Question 1

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

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Question 2

Let  Find  if

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The answer of Let  Find  if ...

Question 3

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

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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.

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Question 5

Let  Find  if

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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.

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A) Do  and...

Question 7

Find the global maximum of  on the unit disk  .

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Question 8

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

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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?

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The answer of Let the functions  and  be...

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.

Question 11

Define the function  by  Is  continuous at the point

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Question 12

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

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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  .

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Question 14

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

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Question 15

Find the partial derivatives  , and  of the function  .

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The answer of Find the partial derivatives  , and...

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.

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Question 17

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

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A) Find the linearization...

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

Question 19

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

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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 ...

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 .