Question 1

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 2

Find the limit.
-

Accepted Answer

A)  -8 
B)    + 7 
C)  -   - 7 
D)  8 
A)  -8 
B)    + 7 
C)  -   - 7 
D)  8

Question 3

Use implicit differentiation to find the specified derivative at the given point.
-Find   at the point (1, 1) for 7   + 7   + 5xy = 0.

Accepted Answer

A)  - 19/12 
B)  - 19/26 
C)  - 1 
D)  19/26 
A)  - 19/12 
B)  - 19/26 
C)  - 1 
D)  19/26

Question 4

Compute the gradient of the function at the given point.
-

Accepted Answer

A)  - 1/15 j + 1/6 k 
B)  - 1/15 j - 1/6 k 
C)  1/6 j - 1/15 k 
D)  - 1/6 j - 1/15 k 
A)  - 1/15 j + 1/6 k 
B)  - 1/15 j - 1/6 k 
C)  1/6 j - 1/15 k 
D)  - 1/6 j - 1/15 k

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

A)  z = 0 
B)  z = 2 
C)  z = 1 
D)  z = -1 
A)  z = 0 
B)  z = 2 
C)  z = 1 
D)  z = -1

Question 7

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

Accepted Answer

A)  f( 6, 6) = 1080, local maximum 
B)  f   =   , local minimum 
C)  f   =   , local minimum 
D)    saddle point 
A)  f( 6, 6) = 1080, local maximum 
B)  f   =   , local minimum 
C)  f   =   , local minimum 
D)    saddle point

Question 8

Determine whether the given function satisfies a Laplace equation.
-f(x, y) =   sin -5x

Accepted Answer

A) True 
 B)False

Question 9

Solve the problem. 
-Amarillo Motors manufactures an economy car called the Citrus, which is notorious for its inability to hold a respectable resale value. The average resale value of a set of 1998 Amarillo Citrus's is summarized in the table below along with the age of the car at the time of resale and the number of cars included in the average. Fit a line of the form ln(V) = m.a + b to the data, where V is the resale value in thousands of dollars and a is the age of the car in years.

Accepted Answer

A)  ln(V) = 0.315a 
B)  ln(V) = 0.314a - 2.986 
C)  ln(V) = 0.299a - 2.951 
D)  ln(V) = -0.303a + 2.953 
A)  ln(V) = 0.315a 
B)  ln(V) = 0.314a - 2.986 
C)  ln(V) = 0.299a - 2.951 
D)  ln(V) = -0.303a + 2.953

Question 10

Find the derivative of the function at the given point in the direction of A.
-

Accepted Answer

A)  - 1/7 
B)  41/7 
C)  20/7 
D)  - 8/7 
A)  - 1/7 
B)  41/7 
C)  20/7 
D)  - 8/7

Question 11

Find the limit.
-

Accepted Answer

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

Question 12

Determine whether the given function satisfies a Laplace equation.
-f(x, y, z) = 5x + 9

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

A)    i +   j 
B)    i +   j 
C)    i -   j 
D)    i -   j 
A)    i +   j 
B)    i +   j 
C)    i -   j 
D)    i -   j

Question 14

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

Accepted Answer

A)   $\pi$/2 
B)   $\pi$ 
C)  1 
D)  No limit 
A)   $\pi$/2 
B)   $\pi$ 
C)  1 
D)  No limit

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

A)  All (x, y) $\neq$ (0, 0) 
B)  All (x, y) satisfying x + y $\neq$ 0 
C)  All (x, y) 
D)  All (x, y) in the first quadrant 
A)  All (x, y) $\neq$ (0, 0) 
B)  All (x, y) satisfying x + y $\neq$ 0 
C)  All (x, y) 
D)  All (x, y) in the first quadrant

Question 17

Determine whether the function is a solution of the wave equation.
-w(x, t) =

Accepted Answer

A) True 
 B)False

Question 18

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

Accepted Answer

A)  f( -10, -8) = 3022, local maximum 
B)  f(10, 8) = -3026. local minimum; f(10, -8) = -978, saddle point; f(-10, 8) = 974, saddle point; f(-10, -8) = 3022, local maximum 
C)  f(10, -8) = -978, saddle point; f(-10, 8) = 974, saddle point 
D)  f (-10, -8) = 3022, local maximum; f(10, 8) = -3026, local minimum 
A)  f( -10, -8) = 3022, local maximum 
B)  f(10, 8) = -3026. local minimum; f(10, -8) = -978, saddle point; f(-10, 8) = 974, saddle point; f(-10, -8) = 3022, local maximum 
C)  f(10, -8) = -978, saddle point; f(-10, 8) = 974, saddle point 
D)  f (-10, -8) = 3022, local maximum; f(10, 8) = -3026, local minimum

Question 19

Solve the problem.
-Find the points on the curve   that are closest to the origin.

Accepted Answer

A)  ( 64, ±   ) 
B)  ( 4, 4   ) 
C)  ( 4, ±4   ) 
D)  ( 64,   ) 
A)  ( 64, ±   ) 
B)  ( 4, 4   ) 
C)  ( 4, ±4   ) 
D)  ( 64,   )

Question 20

Find the limit.
-

Accepted Answer

A)  1 
B)  $\infty$ 
C)  0 
D)  No limit 
A)  1 
B)  $\infty$ 
C)  0 
D)  No limit

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 the limit. -

Use implicit differentiation to find the specified derivative at the given point. -Find at the point (1, 1) for 7 + 7 + 5xy = 0.

Compute the gradient of the function at the given point. -

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

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

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

Determine whether the given function satisfies a Laplace equation. -f(x, y) = sin -5x

Find the derivative of the function at the given point in the direction of A. -

Find the limit. -

Determine whether the given function satisfies a Laplace equation. -f(x, y, z) = 5x + 9

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

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

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

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

Determine whether the function is a solution of the wave equation. -w(x, t) =

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

Solve the problem. -Find the points on the curve that are closest to the origin.

Find the limit. -

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 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 the limit. -

Use implicit differentiation to find the specified derivative at the given point. -Find at the point (1, 1) for 7 + 7 + 5xy = 0.

Compute the gradient of the function at the given point. -

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

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

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

Determine whether the given function satisfies a Laplace equation. -f(x, y) = sin -5x

Find the derivative of the function at the given point in the direction of A. -

Find the limit. -

Determine whether the given function satisfies a Laplace equation. -f(x, y, z) = 5x + 9

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

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

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

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

Determine whether the function is a solution of the wave equation. -w(x, t) =

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

Solve the problem. -Find the points on the curve that are closest to the origin.

Find the limit. -

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