Deck 15: Functions of Several Variables

Question 1

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) = 4x + 3y

A) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c, c > 0
B) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c
C) Domain: all points in the xy-plane; range: real numbers z $\le$ 0; level curves: lines 4x + 3y = c, c $\le$ 0
D) Domain: all points in the xy-plane; range: real numbers z $\le$ 0 ; level curves: lines 4x + 3y = c, c $\le$ 0

Answer 1

Domain: all points in the xy-plane; range: all real numbers; level curves: lines 4x + 3y = c, c > 0

Question 2

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) =

A) Domain: all points in the xy-plane; range: real numbers z ≤ 0; level curves: lines 8x - 10y = c, c ≤ 0

B) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: lines 8x - 10y = c

C) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 8x - 10y = c, c ≥ 0

D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 8x - 10y = c

Answer 2

Domain: all points in the xy-plane; range: all real numbers; level curves: lines 8x - 10y = c

Question 3

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) =

A) Domain: all points in the xy-plane; range: real numbers > 0; level curves: ellipses 3x² + 10y² = c

B) Domain: all points in the xy-plane except (0, 0); range: real numbers > 0; level curves: ellipses 3x² + 10y² = c

C) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses 3x² + 10y² = c

D) Domain: all points in the xy-plane except (0, 0); range: all real numbers; level curves: ellipses 3x² + 10y² = c

Answer 3

Domain: all points in the xy-plane except (0, 0); range: real numbers > 0; level curves: ellipses 3x² + 10y² = c

Question 4

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) =

A) Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas y= cx² - 6

B) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: parabolas y= cx² - 6

C) Domain: all points in the xy-plane excluding x = 0; range: all real numbers; level curves: parabolas y= cx² - 6

D) Domain: all points in the xy-plane excluding x = 0; range: real numbers z ≥ 0; level curves: parabolas y= cx² - 6

Question 5

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) = ln ( 7x + 10y)

A) Domain: all points in the xy-plane; range: all real numbers; level curves: lines 7x + 10y = c

B) Domain: all points in the xy-plane satisfying 7x + 10y > 0; range: all real numbers; level curves: lines 7x + 10y = c

C) Domain: all points in the xy-plane satisfying 7x + 10y ≥ 0; range: all real numbers; level curves: lines 7x + 10y = c

D) Domain: all points in the xy-plane satisfying 7x + 10y > 0; range: real numbers z ≥ 0; level curves: 7x + 10y = c lines

Question 6

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) = ( + )

A) Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at (0, 0)

B) Domain: all points in the xy-plane satisfying x² + y² ≤ 1; range: real numbers ≤ z ≤ ; level curves: circles with centers at (0, 0)

C) Domain: all points in the xy-plane satisfying x² + y² ≤ 1; range: real numbers - 11efabfd_aa12_34a1_af07_d19efc43c81a_TB9662_00 ≤ z ≤11efabfd_aa12_34a1_af07_d19efc43c81a_TB9662_00 ; level curves: circles with centers at (0,0) and radii r, 0 < r ≤ 1

D) Domain: all points in the xy-plane; range: real numbers - 11efabfd_aa12_34a1_af07_d19efc43c81a_TB9662_00 ≤ z ≤ 11efabfd_aa12_34a1_af07_d19efc43c81a_TB9662_00 ; level curves: circles with centers at (0, 0)

Question 7

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) =

A) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: lines x + y = c

B) Domain: all points in the first quadrant of the xy-plane; range: real numbers z > 0; level curves: lines x + y = c

C) Domain: all points in the xy-plane; range: real numbers z > 0; level curves: lines x + y = c

D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines x + y = c

Question 8

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) =

A) Domain: all points in the xy-plane satisfying x² + y² = 100; range: real numbers 0 ≤ z ≤ 10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 10

B) Domain: all points in the xy-plane satisfying x² + y² ≤ 100; range: real numbers 0 ≤ z ≤ 10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 10

C) Domain: all points in the xy-plane; range: real numbers 0 ≤ z ≤ 10; level curves: circles with centers at (0, 0) and radii r, 0 < r ≤ 10

D) Domain: all points in the xy-plane; range: all real numbers; level curves: circles with centers at (0,0)

Question 9

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) =

A) Domain: all points in the xy-plane except y = 0; range: all real numbers; level curves: parabolas y = cx²

B) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: parabolas y = cx²

C) Domain: all points in the xy-plane except y = 0; range: real numbers z ≥ 0 ; level curves: parabolas y = cx²

D) Domain: all points in the xy-plane; range: all real numbers; level curves: parabolas y = cx²

Question 10

Find the domain and range and describe the level curves for the function f(x,y).

-f(x, y) = 3 - 5

A) Domain: all points in the xy-plane; range: all real numbers; level curves: hyperbolas

B) Domain: all points in the xy-plane; range: real numbers z ≥ 0; level curves: ellipses

C) Domain: all points in the xy-plane; range: all real numbers; level curves: ellipses 11efabfe_b7a2_fea3_af07_277d2671e3df_TB9662_00

D) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: hyperbolas 11efabfe_a41f_4462_af07_c3f2d3c3a269_TB9662_00

Question 11

Sketch the surface z = f(x,y).

-f(x, y) =

A)

B)

C)

D)

Question 12

Sketch the surface z = f(x,y).

-f(x, y) = - -

A)

B)

C)

D)

Question 13

Sketch the surface z = f(x,y).

-f(x, y) = 3 -

A)

B)

C)

D)

Question 14

Sketch the surface z = f(x,y).

-f(x, y) = -

A)

B)

C)

D)

Question 15

Sketch the surface z = f(x,y).

-f(x, y) =

A)

B)

C)

D)

Question 16

Sketch the surface z = f(x,y).

-f(x, y) =

A)

B)

C)

D)

Question 17

Sketch the surface z = f(x,y).

-f(x, y) = 4 + 4 + 2

A)

B)

C)

D)

Question 18

Sketch the surface z = f(x,y).

-f(x, y) = 1 -

A)

B)

C)

D)

Question 19

Sketch the surface z = f(x,y).

-f(x, y) = 2 - -

A)

B)

C)

D)

Question 20

Sketch the surface z = f(x,y).

-f(x, y) = 1 - x - 2y

A)

B)

C)

D)

Question 21

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 22

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 23

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 24

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 25

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 26

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 27

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 28

Match the surface show below to the graph of its level curves.

-

A)

B)

C)

D)

Question 29

Solve the problem.

-According to the Gas Kinetic Theory, the average speed of a gas particle is given by where
is the speed in m/s, k is the constant 1.38 × 10^-23 , T is the temperature of the gas in Kelvin, and m is the mass of the gas particle in kg. What is the average speed of an oxygen molecule with a mass of
5.314 X 10^-26 KG at a temperature of 500 K?

A) 1020 m/s
B) 575 m/s
C) 330,700 m/s
D) 174 m/s

Question 30

Write the word or phrase that best completes each statement or answers the question.

-The Ideal Gas Law states that the pressure P of a gas obeys the function where n is the number of particles in the sample, T is the Kelvin temperature of the gas, V is the volume of the gas in liters, and k is the constant 1.38 × 10^-23 . Does the pressure have a local minimum along the line If so, what is the value of the minimum pressure? Give reasons for your answer.

Question 31

Write the word or phrase that best completes each statement or answers the question.
-A sample of matter has a temperature distribution given by Describe the level surfaces and write an equation for the surface T = 300.

Question 32

Find the limit.

-

A) 2
B) 1
C) -1
D) No limit

Question 33

Find the limit.

-

A)
B)
C) 15
D) No limit

Question 34

Find the limit.

- ln

A) 0
B) ln 2
C) -ln 2
D) No limit

Question 35

Find the limit.

-

A) 1
B) $\infty$
C) 0
D) No limit

Question 36

Find the limit.

- sin

A) 1
B) 1/7
C) 0
D) No limit

Question 37

Find the limit.

-

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

Question 38

Find the limit.

-

A) -8
B) + 7
C) - - 7
D) 8

Question 39

Find the limit.

-

A) - 3/5
B) 12
C) 3/5
D) No limit

Question 40

Find the limit.

- x ln y

A) 4
B) ln 4
C) ln ( 4) - 1
D) No limit

Question 41

Find the limit.

-

A) 1
B)
C) 0
D)

Question 42

Find the limit.

-

A) 1
B) 0
C) 1/2
D) No limit

Question 43

Find the limit.

-

A) 13
B) 1
C) 5
D) 0

Question 44

Find the limit.

-

A) 0
B) 10
C) 5
D) No limit

Question 45

Find the limit.

-

A) 1/33
B) 17
C) 0
D) 33

Question 46

Find the limit.

-

A) 0
B) 2
C) 4
D) No limit

Question 47

Find the limit.

-

A) 3
B) -5
C) $\infty$
D) 0

Question 48

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 49

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 50

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 51

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 52

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 53

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 54

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 55

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0).
-f(x, y) =

Question 56

At what points is the given function continuous?

-f(x, y) =

A) All (x, y) $\neq$ (0, 0)
B) All (x, y) satisfying x + y $\neq$ 0
C) All (x, y)
D) All (x, y) in the first quadrant

Question 57

At what points is the given function continuous?

-f(x, y) =

A) All (x, y) such that x $\neq$ y
B) All (x, y) $\neq$ (0, 0)
C) All (x, y) such that x $\neq$ - y
D) All (x, y)

Question 58

At what points is the given function continuous?

-f(x, y) = tan (x + y)

A) All (x, y) $\neq$ (0, 0)
B) All (x, y) $\neq$
C) All (x, y) such that x + y $\neq$ , where n is an integer
D) All (x, y)

Question 59

At what points is the given function continuous?

-f(x, y) =

A)All (x, y) such that x $\neq$ and x $\neq$ -2
B) All (x, y)
C) All (x, y) such that x $\neq$ 0
D) All (x, y) satisfying x - y $\neq$ 0

Question 60

At what points is the given function continuous?

-f(x, y) =

A) All (x, y) such that 10x + 10y $\neq$ 0
B) All (x, y) such that x + y $\neq$ 0
C) All (x, y) such that 10x + 10y $\neq$ 0.
D) All (x, y)

Question 61

Find the limit.

- x - y + z

A) 8
B) 9
C) 6
D) 7

Question 62

Find the limit.

-

A) 2
B) 6
C) -6
D) -4

Question 63

Find the limit.

- ln (z )

A) ln 5
B) 0
C) ln
D) ln 25

Question 64

Find the limit.

-

A) 1
B) $\pi$
C) 0
D) $\pi$ /2

Question 65

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).

-f(x, y) =

A) 1
B) 2
C) 0
D) No limit

Question 66

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).

-f(x, y) =

A) 0
B) 2
C) 1
D) No limit

Question 67

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).

-f(x, y) = cos

A) $\pi$ /2
B) 0
C) 1
D) No limit

Question 68

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).

-f(x, y) =

A) $\pi$
B) 1
C) 0
D) -1

Question 69

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).

-f(x, y) =

A) $\pi$ /2
B) $\pi$
C) 1
D) No limit

Question 70

Use polar coordinates to find the limit of the function as (x, y) approaches (0, 0).

-f(x, y) =

A) 0
B) $\pi$
C) 1
D) No limit

Question 71

Find all the first order partial derivatives for the following function.

-f(x, y) = 10x - 3 - 6

A)

B)

C)

D)

Question 72

Find all the first order partial derivatives for the following function.

-f(x, y) =

A)

B)

C)

D)

Question 73

Find all the first order partial derivatives for the following function.

-f(x, y) =

A)
B)
C)
D)

Question 74

Find all the first order partial derivatives for the following function.

-f(x, y) = ln

A)

B)

C)

D)

Question 75

Find all the first order partial derivatives for the following function.

-f(x, y) = ( -3x - y)

A)
B)
C)
D)

Question 76

Find all the first order partial derivatives for the following function.

-f(x, y) = ln

A)

B)

C)

D)

Question 77

Find all the first order partial derivatives for the following function.

-f(x, y) =

A)

B)
C)
D)

Question 78

Find all the first order partial derivatives for the following function.

-f(x, y) =

A)

B)

C)

D)

Question 79

Find all the first order partial derivatives for the following function.

-f(x, y) = xy

A)

B)

C)

D)

Question 80

Find all the first order partial derivatives for the following function.

-f(x, y) = + 10 y - 2x

A)

B)

C)

D)