Question 1

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

Accepted Answer

The answer of Find the tangent plane to the graph...

Question 2

Find the distance to the origin from the intersection of the two surfaces   and   ; that is, the minimum distance between the origin and a point on the intersection.

Accepted Answer

The distance is 1, o

Question 3

Given that   find   and

Accepted Answer

The answer of Given that   find  ...

Question 4

Evaluate the limit or state that it does not exist.
A)   
B)

Accepted Answer

A) The lim

Question 5

The plane   best fits the points       and   if the sum of   at these points is minimized by     and   The plane that best fits the points   ,   ,   , and   is:

Accepted Answer

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

Question 6

Compute the partial derivatives   and   of the function   .

Accepted Answer

The answer of Compute the partial derivatives   and...

Question 7

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

Accepted Answer

The level curves of the functi

Question 8

Let   .
Define the function   by   Is   continuous on the domain

Accepted Answer

The answer of Let   .
Define the function ...

Question 9

Which of the following functions satisfies the heat equation

Accepted Answer

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

Question 10

The function   satisfies which differential equation?

Accepted Answer

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

Question 11

Find the domain and range of the function   .

Accepted Answer

The answer of Find the domain and range of the...

Question 12

Evaluate the limit or state that it does not exist.
A)   
B)

Accepted Answer

A) The lim

Question 13

Let   . Calculate   , where   ,   , and   .

Accepted Answer

The answer of Let   . Calculate  ...

Question 14

Which of the following statements hold for

Accepted Answer

A)    is continuous, and the partial derivatives   and   exist everywhere. 
B)    is continuous, but the partial derivatives   and   do not exist at the origin. 
C)    is continuous and   exists everywhere, but   does not exist at the origin. 
D)    and   do not exist at points   and   . 
E)    is not continuous at the origin. 
A)    is continuous, and the partial derivatives   and   exist everywhere. 
B)    is continuous, but the partial derivatives   and   do not exist at the origin. 
C)    is continuous and   exists everywhere, but   does not exist at the origin. 
D)    and   do not exist at points   and   . 
E)    is not continuous at the origin.

Question 15

The tangent plane to the surface   is parallel to the plane   and contains the point   .
Find the value of   .

Accepted Answer

The answer of The tangent plane to the surface ...

Question 16

Let   , and compute   and   .

Accepted Answer

The answer of Let   , and compute ...

Question 17

Find the set of all critical points of the function

Accepted Answer

The answer of Find the set of all critical points...

Question 18

The function   has:

Accepted Answer

A)  one local minimum, one local maximum, and two saddle points. 
B)  two local minima, one local maximum, and one saddle point. 
C)  one local minimum, three local maxima, and one saddle point. 
D)  two local minima and two local maxima. 
E)  four saddle points. 
A)  one local minimum, one local maximum, and two saddle points. 
B)  two local minima, one local maximum, and one saddle point. 
C)  one local minimum, three local maxima, and one saddle point. 
D)  two local minima and two local maxima. 
E)  four saddle points.

Question 19

Let   . 
A) Find the parametrization for the plane   using the parameters   and   . 
B) Compute   and   .

Accepted Answer

The answer of Let   . 
A) Find the...

Question 20

Find the global minimum of the function   on the region

Accepted Answer

The answer of Find the global minimum of the function...

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

Find the distance to the origin from the intersection of the two surfaces and ; that is, the minimum distance between the origin and a point on the intersection.

Given that find and

Evaluate the limit or state that it does not exist. A) B)

The plane best fits the points and if the sum of at these points is minimized by and The plane that best fits the points , , , and is:

Compute the partial derivatives and of the function .

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

Let . Define the function by Is continuous on the domain

Which of the following functions satisfies the heat equation

The function satisfies which differential equation?

Find the domain and range of the function .

Evaluate the limit or state that it does not exist. A) B)

Let . Calculate , where , , and .

Which of the following statements hold for

The tangent plane to the surface is parallel to the plane and contains the point . Find the value of .

Let , and compute and .

Find the set of all critical points of the function

The function has:

Let . A) Find the parametrization for the plane using the parameters and . B) Compute and .

Find the global minimum of the function on the region

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 tangent plane to the graph of the function at the point .

Find the distance to the origin from the intersection of the two surfaces and ; that is, the minimum distance between the origin and a point on the intersection.

Given that find and

Evaluate the limit or state that it does not exist. A) B)

The plane best fits the points and if the sum of at these points is minimized by and The plane that best fits the points , , , and is:

Compute the partial derivatives and of the function .

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

Let . Define the function by Is continuous on the domain

Which of the following functions satisfies the heat equation

The function satisfies which differential equation?

Find the domain and range of the function .

Evaluate the limit or state that it does not exist. A) B)

Let . Calculate , where , , and .

Which of the following statements hold for

The tangent plane to the surface is parallel to the plane and contains the point . Find the value of .

Let , and compute and .

Find the set of all critical points of the function

The function has:

Let . A) Find the parametrization for the plane using the parameters and . B) Compute and .

Find the global minimum of the function on the region

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