Question 1

Find the extreme values of the function subject to the given constraint.
-

Accepted Answer

A)  Maximum: 8 at (2, 1, -2); minimum: -8 at (-2, -1, 2) 
B)  Maximum: 1 at (-1, -2, -3); minimum: -1 at (1, 2, 3) 
C)  Maximum: 9 at (1, 2, -2); minimum: -9 at (-1, -2, 2) 
D)  Maximum: 1 at (1, -2, -2); minimum: -1 at (-1, 2, 2) 
A)  Maximum: 8 at (2, 1, -2); minimum: -8 at (-2, -1, 2) 
B)  Maximum: 1 at (-1, -2, -3); minimum: -1 at (1, 2, 3) 
C)  Maximum: 9 at (1, 2, -2); minimum: -9 at (-1, -2, 2) 
D)  Maximum: 1 at (1, -2, -2); minimum: -1 at (-1, 2, 2)

Question 2

Sketch the surface z = f(x,y).
-f(x, y) =

Accepted Answer

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

Question 3

Determine whether the given function satisfies a Laplace equation.
-f(x, y, z) = cos ( -7x) sin ( 3y)

Accepted Answer

A) True 
 B)False

Question 4

Find the limit.
-

Accepted Answer

A)    
B)    
C)  15 
D)  No limit 
A)    
B)    
C)  15 
D)  No limit

Question 5

Write a chain rule formula for the following derivative.
-  for w = f(x, y, z); x = g(r, s), y = h(t), z = k(r, s, t)

Accepted Answer

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

Question 6

Match the surface show below to the graph of its level curves.
-

Accepted Answer

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

Question 7

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).
-f(x, y) =

Accepted Answer

A)  $\pi$ 
B)  1 
C)  0 
D)  -1 
A)  $\pi$ 
B)  1 
C)  0 
D)  -1

Question 8

Find the domain and range and describe the level curves for the function f(x,y).
-f(x, y) =

Accepted Answer

A)   Domain: all points in the xy-plane except y = 0; range: all real numbers; level curves: parabolas  y = cxS1U1P12S1S1P0 
B)   Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: parabolas  y = cxS1U1P12S1S1P0 
C)   Domain: all points in the xy-plane except y = 0; range: real numbers z ≥ 0 ; level curves: parabolas  y = cxS1U1P12S1S1P0 
D)   Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas   y = cxS1U1P12S1S1P0 
A)   Domain: all points in the xy-plane except y = 0; range: all real numbers; level curves: parabolas  y = cxS1U1P12S1S1P0 
B)   Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: parabolas  y = cxS1U1P12S1S1P0 
C)   Domain: all points in the xy-plane except y = 0; range: real numbers z ≥ 0 ; level curves: parabolas  y = cxS1U1P12S1S1P0 
D)   Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas   y = cxS1U1P12S1S1P0

Question 9

Find the absolute maxima and minima of the function on the given domain.
-  on the square

Accepted Answer

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

Question 10

Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle point.
-

Accepted Answer

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

Question 11

Match the surface show below to the graph of its level curves.
-

Accepted Answer

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

Question 12

Find all the first order partial derivatives for the following function.  
-f(x, y, z) =

Accepted Answer

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

Question 13

Solve the problem.
-Find the derivative of the function   at the point   in the direction in which the function decreases most rapidly.

Accepted Answer

A)  - 5/52   
B)  - 5/68   
C)  - 5/44   
D)  -5/28   
A)  - 5/52   
B)  - 5/68   
C)  - 5/44   
D)  -5/28

Question 14

Solve the problem.
-Evaluate   at t =  $\pi$ for the function w(x, y, z) =   ; x = sin t, y = cost, z =   .

Accepted Answer

A)  -     
B)  - 2   
C)  - 2   
D)  2   
A)  -     
B)  - 2   
C)  - 2   
D)  2

Question 15

Find the limit.
-

Accepted Answer

A)  13 
B)  1 
C)  5 
D)  0 
A)  13 
B)  1 
C)  5 
D)  0

Question 16

Solve the problem.
-Find the equation for the tangent plane to the surface   at the point

Accepted Answer

A)  7x + 7y - 5z = -9 
B)  7x + 7y - 5z = 1 
C)  x + y + z = 1 
D)  x + y + z = -9 
A)  7x + 7y - 5z = -9 
B)  7x + 7y - 5z = 1 
C)  x + y + z = 1 
D)  x + y + z = -9

Question 17

Use implicit differentiation to find the specified derivative at the given point.
-Find   at the point ( 5, 5, 6) for -3   - 6 ln xz + 2y   - 2   = 0.

Accepted Answer

A)  2/5 
B)  13/5 
C)  13/30 
D)  - 13/30 
A)  2/5 
B)  13/5 
C)  13/30 
D)  - 13/30

Question 18

At what points is the given function continuous?
-f(x, y) =

Accepted Answer

A) All (x, y) such that x $\neq$   and x $\neq$ -2 
B)  All (x, y) 
C)  All (x, y) such that x $\neq$ 0 
D)  All (x, y) satisfying x - y $\neq$ 0 
A) All (x, y) such that x $\neq$   and x $\neq$ -2 
B)  All (x, y) 
C)  All (x, y) such that x $\neq$ 0 
D)  All (x, y) satisfying x - y $\neq$ 0

Question 19

Find the limit.
-

Accepted Answer

A)  - 3/5 
B)  12 
C)  3/5 
D)  No limit 
A)  - 3/5 
B)  12 
C)  3/5 
D)  No limit

Question 20

Solve the problem.
-Find the derivative of the function   at the point   in the direction in which the function increases most rapidly.

Accepted Answer

A)  3/8   
B)  1/4   
C)  1/4   
D)  3/8   
A)  3/8   
B)  1/4   
C)  1/4   
D)  3/8

Find the extreme values of the function subject to the given constraint. -

Sketch the surface z = f(x,y). -f(x, y) =

Determine whether the given function satisfies a Laplace equation. -f(x, y, z) = cos ( -7x) sin ( 3y)

Find the limit. -

Write a chain rule formula for the following derivative. - for w = f(x, y, z); x = g(r, s), y = h(t), z = k(r, s, t)

Match the surface show below to the graph of its level curves. -

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). -f(x, y) =

Find the domain and range and describe the level curves for the function f(x,y). -f(x, y) =

Find the absolute maxima and minima of the function on the given domain. - on the square

Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle point. -

Match the surface show below to the graph of its level curves. -

Find all the first order partial derivatives for the following function. -f(x, y, z) =

Solve the problem. -Find the derivative of the function at the point in the direction in which the function decreases most rapidly.

Find the limit. -

Solve the problem. -Find the equation for the tangent plane to the surface at the point

Use implicit differentiation to find the specified derivative at the given point. -Find at the point ( 5, 5, 6) for -3 - 6 ln xz + 2y - 2 = 0.

At what points is the given function continuous? -f(x, y) =

Find the limit. -

Solve the problem. -Find the derivative of the function at the point in the direction in which the function increases most rapidly.

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Calculus

Filters

Exam 15: Functions of Several Variables

Find the extreme values of the function subject to the given constraint. -

Sketch the surface z = f(x,y). -f(x, y) =

Determine whether the given function satisfies a Laplace equation. -f(x, y, z) = cos ( -7x) sin ( 3y)

Find the limit. -

Write a chain rule formula for the following derivative. - for w = f(x, y, z); x = g(r, s), y = h(t), z = k(r, s, t)

Match the surface show below to the graph of its level curves. -

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0). -f(x, y) =

Find the domain and range and describe the level curves for the function f(x,y). -f(x, y) =

Find the absolute maxima and minima of the function on the given domain. - on the square

Find all local extreme values of the given function and identify each as a local maximum, local minimum, or saddle point. -

Match the surface show below to the graph of its level curves. -

Find all the first order partial derivatives for the following function. -f(x, y, z) =

Solve the problem. -Find the derivative of the function at the point in the direction in which the function decreases most rapidly.

Solve the problem. -Evaluate at t = π\piπ for the function w(x, y, z) = ; x = sin t, y = cost, z = .

Find the limit. -

Solve the problem. -Find the equation for the tangent plane to the surface at the point

Use implicit differentiation to find the specified derivative at the given point. -Find at the point ( 5, 5, 6) for -3 - 6 ln xz + 2y - 2 = 0.

At what points is the given function continuous? -f(x, y) =

Find the limit. -

Solve the problem. -Find the derivative of the function at the point in the direction in which the function increases most rapidly.

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Calculus

Filters