Question 1

Describe in words the intersection of the surfaces $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 7 - 6 \left( x ^ { 2 } + y ^ { 2 } \right)$

Accepted Answer

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.

Accepted Answer

The function C(s, t) = 250 means that when you borrow $7,000 (s=7, in thousands of dollars) for a period of 48 months (t=48), your monthly payment will be $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) $z = \sin y$

Accepted Answer

A

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 $C \left( q _ { 1 } , q _ { 2 } \right) = 2 q _ { 1 } ^ { 2 } + q _ { 1 } q _ { 2 } + q _ { 2 } ^ { 2 } + 420$ (a)Find $C ( 10,25 )$ (b)By comparing the terms $2 q _ { 1 } ^ { 2 }$ and $q _ { 2 } ^ { 2 }$ 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?

Accepted Answer

The answer of Yummy Potato Chip Company has manufacturing plants...

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?

Accepted Answer

The answer of The points A = (-4, 5, -3),...

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)?

Accepted Answer

The answer of A certain piece of electronic surveying equipment...

Question 7

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

Accepted Answer

The answer of Find a possible formula for a function...

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?

Accepted Answer

The answer of The following table gives the number f(x,...

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.

Accepted Answer

The answer of The following table gives the number f(x,...

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.

Accepted Answer

The answer of A certain piece of electronic surveying equipment...

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) $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Accepted Answer

The answer of Match the graph with the function. 
A)...

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

Accepted Answer

The temperature T is likely to be higher further away from the window, as the cold air entering through the window will make the area near the window cooler. Thus, T increases with distance x from the window.

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?

Accepted Answer

The answer of (a)Find an 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?

Accepted Answer

The answer of You are at (4, 2, 4)facing the...

Question 15

Describe in words, write equations, and give a sketch for the following set of points. $$\text { Consider the set of points } ( x , y , z ) \text { whose distance from the } y z \text {-plane is } 3 \text {. }$$

Accepted Answer

The answer of Describe in words, write equations, and give...

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

Accepted Answer

C is a decreasing function of t because as the time, t, to pay back the loan increases, the monthly payment, C, generally decreases since the loan amount is spread over more months.

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.)

Accepted Answer

The answer of A spherical ball of radius four units...

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

Accepted Answer

The yz-plane is the plane that contains the y-axis and the z-axis. The point closest to the yz-plane is the point with the smallest x-coordinate. The x-coordinate of point A is 4, the x-coordinate of point B is 3, and the x-coordinate of point C is -2. Therefore, point C is closest to the yz-plane.

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

Accepted Answer

The distance of a point (x, y, z) from the origin (0, 0, 0) is given by the formula $\sqrt{x^2 + y^2 + z^2}$. Calculating the distances for each point: A = $\sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$, B = $\sqrt{2^2 + 4^2 + 2^2} = \sqrt{24}$, and C = $\sqrt{3^2 + 2^2 + 2^2} = \sqrt{17}$. Therefore, point A is closest to the origin.

Question 20

Consider the sphere $( x + 1 ) ^ { 2 } + ( y - 0 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 4$ (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.

Accepted Answer

The answer of Consider the sphere \(( x + 1...

Quiz 12: Functions of Several Variables

Describe in words the intersection of the surfaces $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 7 - 6 \left( x ^ { 2 } + y ^ { 2 } \right)$

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?

Match the graph with the function. $Match the graph with the function. A) z = - \sin x \sin y B) z = 3 e ^ { - \frac { x } { 3 } } \sin y C) z = \frac { 1 } { 2 } x ^ { 2 } + \sin ^ { 2 } y D) z = \sin y$

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?

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

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?

Match the graph with the function. $Match the graph with the function. A) f ( x , y ) = 6 - 2 x + 3 y B) f ( x , y ) = 6 - 2 x - 3 y C) f ( x , y ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } D) f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

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)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?

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?

Describe in words, write equations, and give a sketch for the following set of points. $\text { Consider the set of points } ( x , y , z ) \text { whose distance from the } y z \text {-plane is } 3 \text {. }$

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 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.)

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?

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?

Consider the sphere $( x + 1 ) ^ { 2 } + ( y - 0 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 4$ (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.

Quiz 12: Functions of Several Variables

Describe in words the intersection of the surfaces z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } }z=x2+y2​ and z=7−6(x2+y2)z = 7 - 6 \left( x ^ { 2 } + y ^ { 2 } \right)z=7−6(x2+y2)

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?

Match the graph with the function.

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?

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

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?

Match the graph with the function.

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)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?

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?

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 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.)

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?

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?

Consider the sphere (x+1)2+(y−0)2+(z−1)2=4( x + 1 ) ^ { 2 } + ( y - 0 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 4(x+1)2+(y−0)2+(z−1)2=4 (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.

Describe in words the intersection of the surfaces $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 7 - 6 \left( x ^ { 2 } + y ^ { 2 } \right)$

Match the graph with the function. $Match the graph with the function. A) z = - \sin x \sin y B) z = 3 e ^ { - \frac { x } { 3 } } \sin y C) z = \frac { 1 } { 2 } x ^ { 2 } + \sin ^ { 2 } y D) z = \sin y$

Match the graph with the function. $Match the graph with the function. A) f ( x , y ) = 6 - 2 x + 3 y B) f ( x , y ) = 6 - 2 x - 3 y C) f ( x , y ) = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } D) f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Consider the sphere $( x + 1 ) ^ { 2 } + ( y - 0 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 4$ (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.