Deck 12: Functions of Several Variables

Question 1

Describe in words the intersection of the surfaces and

Answer 1

A circle (of radius 1)in the plane z = 1.

Question 2

Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in thousands of dollars), and the time, t, required to pay it back (in months).What is the meaning of C(7, 48)= 250?

A)If you borrow $7,000 from the bank for 48 months (4 year loan), your monthly car loan payment is $250.
B)If you borrow $4,000 from the bank for 48 months (7 year loan), your monthly car loan payment is $250.
C)If you borrow $250 from the bank for 48 months (4 year loan), your monthly car loan payment is $7.
D)If you borrow $7 from the bank for 48 months (4 year loan), your monthly car loan payment is $250.

Answer 2

If you borrow $7,000 from the bank for 48 months (4 year loan), your monthly car loan payment is $250.

Question 3

Match the graph with the function.

A) $z = - \sin x \sin y$
B) $z = 3 e ^ { - \frac { x } { 3 } } \sin y$
C) $z = \frac { 1 } { 2 } x ^ { 2 } + \sin ^ { 2 } y$
D) $z = \sin y$

Answer 3

$z = - \sin x \sin y$

Question 4

Yummy Potato Chip Company has manufacturing plants in N.Y.and N.J.The cost of manufacturing depends on the quantities (in thousand of bags), q₁ and q₂, produced in the N.Y.and N.J.factories respectively.Suppose the cost function is given by (a)Find (b)By comparing the terms and in the above expression, the manager concluded that it is more expensive to produce in the N.Y.factory.Will shifting all the production to the N.J.factory minimize the production cost?

Question 5

The points A = (-4, 5, -3), B = (-1, -3, -4), and C = (-2, 4, -4)are the vertices of a triangle in space.
What is the length of the longest side of the triangle?

Question 6

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C.Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment.Values of the function p = f(t, h)are given in the following table.(The higher the value of p, the better the performance.)
What is the value of p(0, 25)?

Question 7

Find a possible formula for a function f(x, y)with the given values.

Question 8

The following table gives the number f(x, y)of grape vines, in thousands, of age x in year y. In one year a fungal disease killed most of the older grapevines, and in the following year a long freeze killed most of the young vines.Which are these years?

Question 9

The following table gives the number f(x, y)of grape vines, in thousands, of age x in year y. In 1986 a successful advertising campaign led to a dramatic increase in demand for premium wines.The growers followed by adding many more plants.Suppose a vine (the plant)produces the first harvestable grapes at age five, and is removed after sixteen years.How many (thousand)grape vines that bear fruit were there in the year 1986 and how many will be there in the year 1992 (assuming that no current vines die before 1992)?
Enter your answers separated by a semi-colon.

Question 10

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C.Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment.Values of the function p = f(t, h)are given in the following table.(The higher the value of p, the better the performance.) Describe the function p(10, h)and explain its meaning.

Question 11

Match the graph with the function.

A) $f ( x , y ) = 6 - 2 x + 3 y$
B) $f ( x , y ) = 6 - 2 x - 3 y$
C) $f ( x , y ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$
D) $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Question 12

You are in a nicely heated cabin in the winter.Deciding that it's too warm, you open a small window.Let T be the temperature in the room, t minutes after the window was opened, x feet from the window.Is T an increasing or decreasing function of x?

A) $\text { Increasing }$
B) $\text { Decreasing }$
C)Neither

Question 13

(a)Find an equation of the largest sphere that can fit inside the cubical space enclosed by the planes x = 1, x = 5, y = 2, y = 6, z = 2 and z = 6.
(b)If we replace the plane z = 6 in part (a)with z = 7, what will be the new equation of the largest sphere?

Question 14

You are at (4, 2, 4)facing the yz-plane.You walk 3 units, turn right and walk for another 2 units.What are your coordinates now? Are you above or below the xy-plane?

Question 15

Describe in words, write equations, and give a sketch for the following set of points.

Question 16

Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in thousands of dollars), and the time, t, required to pay it back (in months).Is C an increasing or decreasing function of t?

A)Decreasing
B)Increasing

Question 17

A spherical ball of radius four units is in a corner touching both walls and floor.What is the radius of the largest spherical ball that can be fit into the corner behind the given ball? (Hint: The smaller ball will not touch the corner point where the walls meet the floor.)

Question 18

The points A = (4, 1, 2), B = (3, -2, 3), and C = (-2, 3, -4)are the vertices of a triangle in space. Which of the vertices is closest to the yz-plane?

A)C
B)A
C)B

Question 19

The points A = (1, 1, 1), B = (2, 4, 2), and C = (3, 2, 2)are the vertices of a triangle in space. Which of the vertices is closest to the origin?

A)A
B)B
C)C

Question 20

Consider the sphere (a)What are the center and radius of this sphere?
(b)Find an equation of the circle (if any)where the sphere intersects the plane x = -2.

Question 21

Match the function with the graph below.

A) $z = \sin x ^ { 2 }$
B) $z = \cos x ^ { 2 }$

Question 22

The contours of the function f(x, y)= 8x + 4y are all parallel lines with slope 2.

Question 23

Consider the function Suppose you are standing on the surface at the point where x = 3, y = 1.If you start to move on the surface parallel to the y-axis in the direction of increasing y, does your height increase or decrease?

Question 24

The graph of the function f(x, y)is shown below. Draw graph of cross-sections with y fixed at y = 0, and y = 1.

Question 25

The diagram below shows the contour map for a circular island.Sketch the vertical cross-section of the island through the center.Your sketch should show concavity clearly.

Question 26

The following figure contains the graphs of the cross sections z = f(a, y)for a = -2, -1, 0, 1, 2.Which of the graphs of z = f(x, y)in A and B best fits this information?

A)
B)

Question 27

Consider the function Suppose you are standing on the surface at the point where x = 2, y = -1.What is your altitude?

Question 28

Which of the following is a contour diagram for f(x, y)= sin x?

A)
B)
C)
D)

Question 29

Draw the level curves for z = 2ln(x)- ln(y).

A)
B)
C)
D)

Question 30

What is the slope of the contour lines of the function f(x, y)=

Question 31

Draw a possible contour diagram for the function whose graph is shown below.Label your contours with reasonable z-values.

Question 32

Suppose that the temperature T of any point (x, y)is given by .
Sketch isothermal curves (i.e.contours)for T = 100, T = 75, T = 50 and T = 0.Be sure to label each contour.
What does the graph of T(x, y)look like if it is sliced by the plane x = 4?

Question 33

Match the following function with the graphs below. The function z = f(x, y)giving happiness as a function of health y and money x according to the statement of a fortune cookie: 'Whoever said money cannot buy happiness does not know where to shop.'

A)
B)
C)

Question 34

The picture below is the contour diagram of f(x, y).The areas between contours have been shaded.Lighter shades represent higher levels, while darker shades represent lower levels. Sketch the cross section of f(x, y)with x fixed at x = 0.5.

Question 35

Let Find the contour of f that passes through the point .

Question 36

The contour diagram below shows the level curves of the difference between July and January mean temperatures in ° F. Does this graph support or contradict the claim that the largest annual temperature variations are found on the coasts of continents?

Question 37

Two contours of the function f(x, y)corresponding to different values of f cannot ever cross.

Question 38

Find a formula for a function f(x, y), given that its contour at level 9 has equation

Question 39

For what values of the constant k is the intersection between the set of points and the graph of a straight line?

Question 40

A soft drink company is interested in seeing how the demand for its products is affected by price.The company believes that the quantity, q, of soft drinks sold depends on $p _ { 1 }$ , the average price of the company's soft drinks, and $p _ { 2 }$ , the average price of competing soft drinks.Which of the graphs below is most likely to represent q as a function of $p _ { 1 }$ and $p _ { 2 }$ ?

A)
B)
C)
D)

Question 41

A plane passes through the points (1, 3, 7), (-1, 0, 6), and (2, 1, -3).Determine the equation of the plane.

Question 42

A linear function f(x,y)has cross-sections f(x,4)= 2x - 14 and f(2,y)= -4y + 6.Find an equation for f.

Question 43

Sketch the level set of corresponding to f = 16 and describe it in words.

Question 44

Consider the plane that passes through the points (1, 3, 10), (-1, -1, 0), and (2, 1, -1).
If you were walking on this plane with no change in altitude, what would be the slope of your path in the xy-plane?

Question 45

A linear function f(x, y)has the values f(4, 2)= 10, f(1, 2)= 4 and f(4, 1)=7.Find an equation for f.

Question 46

Given the table of some values of a linear function determine a formula for the function.

Question 47

Find a formula for the linear function whose contours are shown below.

Question 48

Consider the (partial)contour diagram below for a linear function Find an equation z = f(x, y)for the function.

Question 49

If f is a linear function, then $f ( 4,2 ) - f ( 4,1 ) = f ( 0,2 ) - f ( 0,1 )$

Question 50

Determine a formula for the linear function f(x, y), such that its cross-section with y fixed at y = 1 has equation z = 4x - 1, and its contour at level 0 is the line y = (4 + 4x)/5.

Question 51

If f is any linear function of two variables, then f(x, y + 1)= f(x + 1, y).

Question 52

Find a formula for a linear function z = f(x, y)whose f = 0 contour is the line y = 2x + 2.

Question 53

Given the table of some values of a linear function complete the table:

Question 54

Does the point (-5, -1, -5)lie on any of the level surfaces of $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 }$ ? If so, what is the equation of that level surface?

Question 55

Determine the equation of the plane which passes through the point (1, 3, -1), has slope 5 in the x-direction and slope -3 in the y-direction.

Question 56

Describe the set of points whose distance from the y-axis equals the distance from the xz-plane.
Write an equation for this set in the form G(x, y, z)= 0.
Can the set be described as the graph of a function of two variables?

Question 57

Find an equation for the plane passing through (1, -5, -2)and containing the x-axis.

Question 58

What is the domain of Describe the level surface g =5.

Question 59

Below is a contour diagram depicting D, the average fox population density as a function of , kilometers east of the western end of England, and , kilometers north of the same point. Is D increasing or decreasing at the point (120, 25)in the northern direction?

Question 60

If all of the contours of a function f(x, y)are parallel lines, then the function must be linear.

Question 61

Let What is the domain of f? What are the possible values of f?

Question 62

Consider the figure shown below. Match the level curve with the function.

A) $f _ { 1 } ( x , y , z ) = 5 x + y + z$
B) $f _ { 2 } ( x , y , z ) = 5 x ^ { 2 } + y ^ { 2 } + z$
C) $f _ { 2 } ( x , y , z ) = 5 x - y + z$

Question 63

Represent (if possible)the following surfaces as graphs of functions f(x, y), and as level surfaces of the form g(x, y, z)= c.(There are many possible answers.)
(a)The upper half of the sphere of radius 3, centered at (5, 5, 0).
(b)The lower half of the cylinder of radius 4 around the x-axis.
(c)The cone

Question 64

What value of c (if any)makes the following function continuous at (0, 0)?

Question 65

Find a function f(x, y, z)whose level surface at level 4 is the graph of the function .

Question 66

Let f(x, y, z)= ax + by + cz + d be a linear function of three variables, for constants a, b, c and d.
Given that some of the cross sections of f are f(x, 1, 1)= 0 + 4x, f(0, y, -3)= 4 + 4y,
and f(1, 1, z)= 6 - 2z, find a formula for f.

Question 67

A function f(x, y)is defined for (x, y) $\neq$ (0, 0).Does $\lim _ { ( x , y ) \rightarrow ( 0 \rho ) } f ( x , y )$ exist if f has the contour diagram below? Assume that different contours represent different values of the function.Explain your answer.

Question 68

Suppose that .Find A and B if you know that the period of is and the period of is .

Question 69

Some level surfaces of the functions and are shown below, respectively. Which function takes the value 0 at (0, 0, 0)?

Question 70

Suppose that for all x and y the function f satisfies .
Determine, if possible, the values of the following limits.Explain your answer.
Note that the limit may not exist nor be determined from the given information.
(a) (b)

Question 71

Find a linear function with f(-5, 5)= 1, whose graph is parallel to the level surfaces of g(x, y, z)= 10x + 15y + 5z.

Question 72

Given that the equation of the level surface of f(x, y, z)at level -3 is , determine f(2, 3, -3)if possible.

Question 73

The level surfaces of the functions and are shown in below respectively. Which function is is decreasing in the positive z-direction?

Question 74

Classify the following surface as ellipsoid, elliptical paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets, or cone.

Question 75

Let .Describe the level curve of f at level 2.

Question 76

Consider the function , for certain constants a and b.Simplify the following expression.

Question 77

Determine the values of a and b such that the following function is continuous everywhere.

Question 78

Suppose f is a function of two variables. Given a point (x,y)in the domain of f, there is a level curve of f passing through (x,y).

Question 79

Consider the function where a, b are positive numbers.For what values of (x, y)will the function f not be defined?

Question 80

Suppose f is a function of two variables. Given a point (x,y)in the plane, there is a level curve of f passing through (x,y).