Deck 15: Functions of Several Variables

Question 1

Find the volume of the tetrahedron with corners at ( 0, 0, 0 ), ( 3, 0, 0 ), ( 0, 10, 0 ) and ( 0, 0, 8 ). ​

A)40
B) 0
C) 80
D) 120
E) 240

Answer 1

40

Question 2

Compute the integral. $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { 7 } ( x - 5 y ) \mathrm { d } x \mathrm {~d} y$

A)-1,372
B) -2
C) -686
D) 172
E) -196

Answer 2

-686

Question 3

Compute the integral. $\int _ { 0 } ^ { 9 } \int _ { - x ^ { 2 } } ^ { x ^ { 2 } } x \mathrm {~d} y \mathrm {~d} x$

A)3,280.5
B) 9
C) 6,561
D) -40.5
E) 0

Answer 3

3,280.5

Question 4

Compute the integral. $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 5 } \left( y e ^ { x } - x - y \right) \mathrm { d } x \mathrm {~d} y$

A)-55.5
B) $9 e ^ { 5 } - 64.5$
C) $4.5 e ^ { 5 } - 19.5$
D) $4.5 e ^ { - 3 } + 55.5$
E) $4.5 e ^ { 5 } - 64.5$

Question 5

Compute the integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 - y } x \mathrm {~d} x \mathrm {~d} y$

A)19
B) $\frac { 7 } { 36 }$
C) 1
D) 7
E) $\frac { 7 } { 6 }$

Question 6

Compute the integral. $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x } e ^ { x ^ { 2 } } \mathrm {~d} y \mathrm {~d} x$

A) $\frac { 1 } { 2 }$
B) $\frac { e ^ { 8 } } { 3 }$
C) $2 e ^ { 4 }$
D) $\frac { 1 } { 2 } \left( e ^ { 4 } - 1 \right)$
E) $e ^ { 4 }$

Question 7

Write the integral with the order of integration reversed (changing the limits of integration as necessary). $$\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 + y } } f ( x , y ) \mathrm { d } x \mathrm {~d} y$$

A) $\int _ { 0 } ^ { \sqrt { 2 } } \int _ { \sqrt { x } - 1 } ^ { 1 } f ( x , y ) \mathrm { d } y \mathrm {~d} x$
B) $\int _ { 0 } ^ { \sqrt { 2 } } \int _ { x ^ { 2 } - 1 } ^ { 1 } f ( x , y ) \mathrm { d } y \mathrm {~d} x$
C) $\int _ { 0 } ^ { 1 } \int _ { 1 } ^ { x ^ { 2 } - 1 } f ( x , y ) \mathrm { d } y \mathrm {~d} x$
D) $\int _ { - 1 } ^ { 0 } \int _ { x ^ { 2 } - 1 } ^ { 1 } f ( x , y ) \mathrm { d } y \mathrm {~d} x$
E) $\int _ { 0 } ^ { \sqrt { 2 } } \int _ { \sqrt { x - 1 } } ^ { 1 } f ( x , y ) \mathrm { d } y \mathrm {~d} x$

Question 8

Compute the integral. $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 8 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y$

A) $e ^ { 8 } e ^ { 4 }$
B) $\left( e ^ { 8 } - 1 \right) \left( e ^ { 4 } - 1 \right)$
C) $e ^ { 12 } + 12$
D) $e ^ { 12 } - 12$
E) $\frac { e ^ { 4 } } { \left( e ^ { 5 } - 1 \right) }$

Question 9

Find the average value of the given function over the indicated domain. $f ( x , y ) = y$

A) $- 2$
B) $\frac { 4 } { 3 }$
C) $- \frac { 2 } { 3 }$
D) 2
E) $\frac { 2 } { 3 }$

Question 10

Compute the integral. $\int _ { 0 } ^ { 10 } \int _ { 4 - x } ^ { 25 - x } ( x + y ) ^ { \frac { 1 } { 2 } } \mathrm {~d} y \mathrm {~d} x$

A)60
B) 15
C) -52
D) 780
E) 117

Question 11

Find $\iint _ { R } f ( x , y ) d x d y$ , where $f ( x , y ) = x$ and R is the indicated domain. (Remember that you often have a choice as to the order of integration.)

A)13
B) 2,197
C) $\frac { 2,197 } { 6 }$
D) 113
E) $\frac { 169 } { 6 }$

Question 12

Compute the integral.
$\int _ { - 10 } ^ { 5 } \int _ { y - 5 } ^ { y + 5 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y$

A) $\frac { 1 } { 2 } \left( e ^ { 10 } - e ^ { 5 } - e ^ { - 5 } + e ^ { - 15 } \right)$
B) $\frac { 1 } { 2 } \left( e ^ { 15 } - e ^ { 5 } - e ^ { - 10 } + e ^ { - 25 } \right)$
C) $\left( e ^ { 5 } - e ^ { - 5 } - e ^ { - 15 } + e ^ { - 10 } \right)$
D) $\frac { 1 } { 4 } \left( e ^ { 10 } - e ^ { - 10 } + e ^ { 5 } - e ^ { - 5 } \right)$
E) $\frac { 1 } { 2 } \left( e ^ { 15 } - e ^ { 5 } - e ^ { - 15 } + e ^ { - 25 } \right)$

Question 13

Your latest CD - ROM drive is expected to sell between $q = 200,000 - p ^ { 2 }$ and $q = 230,000 - p ^ { 2 }$
Units if priced at p. You plan to set the price between $200 and $400. What are the maximum and minimum possible revenues you can make What is the average of all the possible revenues you can make

A)$36,800,000,000, $8,000,000,000, $140,166,667
B) $38,000,000, $16,000,000, $34,500,000
C) $36,800,000,000, $8,000,000,000, $73,500,000
D) $45,960,000, $79,840,000, $3,120,000
E) $92,000,000, $40,000,000, $774,000,000,000,000

Question 14

The town of West Podunk is shaped like a rectangle 80 miles from west to east and 90 miles from north to south (see the figure). It has a population density of
$P ( x , y ) = e ^ { - 0.1 ( x + y ) }$
hundred people per square mile x miles east and y miles north of the southwest corner of town. What is the total population of the town


Please enter your answer as a number without the units.

Question 15

Find the volume under the graph of $z = 5 - x ^ { 2 }$ over the triangle $0 \leq x \leq 3$ and $0 \leq y \leq 3 - x$ .

A)15.75
B) 42
C) 22.5
D) 7.5
E) 29.25

Question 16

Find $\iint _ { R } f ( x , y ) \mathrm { d } x \mathrm {~d} y$ , where $f ( x , y ) = x y ^ { 2 }$ and R is the indicated domain. (Remember that you often have a choice as to the order of integration.) $$f ( y ) = \sqrt { 9 - y ^ { 2 } }$$

A)32.4
B) 24.3
C) 16.2
D) 27
E) 259.2

Question 17

A productivity model at the Handy Gadget Company is $P = 13,000 x ^ { 0.1 } y ^ { 0.9 }$
Where P is the number of gadgets the company turns out per month, x is the number of employees at the company, and y is the monthly operating budget in thousands of dollars. Because the company hires part - time workers, it uses anywhere between 44 and 54 workers each month, and its operating budget varies from $12,000 to $14,000 per month. What is the average of the possible numbers gadgets it can turn out per month (Round the answer to the nearest 1,000 gadgets.)

A)8,000 gadgets
B) 61,000 gadgets
C) 193,000 gadgets
D) 3,000 gadgets

Question 18

The temperature at the point $( x , y )$ on the square with vertices (0, 0), (0, 1), (1, 0) and (1, 1) is given by $T ( x , y ) = 4 x ^ { 2 } + 8 y ^ { 2 }$
Find the hottest and coldest points on the square.

A)(2, 2), (0, 0)
B) (0, 1), (0, 0)
C) (1, 1), (1, 0)
D) (1, 0), (0, 0)
E) (1, 1), (0, 0)

Question 19

Find $\iint _ { R } f ( x , y ) \mathrm { d } x \mathrm {~d} y$ , where $f ( x , y ) = 9 + y$ and R is the indicated domain. (Remember that you often have a choice as to the order of integration.)
$$f ( y ) = 9 - y ^ { 2 }$$

A)243
B) 729
C) 324
D) 972
E) 81

Question 20

Find the average value of the given function over the indicated domain. $f ( x , y ) = y$

A)0
B) 2
C) 1.3
D) 10
E) 7

Question 21

Solve the given problem by using substitution.
Find the minimum value of $f ( x , y , z ) = 2 x ^ { 2 } + 2 x + y ^ { 2 } - y + z ^ { 2 } - z - 4$ subject to $z = 2 y$ .

A) $f _ { \min } = - 3.775$
B) $f _ { \min } = - 3.5$
C) $f _ { \min } = - 4$
D) $f _ { \min } = - 4.95$
E) $f _ { \min } = - 4.7$

Question 22

Use Lagrange Multipliers to solve the problem.
Find the maximum value of $f ( x , y ) = 3 x y$ subject to $x ^ { 2 } + y ^ { 2 } = 8$ .

A) $f _ { \max } = 24$
B) $f _ { \max } = 12$
C) $f _ { \max } = 17$
D) $f _ { \max } = 7$
E) $f _ { \max } = 32$

Question 23

Solve the given problem by using substitution.
Minimize $S = x y + 25 x z + 5 y z$ subject to $x y z = 1$ with $x > 0$ , $y > 0$ , $z > 0$ .

A) $S _ { \min } = 20$
B) $S _ { \min } = 15$
C) $S _ { \min } = 5$
D) $S _ { \min } = 25$
E) $S _ { \min } = 1$

Question 24

Solve the given problem by using substitution.
Find the minimum value of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4$ subject to $y = 2 x$ .

A) $f _ { \min } = - 5$
B) $f _ { \min } = - 4$
C) $f _ { \min } = - 3$
D) $f _ { \min } = - 2$
E) $f _ { \min } = - 6$

Question 25

What point on the surface z is closest to the origin $z = x ^ { 2 } + y - 5$
Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.

A) $( 2.5,0 , - 2.5 )$
B) $\left( - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$ and $\left( 2 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
C) $( 0,2.5 , - 2.5 )$
D) $\left( 0 , \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
E) $( 0,2.5 , - 2.5 )$ and $( 0,2.5,2.5 )$

Question 26

Classify the highlighted value in the table. ​

A)a relative maximum
B) a saddle point
C) a relative minimum
D) none of these

Question 27

Solve the given problem by using substitution.
Find the maximum value of $f ( x , y , z ) = 3 - x ^ { 2 } - x - y ^ { 2 } + y - z ^ { 2 } + z$ subject to $y = 3 x$ .

A) $f _ { \max } = 3.225$
B) $f _ { \max } = 3$
C) $f _ { \max } = 3.35$
D) $f _ { \max } = 2.775$
E) $f _ { \max } = 2.45$

Question 28

Locate the critical point of the function. $f ( x , y ) = 5 x ^ { 2 } + 2 y ^ { 2 } + 4$

A) $( 5,2,137 )$
B) $( 5,2,4 )$
C) $( 0,0,4 )$
D) $( 0,4,4 )$
E) $( 0,0,0 )$

Question 29

Use Lagrange Multipliers to solve the problem.
Find the maximum value of $f ( x , y ) = x y$ subject to $x + 2 y = 80$ .

A) $f _ { \max } = 800$
B) $f _ { \max } = 40$
C) $f _ { \max } = 400$
D) $f _ { \max } = 2$
E) $f _ { \max } = 10$

Question 30

At what point on the given surface is the quantity $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ a minimum (The method of Lagrange multipliers can be used here.) $z = \left( 4 x ^ { 2 } + 10 x + y ^ { 2 } + 22 \right) ^ { \frac { 1 } { 2 } }$

A) $( - 1,0,5 )$
B) $( 1,0,4 )$
C) $( - 1,0,4 )$
D) $( - 1,0,6 )$
E) $( - 3,0,3 )$

Question 31

Your latest CD - ROM drive is expected to sell between
$q = 190,000 - p ^ { 2 }$ and $q = 210,000 - p ^ { 2 }$
units if priced at p. You plan to set the price between $200 and $300. What are the maximum and minimum possible revenues you can make What is the average of all the possible revenues you can make

Please enter your answer as three numbers, separated by commas.

Question 32

Find the point on the given plane closest to .
​ ​
NOTE: Please enter your answer in the form .

Question 33

In the table below, classify each highlighted value. ​ ​

A)a saddle point
B) a relative maximum
C) a relative minimum
D) none of these

Question 34

The temperature at the point on the square with vertices (0, 0), (0, 2), (2, 0) and (2, 2) is given by ​ ​
Find the hottest and coldest points on the square.
​
NOTE: Please enter your answers in the form , separated by commas.

Question 35

Use Lagrange Multipliers to solve the given problem.
Find the maximum value of $f ( x , y ) = x y$ subject to $2 x + y = 60$ .

A) $f _ { \max } = 10$
B) $f _ { \max } = 450$
C) $f _ { \max } = 30$
D) $f _ { \max } = 2$
E) $f _ { \max } = 225$

Question 36

Solve the given problem by using substitution.
Find the maximum value of $f ( x , y , z ) = 8 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 }$ subject to $z = 3 y$ .

A) $f _ { \max } = 8$
B) $f _ { \max } = 9$
C) $f _ { \max } = 3$
D) $f _ { \max } = 10$
E) $f _ { \max } = 7$

Question 37

Solve the given problem by using substitution.
Minimize $S = x y + x z + y z$ subject to $x y z = 5$ with $x > 0$ , $y > 0$ , $z > 0$ .

A) $S _ { \min } = 3 \left( 5 ^ { \frac { 2 } { 3 } } \right)$
B) $S _ { \min } = 3 \left( 6 ^ { \frac { 2 } { 3 } } \right)$
C) $S _ { \min } = 3 \left( 4 ^ { \frac { 2 } { 3 } } \right)$
D) $S _ { \min } = 3 \left( 5 ^ { \frac { 1 } { 3 } } \right)$
E) $S _ { \min } = 3 \left( 6 ^ { \frac { 1 } { 3 } } \right)$

Question 38

Find the point on the given plane closest to $( 1,1,0 )$ . $2 x - 2 y - 4 z + 96 = 0$

A) $( - 3,5,16 )$
B) $( - 7,9,16 )$
C) $( - 8,8,4 )$
D) $( 7,9,16 )$
E) $( - 7,9,4 )$

Question 39

What point on the surface z is closest to the origin
$z = x ^ { 2 } + y - 2$
Hint : Minimize the square of the distance from $( x , y , z )$ to the origin.

NOTE: Please enter your answer(s) in the form $( x , y , z )$ . If there is more than one answer, separate them with commas.

Question 40

A productivity model at the Handy Gadget Company is
$P = 15,000 x ^ { 0.1 } y ^ { 0.9 }$
where P is the number of gadgets the company turns out per month, x is the number of employees at the company, and y is the monthly operating budget in thousands of dollars. Because the company hires part - time workers, it uses anywhere between 43 and 53 workers each month, and its operating budget varies from $13,000 to $17,000 per month. What is the average of the possible numbers gadgets it can turn out per month (Round the answer to the nearest 1,000 gadgets.)

Please enter your answer as a number without the units.

Question 41

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 57 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight
Enter your answer as a number without the units.

Question 42

The US Postal Service (USPS) will accept only packages with length plus girth no more than 120 inches. (See the figure.)
What are the dimensions of the largest volume package the USPS will accept

A) $24 \times 24 \times 24$ inches
B) $24 \times 24 \times 48$ inches
C) $20 \times 20 \times 40$ inches
D) $24 \times 48 \times 48$ inches
E) $40 \times 40 \times 40$ inches

Question 43

Locate the local minimum of the function. $f ( x , y ) = x y + \frac { 4 } { x } + \frac { 4 } { y }$

A) $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 64 ^ { \frac { 2 } { 3 } } \right)$
B) $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 12 ^ { \frac { 2 } { 3 } } \right)$
C) $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 2 } { 3 } } \right)$
D) $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 4 ^ { \frac { 2 } { 3 } } \right)$
E) $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 4 ^ { \frac { 1 } { 3 } } \right)$

Question 44

Locate the local maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 6 } + y ^ { 2 } \right) }$

A) $( 0,0,1 )$
B) $\left( 1,0 , \frac { 1 } { e } \right)$
C) $( 0,0,0 )$
D) $( 1,0 , e )$
E) $\left( 1,0 , \frac { 1 } { e ^ { 2 } } \right)$

Question 45

Find x and y values of the relative extrema of the function. $f ( x , y ) = x ^ { 4 } + 8 x y ^ { 2 } + 2 y ^ { 4 }$

A) $( 0,0,0 )$
B) $( 0 , - 2 , - 4 )$
C) $( - 2 , - 2 , - 16 )$
D) $( 2 , - 2 , - 16 )$
E) $( - 2,2 , - 16 )$

Question 46

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 66 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight

A)10,648 in.³
B) 66 in.³
C) 287,496 in.³
D) 132 in.³
E) 22 in.³

Question 47

Locate the maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 2 x \right) }$

A) $\left( - 1,0 , e ^ { - 1 } \right)$
B) $( 2,0 , e )$
C) $( 1,0 , e )$
D) $( - 1,0 , e )$
E) $( - 2,0 , e )$

Question 48

How many critical points does the following function have $f ( x , y ) = 4 - \left( y ^ { 2 } + x ^ { 2 } \right)$

A)four
B) one
C) two
D) three
E) none

Question 49

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model. $C ( x , y ) = 6,000 + 100 x ^ { 2 } + 25 y ^ { 2 }$
Where x is the reduction in sulfur emissions, y is the reduction in lead emissions (in pounds of pollutant per day), and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $600 per pound of sulfur and $50 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize the net cost (cost minus subsidy)

A)3.5 lb of sulfur and 2 lb of lead
B) 6 lb of sulfur and 1 lb of lead
C) 3 lb of sulfur and 1 lb of lead
D) 1.9 lb of sulfur and 3.5 lb of lead
E) 1 lb of sulfur and 3 lb of lead

Question 50

Your company manufactures two models of stereo speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If one is expensive then more people will buy the other. If p₁ is the price per pair of the Ultra Mini and p₂ is the price of the Big Stack, demand for the Ultra Mini is given by $q _ { 1 } \left( p _ { 1 } , p _ { 2 } \right) = 50,000 - 100 p _ { 1 } + 10 p _ { 2 }$
Where q₁ represents the number of pairs of Ultra Minis that will be sold in a year. The demand for the Big Stack is given by
$q _ { 2 } \left( p _ { 1 } , p _ { 2 } \right) = 150,000 + 10 p _ { 1 } - 100 p _ { 2 }$
Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue.

Round your answer to the nearest dollar.

A)$798 for Ultra Mini and $275 for Big Stack
B) $428 for Ultra Mini and $783 for Big Stack
C) $275 for Ultra Mini and $798 for Big Stack
D) $328 for Ultra Mini and $783 for Big Stack
E) $783 for Ultra Mini and $328 for Big Stack

Question 51

Locate the saddle point of the function. $f ( x , y ) = x ^ { 2 } - 6 x + y - e ^ { y }$

A) $( 6,1 , - 10 )$
B) $( 6,1 , - 1 )$
C) $( 3,0 , - 10 )$
D) $( 6,0 , - 10 )$
E) $( 3,1 , - 10 )$

Question 52

Your company manufactures two models of stereo speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If one is expensive then more people will buy the other. If p₁ is the price per pair of the Ultra Mini and p₂ is the price of the Big Stack, demand for the Ultra Mini is given by $q _ { 1 } \left( p _ { 1 } , p _ { 2 } \right) = 200,000 - 100 p _ { 1 } + 10 p _ { 2 }$
Where q₁ represents the number of pairs of Ultra Minis that will be sold in a year. The demand for the Big Stack is given by
$q _ { 2 } \left( p _ { 1 } , p _ { 2 } \right) = 150,000 + 10 p _ { 1 } - 100 p _ { 2 }$
Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue.

Round your answer to the nearest dollar.

A)$874 for Ultra Mini and $1,033 for Big Stack
B) $1,033 for Ultra Mini and $874 for Big Stack
C) $1,186 for Ultra Mini and $859 for Big Stack
D) $1,086 for Ultra Mini and $859 for Big Stack
E) $859 for Ultra Mini and $1,086 for Big Stack

Question 53

Locate the maximum of the function.
​ ​
Express your answer as an ordered triple. If there is more than one answer, separate them by commas.

Question 54

The US Postal Service (USPS) will accept only packages with length plus girth no more than 102 inches. (See the figure.)

What are the dimensions of the largest volume package the USPS will accept

Express your answer as an ordered triple in the form $( x , y , z )$ .

Question 55

Locate the local minimum of the function.
​ ​
Express your answer as an ordered triple.

Question 56

Let $H = f _ { x x } ( a , b ) f _ { y y } ( a , b ) - f ^ { 2 } x y ( a , b )$
What condition on H guarantees that f has a relative extremum at the point $( a , b )$

Question 57

Locate the critical point of the function. $f ( x , y ) = 4 - x ^ { 2 } - 6 x - y ^ { 2 } + 4 y$

A) $( 3,2 , - 17 )$
B) $( - 3,2,17 )$
C) $( 6,4,4 )$
D) $( - 6,4,4 )$
E) $( 2 , - 3,17 )$

Question 58

Locate the local maximum of the function. $f ( x , y ) = x ^ { 2 } y - 8 x ^ { 2 } - 4 y ^ { 2 }$

A) $( 8,4 , - 256 )$
B) $( 0,0,0 )$
C) $( - 8,8 , - 256 )$
D) $( - 8 , - 4,0 )$
E) $( 8,8 , - 256 )$

Question 59

How many critical points does the function have $f ( x , y ) = 2 x e ^ { y }$

A)three
B) two
C) four
D) one
E) none

Question 60

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model.
$C ( x , y ) = 6,000 + 200 x ^ { 2 } + 75 y ^ { 2 }$
where x is the reduction in sulfur emissions, y is the reduction in lead emissions (in pounds of pollutant per day), and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $300 per pound of sulfur and $50 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize the net cost (cost minus subsidy)

Enter your answer as an ordered pair in the form $( x , y )$ . Round your answer to three decimal places if necessary.

Question 61

Classify each labeled point on the graph.
​ ​
Choose the correct letter for each question.
​
a relative minimum

A)R
B)P
C)Q

Question 62

Calculate $\frac { \partial ^ { 2 } f } { \partial y ^ { 2 } }$ , $\frac { \partial ^ { 2 } f } { \partial x y }$ and evaluate each at $( 5,7 )$ . $f ( x , y ) = 950 - 16 x + 14 y + 6 x y$

A) $\left. \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } \right| _ { ( 5,7 ) } = 6$ , $\left. \frac { \partial ^ { 2 } f } { \partial x y } \right| _ { ( 5,7 ) } = 0$
B) $\left. \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } \right| _ { ( 5,7 ) } = 0$ , $\left. \frac { \partial ^ { 2 } f } { \partial x y } \right| _ { ( 5,7 ) } = 0$
C) $\left. \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } \right| _ { ( 5,7 ) } = 16$ , $\left. \frac { \partial ^ { 2 } f } { \partial x y } \right| _ { \{ 5,7 ) } = 14$
D) $\left. \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } \right| _ { ( 5,7 ) } = 6$ , $\left. \frac { \partial ^ { 2 } f } { \partial x y } \right| _ { ( 5,7 ) } = 6$
E) $\left. \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } \right| _ { ( 5,7 ) } = 0$ , $\left. \frac { \partial ^ { 2 } f } { \partial x y } \right| _ { ( 5,7 ) } = 6$

Question 63

In the table below, classify each highlighted value.


-relative maximum

A)2
B)8
C)9
D)12

Question 64

Classify each labeled point on the graph. Choose the correct letter for each question.
none

A)Q
B)P
C)R

Question 65

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) }$ when defined. $f ( x , y ) = x ^ { 2 } y ^ { 3 } - x ^ { 3 } y ^ { 2 } - x y$

A) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = - 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2048$
B) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = - 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = - 2048$
C) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 2044$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 1016$
D) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 1024$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2048$
E) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2044$

Question 66

In the table below, classify each highlighted value.



-none

A)2
B)8
C)9
D)12

Question 67

Classify each labeled point on the graph. Choose the correct letter for each question.
local maximum

A)Q
B)P
C)R

Question 68

In the table below, classify each highlighted value.


-relative minimum

A)2
B)8
C)9
D)12

Question 69

Calculate $\frac { \partial f } { \partial x }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,2 ) }$ when defined. $f ( x , y ) = 1100 - 16 x + 11 y$

A) $\frac { \partial f } { \partial x } = - 16$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,2 ) } = 11$
B) $\frac { \partial f } { \partial x } = 8$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,2 ) } = 2$
C) $\frac { \partial f } { \partial x } = - 16 y$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,2 ) } = 11 x$
D) $\frac { \partial f } { \partial x } = - 27$ , $\left. \frac { \partial f } { \partial y } \right| _ { \{ 8,2 ) } = 27$
E) $\frac { \partial f } { \partial x } = 11$ , $\left. \frac { \partial f } { \partial y } \right| _ { \{ 8,2 ) } = 16$

Question 70

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 9 x ^ { 0.7 } y ^ { 0.9 }$

A) $\frac { \partial f } { \partial x } = 6.3 x ^ { 1.7 } y ^ { 0.9 }$ , $\frac { \partial f } { \partial y } = 8.1 x ^ { 0.7 } y ^ { 1.9 }$
B) $\frac { \partial f } { \partial x } = - 6.3 x ^ { - 0.3 } y ^ { 0.1 }$ , $\frac { \partial f } { \partial y } = - 8.1 x ^ { 0.3 } y ^ { - 0.1 }$
C) $\frac { \partial f } { \partial x } = 6.3 x ^ { - 0.3 }$ , $\frac { \partial f } { \partial y } = 8.1 y ^ { - 0.1 }$
D) $\frac { \partial f } { \partial x } = 6.3 x ^ { - 0.3 } y ^ { 0.9 }$ , $\frac { \partial f } { \partial y } = 8.1 x ^ { 0.7 } y ^ { - 0.1 }$
E) $\frac { \partial f } { \partial x } = 6.3 x ^ { - 0.3 } y ^ { 0.9 }$ , $\frac { \partial f } { \partial y } = 8.1 x ^ { 1.7 } y ^ { - 1.9 }$

Question 71

Calculate $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ when defined.
$f ( x , y ) = 8 x ^ { 2 } y$

A) $\frac { \partial f } { \partial x } = 8 x ^ { 2 }$ , $\frac { \partial f } { \partial x } = 16 x y$
B) $\frac { \partial f } { \partial x } = 16 x$ , $\frac { \partial f } { \partial y } = 8 x y$
C) $\frac { \partial f } { \partial x } = 16 x y$ , $\frac { \partial f } { \partial y } = 8 x ^ { 2 }$
D) $\frac { \partial f } { \partial x } = 16 x y$ , $\frac { \partial f } { \partial y } = 8 x ^ { 2 } y$
E) $\frac { \partial f } { \partial x } = - 8 y ^ { 2 }$ , $\frac { \partial f } { \partial y } = - 16 x$

Question 72

Classify each labeled point on the graph.
​ ​
Choose the correct letter for each question.
​
a relative maximum

A)R
B)P
C)Q

Question 73

In the table below, classify each highlighted value.



-saddle point

A)2
B)8
C)9
D)12

Question 74

Classify each labeled point on the graph.
​ ​
Choose the correct letter for each question.
​
none

A)R
B)P
C)Q

Question 75

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) }$ when defined. $f ( x , y ) = 960 - 13 x + 6 y + 7 x y$

A) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = - 22$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = - 34$
B) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 110$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 136$
C) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 28$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 5$
D) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 22$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 34$
E) $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,5 ) } = 34$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,5 ) } = 22$

Question 76

Calculate the values of $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ at $( 8,3,4 )$ . $f ( x , y , z ) = x y z$

A) $\left. \frac { \partial f } { \partial x } \right| _ { ( 8,3,4 ) } = 12$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 32$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 0$
B) $\left. \frac { \partial f } { \partial x } \right| _ { ( 8,3,4 ) } = 24$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 32$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 12$
C) $\left. \frac { \partial f } { \partial x } \right| _ { ( 8,3,4 ) } = 12$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 24$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 32$
D) $\left. \frac { \partial f } { \partial x } \right| _ { ( 8,3,4 ) } = 12$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 32$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 24$
E) $\left. \frac { \partial f } { \partial x } \right| _ { ( 8,3,4 ) } = 32$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 12$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,3,4 ) } = 24$

Question 77

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = e ^ { 2 x y }$

A) $\frac { \partial f } { \partial x } = 2 y e ^ { 2 x }$ , $\frac { \partial f } { \partial y } = 2 x e ^ { 2 y }$
B) $\frac { \partial f } { \partial x } = 2 y e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = 2 x e ^ { 2 x y }$
C) $\frac { \partial f } { \partial x } = e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = e ^ { 2 x y }$
D) $\frac { \partial f } { \partial x } = 2 e ^ { 2 x y }$ , $\frac { \partial f } { \partial y } = 2 e ^ { 2 x y }$
E) $\frac { \partial f } { \partial x } = 2 y$ , $\frac { \partial f } { \partial y } = 2 x$

Question 78

Calculate $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } }$ , $\frac { \partial ^ { 2 } f } { \partial y x }$ when defined. $f ( x , y ) = 4 x ^ { 0.9 } y ^ { 0.7 }$

A) $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = - 0.36 y ^ { 0.7 } x ^ { - 1.1 }$ , $\frac { \partial ^ { 2 } f } { \partial y x } = - 2.52 y ^ { 0.3 } x ^ { 0.1 }$
B) $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = - 3.6 y ^ { 0.7 } x ^ { - 0.1 }$ , $\frac { \partial ^ { 2 } f } { \partial y x } = 2.8 y ^ { - 0.3 } x ^ { 0.9 }$
C) $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = - 3.6 y ^ { 0.7 } x ^ { - 1.1 }$ , $\frac { \partial ^ { 2 } f } { \partial y x } = 3.6 y ^ { - 0.3 } x ^ { - 0.1 }$
D) $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = - 0.36 y ^ { 0.7 } x ^ { - 1.1 }$ , $\frac { \partial ^ { 2 } f } { \partial y x } = 2.52 y ^ { - 0.3 } x ^ { - 0.1 }$
E) $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = - 4 y ^ { - 0.3 } x ^ { - 1.1 }$ , $\frac { \partial ^ { 2 } f } { \partial y x } = 4 y ^ { - 1.3 } x ^ { - 0.1 }$

Question 79

Classify each labeled point on the graph. Choose the correct letter for each question.
saddle point

A)Q
B)P
C)R

Question 80

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 6 x ^ { 3 } - y ^ { 3 } + x - 12$

A) $\frac { \partial f } { \partial x } = 18 x ^ { 2 } + 1 - 3 y ^ { 2 }$ , $\frac { \partial f } { \partial y } = 18 x ^ { 2 } + 1$
B) $\frac { \partial f } { \partial x } = 18 x ^ { 2 } + 1$ , $\frac { \partial f } { \partial y } = - 3 y ^ { 2 }$
C) $\frac { \partial f } { \partial x } = - 3 y ^ { 2 }$ , $\frac { \partial f } { \partial y } = 18 x ^ { 2 } + 1$
D) $\frac { \partial f } { \partial x } = 6 x ^ { 2 } + 1$ , $\frac { \partial f } { \partial y } = - 3 y ^ { 2 }$
E) $\frac { \partial f } { \partial x } = 6 x ^ { 2 }$ , $\frac { \partial f } { \partial y } = y ^ { 2 }$