
C)    The volume increases by 0.044 L when the temperature increases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume increases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​ 
D)  ​   The volume increases by 0.044 L when the temperature increases by 1 degree (beyond 260 K) and the pressure is fixed at 700 mm of mercury. The volume increases by 0.016 L when the pressure increases by 1 mm of mercury (beyond 700 mm) and the temperature is fixed at 260 K. ​

Question 5

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

Accepted Answer

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

Question 6

Use a double integral to find the volume of the solid shown in the figure. Enter your answer as a formula.
​

Accepted Answer

The answer of Use a double integral to find the...

Question 7

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

Accepted Answer

The answer of Use a double integral to find the...

Question 8

Find the volume of the solid bounded above by the surface   and below by the plane region R.   ; R is the region bounded by   .

Accepted Answer

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

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)    is the critical point, it is impossible to determine the relative extrema of the function 
B)    is the point of minimum, 9 is the relative minimum 
C)    is the critical point, the function has neither a relative maximum nor a relative minimum at this point 
D)    is the point of maximum, 9 is the relative maximum 
E)  no critical points 
A)    is the critical point, it is impossible to determine the relative extrema of the function 
B)    is the point of minimum, 9 is the relative minimum 
C)    is the critical point, the function has neither a relative maximum nor a relative minimum at this point 
D)    is the point of maximum, 9 is the relative maximum 
E)  no critical points

Question 10

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
5
6
Y
4)5
5
3
2
3)5
1
​
Please round the coefficients in your equation to two decimal places.
​

Accepted Answer

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

Question 11

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 12

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. ​   ​
Please round the coefficients in your equation to three decimal places.
​

Accepted Answer

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

Question 13

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 maximum value:   
B)    ; relative maximum value:   
C)    ; relative maximum value:   
A)    ; relative maximum value:   
B)    ; relative maximum value:   
C)    ; relative maximum value:

Question 14

Find the domain of the function. ​   ​

Accepted Answer

A)  All real values of   and   
B)  All real values of   and   satisfying   
C)  All real values of   and   satisfying   
A)  All real values of   and   
B)  All real values of   and   satisfying   
C)  All real values of   and   satisfying

Question 15

Minimize the function
​   ​
subject to the constraint   .

Accepted Answer

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

Question 16

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

Accepted Answer

A)  36 
B)  24 
C)  144 
D)  12 
E)  48 
A)  36 
B)  24 
C)  144 
D)  12 
E)  48

Question 17

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...

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 2 ft and height 4 ft.

Accepted Answer

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

Question 19

Find the first partial derivatives of the function. ​   ​

Accepted Answer

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

Question 20

An open rectangular box having a volume of 4in3 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

Find the first partial derivatives of the function.

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

Minimize the function Subject to the constraint .

The volume V (in liters) of a certain mass of gas is related to its pressure P (in millimeters of mercury) and its temperature T (in degrees Kelvin) by the law Compute and when T = 260 and P = 700.

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

Use a double integral to find the volume of the solid shown in the figure. Enter your answer as a formula.

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

Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .

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 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 5 6 Y 4)5 5 3 2 3)5 1 Please round the coefficients in your equation to two decimal places.

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

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. Please round the coefficients in your equation to three decimal places.

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 domain of the function.

Minimize the function subject to the constraint .

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

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

The volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 2 ft and height 4 ft.

Find the first partial derivatives of the function.

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

Find the first partial derivatives of the function. ​ ​

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

Minimize the function ​ ​ Subject to the constraint . ​

The volume V (in liters) of a certain mass of gas is related to its pressure P (in millimeters of mercury) and its temperature T (in degrees Kelvin) by the law ​ ​ Compute and when T = 260 and P = 700. ​

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

Use a double integral to find the volume of the solid shown in the figure. Enter your answer as a formula. ​

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

Find the volume of the solid bounded above by the surface and below by the plane region R. ; R is the region bounded by .

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 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 5 6 Y 4)5 5 3 2 3)5 1 ​ Please round the coefficients in your equation to two decimal places. ​

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

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. ​ ​ Please round the coefficients in your equation to three decimal places. ​

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 domain of the function. ​ ​

Minimize the function ​ ​ subject to the constraint .

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

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

The volume of a cylindrical tank of radius r and height h is given by ​ ​ Find the volume of a cylindrical tank of radius 2 ft and height 4 ft.

Find the first partial derivatives of the function. ​ ​

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

Find the first partial derivatives of the function.

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

Minimize the function Subject to the constraint .

The volume V (in liters) of a certain mass of gas is related to its pressure P (in millimeters of mercury) and its temperature T (in degrees Kelvin) by the law Compute and when T = 260 and P = 700.

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

Use a double integral to find the volume of the solid shown in the figure. Enter your answer as a formula.

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

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 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 5 6 Y 4)5 5 3 2 3)5 1 Please round the coefficients in your equation to two decimal places.

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

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. Please round the coefficients in your equation to three decimal places.

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 domain of the function.

Minimize the function subject to the constraint .

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

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

The volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 2 ft and height 4 ft.

Find the first partial derivatives of the function.