Question 1

Use Lagrange Multipliers to solve the problem. Find the maximum value of  subject to  .

Accepted Answer

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Question 2

Find the point on the given plane closest to (1,1,0).  NOTE: Please enter your answer in the form ( x , y , z ).

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Question 3

A productivity model at the Handy Gadget Company is  where P is the number of gadgets the company turns out per month, x is the number of employees at the company, and y is the monthly operating budget in thousands of dollars. Because the company hires part - time workers, it uses anywhere between 39 and 49 workers each month, and its operating budget varies from $15,000 to $19,000 per month. What is the average of the possible numbers gadgets it can turn out per month? (Round the answer to the nearest 1000 gadgets.)Please enter your answer as a number without the units.

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Question 4

Find the point on the given plane closest to (1,1,0).

Accepted Answer

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Question 5

Compute the integral.

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The answer of Compute the integral. 
...

Question 6

Write the integral with the order of integration reversed (changing the limits of integration as necessary).

Accepted Answer

Since we are reversing the order of integration, we need to switch the limits of integration and then switch the order of the variables. The given integral is written as:

$\int_0^1\int_0^{\ln(1+x)}f(x,y)dydx$

To reverse the order of integration, we write the integral as:

$\int_0^{\ln(2)}\int_{e^y-1}^1f(x,y)dxdy$

We can see that the limits of integration have been reversed and the order of variables has been switched from $x$ and $y$ to $y$ and $x$.

Thus, the answer is D.

Question 7

Solve the given problem by using substitution. Find the minimum value of  subject to  .

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Question 8

The town of West Podunk is shaped like a rectangle 80 miles from west to east and 100 miles from north to south (see the figure). It has a population density of  hundred people per square mile x miles east and y miles north of the southwest corner of town. What is the total population of the town?  Please enter your answer as a number without the units.

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Question 9

Your latest CD - ROM drive is expected to sell between  units if priced at p . You plan to set the price between $100 and $400. What are the maximum and minimum possible revenues you can make? What is the average of all the possible revenues you can make? Please enter your answer as three numbers, separated by commas.

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Question 10

What point on the surface  is closest to the origin?  Hint : Minimize the square of the distance from ( x , y , z )to the origin. NOTE: Please enter your answer(s)in the form ( x , y , z ). If there is more than one answer, separate them with commas.

Accepted Answer

The answer of What point on the surface  is...

Question 11

Use Lagrange Multipliers to solve the given problem. Find the maximum value of  subject to  .

Accepted Answer

We need to maximize the function 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 subject to the constraint 11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11. Using Lagrange Multipliers method, we form the Lagrangian function:

L(x,y,λ) = 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 - λ(11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11)

Now, we find the partial derivatives of L with respect to x, y and λ and equate them to 0:

∂L/∂x = 2x = λ*2y = ∂L/∂y
∂L/∂λ = 11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11 - 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11*y = 0

Solving these equations simultaneously, we get:

x = y
11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11 - 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11*y = 0

Substituting y = x in the second equation, we get:

y = 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 /11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11

Substituting this value of y in the expression for x=y, we get:

x = 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 /11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11

Therefore, the maximum value of the function is:

11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11*(11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 /11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11 )^2

Simplifying this expression, we get:

(11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11)^3 /(11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11)^2

Looking at the answer options, we can see that choice A matches with our answer.

Question 12

Solve the given problem by using substitution. Find the maximum value of  subject to  .

Accepted Answer

The answer of Solve the given problem by using substitution....

Question 13

Use Lagrange Multipliers to solve the problem. Find the maximum value of  subject to  .

Accepted Answer

The answer of Use Lagrange Multipliers to solve the problem....

Question 14

Compute the integral.

Accepted Answer

The answer of Compute the integral. 
...

Question 15

Solve the given problem by using substitution. Find the maximum value of  subject to  .

Accepted Answer

The answer of Solve the given problem by using substitution....

Question 16

At what point on the given surface is the quantity  a minimum? (The method of Lagrange multipliers can be used here.)

Accepted Answer

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Question 17

Solve the given problem by using substitution. Find the minimum value of  subject to  .

Accepted Answer

The answer of Solve the given problem by using substitution....

Question 18

Solve the given problem by using substitution. Minimize  subject to  with  .

Accepted Answer

The answer of Solve the given problem by using substitution....

Question 19

What point on the surface  is closest to the origin?  Hint : Minimize the square of the distance from ( x , y , z )to the origin.

Accepted Answer

The answer of What point on the surface  is...

Question 20

Solve the given problem by using substitution. Minimize  subject to  with  .

Accepted Answer

The answer of Solve the given problem by using substitution....

Quiz 8: Functions of Several Variables

Use Lagrange Multipliers to solve the problem. Find the maximum value of subject to .

Find the point on the given plane closest to (1,1,0). NOTE: Please enter your answer in the form ( x , y , z ).

Find the point on the given plane closest to (1,1,0) .

Compute the integral.

Write the integral with the order of integration reversed (changing the limits of integration as necessary) .

Solve the given problem by using substitution. Find the minimum value of subject to .

What point on the surface is closest to the origin? Hint : Minimize the square of the distance from ( x , y , z )to the origin. NOTE: Please enter your answer(s)in the form ( x , y , z ). If there is more than one answer, separate them with commas.

Use Lagrange Multipliers to solve the given problem. Find the maximum value of subject to .

Solve the given problem by using substitution. Find the maximum value of subject to .

Use Lagrange Multipliers to solve the problem. Find the maximum value of subject to .

Compute the integral.

Solve the given problem by using substitution. Find the maximum value of subject to .

At what point on the given surface is the quantity a minimum? (The method of Lagrange multipliers can be used here.)

Solve the given problem by using substitution. Find the minimum value of subject to .

Solve the given problem by using substitution. Minimize subject to with .

What point on the surface is closest to the origin? Hint : Minimize the square of the distance from ( x , y , z ) to the origin.

Solve the given problem by using substitution. Minimize subject to with .