Question 1

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

Accepted Answer

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

Question 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) =

Accepted Answer

Answers will vary. O

Question 3

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

Accepted Answer

A)  x/4 + y/2 =   
B)  x/4 + y/2 =1 
C)  x/2 + y/4 = 1 
D)  x/2 + y/4 =   
A)  x/4 + y/2 =   
B)  x/4 + y/2 =1 
C)  x/2 + y/4 = 1 
D)  x/2 + y/4 =

Question 4

Solve the problem.
-A rectangular box is to be inscribed inside the ellipsoid   Find the largest possible volume for the box.

Accepted Answer

A)  16   
B)  18   
C)  15   
D)  12   
A)  16   
B)  18   
C)  15   
D)  12

Question 5

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; range: real numbers z ≤ 0; level curves: lines 8x - 10y = c,  c ≤ 0 
B)   Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: lines  8x - 10y = c 
C)   Domain: all points in the xy-plane; range: all real numbers; level curves: lines  8x - 10y = c,   c ≥ 0 
D)   Domain: all points in the xy-plane; range: all real numbers; level curves: lines   8x - 10y = c 
A)   Domain: all points in the xy-plane; range: real numbers z ≤ 0; level curves: lines 8x - 10y = c,  c ≤ 0 
B)   Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: lines  8x - 10y = c 
C)   Domain: all points in the xy-plane; range: all real numbers; level curves: lines  8x - 10y = c,   c ≥ 0 
D)   Domain: all points in the xy-plane; range: all real numbers; level curves: lines   8x - 10y = c

Question 6

Use implicit differentiation to find the specified derivative at the given point.
-Find   at the point ( 4, 1, -1) for ln   -   = 0.

Accepted Answer

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

Question 7

Find the limit.
-

Accepted Answer

A)  2 
B)  6 
C)  -6 
D)  -4 
A)  2 
B)  6 
C)  -6 
D)  -4

Question 8

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

Accepted Answer

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

Question 9

Solve the problem.
-Evaluate   at t =   $\pi$for the function w(x, y) =   -   + 8x; x = cost, y = sin t.

Accepted Answer

A)  4 
B)  10 
C)  -8 
D)  6 
A)  4 
B)  10 
C)  -8 
D)  6

Question 10

Find the limit.
-

Accepted Answer

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

Question 11

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

Accepted Answer

A)  1/2 
B)  1 
C)  1/3 
D)  2/3 
A)  1/2 
B)  1 
C)  1/3 
D)  2/3

Question 12

Determine whether the function is a solution of the wave equation.
-w(x, t) = sin ( 6x + 6ct)

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

A)  Maximum: 4 at   minimum: -5 at   
B)  Maximum: 7 at   minimum: -5 at   
C)  Maximum: 7 at   minimum: -2 at   
D)  Maximum: 4 at   minimum: -2 at   
A)  Maximum: 4 at   minimum: -5 at   
B)  Maximum: 7 at   minimum: -5 at   
C)  Maximum: 7 at   minimum: -2 at   
D)  Maximum: 4 at   minimum: -2 at

Question 14

Find all the first order partial derivatives for the following function.       
-f(x, y) =   ( -3x   - y)

Accepted Answer

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

Question 15

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

Accepted Answer

A)  Maximum: 400 at (20, 20); minimum: -400 at (-20, -20) 
B)  Maximum: 400 at (20, 20) and (-20, -20); minimum: -400 at (20, -20) and (-20, 20) 
C)  Maximum: 400 at (20, -20) and (-20, 20); minimum: -400 at (20, 20) and (-20, -20) 
D)  Maximum: 400 at (20, 20); minimum: 0 at (0, 0) 
A)  Maximum: 400 at (20, 20); minimum: -400 at (-20, -20) 
B)  Maximum: 400 at (20, 20) and (-20, -20); minimum: -400 at (20, -20) and (-20, 20) 
C)  Maximum: 400 at (20, -20) and (-20, 20); minimum: -400 at (20, 20) and (-20, -20) 
D)  Maximum: 400 at (20, 20); minimum: 0 at (0, 0)

Question 16

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 17

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

Accepted Answer

A)  All (x, y) $\neq$ (0, 0) 
B)  All (x, y) $\neq$   
C)  All (x, y) such that x + y $\neq$  , where n is an integer 
D)  All (x, y) 
A)  All (x, y) $\neq$ (0, 0) 
B)  All (x, y) $\neq$   
C)  All (x, y) such that x + y $\neq$  , where n is an integer 
D)  All (x, y)

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( 0, 0) = 3 , local maximum; f(3, 3) = -6558, local minimum 
B)  f(0, 0) = 3, local maximum 
C)  f(3, 0) = 3, saddle point; f(0, 3) = 3, saddle point 
D)  f (3, 3) = -6558, local minimum 
A)  f( 0, 0) = 3 , local maximum; f(3, 3) = -6558, local minimum 
B)  f(0, 0) = 3, local maximum 
C)  f(3, 0) = 3, saddle point; f(0, 3) = 3, saddle point 
D)  f (3, 3) = -6558, local minimum

Question 19

Find the limit.
-

Accepted Answer

A)  0 
B)  10 
C)  5 
D)  No limit 
A)  0 
B)  10 
C)  5 
D)  No limit

Question 20

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

Accepted Answer

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

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

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) =

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

Solve the problem. -A rectangular box is to be inscribed inside the ellipsoid Find the largest possible volume for the box.

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

Use implicit differentiation to find the specified derivative at the given point. -Find at the point ( 4, 1, -1) for ln - = 0.

Find the limit. -

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

Find the limit. -

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

Determine whether the function is a solution of the wave equation. -w(x, t) = sin ( 6x + 6ct)

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

Find all the first order partial derivatives for the following function. -f(x, y) = ( -3x - y)

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

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

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

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

Find the limit. -

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

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

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

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) =

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

Solve the problem. -A rectangular box is to be inscribed inside the ellipsoid Find the largest possible volume for the box.

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

Use implicit differentiation to find the specified derivative at the given point. -Find at the point ( 4, 1, -1) for ln - = 0.

Find the limit. -

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

Solve the problem. -Evaluate at t = π\piπ for the function w(x, y) = - + 8x; x = cost, y = sin t.

Find the limit. -

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

Determine whether the function is a solution of the wave equation. -w(x, t) = sin ( 6x + 6ct)

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

Find all the first order partial derivatives for the following function. -f(x, y) = ( -3x - y)

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

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

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

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

Find the limit. -

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

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