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 family of level

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

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

Question 7

Let   ;   Find   .

Accepted Answer

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

Question 9

Evaluate   .

Accepted Answer

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

Question 10

Let   . Find   .

Accepted Answer

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

Question 11

Find an equation for the tangent plane to   at   .

Accepted Answer

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

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)    
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

x = 12, y

Question 18

Let   . Compute the differential dz.

Accepted Answer

The answer of Let   . Compute the differential...

Question 19

Let   . Find   if   .

Accepted Answer

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

Question 20

Let   ;   Find   .

Accepted Answer

The answer of 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.

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

Let . is

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.

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 .

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Multiple Integrals

Topics in Vector Calculus

Filters

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 .

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Multiple Integrals

Topics in Vector Calculus

Filters

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.