Question 1

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

Accepted Answer

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

Question 2

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 3

Sketch the surface z = f(x,y).
-f(x, y) = 3 -

Accepted Answer

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

Question 4

Find the limit.
-

Accepted Answer

A)  3 
B)  -5 
C)  $\infty$ 
D)  0 
A)  3 
B)  -5 
C)  $\infty$ 
D)  0

Question 5

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)  5/39   
B)  5/51   
C)  5/21   
D)  5/33   
A)  5/39   
B)  5/51   
C)  5/21   
D)  5/33

Question 6

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

Accepted Answer

A)  Maximum: 18 at   minimum: -18 at   
B)  Maximum: 18 at   minimum: -9 at   
C)  Maximum: 9 at   minimum: -18 at   
D)  Maximum: 9 at   minimum: -9 at   
A)  Maximum: 18 at   minimum: -18 at   
B)  Maximum: 18 at   minimum: -9 at   
C)  Maximum: 9 at   minimum: -18 at   
D)  Maximum: 9 at   minimum: -9 at

Question 7

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)    
D)    
A)    
B)    
C)    
D)

Question 8

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

Accepted Answer

A)  All (x, y) such that x $\neq$ y 
B)  All (x, y) $\neq$(0, 0) 
C)  All (x, y) such that x $\neq$ - y 
D)  All (x, y) 
A)  All (x, y) such that x $\neq$ y 
B)  All (x, y) $\neq$(0, 0) 
C)  All (x, y) such that x $\neq$ - y 
D)  All (x, y)

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Find the limit.
-

Accepted Answer

A)  1/2 
B)  2 
C)  0 
D)    
A)  1/2 
B)  2 
C)  0 
D)

Question 13

Write a chain rule formula for the following derivative.
-  for w = f(p, q, r); p = g(t), q = h(t), r = k(t)

Accepted Answer

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

Question 14

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

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.
-  on the diamond-shaped region

Accepted Answer

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

Question 16

Solve the problem.
-According to the Gas Kinetic Theory, the average speed of a gas particle is given by    where
    is the speed in m/s, k is the constant 1.38 × 10S1U1P1-23 S1S1P0 , T is the temperature of the gas in Kelvin, and m is the mass of the gas particle in kg. What is the average speed of an oxygen molecule with a mass of
5.314 X 10S1U1P1-26S1S1P0 KG  at a temperature of 500 K?

Accepted Answer

A)  1020 m/s 
B)  575 m/s 
C)  330,700 m/s 
D)  174 m/s 
A)  1020 m/s 
B)  575 m/s 
C)  330,700 m/s 
D)  174 m/s

Question 17

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

Accepted Answer

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

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

Question 19

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: all real numbers; level curves: circles with centers at (0, 0) 
B)   Domain: all points in the xy-plane satisfying  xS1U1P12 S1S1P0 + yS1U1P12S1S1P0  ≤ 1; range: real numbers       ≤ z ≤   ; level curves: circles with centers at (0, 0) 
C)   Domain: all points in the xy-plane satisfying    xS1U1P12 S1S1P0 + yS1U1P12S1S1P0   ≤ 1; range: real numbers  -    ≤ z ≤   ; level curves: circles with centers at (0,0) and radii r, 0 < r ≤ 1 
D)   Domain: all points in the xy-plane; range: real numbers  -    ≤ z ≤   ; level curves: circles with centers at (0, 0) 
A)   Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at (0, 0) 
B)   Domain: all points in the xy-plane satisfying  xS1U1P12 S1S1P0 + yS1U1P12S1S1P0  ≤ 1; range: real numbers       ≤ z ≤   ; level curves: circles with centers at (0, 0) 
C)   Domain: all points in the xy-plane satisfying    xS1U1P12 S1S1P0 + yS1U1P12S1S1P0   ≤ 1; range: real numbers  -    ≤ z ≤   ; level curves: circles with centers at (0,0) and radii r, 0 < r ≤ 1 
D)   Domain: all points in the xy-plane; range: real numbers  -    ≤ z ≤   ; level curves: circles with centers at (0, 0)

Question 20

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

Accepted Answer

A)  Maximum: 77 at (-4, -4, -2); minimum: 13 at (4, 4, 2) 
B)  Maximum: 77 at (-4, -2, -4); minimum: 13 at (4, 2, 4) 
C)  Maximum: 81 at (-2, -4, -4); minimum: 9 at (2, 4, 4) 
D)  Maximum: 49 at (-2, 4, -4); minimum: 41 at (2, -4, 4) 
A)  Maximum: 77 at (-4, -4, -2); minimum: 13 at (4, 4, 2) 
B)  Maximum: 77 at (-4, -2, -4); minimum: 13 at (4, 2, 4) 
C)  Maximum: 81 at (-2, -4, -4); minimum: 9 at (2, 4, 4) 
D)  Maximum: 49 at (-2, 4, -4); minimum: 41 at (2, -4, 4)

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

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

Sketch the surface z = f(x,y). -f(x, y) = 3 -

Find the limit. -

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. -

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

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

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

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

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

Find the limit. -

Write a chain rule formula for the following derivative. - for w = f(p, q, r); p = g(t), q = h(t), r = k(t)

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

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

Write a chain rule formula for the following derivative. - for w = f(p, q); p = g(x, y), q = h(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 domain and range and describe the level curves for the function f(x,y). -f(x, y) = ( + )

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

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

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

Sketch the surface z = f(x,y). -f(x, y) = 3 -

Find the limit. -

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. -

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

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

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

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

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

Find the limit. -

Write a chain rule formula for the following derivative. - for w = f(p, q, r); p = g(t), q = h(t), r = k(t)

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

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

Write a chain rule formula for the following derivative. - for w = f(p, q); p = g(x, y), q = h(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 domain and range and describe the level curves for the function f(x,y). -f(x, y) = ( + )

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

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