Question 1

Find the derivative of  at the point  with respect to the vector  , where  ,  , and  .

Accepted Answer

The answer of Find the derivative of  at the...

Question 2

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

Accepted Answer

The answer of Find the linearization of  at the...

Question 3

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 the...

Question 4

Find an equation of the tangent plane to the graph of the function  at the point  .
A) 
B) 
C) 
D) 
E) None of the above

Accepted Answer

The answer of Find an equation of the tangent plane...

Question 5

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

Accepted Answer

The answer of Find the linearization of  at the...

Question 6

Let  . 
A) Find  and  , if they exist. 
B) Use the definition of differentiability at a point to determine whether  is differentiable at the origin.

Accepted Answer

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

Question 7

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

Accepted Answer

The answer of Find the linearization of  at the...

Question 8

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 linearization...

Question 9

Find  and  if  .

Accepted Answer

The answer of Find  and  if  ....

Question 10

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

Accepted Answer

The answer of Find the linearization of  at the...

Question 11

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

Accepted Answer

The answer of Let  be the function defined by...

Question 12

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 ...

Question 13

Find the gradient of the function

Accepted Answer

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

Question 14

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  in...

Question 15

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

Accepted Answer

The answer of Find the linearization of  at the...

Question 16

Let  . 
A) Is  differentiable at the origin? 
B) Does  have a tangent plane at the origin? If so, find its equation.

Accepted Answer

The answer of Let  . 
A) Is  differentiable...

Question 17

Which of the following functions is differentiable at the origin?
A) 
B) 
C) 
D) 
E) None of the above

Accepted Answer

The function represented by option B is of the form $f(x)=x\sin(\frac{1}{x})$ for $x 
eq 0$, and $f(0)=0$. It can be shown using the limit definition of derivative that $f'(0)=0$, hence the function is differentiable at the origin.

Option A is not differentiable at the origin since the limit defining the derivative does not exist. 
Option C is not even continuous at the origin, hence not differentiable at the origin. 
Option D is also not continuous at the origin, hence not differentiable at the origin.

Question 18

If  and  change by  and  , respectively, from  and  , then the approximate change  in the angle  is which of the following?
A) 
B) 0.464
C) 0.003
D) 1.373
E) 0.002

Accepted Answer

The answer of If  and  change by ...

Question 19

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 20

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 ...

Quiz 15: Differentiation in Several Variables

Find the derivative of at the point with respect to the vector , where , , and .

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

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

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

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

Let . A) Find and , if they exist. B) Use the definition of differentiability at a point to determine whether is differentiable at the origin.

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

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

Find and if .

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

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

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

Find the gradient of the function

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.

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

Let . A) Is differentiable at the origin? B) Does have a tangent plane at the origin? If so, find its equation.

Which of the following functions is differentiable at the origin?

If and change by and , respectively, from and , then the approximate change in the angle is which of the following?

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

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