Exam Prep Chapter 8 Functions Of Several Variables - Final Test Bank | Applied Calculus 7e by Hughes Hallett. DOCX document preview.

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Exam Prep Chapter 8 Functions Of Several Variables

Applied Calculus, 7e (Hughes-Hallett)

Chapter 8 Functions of Several Variables

8.1 Understanding Functions of Two Variables

1) The profit, P, from producing a product is expressed as a function of the cost, C, of producing the product and the revenue, R, from selling the product. Do you expect P to be an increasing or decreasing function of C?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

2) Lift ticket sales, S, at a ski resort are a function of the price, p, of a ticket and the current number, n, of inches of snowpack on the mountain. Thus, S = f ( p, n). Do you expect f to be an increasing or a decreasing function of p?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

3) The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, R = g(t, f ). Find g(100, 600).

A table has 5 rows and 5 columns. The columns represent t, with the following headers from left to right: 100 to 500, in increments of 100. The rows represent f, and have the following headers from top to bottom: 200 to 1000, in increments of 200. The row entries are as follows: Row 1, 200: 11, 19, 27, 35, 43; Row 2, 400: 14, 22, 30, 38, 46; Row 3, 600: 17, 25, 33, 41, 49; Row 4, 800: 20, 28, 36, 44, 52; Row 5, 1000: 23, 31, 39, 47, 55.

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

4) The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, R = g(t, f ). What is the revenue when ticket sales are 500 and food sales are 200?

A table has 5 rows and 5 columns. The columns represent t, with the following headers from left to right: 100 to 500, in increments of 100. The rows represent f, and have the following headers from top to bottom: 200 to 1000, in increments of 200. The row entries are as follows: Row 1, 200: 11, 19, 27, 35, 43; Row 2, 400: 14, 22, 30, 38, 46; Row 3, 600: 17, 25, 33, 41, 49; Row 4, 800: 20, 28, 36, 44, 52; Row 5, 1000: 23, 31, 39, 47, 55.

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

5) The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, R = g(t, f ). What is the revenue when ticket sales are 400 and food sales are 800?

A table has 5 rows and 5 columns. The columns represent t, with the following headers from left to right: 100 to 500, in increments of 100. The rows represent f, and have the following headers from top to bottom: 200 to 1000, in increments of 200. The row entries are as follows: Row 1, 200: 11, 19, 27, 35, 43; Row 2, 400: 14, 22, 30, 38, 46; Row 3, 600: 17, 25, 33, 41, 49; Row 4, 800: 20, 28, 36, 44, 52; Row 5, 1000: 23, 31, 39, 47, 55.

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

6) The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of snacks sold, f. Thus, R = g(t, f ). Assuming there is only one price for snacks, what is that price?

A table has 5 rows and 5 columns. The columns represent t, with the following headers from left to right: 100 to 500, in increments of 100. The rows represent f, and have the following headers from top to bottom: 200 to 1000, in increments of 200. The row entries are as follows: Row 1, 200: 11, 19, 27, 35, 43; Row 2, 400: 14, 22, 30, 38, 46; Row 3, 600: 17, 25, 33, 41, 49; Row 4, 800: 20, 28, 36, 44, 52; Row 5, 1000: 23, 31, 39, 47, 55.

Diff: 3 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

7) The cost, C, of renting a car is $50 per day and $0.25 per mile. Write a cost function, C = f (d, m), for the total cost of renting a car where d is the number of days and m is the number of miles.

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

8) A company sells two products. The fixed costs for the company are $3000, the variable costs for the first product are $7 per unit, and the variable costs for the second product are $12 per unit. If the company produces (q) with subscript (1) units of the first product and (q) with subscript (2) units of the second product, which of the following is a formula for the total cost, C, as a function of (q) with subscript (1) and (q) with subscript (2)?

A) C = 3000 + 7(q) with subscript (1) + 12(q) with subscript (2) B) C = 3000 - 7(q) with subscript (1) - 12(q) with subscript (2)

C) C = 3000((q) with subscript (1) + (q) with subscript (2)) + 7(q) with subscript (1) + 12(q) with subscript (2) D) C = -3000 + 7(q) with subscript (1) + 12(q) with subscript (2)

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

9) A company sells two products. The fixed costs for the company are $3000, the variable costs for the first product are $5 per unit, and the variable costs for the second product are $11 per unit. If the company produces (q) with subscript (1) units of the first product and (q) with subscript (2) units of the second product, and the total cost is given by C = f (q) with subscript (1), (q) with subscript (2), what is f (500, 400)?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

10) A company sells two products. The fixed costs for the company are $2600, the variable costs for the first product are $4 per unit, and the variable costs for the second product are $11 per unit. If the company produces (q) with subscript (1) units of the first product and (q) with subscript (2) units of the second product, and the total cost is given by C = f (q) with subscript (1), (q) with subscript (2), is C an increasing or decreasing function of (q) with subscript (1)?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

11) The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and fuel economy in miles per gallon, m, of the car according to the formula C = (3000p/m). What is the cost when p = $2 and m = 40?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

12) The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and fuel economy in miles per gallon, m, of the car according to the formula C = (3000p/m). On the same set of axes, sketch a graph of C as a function of m with fixed gas prices of $2.00, $2.25, $2.50, and $2.75. What do you observe as gas prices increase?

A) The graphs are similarly shaped curves that get progressively higher.

B) The graphs are all straight lines from the origin with slopes that get progressively less steep.

C) The graphs are all straight lines from the origin with slopes that get progressively steeper.

D) The graphs are similarly shaped curves that get progressively lower.

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Graph and interpret cross-sections of functions of two variables.

13) 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 t?

A) Decreasing B) Increasing C) Neither

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

14) The following table gives the number f (x, y) of grape vines, in thousands, of age x in year y.

Age of Vine

Year

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

2000

3

3

3

3

3

2

2

0

0

2

2

2

1

0

0

0

0

2001

4

3

3

3

3

3

2

2

0

0

2

2

2

1

0

0

0

2002

4

4

3

3

3

1

1

1

1

0

0

1

1

1

0

0

0

2003

0

0

0

1

1

3

1

1

1

1

0

0

1

1

1

0

0

2004

7

0

0

0

1

1

3

1

1

1

1

0

0

1

1

1

0

2005

4

7

0

0

0

1

1

3

1

1

1

1

0

0

1

1

1

2006

4

4

7

0

0

0

1

1

3

1

1

1

1

0

0

1

1

2007

7

4

4

7

0

0

0

1

1

3

1

1

1

1

0

0

1

2008

12

7

4

4

7

0

0

0

1

1

3

1

1

1

1

0

0

2009

12

12

7

4

4

7

0

0

0

1

1

3

1

1

1

1

0

2010

9

12

12

7

4

4

7

0

0

0

1

1

3

1

1

1

1

2011

8

9

12

12

7

4

4

7

0

0

0

1

1

3

1

1

1

2012

8

8

9

12

12

7

4

4

7

0

0

0

1

1

3

1

1

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?

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

15) 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.)

The rows represent temperature, t in degree Celsius, and have the following headers from top to bottom: 0 to 30, in increments of 10. The row entries are as follows: Row 1, 0 degree Celsius: 0.38, 0.46, 0.43, 0.28, 0; Row 2, 10 degrees Celsius: 0.65, 0.79, 0.73, 0.47, 0.01; Row 3, 20 degrees Celsius: 0.81. 0.99. 0.91. 0.59. 0.02; Row 4, 30 degrees Celsius: 0.71, 0.87, 0.81, 0.52, 0.01.

What is the value of p(0, 50)?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

16) 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), (q) with subscript (1) and (q) with subscript (2), produced in the N.Y. and N.J. factories respectively. Suppose the cost function is given by

C ((q) with subscript (1), (q) with subscript (2)) = 3((q) with subscript (1)) with superscript (2) + (q) with subscript (1)(q) with subscript (2) + ((q) with subscript (2)) with superscript (2) + 240

Find C(100, 50).

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

17) 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(8, 60) = 190?

A) If you borrow $8,000 from the bank for 60 months (5 year loan), your monthly car loan payment is $190 .

B) If you borrow $5,000 from the bank for 60 months (8 year loan), your monthly car loan payment is $190.

C) If you borrow $190 from the bank for 60 months (5 year loan), your monthly car loan payment is $8.

D) If you borrow $8 from the bank for 60 months (5 year loan), your monthly car loan payment is $190.

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

18) 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 s?

A) Increasing B) Decreasing

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

19) A television salesman earns a fixed salary of $9.50 per hour plus a $35 commission for each television he sells. If h is the number of hours worked and s is the number of televisions sold, find a formula for E(h, s), the salesman's total earnings, and use it to calculate E(40, 32).

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

20) A television salesman earns a fixed salary of $9.50 per hour plus a $35 commission for each television he sells. If h is the number of hours worked and s is the number of televisions sold, find a formula for E(h, s), the salesman's total earnings. Is E an increasing or decreasing function of h?

Diff: 2 Var: 1

Section: 8.1

Learning Objectives: Evaluate and interpret functions of two variables represented by a formula, a table, or words.

21) The monthly car payment for a new car is a function of three variables, P = f (r, A, t), where r is the annual interest rate, A is the amount borrowed in dollars, and t is time in months before the car is paid off. Would you expect ( f ) with subscript (t) (r, A, t) to be increasing or decreasing?

Diff: 1 Var: 1

Section: 8.1

Learning Objectives: Estimate and interpret partial derivatives.

8.2 Contour Diagrams

1) The following figure shows contours for the function z = f (x, y). Is z an increasing or decreasing function of y?

Seven contour lines are graphed on an x y coordinate plane. The x axis ranges from 0 to 50, in increments of 10, and the y axis ranges from 0 to 5, in increments of 1. A line labeled negative 3 falls from (8, 0.75) to (20, 0). A line labeled negative 2 falls from (8, 1.5) to (25, 0). A line labeled negative 1 falls from (8, 2.25) to (30, 0). A line labeled 0 falls from (8, 3.2) to (35, 0). A line labeled 1 falls from (8, 4.1) to (40, 0). A line labeled 2 falls from (8, 5) to (45, 0). A line labeled 3 falls from (15, 5) to (50, 0). All values are estimated.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

2) The following figure is a contour diagram for the demand for pork as a function of the price of pork and the price of beef. Which axis corresponds to the price of beef?

Five contour lines are graphed on an x y coordinate plane. The x axis ranges from 0 to 5, in increments of 1, and the y axis ranges from 0 to 5, in increments of 1. A line labeled 10 increases from (0, 0) through (5, 1). A line labeled 20 increases from (0, 0) through (4, 2). A line labeled 30 increases from (0, 0) through (3, 3). A line labeled 40 increases from (0, 0) through (2, 4). A line labeled 50 increases from (0, 0) through (1, 5). All values are estimated.

A) The x-axis B) The y-axis

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

3) Which of the following describes the contour diagram of f (x, y) = 5x - y + 1?

A) A set of lines extending from the point (0, 1)

B) A set of lines extending from the origin

C) A set of parallel lines, all with slope 5

D) A set of parallel lines, all with slope (1/5)

Diff: 2 Var: 1

Section: 8.2

Learning Objectives: Construct a contour diagram of a function of two variables represented by a formula, a table, or in words.

4) The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. If the humidity is 60%, what temperature feels like 90°?

Nine contour curves are graphed on a coordinate plane with equally spaced gridlines. The horizontal axis labeled temperature in degrees Fahrenheit, ranges from 65 to 115, in increments of 5. The vertical axis labeled humidity in percentage, ranges from 0 to 60, in increments of 10. A curve labeled 70 decreases concave up from (70, 60) to (77, 0) through (73, 30). A curve decreases concave up from (78, 60) to (87, 0) through (82, 30). A curve labeled 90 decreases concave down from (85, 60) to (90, 30) and then decreases concave up to (99, 0). A curve decreases concave up from (90, 60) to (111, 0) through (98, 30). A curve labeled 110 decreases concave up from (93, 60) to (115, 7) through (100, 40). A curve decreases concave up from (97.5, 60) to (115, 20) through (100, 50). A curve labeled 130 decreases concave up from (100, 60) to (115, 30) through (105, 45). A curve decreases concave up from (103, 60) to (115, 35) through (106.5, 50). A curve labeled 150 decreases concave up from (107, 60) to (115, 40) through (110, 50). All values are estimated.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

5) The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Heat exhaustion is likely to occur when the heat index is 105° or higher. If the temperature is about 100°, at about what humidity level does heat exhaustion become likely?

Nine contour curves are graphed on a coordinate plane with equally spaced gridlines. The horizontal axis labeled temperature in degrees Fahrenheit, ranges from 65 to 115, in increments of 5. The vertical axis labeled humidity in percentage, ranges from 0 to 60, in increments of 10. A curve labeled 70 decreases concave up from (70, 60) to (77, 0) through (73, 30). A curve decreases concave up from (78, 60) to (87, 0) through (82, 30). A curve labeled 90 decreases concave down from (85, 60) to (90, 30) and then decreases concave up to (99, 0). A curve decreases concave up from (90, 60) to (111, 0) through (98, 30). A curve labeled 110 decreases concave up from (93, 60) to (115, 7) through (100, 40). A curve decreases concave up from (97.5, 60) to (115, 20) through (100, 50). A curve labeled 130 decreases concave up from (100, 60) to (115, 30) through (105, 45). A curve decreases concave up from (103, 60) to (115, 35) through (106.5, 50). A curve labeled 150 decreases concave up from (107, 60) to (115, 40) through (110, 50). All values are estimated.

A) 24% B) 28% C) 32% D) 36%

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

6) The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Fix the humidity level at 30%. As the temperature rises, does the heat index rise more rapidly at milder temperatures (70° to 90°) or at very hot temperatures (90° to 110°)?

Nine contour curves are graphed on a coordinate plane with equally spaced gridlines. The horizontal axis labeled temperature in degrees Fahrenheit, ranges from 65 to 115, in increments of 5. The vertical axis labeled humidity in percentage, ranges from 0 to 60, in increments of 10. A curve labeled 70 decreases concave up from (70, 60) to (77, 0) through (73, 30). A curve decreases concave up from (78, 60) to (87, 0) through (82, 30). A curve labeled 90 decreases concave down from (85, 60) to (90, 30) and then decreases concave up to (99, 0). A curve decreases concave up from (90, 60) to (111, 0) through (98, 30). A curve labeled 110 decreases concave up from (93, 60) to (115, 7) through (100, 40). A curve decreases concave up from (97.5, 60) to (115, 20) through (100, 50). A curve labeled 130 decreases concave up from (100, 60) to (115, 30) through (105, 45). A curve decreases concave up from (103, 60) to (115, 35) through (106.5, 50). A curve labeled 150 decreases concave up from (107, 60) to (115, 40) through (110, 50). All values are estimated.

A) milder temperatures B) very hot temperatures

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

7) The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Is the heat index an increasing or decreasing function of temperature?

Nine contour curves are graphed on a coordinate plane with equally spaced gridlines. The horizontal axis labeled temperature in degrees Fahrenheit, ranges from 65 to 115, in increments of 5. The vertical axis labeled humidity in percentage, ranges from 0 to 60, in increments of 10. A curve labeled 70 decreases concave up from (70, 60) to (77, 0) through (73, 30). A curve decreases concave up from (78, 60) to (87, 0) through (82, 30). A curve labeled 90 decreases concave down from (85, 60) to (90, 30) and then decreases concave up to (99, 0). A curve decreases concave up from (90, 60) to (111, 0) through (98, 30). A curve labeled 110 decreases concave up from (93, 60) to (115, 7) through (100, 40). A curve decreases concave up from (97.5, 60) to (115, 20) through (100, 50). A curve labeled 130 decreases concave up from (100, 60) to (115, 30) through (105, 45). A curve decreases concave up from (103, 60) to (115, 35) through (106.5, 50). A curve labeled 150 decreases concave up from (107, 60) to (115, 40) through (110, 50). All values are estimated.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

8) Which of the following describes the contour diagram for the function C = 50d + 0.25m (with m on the vertical axis and d on the horizontal axis)?

A) A set of parallel lines, all with slopes -200

B) A set of parallel lines, all with slopes 12.5

C) A set of parallel lines, all with slopes -0.005

D) A set of parallel lines, all with slopes 50

Diff: 2 Var: 1

Section: 8.2

Learning Objectives: Construct a contour diagram of a function of two variables represented by a formula, a table, or in words.

9) Plants need varying amount of water and sunlight to grow. The following contour diagrams each show the growth of a different plant as a function of the amount of water and sunlight. Which diagram would represent a plant that does best in a temperate climate(moderate water and sunshine)?

Three graphs plot contour diagrams between the horizontal axis labeled water and the vertical axis labeled sunlight. The first graph plots 4 concentric circles in the first quadrant of a coordinate plane. The circles are labeled 10, 8, 6, and 4, from inside to outside. The second graph plots 4 U-shaped curves in the first quadrant of a coordinate plane. The vertices of the curves are one below the other. The curves are labeled 10, 8, 6, and 4, from inside to outside. The third graph plots 4 curves in the first quadrant of a coordinate plane. The vertices of the curves are one beside the other. The curves are labeled 10, 8, 6, and 4, from inside to outside.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

10) You build a campfire while up in the mountains. It is 45°F when you start the fire. Let H(x, t) be the temperature x feet from the fire t minutes after you start it. The following figure shows the contour diagram for H. How many °F is it 8 feet from the fire after 15 minutes?

Four contour curves are graphed on a coordinate plane. The horizontal axis labeled x in feet, ranges from 0 to 10, in increments of 2, and the vertical axis labeled t in minutes, ranges from 0 to 30, in increments of 5. A concave down curve labeled 45 increases from (0, 0) through (6, 5). A concave down curve labeled 55 increases from (0, 0) through (8, 15). A concave down curve labeled 65 increases from (0, 0) through (7.8, 25). A concave down curve labeled 75 increases from (0, 0) through (6, 30). All values are estimated.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

11) You build a campfire while up in the mountains. It is 45°F when you start the fire. Let H(x, t) be the temperature x feet from the fire t minutes after you start it. The following figure shows the contour diagram for H. Is H an increasing or decreasing function of x?

Four contour curves are graphed on a coordinate plane. The horizontal axis labeled x in feet, ranges from 0 to 10, in increments of 2, and the vertical axis labeled t in minutes, ranges from 0 to 30, in increments of 5. A concave down curve labeled 45 increases from (0, 0) through (6, 5). A concave down curve labeled 55 increases from (0, 0) through (8, 15). A concave down curve labeled 65 increases from (0, 0) through (7.8, 25). A concave down curve labeled 75 increases from (0, 0) through (6, 30). All values are estimated.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

12) Below is a contour diagram depicting D, the average fox population density as a function of (x) with subscript (E), kilometers east of the western end of England, and (x) with subscript (N), kilometers north of the same point.

A graph plots kilometers north along the vertical axis versus kilometers east along the horizontal axis. The horizontal axis ranges from 0 to 180, in increments of 60. The vertical axis ranges from 0 to 150, in increments of 50. The contour line labeled 0.5 is through (90, 150), (155, 125), (105, 120), (30, 20), (125, 22), and (180, 65). The contour line labeled 1 is through (130, 150), (170, 130), (160, 110), (105, 110), (50, 22), (65, 35), (135, 25), and (180, 70). The contour lines labeled 1.5 are a loop through (65, 40), (110, 100), (165, 80), and (120, 30), another loop at (120, 55), and a line between (180, 80) and (180, 130). The contour lines labeled 2 are loops between (95, 60) and (115, 85), between (120, 50) and (130, 65), and at (160, 75). All values are estimated.

Is D increasing or decreasing at the point (120, 100) in the southern direction?

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Read and interpret properties of a function of two variables using a contour diagram.

13) Let h(x, y) = (x) with superscript (3) + (y) with superscript (3) + 3xy + 5.

Which figure best represents the contours of this function?

A)

A graph shows contour lines on an x y coordinate plane. Parallel hyperbolas oriented upward to the right have their left branches extending from the bottom left of the second quadrant into the third quadrant, and have their right branches extending from the top right of the fourth quadrant into the first quadrant. Between the left and right branches of these parabolas oriented upward to the right, parallel hyperbolas oriented upward to the left exist in the second and fourth quadrants, such that the left branches are in the second quadrant and the right branches are in the fourth quadrant.

B)

A graph shows contour lines on an x y coordinate plane. Lines slope upward to the right from the top left of the third quadrant to the top left of the first quadrant, through the second quadrant. Lines from the bottom left of the third quadrant rise upward to the right initially before falling down and to the right to the bottom left of the fourth quadrant. Lines from the top right of the first quadrant fall downward to the left initially before falling down and to the right to the top right of the fourth quadrant. Elliptical contours, inclined up and to the left are between the contour lines in the fourth quadrant.

C)

A graph shows contour lines on an x y coordinate plane. Horizontally oriented hyperbolas, parallel to each other are along the x axis. Hyperbolas inclined upward to the left and parallel to each other are in the second and fourth quadrants. Hyperbolas inclined upward to the right and parallel to each other are in the first and third quadrants.

D)

A graph shows contour lines on an x y coordinate plane. Lines slope upward to the left from the top right of the fourth quadrant to the top right of the second quadrant, through the first quadrant. Lines from the bottom right of the fourth quadrant rise upward to the left initially before falling down and to the left to the bottom right of the third quadrant. Lines from the top left of the second quadrant fall downward to the right initially before falling down and to the left to the top left of the third quadrant. Elliptical contours, inclined up and to the right are between the contour lines in the third quadrant.

Diff: 1 Var: 1

Section: 8.2

Learning Objectives: Construct a contour diagram of a function of two variables represented by a formula, a table, or in words.

14) Which of the following contour diagrams is more likely to show the population density of a region of a small town where the center of the diagram is the town center?

table ( (I. graphic(Miscellaneous:hh-appli-Z6286143047 - A contour diagram has 3 concentric circles. The circles are labeled 50, 150, and 300, from inside to outside. ) )(II. graphic(Miscellaneous:hh-appli-Z8286143047 - A contour diagram has 3 concentric circles. The circles are labeled 50,0 300, and 100, from inside to outside. )) )

Diff: 2 Var: 1

Section: 8.2

Learning Objectives: Construct a contour diagram of a function of two variables represented by a formula, a table, or in words.

8.3 Partial Derivatives

1) The demand, D, for gasoline at Station A is a function of the price, a, of gas at Station A and the price, b, of gas at Station B across the street. Thus, D = f (a, b). Do you expect ∂D/∂a to be positive or negative?

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

2) The amount, A = f (w, h), of your paycheck is a function of your hourly wage, w, and the number of hours, h, you work. Would you expect ( f ) with subscript (h)(w, h) to be positive or negative?

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

3) The amount, A = f (w, h), of your paycheck is a function of your hourly wage, w, and the number of hours, h, you work. What does the statement ( f ) with subscript (h)(10, 25) = 10 mean in terms of your paycheck?

A) An increase of 1 hour worked from 25 hours results in a wage increase of $10.

B) An increase of 1 hour worked from 10 hours results in a wage increase of $25.

C) A wage increase of $1 per hour from $25 per hour results in a wage increase of $10.

D) A wage increase of $1 per hour from $10 per hour results in a wage increase of $25.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

4) The number of tons, N, of a product produced at a factory in a day is a function of the number of workers, w, and the number of hours worked, h. We have N = f (w, h). Which of the following equations tells us that when the factory has 5 workers who work for 11 hours, adding one additional worker will cause the amount produced to go up by about 37 tons?

A) ( f ) with subscript (h)(5, 11) = 37 B) ( f ) with subscript (w)(5, 11) = 37

C) ( f ) with subscript (h)(5, 11) = -37 D) ( f ) with subscript (w)(5, 11) = -37

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

5) The following is a contour diagram for z = f (x, y). Is ( f ) with subscript (x)(x, y) positive or negative?

Four contour lines are graphed on an x y coordinate plane with equally spaced gridlines. The x axis ranges from 0 to 4, in increments of 1, and the y axis ranges from 0 to 8, in increments of 2. A line labeled 3 falls from (0, 2) to (1, 0). A line labeled 6 falls from (0, 4) to (2, 0). A line labeled 9 falls from (0, 6) to (3, 0). A line labeled 12 falls from (0, 8) to (4, 0).

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

6) The following is a contour diagram for z = f (x, y). Estimate ( f ) with subscript (y)(1, 4).

Four contour lines are graphed on an x y coordinate plane with equally spaced gridlines. The x axis ranges from 0 to 4, in increments of 1, and the y axis ranges from 0 to 8, in increments of 2. A line labeled 3 falls from (0, 2) to (1, 0). A line labeled 6 falls from (0, 4) to (2, 0). A line labeled 9 falls from (0, 6) to (3, 0). A line labeled 12 falls from (0, 8) to (4, 0).

A) -1.5 B) 0 C) 1.5 D) 3

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

7) A table of values for f (x, y) is given below. Is ( f ) with subscript (y) positive or negative?

A table has 5 rows and 5 columns. The columns represent x, with the following headers from left to right: 0 to 10, in increments of 2. The rows represent y, and have the following headers from top to bottom: 0 to 20, in increments of 5. The row entries are as follows: Row 1, 0: 500, 510, 525, 560, 590, 640; Row 2, 5: 440, 450, 470, 500, 540, 610; Row 3, 10: 410, 420, 445, 480, 520, 575; Row 4, 15: 390, 405, 430, 460, 490, 525; Row 5, 20: 375, 385, 410, 435, 475, 500.

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

8) A table of values for f (x, y) is given below. Estimate ( f ) with subscript (y)(8, 10). Use the next higher point to make your estimate.

A table has 5 rows and 5 columns. The columns represent x, with the following headers from left to right: 0 to 10, in increments of 2. The rows represent y, and have the following headers from top to bottom: 0 to 20, in increments of 5. The row entries are as follows: Row 1, 0: 500, 510, 525, 560, 590, 640; Row 2, 5: 440, 450, 470, 500, 540, 610; Row 3, 10: 410, 420, 445, 480, 520, 575; Row 4, 15: 390, 405, 430, 460, 490, 525; Row 5, 20: 375, 385, 410, 435, 475, 500.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

9) Estimate ( f ) with subscript (x)(3, 3) from the contour diagram shown below. Use the next higher point to make your estimate.

A graph plots 4 U-shaped curves on an x y coordinate plane. The x axis ranges from 0 to 6, in increments of 1, and the y axis ranges from 0 to 8, in increments of 1. The first curve labeled 5 passes through the points (1, 8), (3, 4), and (5, 8). The second curve labeled 10 passes through the points (0.75, 8), (3, 3), and (5.25, 8). The third curve labeled 15 passes through the points (0.5, 8), (3, 2), and (5.5, 8). The fourth curve labeled 20 passes through the points (0.25, 8), (3, 1), and (5.75, 8). All values are estimated.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

10) For a function F = (n, m), we are given f (20, 5) = 3.2, ( f ) with subscript (n)(20, 5) = 0.5, and ( f ) with subscript (m)(20, 5) = -0.2. Estimate f (23, 6).

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate values of a function using partial derivatives and linearization.

11) An airline's revenue, R = f (x, y), is a function of the number of full price tickets, x, and the number of discount tickets, y, sold. When 300 full price and 600 discount tickets are sold, R = $225,000. Use partial derivatives ( f ) with subscript (x)(300, 600) = 350 and ( f ) with subscript (y)(300, 600) = 200 to estimate revenue when x = 304 and y = 603.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate values of a function using partial derivatives and linearization.

12) The amount in dollars, A, spent weekly by a household on food is a function of the household's weekly income in dollars, I, and the number of people in the household, n. Thus, A = f (I, n). If (A) with subscript (I) (1000, 4) = a > 0, then which of the following is true?

A) At a weekly income of $1000 and a household size of 4, an increase of 1 person in the household produces a decrease of $a in the amount spent weekly on food.

B) At a weekly income of $1000 and a household size of 4, an increase of $1 in weekly income produces a decrease of $a in the amount spent weekly on food.

C) At a weekly income of $1000 and a household size of 4, an increase of 1 person in the household produces an increase of $a in the amount spent weekly on food.

D) At a weekly income of $1000 and a household size of 4, an increase of $1 in weekly income produces an increase of $a in the amount spent weekly on food.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

13) Suppose that the price P (in dollars) to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years). So P = f (C, A). What is the sign of (∂P/∂C)?

A) Positive B) Negative

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

14) The consumption of beef, C (in pounds per week per household) is given by the function C = f (I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound.

Do you expect ∂ f/∂p to be positive or negative?

A) Negative B) Positive

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

15) The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f (A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off. It is given that:

f (100000, 7, 20) = 775.29, f (100000, 8, 20) = 836.44, f (100000, 7, 25) = 706.77

f (120000, 7, 20) = 930.35, f (120000, 8, 20) = 1003.72, f (120000, 7, 25) = 848.13

Estimate the value of ((∂ f/∂N)) superscript ( ) subscript ((100000,7,20)).

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

16) The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f (A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off. It is given that:

f (100000, 7, 20) = 775.29, f (100000, 8, 20) = 836.44, f (100000, 7, 25) = 706.77

f (120000, 7, 20) = 930.35, f (120000, 8, 20) = 1003.72, f (120000, 7, 25) = 848.13

Estimate the value of ((∂ f/∂N)) superscript ( ) subscript ((100000,7,20)) and interpret your answer in terms of a mortgage payment. Select all answers that apply.

A) We are currently borrowing $100,000 at 7% interest rate on a 20-year mortgage.

B) The monthly payment will go down by approximately $13.704 for each extra dollar we borrow.

C) The monthly payment will go down by approximately $13.704 for each each extra year of the length of the loan.

D) The monthly payment will go down by approximately $13.704 for each extra percentage point charged.

E) The monthly payment will go up by approximately $13.704 for each each extra year of the length of the loan.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

17) The table below gives values of a function f (x, y) near x = 1, y = 2.

A table has 3 rows and 3 columns. The columns represent y, with the following headers from left to right: 1.5 to 2.5, in increments of 0.5. The rows represent x, and have the following headers from top to bottom: 0 to 2, in increments of 1. The row entries are as follows: Row 1, 0: 36, 35, 34; Row 2, 1: 38, 37, 35; Row 3, 2: 44, 42, 38.

Estimate ( f ) with subscript (x)(0, 1.5).

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

18) The consumption of beef, C (in pounds per week per household) is given by the function C = f (I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Explain the meaning of the statement: ( f ) with subscript (p) (80, 3) = -1.6, and include units in your answer.

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

19) Given the following contour diagram of f (x, y), estimate ( f ) with subscript (x)(1, 2). Use the next higher point to make your estimate.

Four contour curves are graphed on an x y coordinate plane with equally spaced gridlines. The x axis ranges from 0 to 4, in increments of 1. The y axis ranges from 0 to 4, in increments of 1. A curve labeled 2 decreases concave up from (0.25, 4) to (4, 0.25) through (1, 1). A curve labeled 4 decreases concave up from (0.5, 4) to (4, 0.5) through (1, 2) and (2, 1). A curve labeled 6 decreases concave up from (0.75, 4) to (4, 0.75) through (1, 3) and (3, 1). A curve labeled 8 decreases concave up from (1, 4) to (4, 1) through (2, 2). All values are estimated.

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

20) The volume of a cylinder is given by V = f (r, h) - π(r) with superscript (2)h, where r is the radius of the base and h is the height, both in centimeters. At a radius of 5 centimeters and a height of 8 centimeters, how much does the volume increase for each 1 cm increase in the radius?

A) 40 π (cm) with superscript (3) B) 25 π (cm) with superscript (3) C) 50 π (cm) with superscript (3) D) 80 π (cm) with superscript (3)

Diff: 2 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

21) A person's weight, w, is a function of the number of calories consumed, c, and the number of calories burned, b. Thus, w = f (c, b). Would you expect ( f ) with subscript (c)(c, b) to be positive or negative?

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

22) Given the following table of values for f (x, y), estimate ( f ) with subscript (x)(10,0.2). Use the next higher point to make your estimate.

A table has 4 rows and 4 columns. The columns represent y, with the following headers from left to right: 0 to 0.3, in increments of 0.1. The rows represent x, and have the following headers from top to bottom: 0 to 30, in increments of 10. The row entries are as follows: Row 1, 0: 100, 110, 120, 130; Row 2, 10: 97, 106, 115, 124; Row 3, 20: 93, 101, 109, 117; Row 4, 30: 88, 95, 102, 19.

Diff: 1 Var: 1

Section: 8.3

Learning Objectives: Estimate and interpret partial derivatives.

8.4 Computing Partial Derivatives Algebraically

1) Find ( f ) with subscript (x) if f (x, y) = (x) with superscript (2)(y) with superscript (4).

A) 2(x) with superscript ( )(y) with superscript (4) B) 4(x) with superscript (2)(y) with superscript (3) C) 8(x) with superscript ( )(y) with superscript (3) D) 6(x) with superscript ( )(y) with superscript (3)

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

2) Find ( f ) with subscript (y) if (x) with superscript (2) + 6xy + (y) with superscript (2).

A) 6y + 2x B) 6x + 2y C) (x) with superscript (2) + 6x + 2y D) 2x + 6 + 2y

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

3) If V = (1/3)π(r) with superscript (2)h, (∂V/∂r) = (2/3)πrh.

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

4) If V = (4/3)π(r) with superscript (2)h, (∂V/∂r) = (8/3)πr.

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

5) If f (x, y) = (x) with superscript (2)(e) with superscript (xy), find ( f ) with subscript (x)(1, 3).

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

6) The cost of building a new fence is C = 10 f + 120g, where f is the number of linear feet of fence and g is the number of gates. Find ∂C/∂ f.

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

7) Use the values of f (x, y) in the following table to estimate ( f ) with subscript (yx)(2, 3). Use the next higher point to make your estimate.

A table has 4 rows and 4 columns. The columns represent x, with the following headers from left to right: 1 to 4, in increments of 1. The rows represent y, and have the following headers from top to bottom: 1 to 4, in increments of 1. The row entries are as follows: Row 1, 1: 11, 10, 8, 7; Row 2, 2: 12, 11, 9, 8; Row 3, 3: 14, 13, 11, 10; Row 4, 4: 16, 15, 13, 12.

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

8) The amount of principal, $P, needed to obtain a balance of $B after t years at r % interest, compounded continuously, is given by the formula P = B(e) with superscript (-rt). Which of the following gives the amount principal can be changed by and still maintain the same balance if there is an increase of 1% in the interest rate?

A) -Br(e) with superscript (-rt) B) (e) with superscript (-rt) C) -Bt(e) with superscript (-rt) D) -Br(te) with superscript (-rt)

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Interpret first and second order partial derivatives.

9) Suppose the Cobb-Douglas production function for a company is given by P = 50(L) with superscript (0.70)(K) with superscript (0.40), where P is production in tons, L is the number of workers, and K is the capital investment, in thousands of dollars. How many tons are produced if the company has an initial investment of 20 thousand dollars and the company employs 20 workers? Round to the nearest ton.

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Interpret first and second order partial derivatives.

10) Suppose the Cobb-Douglas production function for a company is given by P = 50(L) with superscript (0.70)(K) with superscript (0.40), where P is production in tons, L is the number of workers, and K is the capital investment, in thousands of dollars. Find ∂P/∂K if the company has a capital investment of 25 thousand dollars and the company employs 15 workers. Round to the nearest tenth.

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

11) For V = (s) with superscript (2)h, calculate ∂V/∂(h) with superscript (2).

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

12) For V = (s) with superscript (2)h, calculate (∂) with superscript (2)V/h∂s.

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

13) For f (x, y) = ((x) with superscript (2)/y), calculate ( f ) with subscript (yy)(x, y).

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

14) If $P is invested in a bank account earning r% interest a year, compounded continuously, the balance, $B, at the end of t years is given by

B = f (P, r, t) = P(e) with superscript (rt/100)

Find ∂B/∂r.

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

15) The ideal gas law states that PV = RT for a fixed amount of gas, called a mole of gas, where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin) and R is a positive constant.

Find (∂P/∂V).

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

16) Find the following partial derivative: (H) with subscript (p)(1, 2) if H(P, T) = (3P/5P + T). Give your answer to 4 decimal places.

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

17) Find the following partial derivative: ( f ) with subscript (xy) if f (x, y) = (x) with superscript (8)(y) with superscript (9).

A) ( f ) with subscript (xy) = 9(x) with superscript (8)(y) with superscript (8)

B) ( f ) with subscript (xy) = 8(x) with superscript (7)(y) with superscript (9)

C) ( f ) with subscript (xy) = 9(x) with superscript (7)(y) with superscript (8)

D) ( f ) with subscript (xy) = 8(x) with superscript (7)(y) with superscript (8)

E) ( f ) with subscript (xy) = 72(x) with superscript (7)(y) with superscript (8)

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

18) A manufacturer sells two products. The first sells for $9 and the second sells for $5. The manufacturer's costs are given by the equation

C((q) with subscript (1), (q) with subscript (2)) = 3(q) with subscript (1) + 2(q) with subscript (2) + 90,

where (q) with subscript (1) and (q) with subscript (2) are the respective quantities produced of each product. Let π((q) with subscript (1), (q) with subscript (2)) represent the profit at a production level of (q) with subscript (1) and (q) with subscript (2) units. Find the value of ∂π/∂(q) with subscript (1) .

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

19) If f (x, y) = (x) with superscript (2)y + (y) with superscript (2)x + 3x, find ( f ) with subscript (x)(x, y).

A) 2xy + (x) with superscript (2) B) 2xy + (y) with superscript (2) + 3 C) 4xy + 3 D) 4xy

Diff: 1 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

20) If P = 5(e) with superscript (-rt), find (∂) with superscript (2)P/∂(t) with superscript (2).

A) 5(t) with superscript (2)(e) with superscript (-rt) B) 5(r) with superscript (2)(e) with superscript (-rt) C) -5(t) with superscript (2)(e) with superscript (-rt) D) -5(r) with superscript (2)(e) with superscript (-rt)

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

21) If P = 8(e) with superscript (-rt), find (∂) with superscript (2)P/∂r ∂t.

A) 8(re) with superscript (-rt) B) -8(re) with superscript (-rt) C) 8(rt - 1)(e) with superscript (-rt) D) -8(rt - 1)(e) with superscript (-rt)

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

22) If f (x, y) = (x) with superscript (2) cos y, does ( f ) with subscript (yx)(x, y) = -2x sin y?

Diff: 2 Var: 1

Section: 8.4

Learning Objectives: Algebraically compute or estimate first and second order partial derivatives.

8.5 Critical Points and Optimization

1) A contour diagram of f (x, y) is given in the following figure. Is there a global minimum near the point (5, 3)?

A graph plots 6 curves and 2 circles on an x y coordinate plane. The x axis ranges from 0 to 6, in increments of 1, and the y axis ranges from 0 to 7, in increments of 1. The first curve labeled 1 is U-shaped that passes through the points (1.3, 7), (3, 4), and (4.6, 7). The second curve labeled 3 is U-shaped that passes through the points (1.6, 7), (3, 5), and (4.3, 7). The third curve labeled 5 is U-shaped that passes through the points (2, 7), (3, 6), and (4, 7). The fourth curve labeled 1 is inverted U-shaped that passes through the points (1.3, 0), (3, 3), and (4.6, 0). The fifth curve labeled 2 is inverted U-shaped that passes through the points (1.6, 0), (3, 2), and (4.3, 0). The sixth curve labeled 3 is inverted U-shaped that passes through the points (2, 0), (3, 1), and (4, 0). The first circle of radius 0.8 is centered at (1, 4) and is labeled negative 1. The second circle of radius 0.7 is centered at (5, 3) and is labeled negative 2. All values are estimated.

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

2) A contour diagram of f (x, y) is given in the following figure. Which one of the following statements is true?

Three contour diagrams and a line are graphed on a coordinate plane. The horizontal axis ranges from 0 to 7, in increments of 1. The vertical axis ranges from 0 to 6, in increments of 1. The first contour diagram has three concentric circles, centered at (1, 3.8). The innermost circle is labeled 4, and the outermost circle is labeled 1. The second contour diagram has three concentric circles, centered at (3, 1.6). The innermost circle is labeled negative 4, and the outermost circle is labeled, negative 1. The third contour diagram has seven concentric circles, centered at (5.2, 4.9). The innermost circle is labeled 12, the third circle from the center is labeled 6, and the outermost circle is labeled, 1. The line labeled 0 increases from (0, 0.6) to (3, 3.8) and then decreases to (7, 0.6). All values are estimated.

A) The function has a local maximum of about 5 at (1, 3.8), a global maximum of about 14 at (5.2, 4.9), and a global minimum of about -5 at (3, 1.6).

B) The function has a global maximum of about 5 at (1, 3.8), a local maximum of about 14 at (5.2, 4.9), and a global minimum of about -5 at (3, 1.6).

C) The function has a local minimum of about 5 at (1, 3.8), a global minimum of about 14 at (5.2, 4.9), and a global maximum of about -5 at (3, 1.6).

D) The function has a global maximum of about 5 at (1, 3.8), a global maximum of about 14 at (5.2, 4.9), and a global minimum of about -5 at (3, 1.6).

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

3) The following table gives values for a function f (x, y). For 0 ≤ x ≤ 40 and 0 ≤ y ≤ 20, there is a global minimum of at most ________ near the point ( ________, ________ ).

A table has 5 rows and 5 columns. The columns represent x, with the following headers from left to right: 0 to 40, in increments of 10. The rows represent y, and have the following headers from top to bottom: 0 to 20, in increments of 5. The row entries are as follows: Row 1, 0: 1.5, 1.7, 1.3, 1.1, 0.9; Row 2, 5: 1.8, 1.9, 1.6, 1.0, 0.6; Row 3, 10: 1.3, 1.2, 1.0, 0.7, 0.4; Row 4, 15: 0.9, 1.0, 0.8, 0.3, 0.2; Row 5, 20: 0.8, 0.7, 0.6, 0.4, 0.3.

Part A: 0.2

Part B: 40

Part C: 15

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

4) The function f (x, y) = (x) with superscript (2) + 2xy + 2(y) with superscript (2) has a local ________ (maximum / minimum / neither) at the critical point, where x = ________ and y = ________.

Part A: minimum

Part B: 3

Part C: -3

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

5) The function f (x, y) = (x) with superscript (2) + 3xy - 3y has a local ________(maximum / minimum / neither) at the critical point, where x = ________ and y = ________.

Part A: neither

Part B: 1

Part C: -2

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

6) The function f (x, y) = (x) with superscript (3) + (y) with superscript (2) - 2(x) with superscript (2) + 3y - 9 has a critical point which is a local minimum when x = ________ and y = ________.

Part A: 4/3

Part B: -3/2

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

7) The function f (x, y) = 500 - 2(x) with superscript (2) + 6x + 2xy - 3(y) with superscript (2) + 12y has a local maximum value of ________ when x = ________ and y = ________.

Part A: 527

Part B: 3

Part C: 3

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

8) Two products are manufactured in quantities (q) with subscript (1) and (q) with subscript (2) and sold at prices $8 and $12 respectively. The cost of producing them is given by C = ((q) with subscript (1)) with superscript (2) + 2((q) with subscript (2)) with superscript (2) + 10. Find the maximum profit that can be made.

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

9) A company sells two styles of jeans in quantities (q) with subscript (1) and (q) with subscript (2) at prices (p) with subscript (1) and (p) with subscript (2) respectively, with (p) with subscript (1) and (p) with subscript (2) in dollars. The quantities demanded depend on both (p) with subscript (1) and (p) with subscript (2) according to the formulas

(q) with subscript (1) = 80 - 6(p) with subscript (1) + 4(p) with subscript (2) and (q) with subscript (2) = 140 + 5(p) with subscript (1) - 7(p) with subscript (2).

Write a formula for total sales revenue as a function of (p) with subscript (1) and (p) with subscript (2), and use it to determine what prices the company should charge to maximize sales revenue. They should charge $________ for style (q) with subscript (1) and $________ for style (q) with subscript (2).

Part A: 27.36

Part B: 27.59

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

10) Suppose f (x, y) = (x) with superscript (2) + Ax + (y) with superscript (2) + By + C. Then f (x, y) has a local minimum value of 12 at the point (2, 3) when A = ________, B = ________, and C = ________.

Part A: -4

Part B: -6

Part C: 25

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

11) Find all the critical points of the function f (x, y) = xy + (8/(x) with superscript (2)) + (8/(y) with superscript (2)) .

Classify these critical points as local maxima, local minima, or neither.

Diff: 3 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

12) Is (0, 0) a critical point of the following function?

f (x, y) = -6x2 + 3y4

A) Yes: (global) maximum.

B) Yes: neither a maximum nor a minimum.

C) Yes: (global) minimum.

D) No.

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

13) Let h(x, y) = (x) with superscript (3) + (y) with superscript (3) + 9xy + 8.

Determine all local critical points. Are the local extrema also global extrema?

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

14) Suppose that f (x, y) = (x) with superscript (2) + 5xy + (y) with superscript (2).

Find and classify the critical point(s) as local maxima, local minima, or neither.

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

15) Find all the critical points of f (x, y) = (x) with superscript (3) - 3x + (y) with superscript (2) - 6y and classify each as maximum, minimum, or neither.

Select all possible choices.

A) The point (1, 3) is a local minimum.

B) The point (-1, -3) is neither a maximum nor a minimum.

C) The point (-1, 3) is neither a maximum nor a minimum.

D) The point (1, 3) is a local maximum.

E) The point (1, -3) is a local maximum.

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

16) The function f (x, y) = (x) with superscript (3) - 3x + (y) with superscript (2) has a local minimum at (1, 1). Which of the following is a sketch of the level curves of f near this point?

A)

Three concentric circles with the common center marked by a point.

B)

Two pairs of vertical and horizontally oriented hyperbolas are centered at a marked point. The gap between the branches of the hyperbolas in each pair is small.

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

17) Find the critical points of f (x, y) = (x) with superscript (3) - 9xy + (y) with superscript (2) and classify each as maximum, minimum or neither.

The point ((27/2), (243/4)) is a local minimum.

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

18) The contour diagram of f is shown below. Which of the points A, B, C, D, and E appear to be critical points? Select all that apply.

A graph shows contour lines on an x y coordinate plane. Diagonal lines labeled 1 intersect at the origin labeled C and extend at about 45 degrees in all four quadrants. Above the diagonal line are two almost concave up curves, labeled 2 and 3, from bottom to top, have their maximum points on the vertical axis. The curve labeled 2 intersects the vertical axis at a point D. A point E is marked on this curve in the first quadrant. An almost concave down curve labeled 2 is below the diagonal lines with its maximum point on the negative vertical axis. A horizontal hyperbola has vertices of its branches labeled 0 on the x axis. Two elliptical contours are centered at points A and b on the positive and negative horizontal axis, respectively, beyond the parabola.

A) A B) B C) C D) D E) E

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

19) Consider the function f (x, y) = (x) with superscript (4) + 2(x) with superscript (3)y - 10(x) with superscript (2)(y) with superscript (2) + 7x(y) with superscript (2) + 5.

Check that (0,0) is a critical point of f and classify it as a local minimum, local maximum or neither.

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

20) The point (-2, 1) is a critical point of g(x, y) = 2(x) with superscript (3) - 48xy + 24x(y) with superscript (2).

Classify it either as a local minimum, local maximum, or neither.

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

21) The function f (x, y) = (e) with superscript ((-(x-a)) with superscript (2) - ((y-b)) with superscript (2)) where a and b are constants is sometimes referred to as a "bump function" and is used to construct functions which take on maximum values at certain points. Show that f (x, y) has a maximum at (a, b).

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

22) Find a and b so that f (x, y) = a(x) with superscript (2) + bxy + (y) with superscript (2) has a critical point at (1, 5).

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

23) The Perfect House company produces two types of bathtub, the Hydro Deluxe model and the Singing Bird model. The company noticed that demand and prices are related. In particular,

for Hydro Deluxe: demand = 1600 - price of Hydro Deluxe + price of Singing Bird

for Singing Bird: demand = 1250 + price of Hydro Deluxe - 2(price of Singing Bird).

The costs of manufacturing the Hydro Deluxe and Singing Bird are $500 and $300 per unit respectively. Determine the price of each model that gives the maximum profit.

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

24) A company has two manufacturing plants which manufacture the same item. Suppose the cost function is given by C(q) with subscript (1), (q) with subscript (2) = 6((q) with subscript (1)) with superscript (2) + (q) with subscript (1)(q) with subscript (2) + ((q) with subscript (2)) with superscript (2) where (q) with subscript (1) and (q) with subscript (2) are the quantities (measured in thousands) produced in each plant. The total demand (q) with subscript (1) + (q) with subscript (2) is related to the price, p, by p = 110 - 0.5(q) with subscript (1) + (q) with subscript (2).

How much should each plant produce in order to maximize the company's profit?

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

25) Suppose that f (x, y) = 3(x) with superscript (2) + 3.0000(y) with superscript (2) - x.

Find and classify (as local maxima, minima, or neither) all critical points of f.

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

26) Let f (x, y) = k(x) with superscript (2) - 3kx + (y) with superscript (2), where k ≠ 0. Find the critical points of f.

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

27) The critical point of f (x, y) = (x) with superscript (2) + 6xy + (y) with superscript (2) 10 occurs at x = ________, y = ________, and is a local ________ (maximum / minimum / neither).

Part A: 0

Part B: 0

Part C: neither

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

28) A company's cost function to produce x units of one product and y units of a second product is given by

C(x, y) = 1000 + 4(x) with superscript (2) - 4xy + 3(y) with superscript (2) - 48y.

The minimum cost occurs when x = ________ and y = ________.

Part A: 6

Part B: 12

Diff: 1 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

29) A company manufactures x units of one item and y units of another. The total cost, C, in dollars of producing these two items is given by the function

C = 5(x) with superscript (2) + xy + 4(y) with superscript (2) + 2500.

Use Lagrange multipliers to find the minimum cost subject to the constraint that 100 items (total) must be produced. The minimum cost occurs when x = ________ and y = ________. Round your answers to the nearest whole number.

Part A: 44

Part B: 56

Diff: 2 Var: 1

Section: 8.5

Learning Objectives: Identify critical points and classify as minimum values, maximum values, or neither.

8.6 Constrained Optimization

1) Use Lagrange multipliers to find the maximum or minimum values of f (x, y) = 3xy, subject to the constraint 4x + y = 46. λ = _______ and the maximum/minimum value of f (x, y) is ________.

Part A: 11.5

Part B: 264.5

Diff: 1 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

2) Use Lagrange multipliers to find the maximum or minimum values of f (x, y) = (x) with superscript (2) + (y) with superscript (2), subject to the constraint 2x - y = 30. λ = ________ and the maximum/minimum value of f (x, y) is ________.

Part A: 12

Part B: 180

Diff: 1 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

3) Use Lagrange multipliers to find the maximum and minimum values of f (x, y) = 2x + 2y, subject to the constraint (x) with superscript (2) + (y) with superscript (2) = 8. For the maximum value, λ = ________ (use the decimal form), and the maximum value of f (x, y) is ________.

Part A: 0.5

Part B: 8

Diff: 1 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

4) The following figure shows contours of f (x, y) and the constraint g(x, y) = c . What is the maximum value of f subject to this constraint?

Four contour curves and a line are graphed on an x y coordinate plane. Both the axes range from 0 to 6, in increments of 1. A curve labeled 200 decreases concave up from (0.2, 6) to (6, 0.2) through (1, 1). A curve labeled 400 decreases concave up from (0.35, 6) to (6, 0.35) through (1, 2). A curve labeled 600 decreases concave up from (0.6, 6) to (6, 0.6) through (1, 3) and (3, 1). A curve labeled 800 decreases concave up from (0.7, 6) to (6, 0.7) through (1, 4) and (4, 1). The line falls downward to the right through (0, 4), (1, 2), and (2, 0). The line is labeled g of x and y equals c. All values are estimated.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

5) A company operates two plants that make the same product. If Plant 1 produces quantity x of the product and Plant 2 produces quantity y of the product, the cost functions are (C) with subscript (1) = 8.6 + 0.03(x) with superscript (2) and (C) with subscript (2) = 5.5 + 0.04(y) with superscript (2). The total market demand q = x + y for the product is related to the selling price p by p = 60 - 0.4q. What quantity should Plant 2 produce to maximize the company's profit?

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

6) The quantity, Q, of a good produced depends on the quantities (x) with subscript (1) and (x) with subscript (2) of the two main materials used:

Q = ((x) with subscript (1)) with superscript (0.5)((x) with subscript (2)) with superscript (0.5).

Material (x) with subscript (1) costs $25 per unit, and material (x) with subscript (2) costs $50 per unit. We want to minimize the cost of producing 100 units of the good. What is the objective function?

A) C = 25(x) with subscript (1) + 50(x) with subscript (2) B) ((x) with subscript (1)) with superscript (0.5)((x) with subscript (2)) with superscript (0.5) = 100

C) 25(x) with subscript (1) + 50(x) with subscript (2) = 100 D) C = 25((x) with subscript (1)) with superscript (0.5)50((x) with subscript (2)) with superscript (0.5)

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

7) The quantity, Q, of a good produced depends on the quantities (x) with subscript (1) and (x) with subscript (2) of the two main materials used:

Q = ((x) with subscript (1)) with superscript (0.5)((x) with subscript (2)) with superscript (0.5).

Material (x) with subscript (1) costs $75 per unit, and material (x) with subscript (2) costs $25 per unit. We want to minimize the cost of producing 100 units of the good. What is the constraint function?

A) C = 75(x) with subscript (1) + 25(x) with subscript (2) B) ((x) with subscript (1)) with superscript (0.5)((x) with subscript (2)) with superscript (0.5) = 100

C) 75(x) with subscript (1) + 25(x) with subscript (2) = 100 D) C = 75((x) with subscript (1)) with superscript (0.5)25((x) with subscript (2)) with superscript (0.5)

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

8) The quantity, Q, of a good produced depends on the quantities (x) with subscript (1) and (x) with subscript (2) of the two main materials used:

Q = ((x) with subscript (1)) with superscript (0.5)((x) with subscript (2)) with superscript (0.5).

Material (x) with subscript (1) costs $75 per unit, and material (x) with subscript (2) costs $50 per unit. We want to minimize the cost of producing 100 units of the good. Using Lagrange multipliers, we see that the minimum cost of ________ occurs when (x) with subscript (1) = ________ and (x) with subscript (2) = ________.

Part A: $12,247.45

Part B: 81.65

Part C: 122.47

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

9) The production function for a company is P = 300(x) with superscript (0.75)(y) with superscript (0.25), where P is the amount produced given x units of labor and y units of equipment. Each unit of labor costs $900 and each unit of equipment costs $350. Assuming the goal of the company is to maximize production given a fixed budget of $40,000, what are the objective and constraint functions?

A) objective: f (x, y) = x + y; constraint: 300(900(x) with superscript (0.75)350(y) with superscript (0.25)) = 40,000

B) objective: f (x, y) = 900(x) with superscript (0.75)350(y) with superscript (0.25); constraint: x + y = 40,000

C) objective: f (x, y) = 300(x) with superscript (0.75)(y) with superscript (0.25); constraint: 900x + 350y = 40,000

D) objective: f (x, y) = 900x + 350y; constraint: 300(x) with superscript (0.75)(y) with superscript (0.25) = 40,000

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

10) The production function for a company is P = 300(x) with superscript (0.75)(y) with superscript (0.25), where P is the amount produced given x units of labor and y units of equipment. Each unit of labor costs $800 and each unit of equipment costs $400. Assuming the goal of the company is to maximize production given a fixed budget of $40,000, what is the meaning of the Lagrange multiplier λ?

A) The additional units that can be produced if the budget is increased by $1

B) The amount it would cost the company to produce one more unit

C) The additional units of labor allowed if the budget is increased by $1

D) The amount it would cost the company to obtain one additional unit of equipment

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

11) The production function for a company is P = 300(x) with superscript (0.75)(y) with superscript (0.25), where P is the amount produced given x units of labor and y units of equipment. Each unit of labor costs $900 and each unit of equipment costs $350. Assuming the goal of the company is to minimize cost given a fixed production goal of 9000 units produced, what are the objective and constraint functions?

A) objective: f (x, y) = x + y; constraint: 300(900(x) with superscript (0.75)350(y) with superscript (0.25)) = 9000

B) objective: f (x, y) = 900(x) with superscript (0.75)350(y) with superscript (0.25); constraint: x + y = 9000

C) objective: f (x, y) = 300(x) with superscript (0.75)(y) with superscript (0.25); constraint: 900x + 350y = 9000

D) objective: f (x, y) = 900x + 350y; constraint: 300(x) with superscript (0.75)(y) with superscript (0.25) = 9000

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

12) The production function for a company is P = 300(x) with superscript (0.75)(y) with superscript (0.25), where P is the amount produced given x units of labor and y units of equipment. Each unit of labor costs $1100 and each unit of equipment costs $300. Assuming the goal of the company is to minimize cost given a fixed production goal of 8000 units produced, what is the meaning of the Lagrange multiplier λ?

A) The additional units that can be produced if the budget is increased by $1

B) The amount it would cost the company to produce one more unit

C) The additional units of labor allowed if the budget is increased by $1

D) The amount it would cost the company to obtain one additional unit of equipment

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Interpret lambda, the Lagrange multiplier value.

13) Find the maximum value of f (x, y) = 8xy subject to the constraint equation 2x + y = 14 using Lagrange multipliers.

Diff: 1 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

14) For a production function f (x, y), the maximum production cost of $400 is given by f (15, 32) = 90 units, with λ = 0.3. Estimate the production if the budget cost is raised to $410.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

15) A mop company can produce P(x, y) mops using x units of capital and y units of labor, with production costs C(x, y) dollars. With a budget of $300,000, the maximum production is 65,000 mops, using $250,000 in capital and $50,000 in labor. The Lagrange multiplier is λ = 0.3. Which of the following is true?

A) The objective function is P(x, y) and the constraint is C(x, y) = 300,000.

B) The objective function is P(x, y) and the constraint is C(x, y) = 65,000.

C) The objective function is C(x, y) and the constraint is P(x, y) = 300,000.

D) The objective function is C(x, y) and the constraint is P(x, y) = 65,000.

Diff: 3 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

16) A mop company can produce P(x, y) mops using x units of capital and y units of labor, with production costs C(x, y) dollars. With a budget of $500,000, the maximum production is 65,000, using $350,000 in capital and $150,000 in labor. The Lagrange multiplier is λ = 0.2. What are the units for λ?

A) Number of budget dollars per mop

B) Number of mops per budget dollar

C) Number of units of capital per mop

D) Number of units of labor per budget dollar

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Interpret lambda, the Lagrange multiplier value.

17) A mop company can produce P(x, y) mops using x units of capital and y units of labor, with production costs C(x, y) dollars. With a budget of $500,000, the maximum production is 65,000, using $400,000 in capital and $100,000 in labor. The Lagrange multiplier is λ = 0.3. What is the practical meaning of the statement λ = 0.3?

A) If the number of units of labor is increased by 1, we expect the number of mops to increase by about 0.3.

B) If the number of units of capital is increased by 1, we expect the budget to increase by about $0.3.

C) If the budget is increased by $1, we expect the number of mops to increase by about 0.3.

D) If the number of mops is increased by 1, we expect the budget to increase by about $0.3.

Diff: 3 Var: 1

Section: 8.6

Learning Objectives: Interpret lambda, the Lagrange multiplier value.

18) The quantity, Q, of a good produced depends on the number of workers, W, and the amount of capital invested, K, according to the Cobb-Douglas function

Q = 9(W) with superscript (2/3)(K) with superscript (1/3).

In addition, we know that labor costs are $20 per employee, capital costs are $8 per unit, and the budget is $2500. The maximum production level is about ________ units, where W = ________ and K = ________. Round to the nearest whole number.

Part A: 808 units of the good

Part B: 83 workers

Part C: 104 units of capital

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

19) The quantity, Q, of a good produced depends on the number of workers, W, and the amount of capital invested, K, according to the Cobb-Douglas function

Q = 9(W) with superscript (2/3)(K) with superscript (1/3).

If the maximum production level at a budget of $2500 is 976 and λ = 0.39, estimate the production if the budget is increased by $100.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

20) Suppose that you want to find the maximum and minimum values of f (x, y) = (x) with superscript (2) + (y) with superscript (2) subject to the constraint x + 4y = 1.

Use the method of Lagrange multipliers to find the exact location(s) of any extrema.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

21) Suppose the quantity, q, of a good produced depends on the number of workers, w, and the amount of capital, k, invested and is represented by the Cobb-Douglas function q = 6(w) with superscript ((3/4))(k) with superscript ((1/4)). In addition, labor costs are $12 per worker and capital costs are $20 per unit, and the budget is $1680. Using Lagrange multipliers, find the optimum number of workers.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

22) The owner of a jewelry store has to decide how to allocate a budget of $300,000. He notices that the earnings of the company depend on investment in inventory (x) with subscript (1) (in thousands of dollars) and expenditure (x) with subscript (2) on advertising (in thousands of dollars) according to the function f (x, y) = ((x) with subscript (1)) with superscript ((2/3))((x) with subscript (2)) with superscript ((1/3)).

How should the owner allocate the $300,000 between inventory and advertising to maximize his earnings?

Diff: 1 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

23) The Green Leaf Bakery makes two types of chocolate cakes, Delicious and Extra Delicious. Each Delicious requires 0.1 lb of European chocolate, while each Extra Delicious requires 0.2 lb. Currently there are only 234 lb of chocolate available each month. Suppose the profit function is given by: p(x, y) = 148x - 0.2(x) with superscript (2) + 200y - 0.1(y) with superscript (2),

where x is the number of Delicious cakes and y is the number of Extra Delicious cakes that the bakery produces each month.

(a) How many of each cake should the bakery produce each month to maximize profit?

(b) What is the value of λ? What does it mean?

(c) It will cost $15.00 to get an extra pound of European chocolate. Should the bakery buy it?

(a) x = 366.67, y = 986.67.

(b) λ = 13.33. This means that the profit will increase by $13.33 for every extra pound of European chocolate.

(c) If the bakery has to spend $15.00 to get that extra pound of chocolate, but will get back $13.33 (the value of λ) in return, it is a bad deal.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Interpret lambda, the Lagrange multiplier value.; Set up and solve constrained optimization problems using Lagrange multipliers.

24) A company manufactures x units of one item and y units of another. The total cost, C, in dollars of producing these two items is given by the function

C = 4(x) with superscript (2) + xy + 4(y) with superscript (2) + 3000.

Use Lagrange multipliers to find the minimum cost subject to the constraint that 100 items (total) must be produced. What is the value of λ to one decimal place?

Diff: 1 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

25) The number, N, of children that can be enrolled in a private school is a function of the number of certified teachers, T, and the number of aides, A, available, according to the formula

N(T, A) = 35(T) with superscript ( 0.7)(A) with superscript (0.3).

Teachers average an annual salary of $35,000 and aides average an annual salary of $25,000. The annual budget for salaries is B = $740,000. If we want to maximize enrollment, which of the following is true?

A) The objective function is N(T, A) = 35(T) with superscript ( 0.7)(A) with superscript (0.3), and the constraint is 35,000T + 25,000A = 740,000.

B) The objective function is N(T, A) = 35(T) with superscript ( 0.7)(A) with superscript (0.3), and the constraint is 35,000T + 25,000A = 60,000.

C) The objective function is B(T, A) = 35,000T + 25,000A, and the constraint is 35(T) with superscript ( 0.7)(A) with superscript (0.3) = 740,000.

D) The objective function is B(T, A) = 35,000T + 25,000A, and the constraint is 35(T) with superscript ( 0.7)(A) with superscript (0.3) = 60,000.

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

26) The number, N, of children that can be enrolled in a private school is a function of the number of certified teachers, T, and the number of aides, A, available, according to the formula

N(T, A) = 35(T) with superscript ( 0.7)(A) with superscript (0.3).

Teachers average an annual salary of $32,000 and aides average an annual salary of $20,000. The annual budget for salaries is B = $740,000. Using Lagrange multipliers to maximize enrollment, we find that T = ________, A = ________, and λ = ________. Round T and A to the nearest whole number; round λ to 5 decimal places.

Part A: 16

Part B: 11

Part C: 0.00068

Diff: 2 Var: 1

Section: 8.6

Learning Objectives: Set up and solve constrained optimization problems using Lagrange multipliers.

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Document Type:
DOCX
Chapter Number:
8
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 8 Functions Of Several Variables
Author:
Hughes Hallett

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