Question 1

Describe the family of level curves for z = x² + y², (z $\ge$ 0) and sketch a few of these curves.

Accepted Answer

The answer of Describe the family of level curves for...

Question 2

Find a point on the surface z = 16 - 12x² - y² at which the tangent plane is perpendicular to the line x = 3 + 12t, y = 2t, z = 2 - t.

Accepted Answer

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

Question 3

Sketch the level curves for z = xy for k = 0, 1, -1, and 4.

Accepted Answer

The answer of Sketch the level curves for z =...

Question 4

Let  .  is

Accepted Answer

A)  A relative maximum 
B)  A relative minimum 
C)  A saddle point 
D)  Cannot be determined 
E)  Both a relative maximum and a saddle point

Question 5

Use the chain rule to find,  ,  , and  and if w = 30 + xy + yz, x = r cos $\theta$ , y = r sin $\theta$ , z = z.

Accepted Answer

The answer of Use the chain rule to find, ...

Question 6

Evaluate  .

Accepted Answer

A)  16 
B)  26 
C)  -78 
D)  Does not exist 
E)  6

Question 7

Let  ;  Find  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 8

Use the total differential to approximate the change in  as (x, y) varies from  to  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 9

Evaluate  .

Accepted Answer

A)  6 
B)  7 
C)  1 
D)  Does not exist 
E)  30

Question 10

Let  . Find  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 11

Find an equation for the tangent plane to  at  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 12

Let  . There is a critical point at

Accepted Answer

A)  (0, 0) 
B)   
C)   
D)  none exist 
E)  (1, 1)

Question 13

Let  . Compute the differential dz.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 14

Evaluate at the point whose spherical coordinates are if $ \omega $ = (x² - 2y + z)² and x = $\rho$ sin $\varphi$ cos $\theta$ , y = $\rho$ sin $\varphi$sin $\theta$ , z = $\rho$ cos $\varphi$.

Accepted Answer

The answer of Evaluate  at the point whose spherical...

Question 15

Find the rate of change of  at (1, 6) in the direction of a vector making an angle of 120° with the positive x axis.

Accepted Answer

The answer of Find the rate of change of ...

Question 16

Let  . Find  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 17

An open rectangular box is to contain 864 cubic inches. Use the Lagrange multiplier method to find the dimensions of the box which uses the least amount of material.

Accepted Answer

The answer of An open rectangular box is to contain...

Question 18

Let  . Compute the differential dz.

Accepted Answer

The answer of Let  . Compute the differential dz....

Question 19

Let  . Find  if  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 20

Let  ;  Find  .

Accepted Answer

The answer of Let  ;  Find  ....

Exam 13: Partial Derivatives

Describe the family of level curves for z = x2 + y2, (z ≥\ge≥ 0) and sketch a few of these curves.

Find a point on the surface z = 16 - 12x2 - y2 at which the tangent plane is perpendicular to the line x = 3 + 12t, y = 2t, z = 2 - t.

Sketch the level curves for z = xy for k = 0, 1, -1, and 4.

Let . is

Use the chain rule to find, , , and and if w = 30 + xy + yz, x = r cos θ\thetaθ , y = r sin θ\thetaθ , z = z.

Evaluate .

Let ; Find .

Use the total differential to approximate the change in as (x, y) varies from to .

Evaluate .

Let . Find .

Find an equation for the tangent plane to at .

Let . There is a critical point at

Let . Compute the differential dz.

Evaluate at the point whose spherical coordinates are if ω \omega ω = (x2 - 2y + z)2 and x = ρ\rhoρ sin φ\varphiφ cos θ\thetaθ , y = ρ\rhoρ sin φ\varphiφ sin θ\thetaθ , z = ρ\rhoρ cos φ\varphiφ .

Find the rate of change of at (1, 6) in the direction of a vector making an angle of 120° with the positive x axis.

Let . Find .

An open rectangular box is to contain 864 cubic inches. Use the Lagrange multiplier method to find the dimensions of the box which uses the least amount of material.

Let . Compute the differential dz.

Let . Find if .

Let ; Find .

Describe the family of level curves for z = x² + y², (z $\ge$ 0) and sketch a few of these curves.

Find a point on the surface z = 16 - 12x² - y² at which the tangent plane is perpendicular to the line x = 3 + 12t, y = 2t, z = 2 - t.