Question 1

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

Accepted Answer

A)  0 
B)  1/ e 
C)  1 
D)  e 
A)  0 
B)  1/ e 
C)  1 
D)  e

Question 2

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

Accepted Answer

A)  y = x - 2 
B)  x - y + 2 = 0 
C)  y = x + 1 
D)  x + y = 1 
A)  y = x - 2 
B)  x - y + 2 = 0 
C)  y = x + 1 
D)  x + y = 1

Question 3

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

Accepted Answer

A)  -7x - 5y + 4z = -8 
B)  -7x + 5y + 8z = -8 
C)  -7x - 5y + 4z = 6 
D)  -7x + 5y + 8z = 6 
A)  -7x - 5y + 4z = -8 
B)  -7x + 5y + 8z = -8 
C)  -7x - 5y + 4z = 6 
D)  -7x + 5y + 8z = 6

Question 4

Solve the problem.
-The Van der Waals equation provides an approximate model for the behavior of real gases. The equation is P(V, T) =   -   , where P is pressure, V is volume, T is Kelvin temperature, and a,b , and R are constants. Find the partial derivative of the function with respect to each variable.

Accepted Answer

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

Question 5

Find the domain and range and describe the level curves for the function f(x,y).
-f(x, y) = 4x + 3y

Accepted Answer

A)  Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c, c > 0 
B)  Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c 
C)  Domain: all points in the xy-plane; range: real numbers z $\le$ 0; level curves: lines 4x + 3y = c, c $\le$ 0 
D)  Domain: all points in the xy-plane; range: real numbers z $\le$ 0 ; level curves: lines 4x + 3y = c, c $\le$ 0 
A)  Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c, c > 0 
B)  Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c 
C)  Domain: all points in the xy-plane; range: real numbers z $\le$ 0; level curves: lines 4x + 3y = c, c $\le$ 0 
D)  Domain: all points in the xy-plane; range: real numbers z $\le$ 0 ; level curves: lines 4x + 3y = c, c $\le$ 0

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

A)  -24i - 2j 
B)  72i + 8j 
C)  -24i + 8j 
D)  36i + 8j 
A)  -24i - 2j 
B)  72i + 8j 
C)  -24i + 8j 
D)  36i + 8j

Question 9

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 > 0; level curves: ellipses  3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
B)   Domain: all points in the xy-plane except (0, 0); range: real numbers > 0; level curves: ellipses  3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
C)   Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses   3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
D)   Domain: all points in the xy-plane except (0, 0); range: all real numbers; level curves: ellipses   3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
A)   Domain: all points in the xy-plane; range: real numbers > 0; level curves: ellipses  3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
B)   Domain: all points in the xy-plane except (0, 0); range: real numbers > 0; level curves: ellipses  3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
C)   Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses   3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c 
D)   Domain: all points in the xy-plane except (0, 0); range: all real numbers; level curves: ellipses   3xS1U1P12S1S1P0 + 10yS1U1P12S1S1P0 = c

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

A)  Absolute maximum: 36 at (3, 0) and (-3, 0); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 126 at (3, 3); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 90 at (0, 3) and (0, -3); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 90 at (0, 3) and (0, -3); absolute minimum: 36 at (3, 0) and (-3, 0) 
A)  Absolute maximum: 36 at (3, 0) and (-3, 0); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 126 at (3, 3); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 90 at (0, 3) and (0, -3); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 90 at (0, 3) and (0, -3); absolute minimum: 36 at (3, 0) and (-3, 0)

Question 12

Find the limit.
-

Accepted Answer

A)  1 
B)    
C)  0 
D)    
A)  1 
B)    
C)  0 
D)

Question 13

Determine whether the function is a solution of the wave equation.
-w(x, t) = ln 9cxt

Accepted Answer

A) True 
 B)False

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

A)  Absolute maximum: 4 at (2, 0); absolute minimum: 2 at (1, 0) 
B)  Absolute maximum: 11 at (1, 1); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 11 at (1, 1); absolute minimum: 2 at (1, 0) 
D)  Absolute maximum: 18 at (0, 2); absolute minimum: 0 at (0, 0) 
A)  Absolute maximum: 4 at (2, 0); absolute minimum: 2 at (1, 0) 
B)  Absolute maximum: 11 at (1, 1); absolute minimum: 0 at (0, 0) 
C)  Absolute maximum: 11 at (1, 1); absolute minimum: 2 at (1, 0) 
D)  Absolute maximum: 18 at (0, 2); absolute minimum: 0 at (0, 0)

Question 16

Solve the problem.
-Find the point on the line   that is closest to the point

Accepted Answer

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

Question 17

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

Accepted Answer

A)  2i - 3j + 8k 
B)  -12i + 27j + 56k 
C)  -12i - 27j + 56k 
D)  -6i - 9j + 7k 
A)  2i - 3j + 8k 
B)  -12i + 27j + 56k 
C)  -12i - 27j + 56k 
D)  -6i - 9j + 7k

Question 18

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 19

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

Accepted Answer

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

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

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

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

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

Find the domain and range and describe the level curves for the function f(x,y). -f(x, y) = 4x + 3y

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

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

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

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

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

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

Find the limit. -

Determine whether the function is a solution of the wave equation. -w(x, t) = ln 9cxt

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

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

Solve the problem. -Find the point on the line that is closest to the point

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

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 second order partial derivatives of the given 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. -

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

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

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

Find the domain and range and describe the level curves for the function f(x,y). -f(x, y) = 4x + 3y

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

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

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

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

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

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

Find the limit. -

Determine whether the function is a solution of the wave equation. -w(x, t) = ln 9cxt

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

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

Solve the problem. -Find the point on the line that is closest to the point

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

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 second order partial derivatives of the given 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. -

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