Question 1

Determine the tangent plane for the hyperboloid   at   . ​

Accepted Answer

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

Question 2

Evaluate   , where     . ​

Accepted Answer

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

Question 3

Calculate the line integral along   for   and C is any path starting at the point   and ending at   . ​

Accepted Answer

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

Question 4

Find a vector-valued function whose graph is the plane   . ​

Accepted Answer

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

Question 5

The surface of the dome on a new museum is given by   , where   and   and   is in meters. Find the surface area of the dome. ​

Accepted Answer

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

Question 6

Evaluate the line integral   using the Fundamental Theorem of Line Integrals, where   is the smooth curve from   to   . ​

Accepted Answer

A)  210 
B)  318 
C)  324 
D)  36 
E)  204 
A)  210 
B)  318 
C)  324 
D)  36 
E)  204

Question 7

For the vector field   , find the value of   for which the field is conservative. ​

Accepted Answer

A)  7 
B)  3 
C)  21 
D)  14 
E)    is not conservative for any value of   . 
A)  7 
B)  3 
C)  21 
D)  14 
E)    is not conservative for any value of   .

Question 8

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. ​     -axis
​

Accepted Answer

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

Question 9

Find   for the vector field given by   . ​

Accepted Answer

A)  37 
B)    
C)  73 
D)  0 
E)  9 
A)  37 
B)    
C)  73 
D)  0 
E)  9

Question 10

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the planes   and   . ​

Accepted Answer

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

Question 11

Match the following vector-valued function with its graph. ​

Accepted Answer

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

Question 12

Use Green's Theorem to evaluate the integral   for the path C:   .

Accepted Answer

A)    
B)  0 
C)  10 
D)    
E)  1000 
A)    
B)  0 
C)  10 
D)    
E)  1000

Question 13

Find the total mass of the wire with density   . ​   ,   ,   ​

Accepted Answer

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

Question 14

Use the Divergence Theorem to evaluate   and find the outward flux of   through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.   ​   ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    
E)  0 
A)    ​ 
B)    ​ 
C)    ​ 
D)    
E)  0

Question 15

Use Green's Theorem to evaluate the line integral   where   is   . ​

Accepted Answer

A)  2 
B)  3 
C)  5 
D)  4 
E)  0 
A)  2 
B)  3 
C)  5 
D)  4 
E)  0

Question 16

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

Accepted Answer

A)  conservative with potential function   
B)  conservative with potential function   
C)  conservative with potential function   
D)  conservative with potential function   
E)  not conservative 
A)  conservative with potential function   
B)  conservative with potential function   
C)  conservative with potential function   
D)  conservative with potential function   
E)  not conservative

Question 17

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the sphere   . ​

Accepted Answer

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

Question 18

Use Stokes's Theorem to evaluate   . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above.   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  0 
A)    
B)    
C)    
D)    
E)  0

Question 19

Let   and let S be the surface bounded by   and   . Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral. Round your answer to two decimal places.
​

Accepted Answer

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

Question 20

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Accepted Answer

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

Determine the tangent plane for the hyperboloid at .

Evaluate , where .

Calculate the line integral along for and C is any path starting at the point and ending at .

Find a vector-valued function whose graph is the plane .

The surface of the dome on a new museum is given by , where and and is in meters. Find the surface area of the dome.

Evaluate the line integral using the Fundamental Theorem of Line Integrals, where is the smooth curve from to .

For the vector field , find the value of for which the field is conservative.

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. -axis

Find for the vector field given by .

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and .

Match the following vector-valued function with its graph.

Use Green's Theorem to evaluate the integral for the path C: .

Find the total mass of the wire with density . , ,

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Use Green's Theorem to evaluate the line integral where is .

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the sphere .

Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above.

Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Exam 15: Vector Anal

Determine the tangent plane for the hyperboloid at . ​

Evaluate , where . ​

Calculate the line integral along for and C is any path starting at the point and ending at . ​

Find a vector-valued function whose graph is the plane . ​

The surface of the dome on a new museum is given by , where and and is in meters. Find the surface area of the dome. ​

Evaluate the line integral using the Fundamental Theorem of Line Integrals, where is the smooth curve from to . ​

For the vector field , find the value of for which the field is conservative. ​

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. ​ -axis ​

Find for the vector field given by . ​

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and . ​

Match the following vector-valued function with its graph. ​

Use Green's Theorem to evaluate the integral for the path C: .

Find the total mass of the wire with density . ​ , , ​

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. ​ ​

Use Green's Theorem to evaluate the line integral where is . ​

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the sphere . ​

Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. ​

Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places. ​

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Determine the tangent plane for the hyperboloid at .

Evaluate , where .

Calculate the line integral along for and C is any path starting at the point and ending at .

Find a vector-valued function whose graph is the plane .

The surface of the dome on a new museum is given by , where and and is in meters. Find the surface area of the dome.

Evaluate the line integral using the Fundamental Theorem of Line Integrals, where is the smooth curve from to .

For the vector field , find the value of for which the field is conservative.

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. -axis

Find for the vector field given by .

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and .

Match the following vector-valued function with its graph.

Find the total mass of the wire with density . , ,

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Use Green's Theorem to evaluate the line integral where is .

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the sphere .

Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above.

Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.