Question 1

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

Accepted Answer

A)  -10i - 36j 
B)  -2i - 6j 
C)  -10i + 36j 
D)  5i - 6j 
A)  -10i - 36j 
B)  -2i - 6j 
C)  -10i + 36j 
D)  5i - 6j

Question 2

Find the limit.
-  sin

Accepted Answer

A)  1 
B)  1/7 
C)  0 
D)  No limit 
A)  1 
B)  1/7 
C)  0 
D)  No limit

Question 3

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

Accepted Answer

A) True 
 B)False

Question 4

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 5

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(0, 0) = 7, test is inconclusive 
B)  f( 5, 0) = 7, local minimum; f(0, 5) = 7, local minimum 
C)  f( 5, 5) = 1257, local maximum 
D)  f(0, 0) = 7, saddle point; f( 5, 5) = 1257, local maximum 
A)  f(0, 0) = 7, test is inconclusive 
B)  f( 5, 0) = 7, local minimum; f(0, 5) = 7, local minimum 
C)  f( 5, 5) = 1257, local maximum 
D)  f(0, 0) = 7, saddle point; f( 5, 5) = 1257, local maximum

Question 6

Solve the problem.
-Evaluate   at (u, v) = ( 5, 4) for the function w(x, y) = x   - ln x; x =   , y = uv.

Accepted Answer

A)  -1 
B)  1200   - 1 
C)  500   - 1 
D)  600   - 1 
A)  -1 
B)  1200   - 1 
C)  500   - 1 
D)  600   - 1

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

Solve the problem. 
-The table below summarizes the construction cost of a set of homes (excluding the lot cost) along with the square footage of the home's floor space. Find a linear equation that relates the construction cost in thousands of dollars to the floor space in hundreds of square feet by finding the least squares line for the data.

Accepted Answer

A)  y = 8.13x 
B)  y = 8.13x - 12.79 
C)  y = 7.97x - 13.54 
D)  y = 7.87x - 13.08 
A)  y = 8.13x 
B)  y = 8.13x - 12.79 
C)  y = 7.97x - 13.54 
D)  y = 7.87x - 13.08

Question 9

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

Accepted Answer

A)  Absolute maximum: 81 at (3, 3); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 96 at (2, 4); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 81 at (3, 3); absolute minimum: 36 at (2, 2) 
D)  Absolute maximum: 96 at (2, 4); absolute minimum: 81 at (3, 3) 
A)  Absolute maximum: 81 at (3, 3); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 96 at (2, 4); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 81 at (3, 3); absolute minimum: 36 at (2, 2) 
D)  Absolute maximum: 96 at (2, 4); absolute minimum: 81 at (3, 3)

Question 10

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

Accepted Answer

A)  -1 
B)  4 
C)  - 2 
D)  1 
A)  -1 
B)  4 
C)  - 2 
D)  1

Question 11

Determine whether the function is a solution of the wave equation.
-w(x, t) =   cos 2x

Accepted Answer

A) True 
 B)False

Question 12

Solve the problem.
-The radius r and height h of a cylinder are changing with time. At the instant in question,       At what rate is the cylinder's volume changing at that instant?

Accepted Answer

A)  2.64   /sec 
B)  0.75   /sec 
C)  0.94   /sec 
D)  1.88   /sec 
A)  2.64   /sec 
B)  0.75   /sec 
C)  0.94   /sec 
D)  1.88   /sec

Question 13

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)    
B)    
C)  3   
D)    
A)    
B)    
C)  3   
D)

Question 14

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

Accepted Answer

A)  Minimum: 5 at (±1, 0); maximum: 0 at (0, 0) 
B)  Minimum: 5 at (±1, 0); maximum: 9 at (0, ±1) 
C)  Minimum: 5 at (0, ±1); maximum: 9 at (±1, 0) 
D)  Minimum: 5 at (0, ±1); maximum: 0 at (0, 0) 
A)  Minimum: 5 at (±1, 0); maximum: 0 at (0, 0) 
B)  Minimum: 5 at (±1, 0); maximum: 9 at (0, ±1) 
C)  Minimum: 5 at (0, ±1); maximum: 9 at (±1, 0) 
D)  Minimum: 5 at (0, ±1); maximum: 0 at (0, 0)

Question 15

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 16

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

Accepted Answer

A)  1/3 
B)  - 3/5 
C)  - 1/5 
D)  1/5 
A)  1/3 
B)  - 3/5 
C)  - 1/5 
D)  1/5

Question 17

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

Accepted Answer

A)  - 19/234 
B)  - 2/39 
C)  - 5/234 
D)  - 1/9 
A)  - 19/234 
B)  - 2/39 
C)  - 5/234 
D)  - 1/9

Question 18

Solve the problem.
-Write an equation for the tangent line to the curve   at the point

Accepted Answer

A)  9x + 4y = 72 
B)  4x + 9y = 36 
C)  9x + 4y = 36 
D)  4x + 9y = 72 
A)  9x + 4y = 72 
B)  4x + 9y = 36 
C)  9x + 4y = 36 
D)  4x + 9y = 72

Question 19

Find the absolute maxima and minima of the function on the given domain.
-f(x, y) = 9x + 4y on the trapezoidal region with vertices (0, 0), (1, 0), (0, 2), and (1, 1)

Accepted Answer

A)  Absolute maximum: 13 at (1, 1); absolute minimum: 9 at (1, 0) 
B)  Absolute maximum: 18 at (2, 0); absolute minimum: 9 at (1, 0) 
C)  Absolute maximum: 8 at (0, 2); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 13 at (1, 1); absolute minimum: 0 at (0, 0) 
A)  Absolute maximum: 13 at (1, 1); absolute minimum: 9 at (1, 0) 
B)  Absolute maximum: 18 at (2, 0); absolute minimum: 9 at (1, 0) 
C)  Absolute maximum: 8 at (0, 2); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 13 at (1, 1); absolute minimum: 0 at (0, 0)

Question 20

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

Accepted Answer

A)   $\pi$/2 
B)  0 
C)  1 
D)  No limit 
A)   $\pi$/2 
B)  0 
C)  1 
D)  No limit

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

Find the limit. - sin

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

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

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

Solve the problem. -Evaluate at (u, v) = ( 5, 4) for the function w(x, y) = x - ln x; x = , y = uv.

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

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

Determine whether the function is a solution of the wave equation. -w(x, t) = cos 2x

Solve the problem. -The radius r and height h of a cylinder are changing with time. At the instant in question, At what rate is the cylinder's volume changing at that instant?

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

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

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) =

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

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

Solve the problem. -Write an equation for the tangent line to the curve at the point

Find the absolute maxima and minima of the function on the given domain. -f(x, y) = 9x + 4y on the trapezoidal region with vertices (0, 0), (1, 0), (0, 2), and (1, 1)

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

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

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

Find the limit. - sin

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

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

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

Solve the problem. -Evaluate at (u, v) = ( 5, 4) for the function w(x, y) = x - ln x; x = , y = uv.

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

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

Determine whether the function is a solution of the wave equation. -w(x, t) = cos 2x

Solve the problem. -The radius r and height h of a cylinder are changing with time. At the instant in question, At what rate is the cylinder's volume changing at that instant?

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

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

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) =

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

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

Solve the problem. -Write an equation for the tangent line to the curve at the point

Find the absolute maxima and minima of the function on the given domain. -f(x, y) = 9x + 4y on the trapezoidal region with vertices (0, 0), (1, 0), (0, 2), and (1, 1)

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

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