Question 1

Find an optimal binary compression for a 16-character alphabet with probabilities given in the table below. Use four figures of accuracy.

Accepted Answer

A)   3.0139
B)   0.8398
C)   0.5821
D)   0.2528
E)   0.3607 B

Question 2

Solve the integral equation f(x) = 3 + 2  dt.

Accepted Answer

A)   f(x)  = 2 
B)   f(x)  = 3 
C)   f(x)  = 2 
D)   f(x)  = 3 
E)   f(x)  = 3  D

Question 3

Use Lagrange multipliers to find the point P(  ,  ,  ) on the sphere  +  +  = 9 and the point Q (  ,  ,  ) on the plane x + 2y - 2z = 7 where the distance between P and Q is minimum and determine this minimum distance.

Accepted Answer

A)   P (2, -4, 7) , Q (5, 2, 1) ; minimum distance 9 units
B)   P (4, 0, 3) , Q (3, -2, 5) ; minimum distance 3 units
C)   P (4, 0, 3) , Q (5, 2, 1) ; minimum distance 3 units
D)   P (2, -4, 7) , Q (-1, -10, 13) ; minimum distance 9 units
E)   P (2, -4, 7) , Q (1, -6, 9) ; minimum distance 3 units C

Question 4

Find the value of the constant m so that the line y = mx best fits the experimental data (1, 2.3), (2, 4.2), (3, 7.1), (4, 8.8) in the sense of minimizing the sum of the squares of the vertical distances of the data points from the line.

Accepted Answer

A)   m = 2.36
B)   m = 2.24
C)   m = 2.18
D)   m = 2.11
E)   m = 2.34

Question 5

Find the critical points of f(x, y) = ln (x2 + y2 + 4x - 4y + 8).

Accepted Answer

A)   (-2, 2) 
B)   (2, -2) 
C)   (-2, 2)  and (2, -2) 
D)   (-2, 2) , (2, -2) , and (0, 0) 
E)   There are no critical points.

Question 6

Find the maximum and minimum values of f(x , y) = x2 + xy + 2y2 subject to the constraint x2 + 3y2 = 3.

Accepted Answer

A)   minimum 0 , maximum 3
B)   minimum 2 , maximum 3
C)   minimum 2 + xy + 2y2 subject to the constraint x2 + 3y2 = 3. 
A)   minimum 0 , maximum 3 
B)   minimum 2 , maximum 3 
C)   minimum   , maximum   
D)   minimum 3 , maximum   
E)   minimum   , maximum 2" class="answers-bank-image d-inline" rel="preload" > , maximum 2 + xy + 2y2 subject to the constraint x2 + 3y2 = 3. 
A)   minimum 0 , maximum 3 
B)   minimum 2 , maximum 3 
C)   minimum   , maximum   
D)   minimum 3 , maximum   
E)   minimum   , maximum 2" class="answers-bank-image d-inline" rel="preload" >
D)   minimum 3 , maximum 2 + xy + 2y2 subject to the constraint x2 + 3y2 = 3. 
A)   minimum 0 , maximum 3 
B)   minimum 2 , maximum 3 
C)   minimum   , maximum   
D)   minimum 3 , maximum   
E)   minimum   , maximum 2" class="answers-bank-image d-inline" rel="preload" >
E)   minimum 2 + xy + 2y2 subject to the constraint x2 + 3y2 = 3. 
A)   minimum 0 , maximum 3 
B)   minimum 2 , maximum 3 
C)   minimum   , maximum   
D)   minimum 3 , maximum   
E)   minimum   , maximum 2" class="answers-bank-image d-inline" rel="preload" > , maximum 2

Question 7

Evaluate F(x,y) =  dt for x > 0, y > 0.

Accepted Answer

The answer of Evaluate F(x,y) =  dt for x...

Question 8

Find and classify the critical points of the function f(x, y) = 2y3 - 3x2 - 3xy + 9x.

Accepted Answer

A)   local minimum at (2, -1) , saddle point at 3 - 3x2 - 3xy + 9x. 
A)   local minimum at (2, -1) , saddle point at   
B)   local maximum at (2, -1) , saddle point at   
C)   saddle point at (2, -1) , local maximum at   
D)   saddle point at (2, -1) , local minimum at   
E)   saddle point at (2, -1)  and  " class="answers-bank-image d-inline" rel="preload" >
B)   local maximum at (2, -1) , saddle point at 3 - 3x2 - 3xy + 9x. 
A)   local minimum at (2, -1) , saddle point at   
B)   local maximum at (2, -1) , saddle point at   
C)   saddle point at (2, -1) , local maximum at   
D)   saddle point at (2, -1) , local minimum at   
E)   saddle point at (2, -1)  and  " class="answers-bank-image d-inline" rel="preload" >
C)   saddle point at (2, -1) , local maximum at 3 - 3x2 - 3xy + 9x. 
A)   local minimum at (2, -1) , saddle point at   
B)   local maximum at (2, -1) , saddle point at   
C)   saddle point at (2, -1) , local maximum at   
D)   saddle point at (2, -1) , local minimum at   
E)   saddle point at (2, -1)  and  " class="answers-bank-image d-inline" rel="preload" >
D)   saddle point at (2, -1) , local minimum at 3 - 3x2 - 3xy + 9x. 
A)   local minimum at (2, -1) , saddle point at   
B)   local maximum at (2, -1) , saddle point at   
C)   saddle point at (2, -1) , local maximum at   
D)   saddle point at (2, -1) , local minimum at   
E)   saddle point at (2, -1)  and  " class="answers-bank-image d-inline" rel="preload" >
E)   saddle point at (2, -1)  and 3 - 3x2 - 3xy + 9x. 
A)   local minimum at (2, -1) , saddle point at   
B)   local maximum at (2, -1) , saddle point at   
C)   saddle point at (2, -1) , local maximum at   
D)   saddle point at (2, -1) , local minimum at   
E)   saddle point at (2, -1)  and  " class="answers-bank-image d-inline" rel="preload" >

Question 9

A rectangular box with no top has given surface area S. Find the length, width, and height of the box if the volume is as large as possible.

Accepted Answer

A)   length = width =  , height = 
B)   length = width =  , height = 
C)   length = width =  , height = 
D)   length = width =  , height = 
E)   length = width =  , height =

Question 10

Every one-parameter family of curves in the plane has an envelope.

Accepted Answer

A) True 
 B)False

Question 11

Find the points closest to the origin on the hyperbola in which the cone x2 + y2 = z2 intersects the plane x + y = 2.

Accepted Answer

A)   (1, 1, ± 2 + y2 = z2 intersects the plane x + y = 2. 
A)   (1, 1, ±   )  
B)   (-1, -1, ±   )  
C)   (-1, -1, ±   )  and (1, 1, ±   )  
D)   (2, 0, 2)  and (0, 2, 2)  
E)   (1, 1,   )  and (-1, -1, -   ) " class="answers-bank-image d-inline" rel="preload" > ) 
B)   (-1, -1, ± 2 + y2 = z2 intersects the plane x + y = 2. 
A)   (1, 1, ±   )  
B)   (-1, -1, ±   )  
C)   (-1, -1, ±   )  and (1, 1, ±   )  
D)   (2, 0, 2)  and (0, 2, 2)  
E)   (1, 1,   )  and (-1, -1, -   ) " class="answers-bank-image d-inline" rel="preload" > ) 
C)   (-1, -1, ± 2 + y2 = z2 intersects the plane x + y = 2. 
A)   (1, 1, ±   )  
B)   (-1, -1, ±   )  
C)   (-1, -1, ±   )  and (1, 1, ±   )  
D)   (2, 0, 2)  and (0, 2, 2)  
E)   (1, 1,   )  and (-1, -1, -   ) " class="answers-bank-image d-inline" rel="preload" > )  and (1, 1, ± 2 + y2 = z2 intersects the plane x + y = 2. 
A)   (1, 1, ±   )  
B)   (-1, -1, ±   )  
C)   (-1, -1, ±   )  and (1, 1, ±   )  
D)   (2, 0, 2)  and (0, 2, 2)  
E)   (1, 1,   )  and (-1, -1, -   ) " class="answers-bank-image d-inline" rel="preload" > ) 
D)   (2, 0, 2)  and (0, 2, 2) 
E)   (1, 1, 2 + y2 = z2 intersects the plane x + y = 2. 
A)   (1, 1, ±   )  
B)   (-1, -1, ±   )  
C)   (-1, -1, ±   )  and (1, 1, ±   )  
D)   (2, 0, 2)  and (0, 2, 2)  
E)   (1, 1,   )  and (-1, -1, -   ) " class="answers-bank-image d-inline" rel="preload" > )  and (-1, -1, - 2 + y2 = z2 intersects the plane x + y = 2. 
A)   (1, 1, ±   )  
B)   (-1, -1, ±   )  
C)   (-1, -1, ±   )  and (1, 1, ±   )  
D)   (2, 0, 2)  and (0, 2, 2)  
E)   (1, 1,   )  and (-1, -1, -   ) " class="answers-bank-image d-inline" rel="preload" > )

Question 12

Find and classify all critical points for the function f(x, y) =  .

Accepted Answer

A)   (1, 0) ; minimum
B)   (-1, 0) ; maximum
C)   (0, 0) ; minimum
D)   (0, 0) ; saddle
E)   There are no critical points.

Question 13

Find and classify the critical points of the function f(x, y) = x sin y.

Accepted Answer

A)   local maxima at (0, n  π\piπ )  for even integers n, local minimum at (0, n  π\piπ )  for odd integers n
B)   local minima at (0, n  π\piπ )  for even integers n, local maximum at (0, n  π\piπ )  for odd integers n
C)   saddle points at all points (0, n  π\piπ )  for integers n
D)   only critical point is (0, 0) , a saddle point
E)   local minima at (0, n  π\piπ )  for nonzero integers n, saddle point at (0, 0)

Question 14

Suppose the critical points of f(x,y) = 2x3 - 15x2 + 24x - y2 - 6y + 1 occur at the points(1, -3) and (4, -3).Then the function f has:

Accepted Answer

A)   saddle points at (1, -3)  and (4, -3) 
B)   a local minimum at (4, - 3)  and a saddle point at (1, -3) 
C)   a local maximum at (1, - 3)  and a saddle point at (4, - 3) 
D)   a local maximum at (4, -3)  and a saddle point at (1, -3) 
E)   a local maximum at (4, -3)  and a local minimum at (1, -3)

Question 15

Use Maple's fsolve with Digits : = 6 to find a solution to the system  +  = 3, x sin(x) - y cos(y) = 0 near (1, 2).

Accepted Answer

A)   x = 0.461289, y = 1.42635
B)   x = 0.462289, y = 1.53635
C)   x = 0.465489, y = 1.48635
D)   x = 0.460089, y = 1.42335
E)   x = 0.463445, y = 1.41657

Question 16

Find the absolute maximum and minimum values   on the semi-infinite strip 0 ≤ x < ∞,  -1 ≤ y ≤ 2,  if these extreme values exist.

Accepted Answer

A)   maximum  , minimum - 
B)   maximum  , minimum - 
C)   maximum  , minimum - 
D)   maximum  , minimum 0
E)   no extreme values exist

Question 17

Starfleet Command has given Captain Kirk a subspace amplifier to channel communications among n star systems having galactic Cartesian coordinates (  ,  ,  ) for k = 1, 2,..., n. To what point in space should Captain Kirk take the Enterprise to place the amplifier so as to minimize the sum of the squares of its distances from the n star systems?

Accepted Answer

A)   the point 
B)   the point 
C)   the point 
D)   the point 
E)   the point

Question 18

Use the fact that  =  ln (1 + ab) to evaluate  dx.

Accepted Answer

A)    ln (1 + ab)  + 
B)    ln (1 + ab)  + 
C)    ln (1 + ab)  - 
D)    ln (1 + ab)  - 
E)    ln (1 + ab)  +

Question 19

Find the critical points of f(x, y) = 3x2 - 2xy + 2y2 - 10x + 1.

Accepted Answer

A)   (2, 1) 
B)   (1, 1) 
C)   (0, 0) 
D)   (1, -1)  and (-1, 1) 
E)   (2, 1)  and (1, 2)

Question 20

Given F( ) =  =  , for  >  .
By first differentiating F with respect to , evaluate  .

Accepted Answer

A)   
B)   
C)   - 
D)   
E)

Exam 14: Applications of Partial Derivatives

Find an optimal binary compression for a 16-character alphabet with probabilities given in the table below. Use four figures of accuracy.

Solve the integral equation f(x) = 3 + 2 dt.

Use Lagrange multipliers to find the point P( , , ) on the sphere + + = 9 and the point Q ( , , ) on the plane x + 2y - 2z = 7 where the distance between P and Q is minimum and determine this minimum distance.

Find the value of the constant m so that the line y = mx best fits the experimental data (1, 2.3) , (2, 4.2) , (3, 7.1) , (4, 8.8) in the sense of minimizing the sum of the squares of the vertical distances of the data points from the line.

Find the critical points of f(x, y) = ln (x² + y² + 4x - 4y + 8) .

Find the maximum and minimum values of f(x , y) = x² + xy + 2y² subject to the constraint x² + 3y² = 3.

Evaluate F(x,y) = dt for x > 0, y > 0.

Find and classify the critical points of the function f(x, y) = 2y³ - 3x² - 3xy + 9x.

A rectangular box with no top has given surface area S. Find the length, width, and height of the box if the volume is as large as possible.

Every one-parameter family of curves in the plane has an envelope.

Find the points closest to the origin on the hyperbola in which the cone x² + y² = z² intersects the plane x + y = 2.

Find and classify all critical points for the function f(x, y) = .

Find and classify the critical points of the function f(x, y) = x sin y.

Suppose the critical points of f(x,y) = 2x³ - 15x² + 24x - y² - 6y + 1 occur at the points(1, -3) and (4, -3) .Then the function f has:

Use Maple's fsolve with Digits : = 6 to find a solution to the system + = 3, x sin(x) - y cos(y) = 0 near (1, 2) .

Find the absolute maximum and minimum values of on the semi-infinite strip 0 ≤ x < ∞, -1 ≤ y ≤ 2, if these extreme values exist.

Use the fact that = ln (1 + ab) to evaluate dx.

Find the critical points of f(x, y) = 3x² - 2xy + 2y² - 10x + 1.

Given F( ) = = , for > . By first differentiating F with respect to 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11, evaluate 11ee77e1_77af_06d7_a0f8_1371a5bb3b26_TB9661_11 .

Exam 14: Applications of Partial Derivatives

Find an optimal binary compression for a 16-character alphabet with probabilities given in the table below. Use four figures of accuracy.

Solve the integral equation f(x) = 3 + 2 dt.

Use Lagrange multipliers to find the point P( , , ) on the sphere + + = 9 and the point Q ( , , ) on the plane x + 2y - 2z = 7 where the distance between P and Q is minimum and determine this minimum distance.

Find the value of the constant m so that the line y = mx best fits the experimental data (1, 2.3) , (2, 4.2) , (3, 7.1) , (4, 8.8) in the sense of minimizing the sum of the squares of the vertical distances of the data points from the line.

Find the critical points of f(x, y) = ln (x2 + y2 + 4x - 4y + 8) .

Find the maximum and minimum values of f(x , y) = x2 + xy + 2y2 subject to the constraint x2 + 3y2 = 3.

Evaluate F(x,y) = dt for x > 0, y > 0.

Find and classify the critical points of the function f(x, y) = 2y3 - 3x2 - 3xy + 9x.

A rectangular box with no top has given surface area S. Find the length, width, and height of the box if the volume is as large as possible.

Every one-parameter family of curves in the plane has an envelope.

Find the points closest to the origin on the hyperbola in which the cone x2 + y2 = z2 intersects the plane x + y = 2.

Find and classify all critical points for the function f(x, y) = .

Find and classify the critical points of the function f(x, y) = x sin y.

Suppose the critical points of f(x,y) = 2x3 - 15x2 + 24x - y2 - 6y + 1 occur at the points(1, -3) and (4, -3) .Then the function f has:

Use Maple's fsolve with Digits : = 6 to find a solution to the system + = 3, x sin(x) - y cos(y) = 0 near (1, 2) .

Find the absolute maximum and minimum values of on the semi-infinite strip 0 ≤ x < ∞, -1 ≤ y ≤ 2, if these extreme values exist.

Use the fact that = ln (1 + ab) to evaluate dx.

Find the critical points of f(x, y) = 3x2 - 2xy + 2y2 - 10x + 1.

Given F( ) = = , for > . By first differentiating F with respect to 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11, evaluate 11ee77e1_77af_06d7_a0f8_1371a5bb3b26_TB9661_11 .

Find the critical points of f(x, y) = ln (x² + y² + 4x - 4y + 8) .

Find the maximum and minimum values of f(x , y) = x² + xy + 2y² subject to the constraint x² + 3y² = 3.

Find and classify the critical points of the function f(x, y) = 2y³ - 3x² - 3xy + 9x.

Find the points closest to the origin on the hyperbola in which the cone x² + y² = z² intersects the plane x + y = 2.

Suppose the critical points of f(x,y) = 2x³ - 15x² + 24x - y² - 6y + 1 occur at the points(1, -3) and (4, -3) .Then the function f has:

Find the critical points of f(x, y) = 3x² - 2xy + 2y² - 10x + 1.