Question 1

Find the level curves of the function   , and sketch a contour map.

Accepted Answer

The level curves are

Question 2

Let   be the function defined by   Describe the shape of the level surface

Accepted Answer

The equation of the level surf

Question 3

Find   and   given that   .

Accepted Answer

The answer of Find   and   given...

Question 4

Compute the following limit or explain why it does not exist.

Accepted Answer

The limit does not exist. Appr

Question 5

Which of the following functions is differentiable at the origin?

Accepted Answer

A)    
B)    
C)    
D)    
E)  None of the above 
A)    
B)    
C)    
D)    
E)  None of the above

Question 6

Find the derivative of   at the point   in the direction of the vector   , where   ,   and   .

Accepted Answer

The answer of Find the derivative of   at...

Question 7

Find the global minimum and maximum of   subject to the constraints   and   .

Accepted Answer

Global minimum @#IMG-DLM& @#IMG-DLM& doe

Question 8

Let   be the function defined by   . 
A) Determine whether   is continuous. 
B) Compute whether   and   exist. 
C) Determine whether   is differentiable at   .

Accepted Answer

A) @#IMG-DLM& is continuous fo

Question 9

Find the gradient of the function

Accepted Answer

The answer of Find the gradient of the function ...

Question 10

Find the range of the function   restricted to the domain   .

Accepted Answer

The answer of Find the range of the function ...

Question 11

Find the domain of the function   , and sketch it in the   plane.

Accepted Answer

The answer of Find the domain of the function ...

Question 12

Find an equation of the tangent plane to the graph of the function   at the point   .

Accepted Answer

A)    
B)    
C)    
D)    
E)  None of the above 
A)    
B)    
C)    
D)    
E)  None of the above

Question 13

If   , then   is:

Accepted Answer

A)    . 
B)    . 
C)    . 
D)    . 
E)  none of the above. 
A)    . 
B)    . 
C)    . 
D)    . 
E)  none of the above.

Question 14

Determine whether the following function is continuous.

Accepted Answer

@#IMG-DLM& is not continuous a

Question 15

Find all the functions   such that the function   is a solution of the differential equation   .

Accepted Answer

The answer of Find all the functions   such...

Question 16

The maximum value of   , subject to   is obtained at which of the following point(s)?

Accepted Answer

A)    
B)    
C)    
D)    
E)  The maximum value does not occur at any of these points. 
A)    
B)    
C)    
D)    
E)  The maximum value does not occur at any of these points.

Question 17

Find the level curves of the function   ,
and sketch a contour map.

Accepted Answer

The @#IMG-DLM& axis (except the origin)

Question 18

The equation   defines the function   near the point   .
The point   is:

Accepted Answer

A)  a local maximum of   . 
B)  a local minimum of   . 
C)  a saddle point of   . 
D)  a global maximum of   . 
E)  not a critical point of   . 
A)  a local maximum of   . 
B)  a local minimum of   . 
C)  a saddle point of   . 
D)  a global maximum of   . 
E)  not a critical point of   .

Question 19

Describe and sketch the level curves of the function

Accepted Answer

The answer of Describe and sketch the level curves of...

Question 20

Evaluate the limit or state that it does not exist.

Accepted Answer

The answer of Evaluate the limit or state that it...

Find the level curves of the function , and sketch a contour map.

Let be the function defined by Describe the shape of the level surface

Find and given that .

Compute the following limit or explain why it does not exist.

Which of the following functions is differentiable at the origin?

Find the derivative of at the point in the direction of the vector , where , and .

Find the global minimum and maximum of subject to the constraints and .

Let be the function defined by . A) Determine whether is continuous. B) Compute whether and exist. C) Determine whether is differentiable at .

Find the gradient of the function

Find the range of the function restricted to the domain .

Find the domain of the function , and sketch it in the plane.

Find an equation of the tangent plane to the graph of the function at the point .

If , then is:

Determine whether the following function is continuous.

Find all the functions such that the function is a solution of the differential equation .

The maximum value of , subject to is obtained at which of the following point(s)?

Find the level curves of the function , and sketch a contour map.

The equation defines the function near the point . The point is:

Describe and sketch the level curves of the function

Evaluate the limit or state that it does not exist.

Precalculus Review

Limits

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Calculus of Vector-Valued Functions

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

Filters

Exam 15: Differentiation in Several Variables

Find the level curves of the function , and sketch a contour map.

Let be the function defined by Describe the shape of the level surface

Find and given that .

Compute the following limit or explain why it does not exist.

Which of the following functions is differentiable at the origin?

Find the derivative of at the point in the direction of the vector , where , and .

Find the global minimum and maximum of subject to the constraints and .

Let be the function defined by . A) Determine whether is continuous. B) Compute whether and exist. C) Determine whether is differentiable at .

Find the gradient of the function

Find the range of the function restricted to the domain .

Find the domain of the function , and sketch it in the plane.

Find an equation of the tangent plane to the graph of the function at the point .

If , then is:

Determine whether the following function is continuous.

Find all the functions such that the function is a solution of the differential equation .

The maximum value of , subject to is obtained at which of the following point(s)?

Find the level curves of the function , and sketch a contour map.

The equation defines the function near the point . The point is:

Describe and sketch the level curves of the function

Evaluate the limit or state that it does not exist.

Precalculus Review

Limits

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Calculus of Vector-Valued Functions

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

Filters