Question 1

Use Lagrange multipliers to find the given extremum.In each case,assume that ​x and ​y are positive. ​
Maximize   Constraint   ​

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 2

Sketch the yz-trace of the equation: ​   ​

Accepted Answer

A)    
B)    
C)    
D)    ​ 
E)    
A)    
B)    
C)    
D)    ​ 
E)

Question 3

Use a double integral to find the volume of the indicated solid. ​   ​   ​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  none of the above 
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  none of the above

Question 4

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points   Round your answer to three decimal places. ​

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 5

Find the four second partial derivatives.Observe that the second mixed partials are equal. ​   ​

Accepted Answer

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

Question 6

A manufacturer has an order for 1,100 units of fine paper that can be produced at two locations.Let x​1 and x​2 be the numbers of units produced at the two plants.Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by   ​

Accepted Answer

A)    units and   units 
B)    units and   units 
C)    units and   units 
D)    units and   units 
E)    units and   units 
A)    units and   units 
B)    units and   units 
C)    units and   units 
D)    units and   units 
E)    units and   units

Question 7

A company manufactures two types of sneakers: running shoes and basketball shoes.The total revenue from x1 units of running shoes and x2 units of basketball shoes is: ​   ,
​
Where x1 and x2 are in thousands of units.Find x1 and x2 so as to maximize the revenue.
​

Accepted Answer

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

Question 8

Find the standard equation of the sphere whose center is   and whose radius is 5. ​

Accepted Answer

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

Question 9

Evaluate   and   for the function   at the point   ​

Accepted Answer

A)    and   
B)    and   
C)    and   
D)    and   
E)    and   
A)    and   
B)    and   
C)    and   
D)    and   
E)    and

Question 10

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false.
​A linear regression model with a positive correlation will have a slope that is greater than 0.

Accepted Answer

A) True 
 B)False

Question 11

Examine the function given below for relative extrema and saddle points. ​   ​

Accepted Answer

A)  The function has a saddle point at   
B)  The function has a relative maximum at   
C)  The function has a relative minimum at   
D)  The function has a saddle point at   
E)  The function has a relative maximum at   
A)  The function has a saddle point at   
B)  The function has a relative maximum at   
C)  The function has a relative minimum at   
D)  The function has a saddle point at   
E)  The function has a relative maximum at

Question 12

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model.The cost function for producing x freestanding and y fireplace-insert stoves is   Find the marginal costs (   and   )when   and   Round your answers to two decimal places. ​

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 13

Sketch the graph of the plane given by   ​

Accepted Answer

A)  ​​   
B)  ​ ​   
C)  ​ ​   
D)  ​ ​   
E)  ​ ​   
A)  ​​   
B)  ​ ​   
C)  ​ ​   
D)  ​ ​   
E)  ​ ​

Question 14

For   ,find all values of x and y such that   and   simultaneously. ​   ​

Accepted Answer

A)    
B)  ​   
C)  (0,0) 
D)  Both A and C 
E)  Both A and B 
A)    
B)  ​   
C)  (0,0) 
D)  Both A and C 
E)  Both A and B

Question 15

Use a double integral to find the volume of the solid bounded by the graphs of the equations. ​   ​

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 16

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. ​   ​

Accepted Answer

A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)    
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 17

Sketch the region of integration   ​

Accepted Answer

A)  ​
​   
B)  ​   ​ 
C)  ​
​   
D)  ​
​   
E)  ​
​   
A)  ​
​   
B)  ​   ​ 
C)  ​
​   
D)  ​
​   
E)  ​
​

Question 18

Sketch the region R whose area is given by the following double integral. ​   ​

Accepted Answer

A)  ​​   
B)  ​   
C)  ​
​   
D)  ​   
E)  ​
​   
A)  ​​   
B)  ​   
C)  ​
​   
D)  ​   
E)  ​
​

Question 19

A microbiologist must prepare a culture medium in which to grow a certain type of bacteria.The percent of salt contained in this medium is given by ​   ​
Where x,y,and z are the nutrient solutions to be mixed in the medium.For the bacteria to grow,the medium must be 13% salt.Nutrient solutions x,y,and z cost $1,$2,and $3 per liter,respectively.How much of each nutrient solution should be used to minimize the cost of the culture medium?
​

Accepted Answer

A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​   
A)  ​   
B)  ​   
C)  ​   
D)  ​   
E)  ​

Question 20

Use Lagrange multipliers to maximize the function   subject to the following constraint:   ​ Assume that x,y,and z are positive.
​

Accepted Answer

A)  110 
B)    
C)    
D)  10 
E)  no absolute maximum 
A)  110 
B)    
C)    
D)  10 
E)  no absolute maximum

Use Lagrange multipliers to find the given extremum.In each case,assume that x and y are positive. Maximize Constraint

Sketch the yz-trace of the equation:

Use a double integral to find the volume of the indicated solid.

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points Round your answer to three decimal places.

Find the four second partial derivatives.Observe that the second mixed partials are equal.

A manufacturer has an order for 1,100 units of fine paper that can be produced at two locations.Let x₁ and x₂ be the numbers of units produced at the two plants.Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by

A company manufactures two types of sneakers: running shoes and basketball shoes.The total revenue from x₁ units of running shoes and x₂ units of basketball shoes is: , Where x₁ and x₂ are in thousands of units.Find x₁ and x₂ so as to maximize the revenue.

Find the standard equation of the sphere whose center is and whose radius is 5.

Evaluate and for the function at the point

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. A linear regression model with a positive correlation will have a slope that is greater than 0.

Examine the function given below for relative extrema and saddle points.

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model.The cost function for producing x freestanding and y fireplace-insert stoves is Find the marginal costs ( and )when and Round your answers to two decimal places.

Sketch the graph of the plane given by

For ,find all values of x and y such that and simultaneously.

Use a double integral to find the volume of the solid bounded by the graphs of the equations.

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points.

Sketch the region of integration

Sketch the region R whose area is given by the following double integral.

Use Lagrange multipliers to maximize the function subject to the following constraint: Assume that x,y,and z are positive.

Functions, graphs, and Limits

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Trigonometric Functions

Probability and Calculus

Series and Taylor Polynomials

Differential Equations

Filters

Exam 7: Functions of Several Variables

Use Lagrange multipliers to find the given extremum.In each case,assume that ​x and ​y are positive. ​ Maximize Constraint ​

Sketch the yz-trace of the equation: ​ ​

Use a double integral to find the volume of the indicated solid. ​ ​ ​

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points Round your answer to three decimal places. ​

Find the four second partial derivatives.Observe that the second mixed partials are equal. ​ ​

A manufacturer has an order for 1,100 units of fine paper that can be produced at two locations.Let x​1 and x​2 be the numbers of units produced at the two plants.Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by ​

A company manufactures two types of sneakers: running shoes and basketball shoes.The total revenue from x1 units of running shoes and x2 units of basketball shoes is: ​ , ​ Where x1 and x2 are in thousands of units.Find x1 and x2 so as to maximize the revenue. ​

Find the standard equation of the sphere whose center is and whose radius is 5. ​

Evaluate and for the function at the point ​

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​A linear regression model with a positive correlation will have a slope that is greater than 0.

Examine the function given below for relative extrema and saddle points. ​ ​

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model.The cost function for producing x freestanding and y fireplace-insert stoves is Find the marginal costs ( and )when and Round your answers to two decimal places. ​

Sketch the graph of the plane given by ​

For ,find all values of x and y such that and simultaneously. ​ ​

Use a double integral to find the volume of the solid bounded by the graphs of the equations. ​ ​

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. ​ ​

Sketch the region of integration ​

Sketch the region R whose area is given by the following double integral. ​ ​

Use Lagrange multipliers to maximize the function subject to the following constraint: ​ Assume that x,y,and z are positive. ​

Functions, graphs, and Limits

Differentiation

Applications of the Derivative

Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Trigonometric Functions

Probability and Calculus

Series and Taylor Polynomials

Differential Equations

Filters

Use Lagrange multipliers to find the given extremum.In each case,assume that x and y are positive. Maximize Constraint

Sketch the yz-trace of the equation:

Use a double integral to find the volume of the indicated solid.

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points Round your answer to three decimal places.

Find the four second partial derivatives.Observe that the second mixed partials are equal.

A manufacturer has an order for 1,100 units of fine paper that can be produced at two locations.Let x₁ and x₂ be the numbers of units produced at the two plants.Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by

A company manufactures two types of sneakers: running shoes and basketball shoes.The total revenue from x₁ units of running shoes and x₂ units of basketball shoes is: , Where x₁ and x₂ are in thousands of units.Find x₁ and x₂ so as to maximize the revenue.

Find the standard equation of the sphere whose center is and whose radius is 5.

Evaluate and for the function at the point

Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. A linear regression model with a positive correlation will have a slope that is greater than 0.

Examine the function given below for relative extrema and saddle points.

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model.The cost function for producing x freestanding and y fireplace-insert stoves is Find the marginal costs ( and )when and Round your answers to two decimal places.

Sketch the graph of the plane given by

For ,find all values of x and y such that and simultaneously.

Use a double integral to find the volume of the solid bounded by the graphs of the equations.

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points.

Sketch the region of integration

Sketch the region R whose area is given by the following double integral.

Use Lagrange multipliers to maximize the function subject to the following constraint: Assume that x,y,and z are positive.