Question 1

Find the limit   for the function   . ​

Accepted Answer

A)  2x 
B)  -4 
C)  0 
D)  4x 
E)  4y 
A)  2x 
B)  -4 
C)  0 
D)  4x 
E)  4y

Question 2

Find the directional derivative of the function at P in the direction of   .

Accepted Answer

A)  30/   
B)  -10/   
C)  6/   
D)  46/   
E)  -46/   
A)  30/   
B)  -10/   
C)  6/   
D)  46/   
E)  -46/

Question 3

Find and simplify the function   at the given value   . ​

Accepted Answer

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

Question 4

For   , evaluate   at the point   . ​

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

A cargo container (in the shape of a rectangular solid) must have a volume of 520 cubic feet. The bottom will cost $7 per square foot to construct and the sides and the top will cost $3 per square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this volume that has minimum cost.

Accepted Answer

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

Question 7

Find the partial derivative   for the function   . ​

Accepted Answer

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

Question 8

Differentiate implicitly to find   . ​   ​

Accepted Answer

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

Question 9

Find the limit. ​   ​

Accepted Answer

A)    
B)    
C)  0 
D)    
E)    
A)    
B)    
C)  0 
D)    
E)

Question 10

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

Accepted Answer

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

Question 11

Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 288 cubic feet. The cost of wall paint is $0.06 per square foot and the cost of ceiling paint is $0.16 per square foot. Let x, y, and z be the length, width, and height of a rectangular room respectively. Identify the room dimensions that result in a minimum cost for the paint and use these dimensions to find the minimum cost for the paint. Round your answer to the nearest cent.

Accepted Answer

A)  $30.72 
B)  $7.68 
C)  $9.72 
D)  $17.28 
E)  $12.80 
A)  $30.72 
B)  $7.68 
C)  $9.72 
D)  $17.28 
E)  $12.80

Question 12

Find and simplify the function   at the given value   .

Accepted Answer

A)  -45 
B)  -18 
C)  45 
D)  18 
E)  0 
A)  -45 
B)  -18 
C)  45 
D)  18 
E)  0

Question 13

Suppose 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   when   and   . Round your answer to the nearest integer. ​

Accepted Answer

A)  181 
B)  221 
C)  191 
D)  372 
E)  373 
A)  181 
B)  221 
C)  191 
D)  372 
E)  373

Question 14

According to the Ideal Gas Law,   , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2,500 cubic inches of nitrogen at a pressure of 30 pounds per square inch and a temperature of 700 K. Determine k. ​

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

A)  193.87 
B)  252.00 
C)  262.08 
D)  283.46 
E)  230.51 
A)  193.87 
B)  252.00 
C)  262.08 
D)  283.46 
E)  230.51

Question 17

Find the minimum distance from the point   to the surface   .

Accepted Answer

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

Question 18

Find the limit . ​   ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)  0
​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)  0
​

Question 19

Use polar coordinates to find the limit

Accepted Answer

A)    
B)  0 
C)  1/2 
D)  10 
E)    
A)    
B)  0 
C)  1/2 
D)  10 
E)

Question 20

Describe the domain of the function   .

Accepted Answer

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

Find the limit for the function .

Find the directional derivative of the function at P in the direction of .

Find and simplify the function at the given value .

For , evaluate at the point .

Find using the appropriate Chain Rule for where , and .

Find the partial derivative for the function .

Differentiate implicitly to find .

Find the limit.

Examine the function for relative extrema and saddle points.

Find and simplify the function at the given value .

Suppose 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 when and . Round your answer to the nearest integer.

According to the Ideal Gas Law, , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2,500 cubic inches of nitrogen at a pressure of 30 pounds per square inch and a temperature of 700 K. Determine k.

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

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

Find the minimum distance from the point to the surface .

Find the limit .

Use polar coordinates to find the limit

Describe the domain of the function .

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 the limit for the function . ​

Find the directional derivative of the function at P in the direction of .

Find and simplify the function at the given value . ​

For , evaluate at the point . ​

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

Find the partial derivative for the function . ​

Differentiate implicitly to find . ​ ​

Find the limit. ​ ​

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

Find and simplify the function at the given value .

Suppose 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 when and . Round your answer to the nearest integer. ​

According to the Ideal Gas Law, , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2,500 cubic inches of nitrogen at a pressure of 30 pounds per square inch and a temperature of 700 K. Determine k. ​

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

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

Find the minimum distance from the point to the surface .

Find the limit . ​ ​

Use polar coordinates to find the limit

Describe the domain of the function .

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 the limit for the function .

Find and simplify the function at the given value .

For , evaluate at the point .

Find using the appropriate Chain Rule for where , and .

Find the partial derivative for the function .

Differentiate implicitly to find .

Find the limit.

Examine the function for relative extrema and saddle points.

Suppose 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 when and . Round your answer to the nearest integer.

According to the Ideal Gas Law, , where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2,500 cubic inches of nitrogen at a pressure of 30 pounds per square inch and a temperature of 700 K. Determine k.

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

Find the limit .