Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

A)  0 
B)  2 
C)  1 
D)  No limit 
A)  0 
B)  2 
C)  1 
D)  No limit

Question 3

Use implicit differentiation to find the specified derivative at the given point.
-Find   at the point ( 1, 3, 2) for   +   +   = 0.

Accepted Answer

A)  81/8 
B)  8/81 
C)  - 8/81 
D)  - 81/8 
A)  81/8 
B)  8/81 
C)  - 8/81 
D)  - 81/8

Question 4

Solve the problem.
-Evaluate   at (x, y, z) = ( 1, 2, 1) for the function u(p, q, r) =   -   - r; p = xy, q =   , r = xz.

Accepted Answer

A)  7 
B)  5 
C)  9 
D)  3 
A)  7 
B)  5 
C)  9 
D)  3

Question 5

Solve the problem. 
-Find the least squares line through the points     and

Accepted Answer

A)  y = 48x - 22 
B)  y = 48x - 94 
C)  y = 6x - 22 
D)  y = 6x - 94 
A)  y = 48x - 22 
B)  y = 48x - 94 
C)  y = 6x - 22 
D)  y = 6x - 94

Question 6

Provide an appropriate response.
-Find the direction in which the function is increasing most rapidly at the point   . f(x, y, z) = xy - ln(z),   ( 1, 2, 2)

Accepted Answer

A)    ( 4 i + 2 j - k ) 
B)    ( 4 i - 2 j + k ) 
C)    ( 4 i + 2 j - k ) 
D)    ( 4 i + 2 j - k ) 
A)    ( 4 i + 2 j - k ) 
B)    ( 4 i - 2 j + k ) 
C)    ( 4 i + 2 j - k ) 
D)    ( 4 i + 2 j - k )

Question 7

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Accepted Answer

Answers will vary. O

Question 8

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

Accepted Answer

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

Question 9

Sketch the surface z = f(x,y).
-f(x, y) = 1 -

Accepted Answer

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

Question 10

Find the limit.
-  ln (z   )

Accepted Answer

A)  ln 5   
B)  0 
C)  ln   
D)  ln 25 
A)  ln 5   
B)  0 
C)  ln   
D)  ln 25

Question 11

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

Accepted Answer

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

Question 12

Use implicit differentiation to find the specified derivative at the given point.
-Find   at the point (1, 2,   ) for ln   + 3   = 0.

Accepted Answer

A)  43/2 
B)  - 43/2 
C)  - 2/43 
D)  2/43 
A)  43/2 
B)  - 43/2 
C)  - 2/43 
D)  2/43

Question 13

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

Accepted Answer

A)  Maximum: 6 at (2, 2, 2); minimum: 3 at (1, 1, 1) 
B)  Maximum: 9 at (3, 3, 3); minimum: 3 at (1, 1, 1) 
C)  Maximum: 6 at (2, 2, 2); minimum: -3 at (-1, -1, -1) 
D)  Maximum: 11 at (6, 3, 2), (6, 2, 3), (3, 2, 6), (3, 6, 2), (2, 3, 6), (2, 6, 3); minimum: 1 at (1, 1, -1), (1, -1, 1), (-1, 1, 1) 
A)  Maximum: 6 at (2, 2, 2); minimum: 3 at (1, 1, 1) 
B)  Maximum: 9 at (3, 3, 3); minimum: 3 at (1, 1, 1) 
C)  Maximum: 6 at (2, 2, 2); minimum: -3 at (-1, -1, -1) 
D)  Maximum: 11 at (6, 3, 2), (6, 2, 3), (3, 2, 6), (3, 6, 2), (2, 3, 6), (2, 6, 3); minimum: 1 at (1, 1, -1), (1, -1, 1), (-1, 1, 1)

Question 14

Find the linear approximation of the function at the given point.
-  at

Accepted Answer

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

Question 15

Write a chain rule formula for the following derivative.
-  for u = f(p, q); p = g(x, y, z), q = h(x, y, z)

Accepted Answer

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

Question 16

Find the derivative of the function at the given point in the direction of A.
-    A = 3i- 4j

Accepted Answer

A)  - 116/5 
B)  - 148/5 
C)  - 84/5 
D)  - 36 
A)  - 116/5 
B)  - 148/5 
C)  - 84/5 
D)  - 36

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

A)  Maximum: 4 at (±2 , 1); minimum: -4 at (±2, -1) 
B)  Maximum: 18 at (3, 2); minimum: -18 at (-3, -2) 
C)  Maximum: 4 at (2, 1); minimum: -4 at (-2, -1) 
D)  Maximum: 18 at (±3, 2); minimum: -18 at (±3, -2) 
A)  Maximum: 4 at (±2 , 1); minimum: -4 at (±2, -1) 
B)  Maximum: 18 at (3, 2); minimum: -18 at (-3, -2) 
C)  Maximum: 4 at (2, 1); minimum: -4 at (-2, -1) 
D)  Maximum: 18 at (±3, 2); minimum: -18 at (±3, -2)

Question 19

Find the limit.
-

Accepted Answer

A)  1/33 
B)  17 
C)  0 
D)  33 
A)  1/33 
B)  17 
C)  0 
D)  33

Question 20

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

Accepted Answer

A)    saddle point;   saddle point 
B)    local maximum 
C)    local minimum 
D)    local maximum;   local minimum 
A)    saddle point;   saddle point 
B)    local maximum 
C)    local minimum 
D)    local maximum;   local minimum

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) =

Use implicit differentiation to find the specified derivative at the given point. -Find at the point ( 1, 3, 2) for + + = 0.

Solve the problem. -Evaluate at (x, y, z) = ( 1, 2, 1) for the function u(p, q, r) = - - r; p = xy, q = , r = xz.

Solve the problem. -Find the least squares line through the points and

Provide an appropriate response. -Find the direction in which the function is increasing most rapidly at the point . f(x, y, z) = xy - ln(z), ( 1, 2, 2)

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). -f(x, y) =

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

Sketch the surface z = f(x,y). -f(x, y) = 1 -

Find the limit. - ln (z )

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

Use implicit differentiation to find the specified derivative at the given point. -Find at the point (1, 2, ) for ln + 3 = 0.

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

Find the linear approximation of the function at the given point. - at

Write a chain rule formula for the following derivative. - for u = f(p, q); p = g(x, y, z), q = h(x, y, z)

Find the derivative of the function at the given point in the direction of A. - A = 3i- 4j

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

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

Find the limit. -

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

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

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) =

Use implicit differentiation to find the specified derivative at the given point. -Find at the point ( 1, 3, 2) for + + = 0.

Solve the problem. -Evaluate at (x, y, z) = ( 1, 2, 1) for the function u(p, q, r) = - - r; p = xy, q = , r = xz.

Solve the problem. -Find the least squares line through the points and

Provide an appropriate response. -Find the direction in which the function is increasing most rapidly at the point . f(x, y, z) = xy - ln(z), ( 1, 2, 2)

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). -f(x, y) =

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

Sketch the surface z = f(x,y). -f(x, y) = 1 -

Find the limit. - ln (z )

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

Use implicit differentiation to find the specified derivative at the given point. -Find at the point (1, 2, ) for ln + 3 = 0.

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

Find the linear approximation of the function at the given point. - at

Write a chain rule formula for the following derivative. - for u = f(p, q); p = g(x, y, z), q = h(x, y, z)

Find the derivative of the function at the given point in the direction of A. - A = 3i- 4j

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

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

Find the limit. -

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

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