Question 1

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

Accepted Answer

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

Question 2

Minimize the function
​   ​
subject to the constraint   .
​ 
a.   
b. ​   
c. ​   
d. ​

Accepted Answer

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

Question 3

Let   . Compute the following.   ​

Accepted Answer

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

Question 4

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 5

Find the volume of the solid bounded above by the surface
​
z = f(x, y)
​
and below by the plane region R. Enter your answer as an expression.
​   ; R is the triangle with vertices (0, 0), (4, 0) and (0, 4).

Accepted Answer

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

Question 6

Evaluate the double integral ​   ​
For the given function f(x, y) and the region R.
​
F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.
​

Accepted Answer

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

Question 7

Let   . Compute the following.
​   __________
​   __________
​   __________
​   __________

Accepted Answer

The answer of Let   . Compute the following.
​...

Question 8

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 10. The marginal productivity of capital is 160. The government should have encouraged increased spending on labor rather than on capital. ​ 
B)  The marginal productivity of labor is 7.5. The marginal productivity of capital is 40. The government should not have encouraged increased spending on capital rather than on labor. ​ 
C)  The marginal productivity of labor is 10. The marginal productivity of capital is 160. The government should have encouraged increased spending on capital rather than on labor. ​ 
D)  The marginal productivity of labor is 7.5. The marginal productivity of capital is 40. The government should have encouraged increased spending on capital rather than on labor. ​ 
A)  The marginal productivity of labor is 10. The marginal productivity of capital is 160. The government should have encouraged increased spending on labor rather than on capital. ​ 
B)  The marginal productivity of labor is 7.5. The marginal productivity of capital is 40. The government should not have encouraged increased spending on capital rather than on labor. ​ 
C)  The marginal productivity of labor is 10. The marginal productivity of capital is 160. The government should have encouraged increased spending on capital rather than on labor. ​ 
D)  The marginal productivity of labor is 7.5. The marginal productivity of capital is 40. The government should have encouraged increased spending on capital rather than on labor. ​

Question 9

The radius and height of a right circular cylinder are measured with a maximum error of 0.1 cm in each measurement. Approximate the maximum error in calculating the volume of the cylinder if the measured dimensions r = 8 cm and h = 15 cm are used.

Accepted Answer

The answer of The radius and height of a right...

Question 10

The population density of a certain city is given by the function ​   ​
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
​   ​

Accepted Answer

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

Question 11

An open rectangular box having a volume of 62.5in3 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.
​

Accepted Answer

Question 12

Evaluate the double integral   ​ for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.

Accepted Answer

The answer of Evaluate the double integral   ​...

Question 13

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

A)  2.5 
B)  3.5e 
C)  -3.5 
D)  -2.5 
A)  2.5 
B)  3.5e 
C)  -3.5 
D)  -2.5

Question 14

Evaluate the double integral
​   ​
for the function   and the region R. Enter your answer as an expression.
​   and R is bounded by   ,   ,   and   .

Accepted Answer

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

Question 15

An auxiliary electric power station will serve three communities, A, B, and C, whose relative locations are shown in the accompanying figure. Determine where the power station should be located if the sum of the squares of the distances from each community to the site is minimized. Please round the answer to the nearest hundredth. ​
​   ​

Accepted Answer

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

Question 16

Maximize the function
​   ​
subject to the constraint   .

Accepted Answer

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

Question 17

The population density of a certain city is given by the function
​   ​
where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
​   . Enter your answer as an expression.

Accepted Answer

The answer of The population density of a certain city...

Question 18

​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 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

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

Minimize the function subject to the constraint . a. b. c. d.

Let . Compute the following.

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

Find the volume of the solid bounded above by the surface z = f(x, y) and below by the plane region R. Enter your answer as an expression. ; R is the triangle with vertices (0, 0), (4, 0) and (0, 4).

Evaluate the double integral For the given function f(x, y) and the region R. F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

Let . Compute the following.

The radius and height of a right circular cylinder are measured with a maximum error of 0.1 cm in each measurement. Approximate the maximum error in calculating the volume of the cylinder if the measured dimensions r = 8 cm and h = 15 cm are used.

The population density of a certain city is given by the function Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by

An open rectangular box having a volume of 62.5in³ 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.

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.

Evaluate the double integral for the function and the region R. Enter your answer as an expression. and R is bounded by , , and .

Maximize the function subject to the constraint .

The population density of a certain city is given by the function where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by . Enter your answer as an expression.

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

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

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

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

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

Minimize the function ​ ​ subject to the constraint . ​ a. b. ​ c. ​ d. ​

Let . Compute the following. ​

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

Find the volume of the solid bounded above by the surface ​ z = f(x, y) ​ and below by the plane region R. Enter your answer as an expression. ​ ; R is the triangle with vertices (0, 0), (4, 0) and (0, 4).

Evaluate the double integral ​ ​ For the given function f(x, y) and the region R. ​ F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1. ​

Let . Compute the following. ​ __________ ​ __________ ​ __________ ​ __________

The radius and height of a right circular cylinder are measured with a maximum error of 0.1 cm in each measurement. Approximate the maximum error in calculating the volume of the cylinder if the measured dimensions r = 8 cm and h = 15 cm are used.

The population density of a certain city is given by the function ​ ​ Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by ​ ​

An open rectangular box having a volume of 62.5in3 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. ​

Evaluate the double integral ​ for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 3x + 4y; R is bounded by x = 0, x = 1, y = 0 and y = x.

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. ​

Evaluate the double integral ​ ​ for the function and the region R. Enter your answer as an expression. ​ and R is bounded by , , and .

Maximize the function ​ ​ subject to the constraint .

The population density of a certain city is given by the function ​ ​ where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by ​ . Enter your answer as an expression.

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

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

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

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

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

Minimize the function subject to the constraint . a. b. c. d.

Let . Compute the following.

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

Find the volume of the solid bounded above by the surface z = f(x, y) and below by the plane region R. Enter your answer as an expression. ; R is the triangle with vertices (0, 0), (4, 0) and (0, 4).

Evaluate the double integral For the given function f(x, y) and the region R. F(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

Let . Compute the following.

The population density of a certain city is given by the function Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by

An open rectangular box having a volume of 62.5in³ 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.

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.

Evaluate the double integral for the function and the region R. Enter your answer as an expression. and R is bounded by , , and .

Maximize the function subject to the constraint .

The population density of a certain city is given by the function where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by . Enter your answer as an expression.

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

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

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