Question 1

Find an equation for the tangent plane to   at   .

Accepted Answer

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

Question 2

Let   . Describe the level surface passing through   .

Accepted Answer

A)  Ellipsoid 
B)  Hyperboliod of one sheet 
C)  Elliptical cone 
D)  Elliptic paraboloid 
E)  Hyperbolic paraboloid 
A)  Ellipsoid 
B)  Hyperboliod of one sheet 
C)  Elliptical cone 
D)  Elliptic paraboloid 
E)  Hyperbolic paraboloid

Question 3

Use Lagrange multipliers to find the volume of the largest rectangular box that can be inscribed within the sphere   .

Accepted Answer

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

Question 4

Use Lagrange multipliers to find maximum and minimum values of   subject to   .

Accepted Answer

A)  The maximum value is 0, and there is no minimum value. 2 
B)  The maximum value is   , and the minimum value is
  
C)  The maximum value is   , and the minimum value is
  
D)  The maximum value is   , and the minimum value is
  
E)  There is no maximum value, and the minimum value is 0. 
A)  The maximum value is 0, and there is no minimum value. 2 
B)  The maximum value is   , and the minimum value is
  
C)  The maximum value is   , and the minimum value is
  
D)  The maximum value is   , and the minimum value is
  
E)  There is no maximum value, and the minimum value is 0.

Question 5

Let   . Find   at the point   .

Accepted Answer

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

Question 6

Let   ;   Find   .

Accepted Answer

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

Question 7

Let   ; Find   .

Accepted Answer

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

Question 8

Find the equations of the tangent plane and normal line to x2z - xy2 - yz2 - 18 = 0 at (0, -2, 4).

Accepted Answer

tangent plane 4x + 1

Question 9

Find the equations of the tangent plane and normal line to   at   . Express the equation of the normal line parametrically.

Accepted Answer

The equation of the

Question 10

Let   . There is a critical point at

Accepted Answer

A)  (10, 4) 
B)  (5, 2) 
C)  (-10, -4) 
D)  (0, 0) 
E)  (1, 1) 
A)  (10, 4) 
B)  (5, 2) 
C)  (-10, -4) 
D)  (0, 0) 
E)  (1, 1)

Question 11

Use the chain rule to find   if   and

Accepted Answer

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

Question 12

Let   . Find   .

Accepted Answer

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

Question 13

Find an equation for the tangent plane to   at   .

Accepted Answer

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

Question 14

Sketch the natural domain of   . Shade the region included in the natural domain. Use solid lines for the portion of the boundary included in the natural domain.

Accepted Answer

The answer of Sketch the natural domain of  ...

Question 15

Let f(x, y, z) = 2x cos(yz) - 4z. Then,

Accepted Answer

A)  2xyz cos(yz) 
B)  -2xyz sin(yz) 
C)  2xcos(yz) 
D)  -2zsin(yz) 
E)  2xyz 
A)  2xyz cos(yz) 
B)  -2xyz sin(yz) 
C)  2xcos(yz) 
D)  -2zsin(yz) 
E)  2xyz

Question 16

The level curve of   when   is

Accepted Answer

A)  a circle of radius 4 
B)  an ellipse 
C)  a hyperbola 
D)  a point 
E)  a circle 
A)  a circle of radius 4 
B)  an ellipse 
C)  a hyperbola 
D)  a point 
E)  a circle

Question 17

Let   . Find the slope of the surface   in the x-direction at the point   .

Accepted Answer

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

Question 18

Let   ; Find   .

Accepted Answer

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

Question 19

Locate all relative maxima, relative minima, and saddle points for
f(x, y) = x2 - 2y2 - 6x + 8y + 41.

Accepted Answer

(3, 2); sa

Question 20

The level curve of   when   is

Accepted Answer

A)  a circle of radius 4 
B)  an ellipse 
C)  a circle of radius 16 
D)  a point 
E)  a hyperbola 
A)  a circle of radius 4 
B)  an ellipse 
C)  a circle of radius 16 
D)  a point 
E)  a hyperbola

Find an equation for the tangent plane to at .

Let . Describe the level surface passing through .

Use Lagrange multipliers to find the volume of the largest rectangular box that can be inscribed within the sphere .

Use Lagrange multipliers to find maximum and minimum values of subject to .

Let . Find at the point .

Let ; Find .

Let ; Find .

Find the equations of the tangent plane and normal line to x²z - xy² - yz² - 18 = 0 at (0, -2, 4).

Find the equations of the tangent plane and normal line to at . Express the equation of the normal line parametrically.

Let . There is a critical point at

Use the chain rule to find if and

Let . Find .

Find an equation for the tangent plane to at .

Sketch the natural domain of . Shade the region included in the natural domain. Use solid lines for the portion of the boundary included in the natural domain.

Let f(x, y, z) = 2x cos(yz) - 4z. Then,

The level curve of when is

Let . Find the slope of the surface in the x-direction at the point .

Let ; Find .

Locate all relative maxima, relative minima, and saddle points for f(x, y) = x² - 2y² - 6x + 8y + 41.

The level curve of when is

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

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Exam 13: Partial Derivatives

Find an equation for the tangent plane to at .

Let . Describe the level surface passing through .

Use Lagrange multipliers to find the volume of the largest rectangular box that can be inscribed within the sphere .

Use Lagrange multipliers to find maximum and minimum values of subject to .

Let . Find at the point .

Let ; Find .

Let ; Find .

Find the equations of the tangent plane and normal line to x2z - xy2 - yz2 - 18 = 0 at (0, -2, 4).

Find the equations of the tangent plane and normal line to at . Express the equation of the normal line parametrically.

Let . There is a critical point at

Use the chain rule to find if and

Let . Find .

Find an equation for the tangent plane to at .

Sketch the natural domain of . Shade the region included in the natural domain. Use solid lines for the portion of the boundary included in the natural domain.

Let f(x, y, z) = 2x cos(yz) - 4z. Then,

The level curve of when is

Let . Find the slope of the surface in the x-direction at the point .

Let ; Find .

Locate all relative maxima, relative minima, and saddle points for f(x, y) = x2 - 2y2 - 6x + 8y + 41.

The level curve of when is

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

Find the equations of the tangent plane and normal line to x²z - xy² - yz² - 18 = 0 at (0, -2, 4).

Locate all relative maxima, relative minima, and saddle points for f(x, y) = x² - 2y² - 6x + 8y + 41.