Question 1

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

Accepted Answer

A) 0
B) Th

Question 2

Let   Find   if

Accepted Answer

The answer of Let   Find   if...

Question 3

Find the tangent plane to the surface   at the point   on the surface.

Accepted Answer

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

Question 4

The temperature at the point   in a room is given by   A fly is standing on a table on the plane   at the point   .
Find the direction the fly should move in order to feel the maximum rate of increase in temperature: 
A) if it flies.
B) if it is just walking on the table.

Accepted Answer

The answer of The temperature at the point  ...

Question 5

Let   Find   if

Accepted Answer

The answer of Let   Find   if...

Question 6

Let   . 
A) Do   and   exist? If so, find them. 
B) Use the definition of directional derivative to compute   for   . 
C) Examine whether   holds, and conclude the differentiability of   at the origin.

Accepted Answer

A) @#IMG-DLM& B) @#IMG-DLM& C) @#IMG-DLM& , hence

Question 7

Find the global maximum of   on the unit disk   .

Accepted Answer

The answer of Find the global maximum of  ...

Question 8

Find the linearization of   at the point   and use it to approximate   .

Accepted Answer

The answer of Find the linearization of   at...

Question 9

Let the functions   and   be defined as   and   Does the horizontal trace of   in the plane   intersect the vertical trace of   in the plane   If so, where do the traces intersect?

Accepted Answer

The traces

Question 10

Let   .
Which of the following statements is correct?

Accepted Answer

A)    and   are continuous. 
B)    is continuous but   is not continuous at the origin. 
C)    is continuous but   is not continuous at the origin. 
D)  Both   and   are not continuous at the origin. 
E)  Since   is continuous,   and   are also continuous. 
A)    and   are continuous. 
B)    is continuous but   is not continuous at the origin. 
C)    is continuous but   is not continuous at the origin. 
D)  Both   and   are not continuous at the origin. 
E)  Since   is continuous,   and   are also continuous.

Question 11

Define the function   by   Is   continuous at the point

Accepted Answer

The answer of Define the function   by ...

Question 12

Let   .
Compute the partial derivatives   and   at all the points where they exist.

Accepted Answer

The answer of Let   .
Compute the partial derivatives...

Question 13

Let   be the following function.   
A) Determine whether   is continuous at the origin. 
B) Compute the partial derivatives   and   .

Accepted Answer

The answer of Let   be the following function....

Question 14

Find the linearization of   at the point   and use it to approximate   .

Accepted Answer

The answer of Find the linearization of   at...

Question 15

Find the partial derivatives   , and   of the function   .

Accepted Answer

The answer of Find the partial derivatives   ,...

Question 16

A metal plate is heated so that its temperature at a point   is   .
A bug is placed at the point   . 
A) The bug heads toward the point   . What is the rate of change of temperature in this direction? 
B) In what direction should the bug head in order to warm up at the fastest rate? Find the rate of change of temperature in this direction.
C) In what direction should the bug head in order to cool off at the fastest rate? Find the rate of change of temperature in this direction.

Accepted Answer

A) @#IMG-DLM& per unit length
B) The bug should he

Question 17

Let   . 
A) Find the linearization   of   at the origin. 
B) Estimate   by   and find the error using a calculator.

Accepted Answer

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

Question 18

Which of the following statements is correct for the function   and a nonzero vector

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

Find the point on the surface   where the tangent plane is parallel to the plane   .

Accepted Answer

The answer of Find the point on the surface ...

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

Let Find if

Find the tangent plane to the surface at the point on the surface.

The temperature at the point in a room is given by A fly is standing on a table on the plane at the point . Find the direction the fly should move in order to feel the maximum rate of increase in temperature: A) if it flies. B) if it is just walking on the table.

Let Find if

Let . A) Do and exist? If so, find them. B) Use the definition of directional derivative to compute for . C) Examine whether holds, and conclude the differentiability of at the origin.

Find the global maximum of on the unit disk .

Find the linearization of at the point and use it to approximate .

Let the functions and be defined as and Does the horizontal trace of in the plane intersect the vertical trace of in the plane If so, where do the traces intersect?

Let . Which of the following statements is correct?

Define the function by Is continuous at the point

Let . Compute the partial derivatives and at all the points where they exist.

Let be the following function. A) Determine whether is continuous at the origin. B) Compute the partial derivatives and .

Find the linearization of at the point and use it to approximate .

Find the partial derivatives , and of the function .

Let . A) Find the linearization of at the origin. B) Estimate by and find the error using a calculator.

Which of the following statements is correct for the function and a nonzero vector

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

Find the point on the surface where the tangent plane is parallel to the plane .

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

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

Let Find if

Find the tangent plane to the surface at the point on the surface.

The temperature at the point in a room is given by A fly is standing on a table on the plane at the point . Find the direction the fly should move in order to feel the maximum rate of increase in temperature: A) if it flies. B) if it is just walking on the table.

Let Find if

Let . A) Do and exist? If so, find them. B) Use the definition of directional derivative to compute for . C) Examine whether holds, and conclude the differentiability of at the origin.

Find the global maximum of on the unit disk .

Find the linearization of at the point and use it to approximate .

Let the functions and be defined as and Does the horizontal trace of in the plane intersect the vertical trace of in the plane If so, where do the traces intersect?

Let . Which of the following statements is correct?

Define the function by Is continuous at the point

Let . Compute the partial derivatives and at all the points where they exist.

Let be the following function. A) Determine whether is continuous at the origin. B) Compute the partial derivatives and .

Find the linearization of at the point and use it to approximate .

Find the partial derivatives , and of the function .

Let . A) Find the linearization of at the origin. B) Estimate by and find the error using a calculator.

Which of the following statements is correct for the function and a nonzero vector

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

Find the point on the surface where the tangent plane is parallel to the plane .

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