Question 1

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

Accepted Answer

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

Question 2

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)     D

Question 3

Find all the second order partial derivatives of the given function.
-f(x, y) =

Accepted Answer

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

Question 4

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

Accepted Answer

A)  Maximum: 36 at (3, 4); minimum: 0 at (0, 0) 
B)  Maximum: 26 at (2, 3); minimum: 0 at (0, 0) 
C)  Maximum: 36 at (3, 4); minimum: -36 at (-3, -4) 
D)  Maximum: 26 at (2, 3); minimum: -26 at (-2, -3) 
A)  Maximum: 36 at (3, 4); minimum: 0 at (0, 0) 
B)  Maximum: 26 at (2, 3); minimum: 0 at (0, 0) 
C)  Maximum: 36 at (3, 4); minimum: -36 at (-3, -4) 
D)  Maximum: 26 at (2, 3); minimum: -26 at (-2, -3)

Question 5

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

Accepted Answer

A)  Maximum: 3 at   minimum: -3 at   
B)  Maximum: 3 at   and   minimum: -3 at   and   
C)  Maximum: 3 at   minimum: -3 at   
D)  Maximum: 3 at   minimum: -3 at   
A)  Maximum: 3 at   minimum: -3 at   
B)  Maximum: 3 at   and   minimum: -3 at   and   
C)  Maximum: 3 at   minimum: -3 at   
D)  Maximum: 3 at   minimum: -3 at

Question 6

Find all the second order partial derivatives of the given function.
-f(x, y) = xy

Accepted Answer

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

Question 7

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

Accepted Answer

A)  - 47/5 
B)  - 13 
C)  7/5 
D)  5 
A)  - 47/5 
B)  - 13 
C)  7/5 
D)  5

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

A)  - 1 
B)  1 
C)  3/10 
D)  - 3/10 
A)  - 1 
B)  1 
C)  3/10 
D)  - 3/10

Question 10

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

Accepted Answer

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

Question 11

Solve the problem.
-Find the point on the sphere   that is farthest from the point

Accepted Answer

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

Question 12

Solve the problem.
-A rectangular box with square base and no top is to have a volume of 32   . What is the least amount of material required?

Accepted Answer

A)  48   
B)  36   
C)  42   
D)  40   
A)  48   
B)  36   
C)  42   
D)  40

Question 13

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 14

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

Accepted Answer

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

Question 15

Write a chain rule formula for the following derivative.
-  for u = f(v); v = h(s, t)

Accepted Answer

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

Question 16

Solve the problem.
-Evaluate   at (x, y, z) = ( 3, 4, 3) for the function u(p, q, r) =     - r; p = y - z,

Accepted Answer

A)  -30 
B)  -60 
C)  -120 
D)  84 
A)  -30 
B)  -60 
C)  -120 
D)  84

Question 17

Find all the first order partial derivatives for the following function.       
-f(x, y) =   + 10   y - 2x

Accepted Answer

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

Question 18

Provide an appropriate response.
-Find the direction in which the function is increasing most rapidly at the point   . f(x, y) = x   - ln(x),   ( 2, 0)

Accepted Answer

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

Question 19

Determine whether the function is a solution of the wave equation.
-w(x, t) = cos ( 4ct) sin ( 4x)

Accepted Answer

A) True 
 B)False

Question 20

Find all the second order partial derivatives of the given function.
-f(x, y) = x ln (y - x)

Accepted Answer

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

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. -

Find all the second order partial derivatives of the given function. -f(x, y) =

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

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

Find all the second order partial derivatives of the given function. -f(x, y) = xy

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

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

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

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

Solve the problem. -Find the point on the sphere that is farthest from the point

Solve the problem. -A rectangular box with square base and no top is to have a volume of 32 . What is the least amount of material required?

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) =

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

Write a chain rule formula for the following derivative. - for u = f(v); v = h(s, t)

Solve the problem. -Evaluate at (x, y, z) = ( 3, 4, 3) for the function u(p, q, r) = - r; p = y - z,

Find all the first order partial derivatives for the following function. -f(x, y) = + 10 y - 2x

Provide an appropriate response. -Find the direction in which the function is increasing most rapidly at the point . f(x, y) = x - ln(x), ( 2, 0)

Determine whether the function is a solution of the wave equation. -w(x, t) = cos ( 4ct) sin ( 4x)

Find all the second order partial derivatives of the given function. -f(x, y) = x ln (y - x)

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 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. -

Find all the second order partial derivatives of the given function. -f(x, y) =

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

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

Find all the second order partial derivatives of the given function. -f(x, y) = xy

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

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

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

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

Solve the problem. -Find the point on the sphere that is farthest from the point

Solve the problem. -A rectangular box with square base and no top is to have a volume of 32 . What is the least amount of material required?

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) =

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

Write a chain rule formula for the following derivative. - for u = f(v); v = h(s, t)

Solve the problem. -Evaluate at (x, y, z) = ( 3, 4, 3) for the function u(p, q, r) = - r; p = y - z,

Find all the first order partial derivatives for the following function. -f(x, y) = + 10 y - 2x

Provide an appropriate response. -Find the direction in which the function is increasing most rapidly at the point . f(x, y) = x - ln(x), ( 2, 0)

Determine whether the function is a solution of the wave equation. -w(x, t) = cos ( 4ct) sin ( 4x)

Find all the second order partial derivatives of the given function. -f(x, y) = x ln (y - x)

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