Question 1

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)  -14 
B)  -21 
C)  -8 
D)  -7 
A)  -14 
B)  -21 
C)  -8 
D)  -7

Question 2

Find the linear approximation of the function at the given point.
-  at

Accepted Answer

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

Question 3

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

Accepted Answer

A)  Maximum: 4 at (0, 1); minimum: -4 at (0, -1) 
B)  Maximum: 4 at (0, 1); minimum: -31 at (1, -2) 
C)  Maximum: 8 at (2, 1); minimum: -31 at (1, -2) 
D)  Maximum: 8 at (2, 1); minimum: -4 at (0, -1) 
A)  Maximum: 4 at (0, 1); minimum: -4 at (0, -1) 
B)  Maximum: 4 at (0, 1); minimum: -31 at (1, -2) 
C)  Maximum: 8 at (2, 1); minimum: -31 at (1, -2) 
D)  Maximum: 8 at (2, 1); minimum: -4 at (0, -1)

Question 4

Solve the problem.
-Evaluate   at t = 9 for the function w(x, y) =   - ln x; x =   , y = ln t.

Accepted Answer

A)  8/9 
B)  7 
C)  - 7/9 
D)  7/9 
A)  8/9 
B)  7 
C)  - 7/9 
D)  7/9

Question 5

Find the linear approximation of the function at the given point.
-  at

Accepted Answer

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

Question 6

Find the derivative of the function at the given point in the direction of A.
-    A= 12i- 5j

Accepted Answer

A)  - 183/6500 
B)  - 31 / 1300 
C)  - 141/ 6500 
D)  - 44/1625 
A)  - 183/6500 
B)  - 31 / 1300 
C)  - 141/ 6500 
D)  - 44/1625

Question 7

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

Accepted Answer

A)  268,435,456i + 33,554,432j + 201,326,592k 
B)  67,108,864i + 50,331,648j + 8,388,608k 
C)  268,435,456i + 50,331,648j + 134,217,728k 
D)  67,108,864i + 33,554,432j + 12,582,912k 
A)  268,435,456i + 33,554,432j + 201,326,592k 
B)  67,108,864i + 50,331,648j + 8,388,608k 
C)  268,435,456i + 50,331,648j + 134,217,728k 
D)  67,108,864i + 33,554,432j + 12,582,912k

Question 8

Find the limit.
-  x ln y

Accepted Answer

A)  4 
B)  ln 4 
C)  ln ( 4) - 1 
D)  No limit 
A)  4 
B)  ln 4 
C)  ln ( 4) - 1 
D)  No limit

Question 9

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( -2, 9) = 241, saddle point 
B)  f( 2, 9) = 257, local maximum 
C)  f( 2, -9) = -67, saddle point 
D)  f( -2, -9) = -83, local minimum 
A) f( -2, 9) = 241, saddle point 
B)  f( 2, 9) = 257, local maximum 
C)  f( 2, -9) = -67, saddle point 
D)  f( -2, -9) = -83, local minimum

Question 10

Find the limit.
-  ln

Accepted Answer

A)  0 
B)  ln 2 
C)  -ln 2 
D)  No limit 
A)  0 
B)  ln 2 
C)  -ln 2 
D)  No limit

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Find the absolute maxima and minima of the function on the given domain.
-f(x, y) = 5x + 10y on the closed triangular region with vertices (0, 0), (1, 0), and (0, 1)

Accepted Answer

A)  Absolute maximum: 5 at (1, 0); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 15 at (1, 1); absolute minimum: 5 at (1, 0) 
C)  Absolute maximum: 10 at (0, 1); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 10 at (0, 1); absolute minimum: 5 at (1, 0) 
A)  Absolute maximum: 5 at (1, 0); absolute minimum: 0 at (0, 0) 
B)  Absolute maximum: 15 at (1, 1); absolute minimum: 5 at (1, 0) 
C)  Absolute maximum: 10 at (0, 1); absolute minimum: 0 at (0, 0) 
D)  Absolute maximum: 10 at (0, 1); absolute minimum: 5 at (1, 0)

Question 14

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

Accepted Answer

A)  1/44 i + 1/44 j 
B)  5/22 i - 9/44 j 
C)  2/11 i + 1/11 j 
D)  1/11 i + 2/11 j 
A)  1/44 i + 1/44 j 
B)  5/22 i - 9/44 j 
C)  2/11 i + 1/11 j 
D)  1/11 i + 2/11 j

Question 15

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 satisfying  xS1U1P12S1S1P0  + yS1U1P12S1S1P0  =  100; range: real numbers 0 ≤ z ≤  10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤  10 
B)   Domain: all points in the xy-plane satisfying    xS1U1P12S1S1P0  + yS1U1P12S1S1P0   ≤  100; range: real numbers 0 ≤ z ≤  10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤  10 
C)   Domain: all points in the xy-plane; range: real numbers 0 ≤ z ≤  10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤  10 
D)   Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at   (0,0) 
A)   Domain: all points in the xy-plane satisfying  xS1U1P12S1S1P0  + yS1U1P12S1S1P0  =  100; range: real numbers 0 ≤ z ≤  10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤  10 
B)   Domain: all points in the xy-plane satisfying    xS1U1P12S1S1P0  + yS1U1P12S1S1P0   ≤  100; range: real numbers 0 ≤ z ≤  10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤  10 
C)   Domain: all points in the xy-plane; range: real numbers 0 ≤ z ≤  10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤  10 
D)   Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at   (0,0)

Question 16

Sketch the surface z = f(x,y).
-f(x, y) = 2 -   -

Accepted Answer

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

Question 17

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

Accepted Answer

A)  Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: 0 at (0, 0, 0) 
B)  Maximum: 8 at (2, 0, 0); minimum: 0 at (0, 0, 0) 
C)  Maximum: 8 at (2, 0, 0); minimum: -8 at (-2, 0, 0) 
D)  Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: -8 at (-2, 0, 0), (0, -2, 0), (0, 0, -2) 
A)  Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: 0 at (0, 0, 0) 
B)  Maximum: 8 at (2, 0, 0); minimum: 0 at (0, 0, 0) 
C)  Maximum: 8 at (2, 0, 0); minimum: -8 at (-2, 0, 0) 
D)  Maximum: 8 at (2, 0, 0), (0, 2, 0), (0, 0, 2); minimum: -8 at (-2, 0, 0), (0, -2, 0), (0, 0, -2)

Question 18

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

Accepted Answer

A)  28 / 4001 i + 63/ 4001 j 
B)  28 / 4001 i - 63/ 4001 j 
C)  28 / 3985 i + 63 / 3985 j 
D)  28 / 3985 i - 63 / 3985 j 
A)  28 / 4001 i + 63/ 4001 j 
B)  28 / 4001 i - 63/ 4001 j 
C)  28 / 3985 i + 63 / 3985 j 
D)  28 / 3985 i - 63 / 3985 j

Question 19

Solve the problem.
-Evaluate   at (x, y, z) = ( 1, 3, 0) for the function u(p, q, r) =   cos(r); p =   ,

Accepted Answer

A)  1 
B)  1/3 
C)  0 
D)  3 
A)  1 
B)  1/3 
C)  0 
D)  3

Question 20

Find the limit.
-

Accepted Answer

A)  0 
B)  2 
C)  4 
D)  No limit 
A)  0 
B)  2 
C)  4 
D)  No limit

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

Find the linear approximation of the function at the given point. - at

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

Solve the problem. -Evaluate at t = 9 for the function w(x, y) = - ln x; x = , y = ln t.

Find the linear approximation of the function at the given point. - at

Find the derivative of the function at the given point in the direction of A. - A= 12i- 5j

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

Find the limit. - x ln 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. - ln

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

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

Find the absolute maxima and minima of the function on the given domain. -f(x, y) = 5x + 10y on the closed triangular region with vertices (0, 0), (1, 0), and (0, 1)

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) =

Sketch the surface z = f(x,y). -f(x, y) = 2 - -

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

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

Solve the problem. -Evaluate at (x, y, z) = ( 1, 3, 0) for the function u(p, q, r) = cos(r); p = ,

Find the limit. -

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

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

Find the linear approximation of the function at the given point. - at

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

Solve the problem. -Evaluate at t = 9 for the function w(x, y) = - ln x; x = , y = ln t.

Find the linear approximation of the function at the given point. - at

Find the derivative of the function at the given point in the direction of A. - A= 12i- 5j

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

Find the limit. - x ln 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. - ln

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

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

Find the absolute maxima and minima of the function on the given domain. -f(x, y) = 5x + 10y on the closed triangular region with vertices (0, 0), (1, 0), and (0, 1)

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) =

Sketch the surface z = f(x,y). -f(x, y) = 2 - -

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

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

Solve the problem. -Evaluate at (x, y, z) = ( 1, 3, 0) for the function u(p, q, r) = cos(r); p = ,

Find the limit. -

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