Deck 14: Functions of Several Variables and Partial Differentiation

Question 1

Compute the indicated function value.

A) -53
B) -29
C) -45
D) -98

Answer 1

-53

Question 2

Describe the range of the function.

A)
B)
C)
D)

Answer 2

Question 3

Which of the following is the contour plot of ?

A)
B)
C)
D)

Answer 3

Question 4

Sketch the indicated traces of .

Question 5

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 6

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 . Temperature (degrees Fahrenheit)
Wind speed
(mph)

A)
B)
C)
D)

Question 7

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 8

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 9

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

A)
B)
C)
D)

Question 10

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 11

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 12

Sketch the indicated traces of .

Question 13

Compute the indicated function value.

A)
B)
C) -2
D) undefined

Question 14

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

A)
B)
C)
D)

Question 15

Which of the following is the contour plot of ?

A)
B)
C)
D)

Question 16

Describe the range of the function.

A)
B)
C)
D)

Question 17

Describe the range of the function.

A)
B)
C)
D)

Question 18

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

A)
B)
C)
D)

Question 19

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 30 mph, how much change in wind chill can be expected if the wind speed increases by 10 mph? Temperature (degrees Fahrenheit)
Wind speed
(mph)

A)
B)
C)
D)

Question 20

Describe and sketch the domain of the function.

A)
B)
C)
D)

Question 21

Show that the limit does not exist.

Question 22

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 23

Match the surface to its density plot.

A)
B)
C)
D)

Question 24

Determine all points at which the given function is continuous.

A)
B)
C)
D)

Question 25

Show that the limit does not exist.

Question 26

Show that the limit does not exist.

Question 27

Show that the indicated limit exists.

Question 28

Show that the indicated limit exists.

Question 29

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) 44.2 atm
C) 14.3 atm
D) 15.5 atm

Question 30

Compute the indicated limit.

A)
B)
C) 1
D) undefined

Question 31

Match the surface to its contour plot.

A)
B)
C)
D)

Question 32

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 33

Determine all points at which the given function is continuous.

A)
B)
C)
D)

Question 34

Determine all points at which the given function is continuous.

A)
B)
C)
D)

Question 35

Compute the indicated limit.

A) undefined
B)
C) 0
D)

Question 36

Match the surface to its density plot.

A)
B)
C)
D)

Question 37

Match the surface to its contour plot.

A)
B)
C)
D)

Question 38

Show that the indicated limit exists.

Question 39

Match the surface to its contour plot.

A)
B)
C)
D)

Question 40

Match the surface to its density plot.

A)
B)
C)
D)

Question 41

Find the indicated partial derivative.

A)
B)
C)
D)

Question 42

Find a function with the given properties.

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

Use the contour plot to estimate and at the origin.

A)
B)
C)
D)

Question 45

Find the indicated partial derivative.

A)
B)
C)
D) 0

Question 46

Find the indicated partial derivative.

A) 0
B) 6w²z
C) 12wz
D) 3w²

Question 47

Find all first-order partial derivatives.

A)
B)
C)
D)

Question 48

Use the contour plot to estimate and at the origin.

A)
B)
C)
D)

Question 49

Show that the function satisfies the diffusion equation .

Question 50

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 3205 to 3355 rad/s. [Note: you should be able to answer the question with the given information: no additional information is necessary.]

A) 9%
B) 3%
C) 0.02%
D) 5%

Question 51

Find the indicated partial derivatives.

Question 52

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

A)
B)
C)
D)

Question 53

Use the contour plot to estimate and at the origin.

A)
B)
C)
D)

Question 54

Find all first-order partial derivatives.

A)
B)
C)
D)

Question 55

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

A)
B)
C)
D)

Question 56

Find all points at which .

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

Question 57

Find all points at which .

A)
B)
C)
D)

Question 58

Find all first-order partial derivatives.

A)
B)
C)
D)

Question 59

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

A)
B)
C)
D)

Question 60

Find all first-order partial derivatives.

Question 61

Find the total differential of .

A)
B)
C)
D)

Question 62

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

A)
B)
C)
D)

Question 63

Use implicit differentiation to find .

A)
B)
C)
D)

Question 64

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

A)
B)
C)
D)

Question 65

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 90 and capital is growing at a rate of $200,000 per year. Determine the rate of change of production.

A) Production is increasing at a rate of 13.27 units per year.
B) Production is increasing at a rate of 2.13 units per year.
C) Production is increasing at a rate of 10.15 units per year.
D) Production is increasing at a rate of 21.29 units per year.

Question 66

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

A)
B)
C)
D)

Question 67

Find the total differential of .

A)
B)
C)
D)

Question 68

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

Question 69

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

Question 70

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

A)
B)
C)
D)

Question 71

Use implicit differentiation to find .

A)
B)
C)
D)

Question 72

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 73

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 = 296.6 $\pm$ 0.01 K, P = 1.05 $\pm$ 0.01 atm, and V = 1.302 $\pm$ 0.002 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 74

Find an equation of the tangent plane to the parametric surface at the indicated point. S is defined by ; at u = -4 and v = -1

A)
B)
C)
D)

Question 75

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

A)
B)
C)
D)

Question 76

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 77

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

A)
B)
C)
D)

Question 78

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

A)
B)
C)
D)

Question 79

Find the total differential of .

A)
B)
C)
D)

Question 80

Find the gradient of the given function.

A)
B)
C)
D)