Deck 13: Functions of Several Variables and Partial Differentiation

Question 1

Use a graphing utility to sketch the graph of . Use the viewpoint and (see figure).

A)
B)
C)
D)

Answer 1

B

Question 2

Compute the indicated function value.

A)
B)
C) 1
D) undefined

Answer 2

C

Question 3

Describe and sketch the domain of the function.

A)
B)
C)
D)

Answer 3

B

Question 4

Sketch the indicated traces of .

Question 5

Describe the range of the function.

A)
B)
C)
D)

Question 6

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 7

Describe the range of the function.

A)
B)
C)
D)

Question 8

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. For the function , find .

A)
B)
C)
D)

Question 9

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 10

Use a graphing utility to sketch the graph of . Use the viewpoint and (see figure).

A)
B)
C)
D)

Question 11

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 12

Sketch the indicated traces of .

Question 13

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 14

Use a graphing utility to sketch the graph of . Use the viewpoint and (see figure).

A)
B)
C)
D)

Question 15

Compute the indicated function value.

A) -480
B) -416
C) -484
D) -928

Question 16

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?

A)
B)
C)
D)

Question 17

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 18

Describe the range of the function.

A)
B)
C)
D)

Question 19

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 20

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 21

Match the surface to its density plot.

A)
B)
C)
D)

Question 22

Match the surface to its density plot.

A)
B)
C)
D)

Question 23

Match the surface to its contour plot.

A)
B)
C)
D)

Question 24

Show that the limit does not exist.

Question 25

Show that the limit does not exist.

Question 26

Show that the indicated limit exists.

Question 27

Match the surface to its contour plot.

A)
B)
C)
D)

Question 28

Determine all points at which the given function is continuous.

A)
B)
C)
D)

Question 29

Show that the indicated limit exists.

Question 30

Determine all points at which the given function is continuous.

A)
B)
C)
D)

Question 31

Use polar coordinates to find the indicated limit, if it exists. Note that is equivalent to .

A) 0
B) 1
C) -1
D) The limit is undefined.

Question 32

Match the surface to its contour plot.

A)
B)
C)
D)

Question 33

Compute the indicated limit.

A)
B)
C) 1
D) undefined

Question 34

Match the surface to its density plot.

A)
B)
C)
D)

Question 35

Show that the indicated limit exists.

Question 36

Use polar coordinates to find the indicated limit, if it exists. Note that is equivalent to .

A) 0
B) 1
C) -1
D) The limit is undefined.

Question 37

Show that the limit does not exist.

Question 38

Ammonia (NH₃) is a gas used for manufacturing fertilizers as well as urethane foams and coatings. A particular gas cylinder contains 17.03 grams of ammonia. The pressure P (in atmospheres (atm)), volume V (in liters (L)), and temperature T (in Kelvin (K)) of the sample are related by the equation . For the function , compute .

A) 0.0 atm
B) 45.5 atm
C) 16.5 atm
D) 17.9 atm

Question 39

Compute the indicated limit.

A) undefined
B)
C) 0
D)

Question 40

Determine all points at which the given function is continuous.

A)
B)
C)
D)

Question 41

Find all first-order partial derivatives.

A)
B)
C)
D)

Question 42

Find all points at which .

A)
B)
C)
D)

Question 43

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 .

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

Question 44

Find all first-order partial derivatives.

A)
B)
C)
D)

Question 45

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

A)
B)
C)
D)

Question 46

Find the indicated partial derivative.

A)
B)
C)
D) 0

Question 47

Show that the function satisfies the diffusion equation .

Question 48

Find a function with the given properties.

A)
B)
C)
D)

Question 49

Use the contour plot to estimate and at the origin.

A)
B)
C)
D)

Question 50

Find all points at which .

A)
B)
C)
D) (0, 0)

Question 51

Find all first-order partial derivatives.

A)
B)
C)
D)

Question 52

Find the indicated partial derivative.

A) 0
B) 6w²z
C) 12wz
D) 3w²

Question 53

Use the contour plot to estimate and at the origin.

A)
B)
C)
D)

Question 54

Solid bodies, such as mechanical flywheels, experience stress when rotating. The stress s on a flywheel with uniform density d (kg/m³) and radius r (meters) is approximated by where is the rotational speed of the flywheel in radians per second. Estimate the percent increase in the stress if the rotational speed of a uniform flywheel is increased from 3917 to 4067 rad/s. [Note: you should be able to answer the question with the given information: no additional information is necessary.]

A) 8%
B) 3%
C) 0.01%
D) 4%

Question 55

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

A)
B)
C)
D)

Question 56

Find the indicated partial derivatives.

Question 57

Find all first-order partial derivatives.

Question 58

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

A)
B)
C)
D)

Question 59

Use the contour plot to estimate and at the origin.

A)
B)
C)
D)

Question 60

Find the indicated partial derivative.

A)
B)
C)
D)

Question 61

Use the chain rule to find the indicated derivative. , where

A)
B)
C)
D)

Question 62

Two electrical resistors with resistances R₁ and R₂ (in ohms) are wired into an electronics circuit as shown in the figure below. The pair of resistors behaves as a single resistor R_T whose resistance is given by . Once the circuit is energized, the temperatures of the resistors increase, causing the resistances to increase according to: , , where t is in seconds. Find .

A)
B)
C)
D)

Question 63

Use the given contour plot to estimate the linear approximation of at . Assume each contour line represents a 1 unit increase from left to right with the left-most contour line corresponding to a value of 1.

A)
B)
C)
D)

Question 64

Find the gradient of the given function.

A)
B)
C)
D)

Question 65

Use implicit differentiation to find .

A)
B)
C)
D)

Question 66

Compute the linear approximation of the function at the given point. at

A)
B)
C)
D)

Question 67

Find the total differential of .

A)
B)
C)
D)

Question 68

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

A)
B)
C)
D)

Question 69

State the chain rule for finding for the general composite function involving

A)
B)
C)
D)

Question 70

Find an equation of the tangent plane to the parametric surface at the indicated point. S is defined by ; at u = -4 and v = -2

A)
B)
C)
D)

Question 71

Compute the linear approximation of the function at the given point.

A)
B)
C)
D)

Question 72

Use the given contour plot to estimate the linear approximation of at .

A)
B)
C)
D)

Question 73

Use the chain rule twice to find for the general composite function involving

Question 74

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

Question 75

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.

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

Question 76

Find the total differential of .

A)
B)
C)
D)

Question 77

Use the chain rule to find the indicated derivative. , where

A)
B)
C)
D)

Question 78

Use implicit differentiation to find .

A)
B)
C)
D)

Question 79

Find the total differential of .

A)
B)
C)
D)

Question 80

Suppose the production of a firm is modeled by where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that and the labor force is increasing at the rate of 40 and capital is growing at a rate of $900,000 per year. Determine the rate of change of production.

A) Production is increasing at a rate of 71.67 units per year.
B) Production is increasing at a rate of 7.64 units per year.
C) Production is increasing at a rate of 12.38 units per year.
D) Production is increasing at a rate of 76.41 units per year.