Question 1

Find   using the appropriate Chain Rule for   where   , and   . ​

Accepted Answer

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

Question 2

Use Lagrange multipliers to find the minimum distance from the plane   to the point   . Round your answer to two decimal places.

Accepted Answer

A)  2.45 
B)  6.56 
C)  6.00 
D)  1.73 
E)  3.00 
A)  2.45 
B)  6.56 
C)  6.00 
D)  1.73 
E)  3.00 D

Question 3

Find a unit normal vector to the surface   at the point   .

Accepted Answer

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

Question 4

Find the second partial derivative for the function   with respect to x.

Accepted Answer

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

Question 5

Use Lagrange multipliers to minimize the function   subject to the following two constraints.     Assume that x, y, and z are nonnegative.

Accepted Answer

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

Question 6

Find symmetric equations of the normal line to the surface   at the point   .

Accepted Answer

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

Question 7

Examine the function   for relative extrema and saddle points.

Accepted Answer

A)  saddle point:   
B)  relative minimum:   
C)  relative minimum:   
D)  saddle point:   
E)  saddle point:   
A)  saddle point:   
B)  relative minimum:   
C)  relative minimum:   
D)  saddle point:   
E)  saddle point:

Question 8

Use Lagrange multipliers to find the maximum value of   where   and   subject to the constraint   . ​

Accepted Answer

A)  maxima:   ; minima:  
​ 
B)  maxima:   ; minima:  
​ 
C)  maxima:   ; minima:  
​ 
D)  maxima:   ; minima:  
​ 
E)  maxima:   ; minima:  
​ 
A)  maxima:   ; minima:  
​ 
B)  maxima:   ; minima:  
​ 
C)  maxima:   ; minima:  
​ 
D)  maxima:   ; minima:  
​ 
E)  maxima:   ; minima:  
​

Question 9

Determine the continuity of the function   .

Accepted Answer

A)  continuous except at   
B)  continuous for   
C)  continuous except at   
D)  continuous except at   
E)  continuous for   
A)  continuous except at   
B)  continuous for   
C)  continuous except at   
D)  continuous except at   
E)  continuous for

Question 10

Find the critical points of the function   , and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Accepted Answer

A)  relative maxima at   ,   ,   , where   and   are arbitrary real numbers 
B)  relative minima at   ,   ,   , where   and   are arbitrary real numbers 
C)  relative minimum at   
D)  relative maximum at   
E)  no relative extrema 
A)  relative maxima at   ,   ,   , where   and   are arbitrary real numbers 
B)  relative minima at   ,   ,   , where   and   are arbitrary real numbers 
C)  relative minimum at   
D)  relative maximum at   
E)  no relative extrema

Question 11

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

Accepted Answer

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

Question 12

​Find the partial derivative   for the function   . ​

Accepted Answer

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

Question 13

Find the maximum value of the directional derivative at the point   of the function   . Round your answer to two decimal places.

Accepted Answer

A)  0.32 
B)  0.14 
C)  0.33 
D)  0.10 
E)  0.04 
A)  0.32 
B)  0.14 
C)  0.33 
D)  0.10 
E)  0.04

Question 14

Determine the continuity of the function   .

Accepted Answer

A)  continuous except at   
B)  continuous except at   
C)  continuous for all   
D)  continuous for all   
E)  continuous everywhere 
A)  continuous except at   
B)  continuous except at   
C)  continuous for all   
D)  continuous for all   
E)  continuous everywhere

Question 15

Use Lagrange multipliers to find the minimum distance from the parabola   to the point   . Round your answer to two decimal places. ​

Accepted Answer

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

Question 16

Find the minimum cost of producing 60,000 units of a product   , where x is the number of units of labor (at $80 per unit) and y is the number of units of capital (at $40 per unit). Round your answer to the nearest cent.

Accepted Answer

A)  $190,278.22 
B)  $62,746.50 
C)  $37,020.43 
D)  $112,264.15 
E)  $55,724.88 
A)  $190,278.22 
B)  $62,746.50 
C)  $37,020.43 
D)  $112,264.15 
E)  $55,724.88

Question 17

Determine the continuity of the function   .

Accepted Answer

A)  continuous except at   
B)  continuous except at   
C)  continuous except at   
D)  continuous except at   
E)  continuous everywhere 
A)  continuous except at   
B)  continuous except at   
C)  continuous except at   
D)  continuous except at   
E)  continuous everywhere

Question 18

Use Lagrange multipliers to find the maximum value of   where   and   , subject to the constraint   . ​

Accepted Answer

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

Question 19

Suppose a corporation manufactures candles at two locations. The cost of producing   units at location 1 is   and the cost of producing   units at location 2 is   . The candles sell for $12 per unit. Find the quantity that should be produced at each location to maximize the profit   . ​

Accepted Answer

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

Question 20

The radius of a right circular cylinder is increasing at a rate of 8 inches per minute, and the height is decreasing at a rate of 6 inches per minute. What is the rate of change of the surface area when the radius is 16 inches and the height is 37 inches?

Accepted Answer

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

Find using the appropriate Chain Rule for where , and .

Use Lagrange multipliers to find the minimum distance from the plane to the point . Round your answer to two decimal places.

Find a unit normal vector to the surface at the point .

Find the second partial derivative for the function with respect to x.

Use Lagrange multipliers to minimize the function subject to the following two constraints. Assume that x, y, and z are nonnegative.

Find symmetric equations of the normal line to the surface at the point .

Examine the function for relative extrema and saddle points.

Use Lagrange multipliers to find the maximum value of where and subject to the constraint .

Determine the continuity of the function .

Find the critical points of the function , and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

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

Find the partial derivative for the function .

Find the maximum value of the directional derivative at the point of the function . Round your answer to two decimal places.

Determine the continuity of the function .

Use Lagrange multipliers to find the minimum distance from the parabola to the point . Round your answer to two decimal places.

Find the minimum cost of producing 60,000 units of a product , where x is the number of units of labor (at $80 per unit) and y is the number of units of capital (at $40 per unit). Round your answer to the nearest cent.

Determine the continuity of the function .

Use Lagrange multipliers to find the maximum value of where and , subject to the constraint .

Suppose a corporation manufactures candles at two locations. The cost of producing units at location 1 is and the cost of producing units at location 2 is . The candles sell for $12 per unit. Find the quantity that should be produced at each location to maximize the profit .

The radius of a right circular cylinder is increasing at a rate of 8 inches per minute, and the height is decreasing at a rate of 6 inches per minute. What is the rate of change of the surface area when the radius is 16 inches and the height is 37 inches?

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Exam 13: Functions of Several Variables

Find using the appropriate Chain Rule for where , and . ​

Use Lagrange multipliers to find the minimum distance from the plane to the point . Round your answer to two decimal places.

Find a unit normal vector to the surface at the point .

Find the second partial derivative for the function with respect to x.

Use Lagrange multipliers to minimize the function subject to the following two constraints. Assume that x, y, and z are nonnegative.

Find symmetric equations of the normal line to the surface at the point .

Examine the function for relative extrema and saddle points.

Use Lagrange multipliers to find the maximum value of where and subject to the constraint . ​

Determine the continuity of the function .

Find the critical points of the function , and from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

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

​Find the partial derivative for the function . ​

Find the maximum value of the directional derivative at the point of the function . Round your answer to two decimal places.

Determine the continuity of the function .

Use Lagrange multipliers to find the minimum distance from the parabola to the point . Round your answer to two decimal places. ​

Find the minimum cost of producing 60,000 units of a product , where x is the number of units of labor (at $80 per unit) and y is the number of units of capital (at $40 per unit). Round your answer to the nearest cent.

Determine the continuity of the function .

Use Lagrange multipliers to find the maximum value of where and , subject to the constraint . ​

Suppose a corporation manufactures candles at two locations. The cost of producing units at location 1 is and the cost of producing units at location 2 is . The candles sell for $12 per unit. Find the quantity that should be produced at each location to maximize the profit . ​

The radius of a right circular cylinder is increasing at a rate of 8 inches per minute, and the height is decreasing at a rate of 6 inches per minute. What is the rate of change of the surface area when the radius is 16 inches and the height is 37 inches?

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Find using the appropriate Chain Rule for where , and .

Use Lagrange multipliers to find the maximum value of where and subject to the constraint .

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

Find the partial derivative for the function .

Use Lagrange multipliers to find the minimum distance from the parabola to the point . Round your answer to two decimal places.

Use Lagrange multipliers to find the maximum value of where and , subject to the constraint .

Suppose a corporation manufactures candles at two locations. The cost of producing units at location 1 is and the cost of producing units at location 2 is . The candles sell for $12 per unit. Find the quantity that should be produced at each location to maximize the profit .