Question 1

Compute the directional derivative of f at the given point in the direction of the indicated vector.

Accepted Answer

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

Question 2

Describe and sketch the domain of the function.

Accepted Answer

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

Question 3

Wind chill is a combination of temperature and wind speed that expresses how cold the air feels. The larger the wind speed, the more rapidly heat is lost to the air and thus the colder it feels. The table shows the wind chill for selected temperatures and wind speeds. At   and a wind speed of 10 mph, how much change in wind chill can be expected if the wind speed increases by 10 mph?

Accepted Answer

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

Question 4

Use the contour plot to estimate   and   at the origin.

Accepted Answer

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

Question 5

Find all first-order partial derivatives.

Accepted Answer

The answer of Find all first-order partial derivatives.  ...

Question 6

Use implicit differentiation to find   and   Assume that the equation defines z as a differentiable function near each

Accepted Answer

The answer of Use implicit differentiation to find  ...

Question 7

Sketch the indicated traces of   .

Accepted Answer

The answer of Sketch the indicated traces of  ...

Question 8

The number n of gas particles in a leak-tight container is approximately described by the equation   , where T is the temperature of the gas in kelvins (K), P is the pressure of the gas in atmospheres (atm), and V is the volume of the container in liters (L). For a particular sample, an experimenter finds T = 292.9 $ \pm $ 0.01 K, P = 0.68 $ \pm $ 0.02 atm, and V = 1.731 $ \pm $ 0.005 L. Use a linear approximation to estimate the range of the computed value of n.

Accepted Answer

A)  0 to   
B)    to   
C)    to   
D)    to   
A)  0 to   
B)    to   
C)    to   
D)    to

Question 9

Sketch the path of steepest ascent from the indicated point.

Accepted Answer

Path is ap

Question 10

Compute the directional derivative of   at   in the direction of

Accepted Answer

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

Question 11

Find an equation for the tangent plane at the given point.

Accepted Answer

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

Question 12

Write out the third-order Taylor polynomial for   about (0,0).

Accepted Answer

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

Question 13

Use Lagrange multipliers to find the maximum and minimum of the function   subject to the constraint   .

Accepted Answer

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

Question 14

Describe and sketch the domain of the function.

Accepted Answer

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

Question 15

Which of the following is the contour plot of   ?

Accepted Answer

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

Question 16

The pressure P of a gas in a leak-tight container can be approximated by   , where V is the volume of the container, T is its absolute (Kelvin) temperature, and k is a constant. Find   and   .

Accepted Answer

A)    ,   
B)    ,   
C)    ,   
D)    ,   
A)    ,   
B)    ,   
C)    ,   
D)    ,

Question 17

Find all first-order partial derivatives.

Accepted Answer

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

Question 18

Which of the following is the contour plot of   ?

Accepted Answer

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

Question 19

Match the surface to its contour plot.

Accepted Answer

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

Question 20

Show that the indicated limit exists.

Accepted Answer

Answers may vary.
Find the lim

Compute the directional derivative of f at the given point in the direction of the indicated vector.

Describe and sketch the domain of the function.

Use the contour plot to estimate and at the origin.

Find all first-order partial derivatives.

Use implicit differentiation to find and Assume that the equation defines z as a differentiable function near each

Sketch the indicated traces of .

Sketch the path of steepest ascent from the indicated point.

Compute the directional derivative of at in the direction of

Find an equation for the tangent plane at the given point.

Write out the third-order Taylor polynomial for about (0,0).

Use Lagrange multipliers to find the maximum and minimum of the function subject to the constraint .

Describe and sketch the domain of the function.

Which of the following is the contour plot of ?

The pressure P of a gas in a leak-tight container can be approximated by , where V is the volume of the container, T is its absolute (Kelvin) temperature, and k is a constant. Find and .

Find all first-order partial derivatives.

Which of the following is the contour plot of ?

Match the surface to its contour plot.

Show that the indicated limit exists.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 13: Functions of Several Variables and Partial Differentiation

Compute the directional derivative of f at the given point in the direction of the indicated vector.

Describe and sketch the domain of the function.

Use the contour plot to estimate and at the origin.

Find all first-order partial derivatives.

Use implicit differentiation to find and Assume that the equation defines z as a differentiable function near each

Sketch the indicated traces of .

Sketch the path of steepest ascent from the indicated point.

Compute the directional derivative of at in the direction of

Find an equation for the tangent plane at the given point.

Write out the third-order Taylor polynomial for about (0,0).

Use Lagrange multipliers to find the maximum and minimum of the function subject to the constraint .

Describe and sketch the domain of the function.

Which of the following is the contour plot of ?

The pressure P of a gas in a leak-tight container can be approximated by , where V is the volume of the container, T is its absolute (Kelvin) temperature, and k is a constant. Find and .

Find all first-order partial derivatives.

Which of the following is the contour plot of ?

Match the surface to its contour plot.

Show that the indicated limit exists.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters