TEST BANK for Functions and Change: A Modeling Approach to College Algebra 5th Edition by Crauder, E by StudyGuide

Section 1.1 Functions Given by Formulas TRUE/FALSE 1. The functional notation ANS: F

means

PTS: 1

2. The functional notation ANS: T

DIF:

easy

means the function whose name is

PTS: 1

DIF:

evaluated at .

easy

3. The only valid way of presenting a function is with a formula. ANS: F

PTS: 1

DIF:

easy

4. For functions that model practical phenomena, the variables involved often have units associated with them. ANS: T 5. If a.m.

PTS: 1

represents the temperature

ANS: T

DIF:

easy

hours after midnight, then

PTS: 1

DIF:

indicates the temperature at 7:00

easy

6. If represents the value of a car years after it is purchased from the dealer, then the value of the car on the showroom floor. ANS: T 7. If

PTS: 1

easy

represents the cost of purchasing burgers, orders of fries, and represents 10 burgers, 6 orders of fries, and 12 drinks.

ANS: F 8. If can use

DIF:

PTS: 1

DIF:

drinks, then

easy

represents the cost of purchasing belts, pairs of earrings, and necklaces, then we to indicate the cost of buying 2 belts, 4 pair of earrings, and 1 necklace.

ANS: T

PTS: 1

DIF:

9. If represents federal defense spending defense spending in 2012. ANS: F

PTS: 1

easy

years after 2000, then

DIF:

easy

MULTIPLE CHOICE 1. If

indicates

, calculate the value of

represents federal

a. 7.73 b. 596.48 ANS: A 2. If a. 31.58 b. 42.82 ANS: A 3. If

c. 11.48 d. None of the above PTS: 1 , calculate the value of

PTS: 1

4. If

5. If

PTS: 1

6. If a. –9.6 b. 13.32 ANS: D 7. If

8. If

PTS: 1

DIF:

easy

DIF:

easy

, calculate the value of

. c. 9.05 d. 9.02

PTS: 1

DIF:

easy

, calculate the value of c. –11.2 d. –8.12 PTS: 1

DIF:

easy

, calculate the value of

. c. 2 d. 3.62

PTS: 1

DIF:

, calculate the value of

a. 274 b. 1001.78 ANS: C

easy

c. 1.66 d. 2.32

a. 0.6 b. 6.64 ANS: D

DIF:

, calculate the value of

a. 7.71 b. 15.76 ANS: D

. c. 44.82 d. None of the above

c. 386.09 d. –386.09

a. 5.55 b. 3.41 ANS: C

easy

, calculate the value of

a. 104.59 b. 491.66 ANS: B

DIF:

easy .

c. 273.62 d. –0.38 PTS: 1

DIF:

easy

9. If

, calculate the value of c. –1.65 d. 18.09

a. 2.71 b. 14.79 ANS: A

PTS: 1

10. If

DIF:

easy

, calculate the value of

a. 0.64 b. 1 ANS: A

c. 1.84 d. 4.47 PTS: 1

DIF:

easy

11. The distance, in miles, from me to a moving train is given by . Here represents hours since I heard the train whistle. Calculate the distance to the train 6 hours after I heard the whistle. a. 5 miles c. 63025 miles b. 251.05 miles d. 178.06 miles ANS: B

PTS: 1

DIF:

medium

12. The number of electrical outlets needed in an office building depends on the number of offices and the number of employees. If there are offices and employees, then the number of outlets needed is . Use functional notation to represent the number of outlets needed if there are 36 offices and 61 employees. Then calculate that value. Round your answer to the nearest whole number. a. Functional notation: . Value: 118 outlets. b. Functional notation: . Value: 136 outlets. c. Functional notation: . Value: 173 outlets. d. Functional notation: . Value: 96 outlets. ANS: A

PTS: 1

DIF:

medium

13. For medium-sized dog breeds, the predicted adult weight, in pounds, of a puppy that weighs pounds at age weeks is given by the function . Use functional notation to express the predicted weight of a puppy that weighs 5 pounds at age 15 weeks. Then calculate that value. a. Functional notation: . Value is 17.33 pounds. b. Functional notation: . Value is 156 pounds. c. Functional notation: . Value is 15 pounds. d. None of the above. ANS: A

PTS: 2

DIF:

medium

14. In the event of an emergency stop, the speed , in miles per hour, of a car when brakes are applied can be calculated from the length , in feet, of skid marks. The relationship is . Suppose skid marks are 70.01 feet long. Use functional notation to express the speed of the car, and then calculate that value. a. miles per hour c. miles per hour b. miles per hour d. miles per hour ANS: A

PTS: 1

DIF:

medium

15. If you are driving at a speed of miles per hour and make an emergency stop, you can expect to leave skid marks of length feet. The relationship is . Suppose your speed is 80 miles per hour. Use functional notation to express the length of skid marks an emergency stop will produce, and then calculate that value. a. feet c. feet b. feet d. feet ANS: C

PTS: 1

DIF:

medium

16. The height , in feet, of the winning pole value in the early years of the Olympic games can be modeled by , where is years since 1900. Use functional notation to express what the height of the winning pole vault would have been in 1940. Then calculate that value. a. Functional notation: . Value is 433.36 feet. b. Functional notation: . Value is 21010.36 feet. c. Functional notation: . Value is 17.23 feet. d. Functional notation: . Value is 321.23 feet. ANS: C

PTS: 2

DIF:

easy

17. A rock is tossed upward from the top of a building and allowed to fall to the ground. Its height, in feet, above the ground after seconds is given by . Use functional notation to express the height of the rock after 1.32 seconds . Then calculate that value. a. Functional notation . Value is 84.92 feet. b. Functional notation . Value is feet. c. Functional notation . Value is 108.58 feet. d. None of the above. ANS: A

PTS: 2

DIF:

easy

18. A water source is contaminated with a toxic chemical and is being cleaned. The amount of chemical, in grams, remaining hours after the cleaning process began is given by . How much of the chemical is removed from time to time ? a. 632.51 grams c. 24.34 grams b. 1.13 grams d. 23.03 grams ANS: D

PTS: 1

DIF: medium

19. A desalination process is removing salt from a container of sea water. The amount of salt, in kilograms, remaining hours after the cleaning process began is given by . How much of the salt is removed from time to time ? a. 4.38 kilograms c. 7.18 kilograms b. 1.74 kilograms d. 6.02 kilograms ANS: C

PTS: 1

20. The number of armadillos in a certain area

DIF:

medium

years since observation began is given by .

How much did the armadillo population grow from year 3 to year 4? Round your answer to the nearest whole number. a. 0 c. 384 b. 214 d. 597 ANS: B

PTS: 1

DIF:

medium

21. The balance of a savings account months since it was opened depends on the APR. If the APR is , expressed as a decimal, then the balance is given by . What is the balance after 10 years if the APR is 4 percent? (Be sure first to express the APR as a decimal.) a. $110950.28 c. $6459.42 b. $9314.72 d. None of the above ANS: B

PTS: 1

DIF: medium

SHORT ANSWER 1. If represents the library charges on terms the meaning of .

books that are

weeks overdue, explain in practical

ANS: It is the library charges due on 6 books that are 3 weeks overdue. PTS: 1

DIF: easy

2. If represents the library charges on terms the meaning of .

books that are

weeks overdue, explain in practical

ANS: It is the library charges on 6 books that are 7 weeks overdue. PTS: 1

DIF: easy

3. The balance, in dollars, of an investment after

months is given by

How much money was originally invested? ANS: 2409 dollars PTS: 1

DIF: easy

4. A rock is tossed upward from the top of a building and allowed to fall to the ground. Its height above ground, in feet, seconds after the toss is given by

How tall is the building? ANS: The building is 35 feet tall. PTS: 1

DIF: easy

5. Let denote the traffic fine, in dollars, associated with driving limit. Explain in practical terms the meaning of .

miles per hour over the speed

ANS: It is the traffic fine associated with driving 12 miles per hour over the speed limit. PTS: 1

DIF: easy

6. Let denote the temperature of an oven minutes after it is turned on. Use functional notation to indicate the temperature of the oven one hour and 12 minutes after it is turned on. ANS: PTS: 1

DIF: easy

7. Let denote the balance, in dollars, of an account months after the account is opened. Use functional notation to indicate the balance of the account after 2 years and 7 months. ANS: PTS: 1

DIF: easy

8. Let denote the monthly payment if you borrow dollars at an APR of percent, and the loan is repaid in months. Use functional notation to indicate the monthly payment if you borrow 5437 dollars at an APR of 2.37 percent, and you repay the loan in 5 years. ANS: PTS: 1

DIF: medium

9. Let denote the monthly payment if you borrow dollars at a monthly rate of expressed as a decimal, and the loan is repaid in months. Use functional notation to indicate the monthly payment if you borrow 5444 dollars at an APR of 5.74 percent, and you repay the loan in 5 years. Use 3 decimal places for . ANS: PTS: 1

DIF: medium

10. Let denote the price in dollars of pizzas, sodas, and bags of chips. functional notation to indicate the cost of 3 pizzas, 8 sodas, and 11 bags of chips.

Use

ANS: PTS: 1

DIF: easy

ESSAY 1. Suppose you borrow dollars at an APR of expressed as a decimal. Suppose further that you repay the loan in monthly payments and that interest is compounded continuously. Then your monthly payment, in dollars, is given by

, where is the monthly rate ( ) expressed as a decimal. Suppose you borrow 1122 dollars at an APR of 12% and repay the loan over a period of 3 years. And suppose interest is compounded continuously. A: Use functional notation to express your monthly payment. B: Calculate the value you found in part A. C: Use your answer to part B to answer this question: When the loan is paid off, how much of what you paid is interest? ANS: A: B. $37.3 C. $220.8 PTS: 3

DIF: hard

2. Suppose you borrow dollars at a monthly rate of expressed as a decimal. Suppose further that you repay the loan in monthly payments and that interest is compounded monthly. Then your monthly payment, in dollars, is given by

A: What is your monthly payment if you borrow 10137 dollars at a monthly rate of 0.01 and pay it off over 5 years? B: Use your answer to part A to determine the total amount you pay the bank. C: If you accept a $1000 rebate, you only need to borrow 9137 dollars, but your monthly rate is 0.015. What is the total amount you pay the bank in this circumstance? (Round the monthly payment to two decimal places before you calculate your answer.) ANS: A: $225.49 B. $13529.4 C.$13921.2 PTS: 3

DIF: hard

3. If you have a mortgage, and you make monthly payments, then your equity is the total paid toward the principal at a given time. If your mortgage is for dollars at a monthly rate of 0.01, and if you have paid of a total of t monthly payments due, then your equity in dollars is given by . Suppose your mortgage is for 212308 dollars and that you have made 288 out of 360 monthly payments due. A: Use functional notation to express your equity. B: Calculate the value you found in part a. C: If you have made half of the 360 monthly payments, have you paid off half of the mortgage? ANS: A: B. $100604.4 C. No. PTS: 3

DIF: hard

4. A cup of coffee is poured from a pot that maintains a constant temperature. The fresh coffee is placed on the counter to cool. Its temperature, in degrees Fahrenheit, minutes after it is placed on the counter is given by . A. What is the temperature of the coffee in the pot?

B. Use functional notation to express the temperature of the coffee after 18 minutes. C. Calculate the value you found in part B. ANS: A: 200 degrees B. C. 78.51 degrees PTS: 3

DIF: hard

5. A patient is placed on a diet to improve his blood-cholesterol content. The concentration of cholesterol in the blood, in milligrams per deciliter, after t months on the diet is given by . A. Explain in practical terms the meaning of

B. Calculate the value you found in part A. C. How much did the cholesterol level decline from month 5 to month 8? ANS: A: It is the concentration, in milligrams per deciliter, of cholesterol in the blood after 6 months. B. 145.73 milligrams per deciliter C. 10.2 milligrams per deciliter PTS: 3

DIF: hard

6. The cumulative number of flu cases reported by

days after an epidemic began is given by .

A. Use functional notation to indicate the initial number of flu cases. B. Calculate the value you found in part A. C. How many new cases were reported from day 20 to day 40? Round your final answer to the nearest whole number. ANS: A: B. 125 cases C. 294 new cases PTS: 3

DIF: hard

7. It starts to snow when there is already snow on the ground. The depth, in inches, of the snow later is given by

hours

. A. How much snow was on the ground when the snow started to fall? B. By how much did the depth of snow increase from hour 4 to hour 6? C. Snow ceases to fall after 8 hours. What is the resulting depth of snow on the ground? ANS: A: 7 inches B. 0.8 inches C. 10.2 inches PTS: 3

DIF: hard

8. If a rock is dropped form a tall building, the distance, in feet, that it travels after . A. Explain in practical terms the meaning of

seconds is given by

B. Calculate the value you found in part A. C. How far did the rock fall from 3 seconds after it was dropped to 4 seconds after it was dropped? ANS: A: It is the distance, in feet, the rock falls in the first 3 seconds. B. 144 feet C. 112 feet PTS: 3

DIF: hard

9. The volume, in cubic inches, of a balloon of radius

inches is given by .

A. Use functional notation to indicate the volume, in cubic inches, of a balloon of radius 10 inches. B. Calculate the value you found in part A. C. If the radius of a balloon is doubled from 3 inches to 6 inches, does the volume double? D. By what factor does the volume of the balloon increase if the radius is increased from 3 inches to 6 inches? ANS:

cubic inches.

B. 4188.79 cubic inches C. No. D. The volume increases by a factor of 8. PTS: 4

DIF: hard

10. The surface area, in square inches, of a balloon of radius

inches is given by .

A. Use functional notation to indicate the surface area, in cubic inches, of a balloon of radius 10 inches. B. Calculate the value you found in part A. C. If the radius of a balloon is doubled from 3 inches to 6 inches, does the surface area double? D. By what factor does the surface area of the balloon increase if the radius is increased from 3 inches to 6 inches? ANS: A:

square inches.

B. 1256.64 square inches C. No. D. The surface area increases by a factor of 4. PTS: 4

DIF: hard

Section 1.2 Functions Given by Tables TRUE/FALSE 1. Every function given by a table of values has a limiting value. ANS: F

PTS: 1

DIF:

easy

2. When a function is given by a table of values, it is sometimes reasonable to fill in gaps by averaging nearby function values. ANS: T

PTS: 1

DIF:

easy

3. The average rate of change of an increasing function is a measure of the rate at which the function grows. ANS: T

PTS: 1

DIF:

easy

4. It is never appropriate to use functional notation when dealing with a table of values. ANS: F

PTS: 1

DIF:

easy

5. Limiting values give information about the long-term behavior of functions. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. The following table shows the U.S. population, in millions, in the given year. year

1960 179.32

1970 203.3

1980 226.54

1990 248.71

2000 281.42

population in millions Calculate the average rate of change from 1980 to 1990.

a. 2.22 million people per year b. 22.17 million people per year ANS: A

PTS: 1

c. 1731.4 million people per year d. None of the above DIF:

medium

2. The following table shows the U.S. population, in millions, in the given year. year

1960 179.32

1970 203.3

1980 226.54

1990 248.71

population in millions Use the average rate of change to estimate the U.S. population in 1996.

2000 281.42

a. 268.34 million b. 251.98 million ANS: A

c. 265.07 million d. None of the above DIF: medium

PTS: 1

3. The following table shows the number of items produced when a production company hires employees. 130 150 160 190 200 employees items 621 730 921 1044 1161 produced Use the average rate of change to estimate the number of items produced if 185 employees are hired. Round your answer to the nearest whole number.

a. 1014 b. 1024

c. 983 d. None of the above

ANS: B

PTS: 1

DIF:

medium

4. The following table shows the population of a city in the given year. year

1970 1296

1980 1345

1990 1810

2000 1934

2010 2025

population Use averaging to estimate the population in 1975. Round your answer to the nearest whole number.

a. 1969 people b. 1321 people ANS: B

c. 1345 people d. None of the above PTS: 1

DIF:

medium

5. The following table shows the height, in inches, of a boy who is age height

5 36.84

9 48.34

13 61.96

years old. 17 68.82

21 73.48

Use the averaging to estimate the boy’s height at age 7. a. 61.01 inches b. 42.59 inches ANS: B

c. 42.37 inches d. None of the above PTS: 1

DIF: medium

6. The following table shows the number of bird species found on an island of area is part of a particular Pacific island chain. area 10 number 25

20 31

30 36

40 39

square miles that

50 42

of species Calculate the average rate of change in the number of species as the area changes from 10 to 20 square miles.

a. 1.52 species per square mile b. 31 species per square mile ANS: C

PTS: 1

c. 0.6 species per square mile d. None of the above DIF:

medium

7. The following table shows the length, in inches, of certain animals as a function of their running speed, in feet per second. Animal running speed length

Deermouse 10

Chipmunk 20

Grey squirrel 30

Red fox 40

Cheetah 50

Use the average rate of change to estimate the length of an animal that has a running speed of 47.7 feet per second. a. 39.76 inches b. 41.31 inches ANS: B

c. 39.34 inches d. None of the above PTS: 1

DIF:

medium

8. The following table shows average rice yield, in tons per hectare, in Asia t=years since 1980 yield

years since 1980.

3.32

3.61

3.73

3.95

4.11

Use the average rate of change to estimate the yield in 1992. a. 3.61 tons per hectare b. 3.66 tons per hectare ANS: B

c. 3.75 tons per hectare d. None of the above

PTS: 1

DIF:

medium

9. The following table shows average rice yield, in tons per hectare, in Asia t=years since 1980 yield

3.32

3.61

3.73

3.95

4.11

Use the average rate of change to estimate the yield in 1999. a. 3.73 tons per hectare b. 3.91 tons per hectare ANS: B

years since 1980.

PTS: 1

c. 3.87 tons per hectare d. None of the above DIF:

medium

10. The following table shows the percentage P of the American food dollar that was spent on eating away from home as a function of the date d. date percent

1969 25

1989 30

2009 34

Use functional notation to express the percentage of the American food dollar that was spent eating away from home in 2015. Then use the average rate of change to estimate that value. a. b.

35.35 % 34%

ANS: C

c. =35.2 % d. None of the above

PTS: 1

DIF:

medium

11. The following table gives the number , in millions, of adult Americans with Internet access in year d. year millions

2000 113

2003 166

2009 196

Use functional notation to express the number, in millions, of adult Americans with Internet access in 2013. Then use the average rate of change to estimate that value. a. b.

232.89 million 196 million

ANS: C

PTS: 1

c. =216 million d. None of the above DIF:

medium

12. The resident population of Oklahoma in 1990 was 3.14 million. From 1990 to 2000 the average rate of change in population was 0.21 million people per year. Use these facts to estimate the Oklahoma resident population in 1996. a. 4.4 million b. 5.06 million ANS: A

c. 3.35 million d. none of the above PTS: 1

DIF:

medium

13. The temperature in degrees Fahrenheit can be estimated from the number of cricket chirps per minute. If a cricket chirps 40 times per minute, the temperature is approximately 50 degrees Fahrenheit. The average rate of change in temperature is 0.25 degree per chirp per minute. Use these facts to estimate the temperature when a cricket chirps 56 times per minute. a. 54 degrees b. 58 degrees ANS: A

c. 40.25 degrees d. none of the above PTS: 1

DIF:

medium

14. The speed of sound in air depends on the temperature. When the temperature is 32 degrees Fahrenheit, the speed of sound is 1087.5 feet per second. The average rate of change for is 1.1 feet per second per degree Fahrenheit. Use these facts to estimate the speed of sound in air when the temperature is 49 degrees Fahrenheit. a. 1088.6 feet per second

c. 1106.2 feet per second

b. 1090.5 feet per second ANS: C

d. none of the above

PTS: 1

DIF:

medium

15. The population of a certain state today is 16.7 million people. The average rate of change for is million people per year. Use these facts to estimate the population of the state 3 years from now. a. 16.7 million people b. 18.18 million people ANS: C

c. 16.22 million people d. none of the above

PTS: 1

DIF: medium

16. A balloon that originally holds 13.6 cubic inches of air springs a leak. Let represent the volume, in cubic inches, of air in the balloon minutes after the balloon starts to leak air. The average rate of change of is cubic inches per minute. Use these facts to estimate . a. 13.6 cubic inches b. 20.74 cubic inches ANS: D

c. 12.59 cubic inches d. none of the above

PTS: 1

DIF:

medium

SHORT ANSWER 1. The following table shows the population N of a small town for the given date. date population

1980 558

What is the value of

1990 625

2000 694

2010 755

ANS: 625 PTS: 1

DIF: easy

2. The following table shows the population of a small town for the given date. date population

1980 560

1990 625

2000 695

2010 754

What is the population in 2000? ANS: 695 PTS: 1

DIF: easy

3. The following table shows the height F, in centimeters, of a flower days height

12 3

47 17

57 19

days after it sprouts. 97 25

What is the value of

ANS: 19 centimeters PTS: 1

DIF: easy

4. The following table shows the height, in centimeters, of a flower days height

11 5

46 16

days after it sprouts.

59 20

96 25

What is the height of the flower after 96 days? ANS: 25 centimeters PTS: 1

DIF: easy

5. The following table shows the depth D, in inches, of snow on the ground of the snowfall. hours depth

2 3

What is the value of

3 4

5 8

hours after the beginning

8 10

ANS: 4 inches PTS: 1

DIF: easy

6. The following table shows the depth, in inches, of snow on the ground the snowfall. hours depth

2 3

3 5

5 8

hours after the beginning of

8 10

What is the depth of the snow 5 hours after the snowfall began? ANS: 8 inches PTS: 1

DIF: easy

7. The following table shows the number of magazines sold weeks sales

10 1631

20 1906

30 1978

40 1994

weeks after the first issue was published. 50 1998

Based on the table, what do you expect is the limiting value of magazine sales? ANS:

60 1999

Any answer close to 2000 is reasonable. PTS: 1

DIF: easy

8. The following table shows the number of nesting geese in a protected area began. years

2 516

5 659

10 684

13 692

years after observation

17 697

20 699

nesting geese Based on the table, what do you expect is the limiting value of the number of geese nesting in this area? ANS: Any answer close to 700 is reasonable. PTS: 1

DIF: easy

9. The following table shows the length L, in inches, of a certain type of fish when it is years length

2 7.3

5 13.7

8 18.4

10 19.5

Based on the table, what do you expect is the limiting value of

13 21.7

years old.

15 21.97

ANS: Any answer close to 22 inches is reasonable. PTS: 1

DIF: easy

10. The following table shows the height H, in centimeters, of a certain type of flower when it is old. years 2 height 7.11

5 21.76

8 37.55

10 40.9

Based on the table, what do you expect is the limiting value of

13 42.64

weeks

15 42.96

ANS: Any answer close to 43 centimeters is reasonable. PTS: 1

DIF: easy

11. The following table shows the balance, in dollars, of a savings account years

0 8667

5 10047.43

10 11647.68

15 13502.93

years after it is opened.

20 15653.37

25 18147.67

balance Make a new table of values showing the average rate of change over each five-year period.

ANS: Period Average rate of change

PTS: 1

0 to 5 years

5 to 10 years

276.09 dollars per year

320.05 dollars per year

10 to 15 years 371.05 dollars per year

15 to 20 years 430.09 dollars per year

20 to 25 years 498.86 dollars per year

DIF: easy

12. The following table shows the distance, in feet, that a rock travels downward. seconds 0 D= 0 distance

5 521.75

10 1843.5

15 3965.25

seconds after it is thrown

20 6887

25 10608.75

Make a new table of values showing the average rate of change in distance over each five-second period. ANS: Period Average rate of change PTS: 1

0 to 5 seconds 104.35 feet per second

5 to 10 seconds 264.35 feet per second

10 to 15 seconds 424.35 feet per second

15 to 20 seconds 584.35 feet per second

20 to 25 seconds 744.35 feet per second

DIF: easy

13. If represents the cumulative number of flu cases reported by day , what units are associated with the average rate of change for with respect to ? ANS: Number of new flu cases per day. PTS: 1

DIF: medium

14. If represents the total miles a car travels in rate of change for with respect to ?

hours, what units are associated with the average

ANS: Miles per hour. PTS: 1

DIF: medium

15. If represents enrollment at your university in year rate of change for with respect to ? ANS: Number of students per year. PTS: 1

DIF: medium

, what units are associated with the average

16. If represents the temperature, in degrees Fahrenheit, of a potato in the oven after what units are associated with the average rate of change for with respect to ?

minutes,

ANS: Degrees Fahrenheit per minute. PTS: 1

DIF: medium

17. A hot potato is placed on the kitchen counter to cool. Its temperature after minutes is given by The temperature of the kitchen is 77 degrees Fahrenheit. What is the limiting value of ?

ANS: 77 degrees Fahrenheit PTS: 1

DIF: easy

18. A yam is placed in an oven to bake. Its temperature after minutes is given by degrees Fahrenheit. The temperature of the oven is 372 degrees Fahrenheit. What is the limiting value of

ANS: 372 degrees Fahrenheit PTS: 1

DIF: easy

19. A tire has a leak and is losing air. The air pressure, in pounds per square inch, after by . Assuming the leak is not repaired, what is the limiting value of ?

minutes is given

ANS: 0 pounds per square inch PTS: 1

DIF: easy

ESSAY 1. The following table shows the enrollment, in thousands, at a university years since 1990 enrollment (thousands)

years after 1990.

A. Calculate the average rate of change in enrollment from 1995 to 2000. B. Explain in practical terms the meaning of the number you calculated in part A. C. Use your answer from part A to estimate the enrollment in 1998. ANS: A. 0.6 thousand students per year B. From 1995 to 2000 enrollment increased on average by 0.6 thousand students each year.

C. 25.8 thousand students PTS: 3

DIF: hard

2. The following table shows the cumulative number of swine flu cases days since epidemic began swine flu cases

days after an epidemic began.

128

155

177

A. Calculate the average rate of change in the cumulative number of swine flu cases from day 3 to day 10. B. Explain in practical terms the meaning of the number you calculated in part A. C. Use your answer from part A to estimate the cumulative number of swine flu cases after 9 days. Round your answer to the nearest whole number. ANS: A. 2.57 cases per day B. From 3 to 10 days since the epidemic began, on average there were 2.57 new cases each day. C. 41 cases PTS: 3

DIF: hard

3. The following table shows the value, in dollars, of an investment years since 0 2000 value 561.34

years after 2000.

722.55

928.43

1170.41

1480.66

A. Calculate the average rate of change in the investment value from 2005 to 2010. B. Explain in practical terms the meaning of the number you calculated in part A. C. Use your answer from part A to estimate the value of the investment in 2008. ANS: A. 41.18 dollars per year B. From 2005 to 2010 , on average the value of the investment increased by 41.18 dollars each year. C. $846.09 PTS: 3

DIF: hard

4. The following table shows the amount remaining, in grams, of a radioactive substance after years

years.

amount remaining

500

429.37

368.71

316.63

271.9

A. Calculate the average rate of change in amount of radioactive substance from t= 5 to t= 10. (Be sure to get the sign right.) B. Explain in practical terms the meaning of the number you calculated in part A. C. Use your answer from part A to estimate the amount remaining after 9 years. D. What is the limiting value of amount remaining of this (or any other) radioactive substance? ANS: A. –12.13 grams per year B. From year 5 to year 10 , on average the amount of the radioactive substance remaining decreased by 12.13 grams each year. C. 380.85 grams D. 0 PTS: 4

DIF: hard

5. For a certain island chain, the area , in square miles, can be estimated by counting the number of reptile and amphibian species on the island. The following table shows the relationship. number of species area

525

1921

5155

9456

19562

A. Calculate the average rate of change in area from 50 to 60 species. B. Explain in practical terms the meaning of the number you calculated in part A. C. Use your answer from part A to estimate the area of an island in the chain which has 53 species of reptiles and amphibians. ANS: A. 1010.6 square miles per species B. Between 50 and 60 species, on average the area increases by 1010.6 square miles for each additional species. C. 12487.8 square miles PTS: 3

DIF: hard

6. The following table shows the mass M, in kilograms, of a certain type of fish as a function of its length in centimeters. length mass

80 21.5

100 42.5

120 74.1

140 119

160 179

A. Calculate the average rate of change in mass as the length goes from 100 centimeters to 120 centimeters. B. Calculate the average rate of change in mass as the length goes from 140 centimeters to 160 centimeters. C. Based on your calculations from parts A and B, does an extra centimeter of length make more difference in weight for a smaller fish or a larger fish? ANS: A. 1.58 kilograms per centimeter B. 3 kilograms per centimeter C. A larger fish. PTS: 3

DIF: hard

7. The following table shows the length mass M in kilograms. mass length

14 70.6

31 90.4

, in centimeters, of a certain type of fish as a function of its

57 111

94 132.3

149 151.7

A. Calculate the average rate of change in length as the mass goes from 14 kilograms to 31 kilograms. B. Calculate the average rate of change in length as the mass goes from 94 kilograms to 149 kilograms. C. Based on your calculations from parts A and B, does an extra kilogram of mass make more difference in length for a smaller fish or a larger fish? ANS: A. 1.16 centimeters per kilogram B. 0.35 centimeters per kilogram C. A smaller fish. PTS: 3

DIF: hard

8. The following table shows the population, in millions, of a certain town in the given year. year millions

1985 2397

1987 2213

1989 2046

1991 1877

A. Calculate the average rate of change in population from 1987 to 1989. (Be careful to get the sign right.) B. Explain in practical terms the meaning of the number you calculated in part A.

C. Use averaging to estimate the population in 1988. ANS: A. –83.5 million people per year B. From 1987 to 1989, on average the population decreased by 83.5 million people each year. C. 2129.5 people PTS: 3

DIF: hard

9. The following table shows the running speed, in feet per second, of certain animals as a function of their length, in inches. Animal

Deermouse Chipmunk

length 3.5 running 8.2 speed

6.3 15.7

Grey squirrel 9.8 24.9

Red fox

Cheetah

24 65.6

47 95.1

A. Calculate the average rate of change in running speed from a length of 3.5 inches to a length of 6.3 inches. B. Explain in practical terms the meaning of the number you calculated in part A. C. Use the average rate of change to estimate the running speed of an animal that is 4.5 inches long. ANS: A. 2.68 feet per second per inch B. From a length of 3.5 inches to a length of 6.3 inches, on average each one-inch increase in length corresponds to an increase running speed by 2.68 feet per second. C. 10.88 feet per second PTS: 3

DIF: hard

10. The following table shows the running speed, in centimeters per second, of ants as a function of temperature, in degrees Celsius. T=temperature 25.6 running 2.62 speed

27.5 3.03

30.3 3.55

30.4 3.56

33.8 4.32

A. Calculate the average rate of change in running speed from 30.4 inches to 33.8 inches. B. Explain in practical terms the meaning of the number you calculated in part A.

C. Use the average rate of change to estimate the running speed of ants when the temperature is 41 degrees Celsius. ANS: A. 0.22 centimeters per second per degree Celsius B. From a temperature of 30.4 degrees Celsius to a temperature of 33.8 degrees Celsius, on average each one-degree increase in temperature corresponds to increasing running speed by 0.22 centimeters per second. C. 5.9 centimeters per second PTS: 3

DIF: hard

Section 1.3 Functions Given by Graphs TRUE/FALSE 1. A graph that is concave up represents a function that is increasing. ANS: F

PTS: 1

DIF:

easy

2. A decreasing graph is always concave down. ANS: F

PTS: 1

DIF:

easy

3. The graph of a function that is increasing at an increasing rate is increasing and concave up. ANS: T

PTS: 1

DIF:

easy

4. The graph of a function that is decreasing at a decreasing rate is decreasing and concave up. ANS: T

PTS: 1

DIF:

easy

5. Inflection points may occur where a function is increasing at the fastest rate. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Choose the answer that best completes the following sentence. A graph that is increasing and concave up represents a function that is ... a. increasing at an increasing rate c. decreasing at a decreasing rate b. increasing at a decreasing rate d. decreasing at an increasing rate ANS: A

PTS: 1

DIF:

easy

2. Choose the answer that best completes the following sentence. A graph that is decreasing and concave up represents a function that is ... a. increasing at an increasing rate c. decreasing at a decreasing rate b. increasing at a decreasing rate d. decreasing at an increasing rate ANS: C

PTS: 1

DIF:

easy

3. Choose the answer that best completes the following sentence. A point of inflection occurs where... a. concavity changes c. the graph is increasing b. concavity is at a maximum d. the graph is bent ANS: A

PTS: 1

4. Below is a graph of a function

DIF:

easy

. Find the value of

a. 5 b. 1 ANS: A

c. 12.2 d. None of the above PTS: 1

5. Below is a graph of a function

a. 3 b. 5 ANS: C

DIF:

easy

. Find the smallest value of

so that

c. 1 d. None of the above PTS: 1

DIF: medium

6. Below is a graph of a function

a. 3 b. 2 ANS: B

from 1 to 3 .

c. 4 d. None of the above PTS: 1

7. Below is a graph of a function

a. 20 b. –5

. Find the average rate of change in

DIF: medium . Find the average rate of change in

c. –30 d. None of the above

from 4 to 10 .

ANS: B

PTS: 1

8. Below is a graph of a function

medium

. Find the value of

c. –50 d. None of the above

a. 14 b. 30 ANS: B

DIF:

PTS: 1

9. Below is a graph of a function that value?

DIF:

easy

. At what value of

does

reach its maximum value, and what is

a. 50 b. maximum of 60 at x=2 ANS: B

PTS: 1

10.10.Below is a graph of a function

a. 60 b. 20 ANS: C

c. maximum of 2 at x=60 d. None of the above DIF:

medium

. What is the limiting value for

c. 50 d. None of the above PTS: 1

DIF: medium

11. You put a drink in the freezer to cool. You take it out of the freezer when it is cold. But you forget about the drink and leave it sitting on the kitchen counter. The graph below shows the temperature, in degrees, of the drink minutes after the drink is placed in the freezer. What is the temperature in the kitchen?

a. 20 degrees b. 60 degrees ANS: C

c. 70 degrees d. None of the above PTS: 1

12. Below is a graph of a function

a. from 3 to 6 b. from 0 to 3 ANS: B

DIF:

medium

. Over what region(s) is the function increasing?

c. at x=3 d. None of the above PTS: 1

DIF: easy

13. Below is a graph of a function

a. 9 b. 10 ANS: A

. What is the value of

c. 0 d. None of the above PTS: 1

14. Below is a graph of a function

a. from 0 to 1 and from 4 to 6

DIF: easy . Over what region(s) is the function decreasing?

c. from -20 to 10

b. from 1 to 4 ANS: B

d. None of the above PTS: 1

15. Below is a graph of a function

a. at b. from 0 to 3 ANS: C

DIF:

easy

. Over what region(s) is the graph concave up?

c. from 3 to 6 d. None of the above PTS: 1

DIF:

easy

16. The graph below shows the value, in dollars, of a foreign currency years after 2000. For what two dates between 2000 and 2020 would earn you the most money if you bought the foreign currency on the first date and sold on the second?

a. Buy in 2002 and sell in 2008 b. Buy in 2008 and sell the same year ANS: C

PTS: 1

c. Buy in 2005 and sell in 2008 d. None of the above DIF:

easy

17. The graph below shows the value, in dollars, of a foreign currency years after 2000. In what year from 2000 to 2020 did the value of the foreign currency reach its maximum, and what was that maximum value?

a. The maximum value of $1.80 occurred in 2008.

b. The maximum value of $1.20 occurred in 2005. c. The maximum value of $1.40 occurred in 2002. d. The maximum value of $0.90 occurred in 2020. ANS: A

PTS: 1

DIF:

easy

18. The graph below shows the blood-glucose levels, in milligrams per deciliter, as a function of hours since a meal was ingested. The three graphs are for a healthy person (labeled normal), a prediabetic, and a diabetic person. During the period shown on the graph, what is the shortest time since eating that glucose levels are the same for a healthy person and a prediabetic?

a. About 6 hours after eating b. About 3 hours after eating ANS: B

PTS: 1

c. At about 1 hour after eating d. None of the above DIF:

easy

19. The graph below shows the blood-glucose levels, in milligrams per deciliter, as a function of hours since a meal was ingested. The three graphs are for a healthy person (labeled normal), a prediabetic, and a diabetic person. The graph indicates that regardless of diabetic condition, blood-glucose reaches a maximum concentration about how many hours after eating?

a. About 110 milligrams per deciliter b. About 70 milligrams per deciliter ANS: C

PTS: 1

c. About 1 hour after eating d. About 4.5 hours after eating DIF:

easy

20. The graph below shows the atmospheric pressure, in millibars of mercury, as a function of altitude in meters. According to the graph, which of the following sentences best describes the relationship between pressure and altitude?

a. Pressure decreases with altitude, but at higher altitudes a small change in altitude makes relatively little difference in pressure. b. Pressure decreases at an increasing rate as altitude increases. c. Pressure is concave down with respect to altitude. d. As altitude increases pressure does not change. ANS: A

PTS: 1

DIF:

medium

21. The graph below shows the atmospheric pressure , in millibars of mercury, as a function of altitude in meters. According to the graph, what is the approximate value of the solution for the equation ?

a. b.

meters meters

ANS: C

PTS: 1 DIF:

c. 15000 meters. d. The equation has no solution. medium

22. The graph below shows the fundraising of the Republican National Committee, in millions of dollars, as a function of the date. During the period shown on the graph when was the minimum amount of funds raised, and what was that amount?

a. b. c. d.

The Committee raised 10 million dollars in 1993. The Committee raised 17 million dollars in 1995. The Committee raised 12 million dollars in 1997. The Committee raised 21 million dollars in 2009.

ANS: A

PTS: 1

DIF:

medium

23. The graph below shows the fundraising of the Republican National Committee, in millions of dollars, as a function of the date. The graph indicates that there are two solutions of the equation 24 million. What two dates (approximately) correspond to these solutions?

a. About 2001 and 2009. b. About 1995 and 2005 ANS: A

PTS: 1

c. About 1993 and 2009 d. There is no solution. DIF:

medium

24. The graph below shows the maximum angle above the horizon reached by the moon during the month of June. On what day does the smallest maximum angle occur, and what is that angle?

a. b. c. d.

The smallest maximum angle of about 71 degrees occurs on June 29. The smallest maximum angle occurs at the time of the new moon. The smallest maximum angle of about 22 degrees occurs on June 15. The smallest maximum angle of about 71 degrees occurs on June 2.

ANS: C

PTS: 1

DIF:

medium

25. The graph below shows the maximum angle above the horizon reached by the moon during the month of June. What was the average rate of change in the maximum angle from June 15 to June 23?

a. 3.5 degrees per day b. 18 degrees per day ANS: A

PTS: 1

c. 8 degrees per day d. None of the above. DIF:

medium

26. The graph below shows the growth rate , in water fleas per day, of a population of water fleas as a function of the population size N. Calculate the average rate of change in from 20 to 70 water fleas.

a. 0.08 water fleas per day per water flea. b. 4 water fleas per day per water flea. ANS: A

PTS: 1

c. 50 water fleas per day per water flea. d. None of the above. DIF:

medium

27. The graph below shows the numbers, in thousands, of men and women of a given height in a population. This graph allows us to conclude that:

a. b. c. d.

Women are taller than men. The most common height for women is about 160 centimeters. The most common height for men is about 170 centimeters. There are fewer tall men than tall women.

ANS: B

PTS: 1

DIF:

medium

28. The graph below shows the numbers, in thousands, of men and women of a given height in a population. What are the most common heights for men and for women?

a. The most common height for both men and women is about 170 centimeters. b. The most common height for women is about 160 centimeters, and the most common height for men is about 177 centimeters. c. The most common height for men is about 110 centimeters, and the most common height for women is about 120 centimeters. d. None of the above. ANS: B

PTS: 1

29. Below is a graph of a function

DIF:

medium

. At what value of

does the point of inflection occur?

a. at b. at ANS: A

c. at d. None of the above PTS: 1

DIF:

easy

30. Which of the following stories best fits the graph below?

a. I left home walking to class but forgot my book. I went back home for the book and then went to math class. b. I left home and went to math class. After class I returned home. c. I left English class, walked home for lunch, then went to math class. d. I went from English class to math class. After math I walked home. ANS: A

PTS: 1

DIF:

medium

31. Which of the following stories best fits the graph below?

PTS: 1

DIF:

medium

32. Which of the following stories best fits the graph below?

PTS: 1

DIF:

medium

33. Which of the following stories best fits the graph below?

PTS: 1

DIF:

ESSAY 1. The graph of

is shown.

A. What is the minimum value of B. At what value of

does the minimum occur?

medium

ANS: A. –20 B. x=4 PTS: 2

DIF: medium

2. The graph below shows the value

, in dollars, of an investment

A. What was the value of the investment in 2000? B. Is the graph concave up or concave down?

years after 2000.

C. What does the concavity tell you about how the investment grows? ANS: A. $10,000 B. Concave up C. The value increases at an increasing rate. PTS: 3

DIF: medium

3. The graph below shows the value

, in dollars, of an investment

years after 2000.

A. What was the value of the investment in 2012? B. When will the value of the investment reach $30000? ANS: A. About $20000 B. In about 2019 PTS: 2

DIF: medium

4. The graph below shows the value

, in dollars, of an investment

years after 2000.

A. What is the average rate of change in

from 2007 to 2012?

B. What is the average rate of change from 2016 to 2019? C: Is the investment growing at a faster rate from 2007 to 2012 or from 2016 to 2019? ANS: A. About $1000 per year B. About $1666.67 per year C. From 2016 to 2019 PTS: 3

DIF: medium

5. The graph below shows the population of weasels in a protected area

years after 2000.

A. When was the weasel population growing at the fastest rate? B. What is the limiting value of the weasel population? ANS: A. At about 2009 B. About 2000 weasels PTS: 2

DIF: medium

6. You put a drink in the freezer to cool. You take it out of the freezer when it is cold. But you forget about the drink and leave it sitting on the kitchen counter. The graph below shows the temperature, in degrees Fahrenheit, of the drink minutes after the drink is placed in the freezer.

A. Explain in practical terms the meaning of

B. Calculate the value you found in part A. C. When was the drink removed from the freezer? D. What was the coldest the drink ever got? ANS: A. It is the temperature of the drink 24 minutes after the drink is put in the freezer. B. 60 degrees Fahrenheit C. 10 minutes after it was placed in the freezer D. 20 degrees Fahrenheit PTS: 4

DIF: hard

7. The graph below shows the growth rate G, in water fleas per day, of a population of water fleas as a function of the population size N.

A. At what population level is growth rate a maximum? B. What is the growth rate for the population level you found in part A? C. Explain what is happening to the population when there are 230 water fleas present. D. Explain what is happening to the population when there are 250 water fleas present. ANS: A. A population of about 70 water fleas. B. About 10 water fleas per day. C. The growth rate is 0, so the population level remains constant. D. The population level is declining by about 2 water fleas per day. PTS: 4

DIF: hard

8. The graph below shows the population of mustang ponies in a protected area

years after 2000.

A. Over what time period is the graph concave down? B. Explain what the concavity tells you about population increase over the time period you chose for part A. ANS: A. From about 2009 or 2010 on B. The population is increasing at a decreasing rate. PTS: 2

DIF: medium

9. The graph below shows the net profit S, in dollars per acre, expected from harvesting a certain forest stand as a function of the age t of the stand, in years. The net profit expected is known as the stumpage value of the forest stand.

A. Explain in practical terms the meaning of

B. Use the graph to estimate the value you found in part A. C. At about what age will stumpage value be 20000 dollars per acre? ANS: A. It is the stumpage value, or net profit expected, from harvesting the forest stand at age 100 years. B. About 30000 dollars per acre C. At about age 85 PTS: 3

DIF: medium

10. The graph below shows the temperature , in degrees Fahrenheit, adjusted for windchill as a function of the speed, in miles per hour, of the wind when the thermometer reads 30 degrees Fahrenheit. .

A. At what wind speed is the temperature adjusted for wind chill 0 degrees Fahrenheit?

B. Suppose the wind speed is 45 miles per hour. Judging from the shape of the graph, how significant would you expect the effect on effective temperature to be if wind speed increases? C. At what value of temperature?

would a small increase in wind speed have the greatest effect on effective

ANS: A. About 25 miles per hour B. Little change is expected C. Any answer between 5 and 10 miles per hour is reasonable. PTS: 3

DIF: hard

11. The graph below shows the average wedding costs in thousands of dollars in the given year.

A. Does this graph make weddings costs appear to be increasing more rapidly from 1945 to 1990 or from 1999 to 2002? B. Calculate the average rate of change in wedding costs from 1945 to 1990. C. Calculate the average rate of change in wedding costs from 1999 to 2002. D. Were wedding costs increasing more rapidly from 1945 to 1990 or from 1999 to 2002? ANS: A: From 1945 to 1990 B. 0.29 thousand dollars per year C. 1.17 thousand dollars per year D. From 1999 to 2002 PTS: 4

DIF: hard

12. The graph below shows the average wedding costs in thousands of dollars in the given year.

A. During which year did the average wedding costs achieve the maximum value? B. What was the maximum value of wedding costs from 1945 to 2009? C. During which period was average wedding costs decreasing most rapidly? ANS: A: 2007 B. $28.73 thousand C. From 2007 to 2008 PTS: 3

DIF: medium

Section 1.4 Functions Given by Words TRUE/FALSE 1. Verifying agreement at several points assures that a verbal description and a formula describe the same function. ANS: F

PTS: 1

DIF:

easy

2. When formulas are used to represent verbal descriptions, it is important to write down what the variables represent. ANS: T

PTS: 1

DIF:

easy

3. When variables are introduced to describe real situations, it is important to include proper units. ANS: T

PTS: 1

DIF:

easy

4. The term proportional is used to indicate that one thing is a multiple of another. ANS: T

PTS: 1

DIF:

easy

5. In a proportionality relationship, the constant of proportionality varies with time. ANS: F

PTS: 1

DIF:

easy

6. If a formula does not match a verbal description at one point, the formula and verbal description may still describe the same function. ANS: F

PTS: 1

DIF:

easy

7. The weight of a pizza with fixed diameter is proportional to its thickness. ANS: T

PTS: 1

DIF:

easy

8. The area of a square is proportional to the length of a side. ANS: F

PTS: 1

DIF:

easy

9. A woman’s height is proportional to her age. ANS: F

PTS: 1

DIF:

easy

10. If I make $23 per hour then my pay is proportional to the hours I work. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. If you are driving at a constant speed, the distance you travel is proportional to the time you spend driving. Using for the distance traveled, in miles, for the time, in hours, that you drive, and for the constant of proportionality, express the proportionality relation using a formula.

a. b.

c. d.

ANS: A

PTS: 1

DIF:

easy

2. If you work for $10 per hour, then your pay , in dollars, is proportional to the hours you work. Express this proportionality relationship as a formula using for hours worked. a. b.

c. d. None of the above.

ANS: B

PTS: 1

DIF:

easy

3. If you work 45 hours per week, then your weekly pay , in dollars, is proportional to your hourly wage , in dollars per hour. Express this proportionality relationship as a formula. a. b.

c. d. None of the above.

ANS: B

PTS: 1

DIF:

easy

4. Sugar costs $1.62 per pound, and lemons cost $0.64 each. You sell lemonade for $1.75 per glass. Write for the pounds of sugar used, for the number of lemons used, for the number of glasses of lemonade you sell, and for the profit, in dollars, you make. Express using a formula your profit in terms of pounds of sugar used, number of lemons used, and number of glasses of lemonade sold. a. b.

c. d.

ANS: A

PTS: 1

DIF:

medium

5. With no advertising your newspaper has a circulation of 1638 newspapers. For each dollar you spend on advertising, newspaper circulation increases by 16 newspapers. You sell each paper for 36 cents. You pay $499 to print the paper and have it delivered. Give a formula that shows your net income , in dollars, if you spend dollars on advertising. Suggestion: First calculate the number of newspapers you sell. Don’t forget to include the dollars you spend on advertising in your expenses. a. b.

c. d.

ANS: A

PTS: 1

6. The following table shows the value years value

1 $602.15

DIF:

hard

, in dollars, of an investment after

years.

2 $650.33

4 $758.54

3 $702.35

Which formula below fits these data? a. b. ANS: A

c. d. None of the above PTS: 1

DIF:

7. The following table shows some function values.

easy

1.6 13

3.2 21

6.7 38.5

9.1 50.5

Which formula below fits these data? a.

d. None of the above

ANS: A 8.

PTS: 1

DIF:

easy

The following table shows some function values. 1.3 1.61

2.9 1.84

5 2.14

6.2 2.31

Which formula below fits these data? a.

b. ANS: C 9.

d. PTS: 1

DIF:

easy

The following table shows some function values. 1.6 13.4

4 23

6.1 31.4

8.5 41

8.7 0.36

9.9 0.34

Which formula below fits these data? a.

c. b. ANS: B

10.

d. PTS: 1

DIF:

easy

The following table shows some function values. 1.3 0.65

4.5 0.48

Which formula below fits these data? a.

c. b. ANS: D

d. PTS: 1

DIF:

easy

11. The following table shows some function values. 1.3 11.83

2.1 30.87

4.2 123.48

6 252

Which formula below fits these data? a.

ANS: B

PTS: 1

DIF:

easy

12. The following table shows the cost C, in dollars, of taking number of kids cost in dollars

kids to a water park.

$48

$67

$86

$105

Which formula below fits these data? a. b.

c. d. None of the above

ANS: C

PTS: 1

DIF:

easy

13. The following table shows your monthly payment P, in dollars, if you pay off a loan in number of months monthly payment

$446.11

$236.36

$166.77

$132.22

Which formula below fits these data? a.

d. None of the above

ANS: B

PTS: 1

DIF:

medium

14. The following table shows the cost R, in dollars, of renting a car for number of days rental car cost

days.

$117

$204

$291

$378

Which formula below fits these data?

months.

a. b. ANS: B

c. d. None of the above PTS: 1

DIF:

easy

15. An initial investment of $554.25 grows at a rate of 5 percent per year. What is the value of the investment after 3 years?

a. $581.96 b. $637.39 ANS: C

c. $641.61 d. None of the above PTS: 1

DIF:

medium

16. An initial investment of $559.09 declines at a rate of 6 percent per year. What is the value of the investment after 3 years?

a. $592.64 b. $100.64 ANS: C

c. $464.37 d. None of the above PTS: 1

DIF:

medium

17. The maximum length of a certain type of fish is 65 centimeters. The difference between the maximum length and the length at age years is given by centimeters. Write a formula that gives the length , in centimeters, of the fish at age years. a. b. ANS: B

c. d. PTS: 1

DIF:

medium

18. The temperature of an oven remains constant at 377 degrees Fahrenheit. A potato is placed in the oven to bake. The difference between the temperature of the oven and that of the potato after minutes in the oven is given by degrees Fahrenheit. Write a formula that gives the temperature , in degrees Fahrenheit, of the potato after minutes in the oven. a. b. ANS: A

c. d. PTS: 1

DIF:

medium

19. You start with an investment of $1445.69. The first year the investment grows by 6%. The second year the investment declines by 8%. What is the value of the investment at the end of the two-year period? a. $1474.6 b. $1409.84 ANS: B

c. $28.91 d. None of the above PTS: 1

DIF: medium

20. You start with an investment of $1477.05. The first year the investment declines by 6%. The second year the investment grows by 9%. What is the value of the investment at the end of the two-year period?

a. $1513.39 b. $1424.76 ANS: A

c. $44.31 d. $96.6 PTS: 1

DIF: medium

21. You start with $33.22 in your piggy bank and add $9.28 each week. Write a formula for the amount of money, in dollars, in your piggy bank after weeks.

a. b. ANS: C

c. d. None of the above PTS: 1

DIF:

easy

22. You sell cookies for 60 cents each. You paid $8.31 for ingredients to make the cookies. Which of the following is a correct formula for the profit , in dollars, you make from baking and selling cookies? a. b. ANS: A

c. d. None of the above PTS: 1

DIF:

easy

23. You sell earrings for $9.33 each. You paid $54.21 for supplies to make the earrings. Which of the following is a correct formula for the profit , in dollars, you make from making and selling earrings? a. b. ANS: B

c. d. PTS: 1

DIF:

easy

24. It takes 13 minutes to preheat your oven. Once the oven is preheated, baking time for a roast is 10 minutes per pound. Using as the weight of the roast, in pounds, and for oven time (including preheating time), in minutes, give a formula that shows the oven time for a roast that weighs pounds. a. b. ANS: C

c. d. None of the above PTS: 1

DIF:

easy

25. There are 17783 students at your university. The student population is expected to grow by 7% per year for the next few years. What is the expected student population after 3 years? Round your answer to the nearest whole number. a. 21785 students b. 21517 students ANS: A

c. 19028 students d. None of the above PTS: 1

DIF: medium

26. Assume the median income of a high school graduate is 40,926 dollars per year. For each year of college education, median income increases by 17%. What is the expected median income if 3 years of college are completed?

a. 65,547.61 dollars per year b. 61,798.26 dollars per year ANS: A

PTS: 1

c. 47,883.42 dollars per year d. None of the above DIF:

medium

27. The table below shows grams of fat in selected food items. Item

Hamburger

Grams of fat

Chicken sandwich 14

Fries

Onion Rings

Use a formula to express the number of grams of fat in an order of sandwiches, orders of fries, and orders of onion rings. a. b. ANS: A

hamburgers,

chicken

c. d. None of the above PTS: 1

DIF:

medium

28. The table below shows the cost of office supplies. Item Cost

Box of staples $1.84

Box of pens $3.65

Ream of paper $6.89

For each hour the office staff works, you earn $18.44. Use a formula to express the net income , in dollars, that is the result hours of work by the office staff less the cost of purchasing boxes of staples, boxes of pens, and reams of paper. a. b. ANS: B

c. d. PTS: 1

DIF:

medium

29. The table below shows the initial population and growth rate for several villages. Village Initial population Growth rate

Village A 528

Village B 636

Village C 340

Village D 956

23 people per year

13 people per year

–21 people per year

26 people per year

Use a formula to express the expected combined population a. b. ANS: B

of the four villages after

c. d. None of the above PTS: 1

DIF:

medium

30. The table below shows the initial population and growth rate for several large cities. City Initial population

City 1 6.41

City 2 5.77

City 3 6.44

City 4 3.61

years.

(millions) Growth rate

–0.22 million per year

0.15 million per year

–3.38 million per year

Use a formula to express the expected combined population years. a. b. ANS: B

4.17 million per year

, in millions, of the four cities after

c. d. PTS: 1

DIF:

medium

ESSAY 1. The world record time for a certain track event in 2000 was 75.32 seconds. In the ensuing years, the record time has decreased by 0.47 seconds each year. A: Write a formula for the record

, in seconds,

years after 2000.

B: Use functional notation to express the record in 2008. C. Calculate the value you found in part B. ANS: A. B. C. 71.56 seconds PTS: 3

DIF: medium

2. It costs $611 to rent a dining hall for a dinner you are catering. In addition it costs $29 for each person attending the dinner. A: Write a formula for the total cost

, in dollars, of catering a dinner for

B: Explain in practical terms the meaning of

guests.

C. Calculate the value you found in part B. ANS: A. B. It is the total cost, in dollars, of catering a dinner for 40 guests. C. $1771 PTS: 3

DIF: medium

3. When you secure a mortgage to buy a home, you may have to pay a loan origination fee. One lending institution changes a loan origination fee of $2,696 plus 3% of the mortgage amount.

A: Write a formula for the origination fee is dollars.

, in dollars, charged by this bank if the mortgage amount

B: Explain in practical terms the meaning of

C. Calculate the value you found in part B. ANS: A. B. It is the loan origination fee, in dollars, you pay on a mortgage of 301019 dollars. C. $11,726.57 PTS: 3

DIF: medium

4. The weight , in pounds, of a rock is proportional to its volume proportionality is the density , in pounds per cubic inch.

, in cubic inches. The constant of

A: Write a formula that expresses the proportionality relationship. B: What is the weight of a rock that has a density of 0.68 pounds per cubic inch and a volume of 14 cubic inches? ANS: A. B. 9.52 pounds PTS: 2 5. The total number employees.

DIF: medium of items that a factory can produce in an hour is proportional to the number

A: Using to denote the constant of proportionality, write a formula that expresses the proportionality relationship. B: Explain in practical terms what the constant of proportionality

means in this setting.

ANS: A. B. It is the number of items each employee can produce in an hour. PTS: 2

DIF: medium

6. The cost , in dollars, of operating your car depends on the mileage , in miles per gallon, that your car gets, the distance , in miles, that you drive, and the cost , in dollars per gallon, of gasoline. A. Write a formula that gives the cost miles per gallon, using gasoline that costs

, in dollars, of driving dollars per gallon.

miles in a car that gets

B. Use functional notation to indicate the cost of driving 181 miles in a car that gets 24 miles per gallon using gasoline that costs 4.24 dollars per gallon. C. Calculate the value you found in part B. ANS: A. B. C. $31.98 PTS: 3

DIF: hard

7. The cost , in dollars, of operating your car depends on the mileage , in miles per gallon, that your car gets, the distance , in miles, that you drive, and the cost , in dollars per gallon, of gasoline. A. Write a formula that gives the cost miles per gallon, using gasoline that costs

, in dollars, of driving dollars per gallon.

miles in a car that gets

B. Use functional notation to indicate the cost of driving 183 miles in a car that gets 27 miles per gallon using gasoline that costs 4 dollars per gallon. C. Calculate the value you found in part B. ANS: A. B. C. $27.11 PTS: 3

DIF: hard

8. Gold costs $1584 per ounce, and silver costs $31 dollars per ounce. You want to make a 9% profit on the jewelry you make. A. How much would you charge for jewelry that uses 1 ounce of gold and 1 ounce of silver? B. Write a formula that gives the price of gold and ounces of silver.

, in dollars, you charge for jewelry that uses

ounces

C. Use functional notation to indicate the price you charge for jewelry that uses 1.81 ounces of gold and 2.99 ounces of silver. ANS: A. $1,760.35 B.

C. PTS: 3

DIF: hard

9. You buy cups for $7.16 each and saucers for $6.05 each. To sell cup and saucer sets, you mark up each item by 6 dollars ($6 for each cup and $6 for each saucer). You give preferred customers a 7% discount. A. How much would you charge a preferred customer for 1 cup and saucer set? B. Use your answer to part A to write a formula that gives the price preferred customer for cup and saucer sets.

, in dollars, you charge a

C. Use functional notation to indicate the price you charge a preferred customer for 19 cup and saucer sets. ANS: A. $23.45 B. C. PTS: 3

DIF: hard

10. You sell calculators at a base price of $68 each. To encourage volume purchases, you deduct $2 from the base price for each extra calculator purchased. So, if one calculator is purchased you charge $68. If two are purchased you charge $66 for each calculator. If three are purchased, you charge $64 for each calculator, and so on. A. What is the total amount you charge a customer who buys 4 calculators? B. What is the total amount you charge a customer who buys 5 calculators? C. Write a formula that gives the total amount calculators. ANS: A. $248 B. $300 C. PTS: 3

dollars DIF: hard

, in dollars, you charge a customer who buys

Section 2.1 Table and Trends TRUE/FALSE 1. A formula can be used to generate a table of values. ANS: T

PTS: 1

DIF:

easy

2. A table of values always shows that a function has a limiting value. ANS: F

PTS: 1

DIF:

easy

3. If a function has a maximum value, a table of values can be used to find it. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. If a man jumps from an airplane with no parachute, his velocity after seconds is given by feet per second. Use a table of values to estimate the limiting value for his velocity. a. 172 b. 0.83 ANS: A

c. 0 d. 171.17 PTS: 1

DIF: medium

2. If a man jumps from an airplane with no parachute, his velocity after seconds is given by feet per second. How long will it take for his velocity to reach 92.47 feet per second? a. 6 seconds c. 0.83 second b. 4 seconds d. 175.17 seconds ANS: B

PTS: 1

3. The number of animals present

DIF:

medium

years after their introduction to a game preserve is given by .

Use a table of values to estimate the environmental carrying capacity of this reserve for this particular animal. a. 67.5 b. 270 ANS: B

c. 0 d. 3 PTS: 1

DIF: medium

4. Your pricing structure for canoe rentals encourages group rentals. The total price, in dollars, you charge for renting canoes is . What number of canoe rentals gives the maximum total charge? a. 36 canoes

c. There is no solution.

b. 66 canoes ANS: D 5. Find the solution for

d. 33 canoes PTS: 1

DIF:

medium

of the equation .

a. b.

13 301.6

ANS: D

c. There is no solution. d. 10 PTS: 1

DIF:

medium

6. Your pricing structure for canoe rentals encourages group rentals. The total price, in dollars, you charge for renting canoes is . What is the maximum charge for a group canoe rental? a. $729 c. $732 b. $54 d. $27 ANS: A

PTS: 1

7. The number of animals present

DIF:

medium

years after their introduction to a game preserve is given by .

How long will it take for the population to reach 104 animals? Round your answer to the nearest whole number. a. 7 years. b. 270 years. ANS: A

c. 10 years d. It never happens. PTS: 1

DIF:

medium

8. For certain paragraphs, the Flesch Reading Ease score is given by , where is the number of words in the paragraph. For a certain age group, a score of 44 is desired. Which of the following number of words will produce this Flesch score when the score is rounded to the nearest whole number? Round your answer to the nearest whole number. a. 208 words b. 44 words ANS: A 9.

c. This score is not possible. d. 218 words PTS: 1

DIF: medium

On a certain island chain, the number of species present relationship is given by

is related to the area of the island. The

, where is the area of the island measured in square miles. Which one of the following island areas will have 294 species present (rounded to the nearest whole number)?

a. 294 square miles b. 21582 square miles ANS: B

PTS: 1

c. 299 square miles d. 21825 square miles DIF:

10. The number of geese nesting in a protected area

medium years after their introduction to the area is given by .

When will the number of nesting geese reach 91? a. After 11 years b. After 121 years ANS: A

c. Never d. After 5 years PTS: 1

DIF: medium

11. The concentration C, in milligrams per deciliter, of a drug in the bloodstream administered is .

hours after it is

What is the maximum concentration ever achieved? a. 5 milligrams per deciliter b. 2.47 milligrams per deciliter ANS: D

PTS: 1

c. 4 milligrams per deciliter d. 1.47 milligrams per deciliter DIF:

medium

12. The length , in centimeters, of a certain type of fish is a function of the age relationship is

in years. The

. At what age will the fish be 52 centimeters long? a. 38 years b. 52 years ANS: A

c. 54 years d. 51 years PTS: 1

DIF: medium

13. Find the minimum value of .

a. b.

19.14 16

ANS: B

PTS: 1

c. d.

20 64

DIF:

medium

14. The length , in centimeters, of a certain type of snake is a function of the age relationship is . What is the longest snake of this variety that can be found?

in years. The

a. 47 centimeters b. 43 centimeters ANS: A

c. 22.04 centimeters d. 53 centimeters PTS: 1

DIF: medium

15. The concentration, in milligrams per deciliter, of a drug in the bloodstream administered is . When is the maximum concentration achieved? a. After 6 hours b. After 2.84 hours ANS: C 16. Find the solution for

c. After 5 hours d. After 1.84 hours PTS: 1

DIF:

medium

of the equation .

a. b. ANS: A 17. Find the solution for

c. d. There is no solution. PTS: 1

DIF:

medium

of the equation .

a. b. ANS: B 18. Find the solution for

c. d. There is no solution. PTS: 1

DIF:

medium

of the equation .

a. b. ANS: C 19. Find the solution for

c. d. There is no solution. PTS: 1

DIF:

medium

of the equation .

a. b.

c. d. There is no solution.

hours after it is

ANS: C

PTS: 1

20. Find the solution for

DIF:

medium

of the equation .

a. b. ANS: B

c. d. There is no solution. PTS: 1

DIF:

medium

21. Find the minimum value of . Consider only positive -values.

a. b. ANS: B 22. What value of

c. d. There is no solution. PTS: 1

DIF:

medium

gives the minimum value of ?

Consider only positive -values.

a. b. ANS: C 23. What value of

c. d. There is no solution. PTS: 1

DIF:

medium

gives the minimum value of ?

Consider only positive -values.

a. b. ANS: C

c. d. There is no solution. PTS: 1

DIF:

medium

24. What is the minimum value of ? Consider only positive -values.

a. b. ANS: A

c. d. There is no solution. PTS: 1

DIF:

medium

25. What is the maximum value of ? Consider only positive -values.

a. b. ANS: A 26. What value of

c. d. There is no solution. PTS: 1

DIF:

medium

gives the maximum value of ?

Consider only positive -values.

a. b. ANS: C

c. d. There is no solution. PTS: 1

DIF:

medium

27. What is the maximum value of ? Consider only positive -values.

a. b. ANS: C

c. d. There is no solution. PTS: 1

DIF:

medium

28. What is the maximum value of ? Consider only positive -values.

a. b. ANS: C

c. d. There is no solution. PTS: 1

DIF:

medium

29. What -value gives the maximum value of ? Consider only positive -values.

a. b.

c. d. There is no solution.

ANS: A

PTS: 1

DIF:

medium

30. What is the maximum value of ? Consider -values between 2 and 11 only.

a. b.

c. d. There is no solution.

ANS: C

PTS: 1

DIF:

medium

31. What is the maximum value of ? Consider -values between 6 and 15 only.

a. b.

c. d. There is no solution.

ANS: C

PTS: 1

DIF:

medium

SHORT ANSWER 1. Complete the following table of values for x f(x) ANS: x f(x) PTS: 1

1 2.04

2 2.68

3 3.49

4 4.38

5 5.31

DIF: easy

2. Complete the following table of values for x f(x) ANS: x f(x)

23 2.07

24 2.03

25 1.99

26 1.95

27 1.92

PTS: 1

DIF: easy

3. Complete the following table of values for x f(x) ANS: x f(x)

18 23.45

25 16.17

32 10.28

39 6.22

46 3.64

PTS: 1

DIF: easy

4. Complete the following table of values for x f(x) ANS: x f(x)

2 1.21

5 0.99

8 1.18

11 1.74

14 2.95

PTS: 1

DIF: easy

5. Complete the following table of values for x f(x) ANS: x f(x) PTS: 1

2 4.7

5 16.96

8 34.77

11 57.44

14 84.52

DIF: easy

ESSAY 1. If you roll

dice, then the probability of getting exactly 3 ones is

. A: Find the probability of getting exactly 3 ones if you roll 15 dice. Report your answer correct to 3 decimal places. B. How many dice should you roll in order to maximize your chances of getting exactly 3 ones? C: What probability corresponds to the number(s) you found in part B? Report your answer correct to 3 decimal places. ANS: A. 0.236 B. 17 or 18 dice C. 0.245 PTS: 2

DIF: medium

2. You sell calculators, and the price you charge for each calculator depends on the number purchased. If a customer orders calculators, then you charge dollars for each calculator.

A: Write a formula that gives your income

, in dollars, if you sell

calculators.

B. What is your income if you sell 11 calculators? C: What number of calculators sold will provide you with the most income? ANS: A. B. $891 C. 46 calculators PTS: 3

DIF: hard

3. You sell books, and the price you charge a single customer for each book depends on the number purchased. If a customer orders books then you charge dollars for each book. You have to pay the wholesaler $168 for each book you sell.

A: Write a formula that gives your income , in dollars, if you sell not consider the price you pay for each book.) B. Write a formula that gives your net profit must consider both income and cost.)

, in dollars, if you sell

books. (For income, you do

books. (For net profit you

C: What number of books sold will provide you with the most profit? ANS: A. B. C. 28 books PTS: 3 4. Sales, in dollars,

DIF: hard years after the binaural issue of a magazine are given by

A: What were the sales for the inaugural issue? B. When did sales reach $1001.29? Round your answer to the nearest whole number. C: What is the largest amount sales will ever reach? Round your answer to the nearest whole number. ANS: A. $374.17 B. After 10 years C. $2245 PTS: 3 5. The temperature

DIF: hard , in degrees, of a cup of coffee

minutes after it is poured is given by .

A: What was the temperature of the coffee when it was poured? B. When did the temperature reach 129.99 degrees? Round your answer to the nearest minute. C: What is the temperature of the room in which the coffee sits? ANS: A. 194 degrees B.After 18 minutes C. 71 degrees PTS: 3

DIF: hard

6. The growth rate , in hundreds of new cases per day, in the spread of an epidemic depends on the number , measured in hundreds, of currently sick individuals. The relationship, which is valid for up to 4000 currently sick people (that is, ), is

A: What is the smallest value of for which the growth rate is 1359 new cases per day (that is, )? Round your answer for S to the nearest whole number. B. For between 0 and 40, what number of currently sick individuals gives the maximum growth rate, and what is that growth rate? Round the growth rate to the nearest whole number, and remember that G is measured in hundreds of new cases per day. C: Explain what is happening at value you found in part B. ANS: A.

hundred cases

B. When there are 4000 currently sick individuals, the maximum growth rate of 8626 new cases per day occurs. C. There are 8626 new cases on that day. PTS: 3

DIF: hard

7. The circulation , in thousands, of a magazine is a function of the time of 2000. The formula

is valid over the interval from

in years since the beginning

A. Express in functional notation the circulation of the magazine in 2003, and then calculate that value. B. When did the circulation reach 130 thousand? Round your answer to the nearest whole number year. C. What was the maximum circulation, and when did that occur? Round both answers to the nearest whole number. ANS: A.

thousand

B. In 2004 C. The maximum circulation of 258 thousand occurred in 2008. PTS: 3

DIF: hard

8. For a certain consumer loan, the balance you still owe the bank after

monthly payments is given by

A. Express in functional notation the amount you owe the bank after 1 year of payments, and then calculate that value. B. How many payments do you need to make to reduce the balance owed to $12,043.98? C. When is the balance a maximum, and what is that maximum value? ANS: A.

dollars

B. 15 payments C. The maximum balance of $18,852 occurs at PTS: 3

, before any payments are made.

DIF: hard

9. If an average-sized man jumps from an airplane with an open parachute, his downward velocity seconds into the fall is

feet per second.

A. Find the average rate of change in velocity from B. Find the average rate of change from

C. What do your answers from parts A and B tell you about how velocity increases with time? D. What is the limiting value for velocity? ANS: A. 3.2 feet per second per second B. 0.01 feet per second per second C. Velocity increases rapidly at first, but the rate of increase slows with time. D. 20 feet per second PTS: 4

DIF: hard

10. A child has 121 blocks that are 1-inch cubes. She arranges the blocks into a solid rectangle of length and width . The relationship between and is given by

. A. Use a formula to express the perimeter

in terms of

B. Use the relationship between

to express the perimeter in terms of

C. What values of

and

give the smallest possible perimeter?

ANS: A. B. C. PTS: 3

and

DIF: hard

only.

Section 2.2 Graphs TRUE/FALSE 1. If we are given a horizontal span for the graph of a function, a table of values can help us choose a suitable vertical span for the graph. ANS: T

PTS: 1

DIF:

easy

2. The graph of a function that is increasing at a decreasing rate is concave down. ANS: T

PTS: 1

DIF:

easy

3. Concavity can be discerned from a table of values but not from a graph. ANS: F

PTS: 1

DIF:

easy

4. Inflection points often occur where the graph of a function is increasing most rapidly. ANS: T

PTS: 1

DIF:

easy

5. A graph can show where a function is increasing. ANS: T

PTS: 1

DIF:

easy

6. A graph of a function that is decreasing at a decreasing rate is concave down. ANS: F

PTS: 1

DIF:

easy

7. A graph of a function can show limiting values if they exist. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. For a satellite orbiting Earth, the time required to complete a single orbit is known as the period. The period , in hours, is related to the distance , in miles, from the center of Earth by the formula . What happens to the distance as the period increases? A graph of this question. a. b. c. d.

versus

can help you answer

Distance increases as period increases. Distance decreases as period increases. Distance is at a maximum. Concavity changes as the period increases.

ANS: A

PTS: 1

DIF:

medium

2. For a satellite orbiting Earth, the time required to complete a single orbit is known as the period. The period , in hours, is related to the distance , in miles, from the center of Earth by the formula

. As the period increases, so does the distance. What can you say about the rate of increase in distance as period increases? A graph of versus can help you answer this question. a. b. c. d.

Distance increases at an increasing rate as period increases. Distance increases at a decreasing rate as period increases. A point of inflection is reached. None of the above.

ANS: B 3. The life expectancy The relationship is

PTS: 1

DIF:

medium

, in solar lifetimes, of certain stars depends on their mass

, in solar masses.

. How does the life expectancy of larger stars compare with that of smaller stars? A graph of versus can help you answer this question. a. b. c. d.

Larger stars live 2.5 times as long as smaller stars. Larger stars have longer life expectancy. Larger stars have shorter life expectancy. All of the above.

ANS: C 4. The life expectancy The relationship is

PTS: 1

DIF:

medium

, in solar lifetimes, of certain stars depends on their mass

, in solar masses.

. As mass increases, life expectancy decreases. What can you say about the rate of decrease? A graph of versus can help you answer this question. a. b. c. d.

Life expectancy decreases at a decreasing rate. Life expectancy decreases at an increasing rate. Life expectancy reaches a point of inflection. None of the above.

ANS: A

PTS: 1

5. When a car skids to a stop, the length per hour, of the car by the equation

DIF:

medium

, in feet, of the skid marks is related to the speed

, in miles

. As speed increases, so does the length of the skid marks. What can you say about the rate of increase? A graph of versus can help you answer this question. a. Skid mark length increases more rapidly as speed increases. b. Skid mark length increases rapidly at first but slows as speed increases. c. Skid mark length increases at a decreasing rate.

d. None of the above. ANS: A

PTS: 1

6. The height h, in centimeters, of a sunflower

DIF:

medium

days after the seed emerges is given by

Use a graph to estimate the tallest the sunflower will ever be. a. About 360 centimeters tall b. About 18 centimeters tall ANS: A

PTS: 1

c. About 378 centimeters tall d. About 364.3 centimeters tall DIF:

medium

7. The amount of mercury, in milligrams per deciliter, in the blood of a man eating contaminated food is given by , where is the time in months since observation began. What is the limiting value of mercury in the bloodstream? a. About 0.88 milligrams per deciliter b. About 1.75 milligrams per deciliter ANS: B

PTS: 1

8. The concentration given by

c. About 0.91 milligrams per deciliter d. About 2.58 milligrams per deciliter DIF:

medium

of a drug, in milligrams per deciliter, in the blood

hours after an injection is

. What is the eventual concentration of the drug in the blood? a. About 4.03 milligrams per deciliter b. About 8.54 milligrams per deciliter ANS: D

PTS: 1

c. About 4.51 milligrams per deciliter d. 0 milligrams per deciliter DIF:

medium

9. The amount of adrenaline, in nanograms per liter, in a man’s bloodstream been frightened is given by

minutes after he has

. What is the eventual amount of adrenaline in the bloodstream? a. About 4.01 nanograms per liter b. About 1.81 nanograms per liter ANS: B 10. The amount

PTS: 1

c. About 5.82 nanograms per liter d. 0 nanograms per liter DIF:

medium

, in grams, of a radioactive substance remaining after

years is given by

. What is the limiting value for the amount of the radioactive substance? a. About 0.54 grams b. About 19 grams ANS: D

c. About 19.54 grams d. 0 grams PTS: 1

DIF:

medium

11. Dye is being added to a liquid mixture. The amount minutes is given by

, in grams, of dye in the mixture after .

What is the total amount of dye that is added to the solution?

a. About 171.32 grams b. About 184 grams ANS: B

c. About 0.88 grams d. 178.12 grams

PTS: 1

DIF:

12. Which of the following is the graph of

on a horizontal span of 0 to 4.5?

medium

4.5

3.5

2.5

1.5

0.5

0.5 –0.5 –0.5 –0.5

0.5

1.5

2.5

3.5

4.5

1.5

2.5

3.5

4.5

0.5

1.5

2.5

3.5

4.5

0.5

4.5

3.5

2.5

1.5

1 0.5

–0.5 –10

0.5

1.5

2.5

3.5

4.5

x –0.5 –0.5

–20

ANS: C

PTS: 1

DIF:

medium

13. Which of the following is the graph of y a.

on a horizontal span of 0 to 4.5? c.

120

y 30 25

105

45 30

–0.5 –5

0.5

1.5

2.5

3.5

4.5

0.5

1.5

2.5

3.5

4.5

–10 –0.5 –15

0.5

1.5

2.5

3.5

4.5

–15 –20

–30

–25

–45 y

y 30 25

30 –0.5 –5

20 10 –0.5 –10

–10 0.5

1.5

2.5

3.5

4.5

–15

–20

ANS: D

–25

PTS: 1

14. Sales , in thousands of dollars,

DIF:

medium

months after the beginning of the year are given by .

The formula is valid over a 12-month period. Over what period were sales increasing? A graph of versus can help you answer this question. a. From month 3 to the end of the year b. From month 6 to the end of the year ANS: C 15. The population

PTS: 1

c. Over the first 3 months d. Over the first 6 months DIF:

, in thousands, of a certain city

medium years after 2000 is given by .

The formula is valid over a 10-year period. Over what period was the population decreasing? A graph of versus can help you answer this question. a. From 2005 to 2010

c. From 2000 to 2005

b. From 2007 to 2010 ANS: A

d. From 2000 to 2007

PTS: 1

DIF:

medium

SHORT ANSWER 1. Make the graph of

. Use a horizontal span of 0 to 10.

ANS: y 5 4 3 2 1

–1

–2 –3 –4 –5

PTS: 1

DIF: easy

2. Make the graph of

. Use a horizontal span of 0 to 5.

ANS: y 3

–1

–2

–3

PTS: 1 3. Make the graph of ANS:

DIF: easy . Use a horizontal span of 0 to 5.

y 3

–1

–2

–3

PTS: 1

DIF: easy

4. Make the graph of

. Use a horizontal span of 0 to 5.

ANS: y 7 6 5 4 3 2 1 –1

–1

–2 –3

PTS: 1 5. Make the graph of ANS:

DIF: easy . Use a horizontal span of 0 to 100.

y 8 7 6 5 4 3 2 1

–1

100

–2

PTS: 1

DIF: medium

6. Make the graph of

. Use a horizontal span of

to 5.

ANS: y 100 80 60 40 20 –5 –4 –3

–2

–1 –20

–40 –60 –80 –100

PTS: 1 7. Make the graph of ANS:

DIF: medium . Use a horizontal span of 0 to 0.9.

y 80 70 60 50 40 30 20 10 –0.1 –10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

–20

PTS: 1

DIF: medium

8. Make the graph of

. Use a horizontal span of 0 to 5.

ANS: y 16 14 12 10 8 6 4 2 –1

–2

–4

PTS: 1 9. Make the graph of ANS:

DIF: easy . Use a horizontal span of 0 to 3.

y 9 8 7 6 5 4 3 2 1

–1

PTS: 1

DIF: easy

10. Make the graph of

. Use a horizontal span of 0 to 5.

ANS: y 1 0.8 0.6 0.4 0.2 –1

–0.2

–0.4 –0.6 –0.8 –1

PTS: 1

DIF: easy

11. Sketch a graph that is increasing and concave up. ANS:

y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

PTS: 1

DIF: easy

12. Sketch a graph that is decreasing and concave down. ANS: y 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

PTS: 1

DIF: easy

13. When a car skids to a stop, the length per hour, of the car by the equation

, in feet, of the skid marks is related to the speed

, in miles

. Does an increase in speed have a greater effect on the length of the skid marks for a car going slow or for a car going fast? A graph of versus can help you answer this question. ANS: For a fast car. PTS: 1

DIF: hard

14. For a satellite orbiting Earth, the time required to complete a single orbit is known as the period. The period , in hours, is related to the distance , in miles, from the center of Earth by the formula

. Does an increase in period have a greater effect on distance for a satellite with a longer period or for a satellite with a shorter period? A graph of versus can help you answer this question. ANS: The effect is greater for a satellite with a shorter period. PTS: 1 15. The life expectancy The relationship is

DIF: medium , in solar lifetimes, of certain stars depends on their mass

, in solar masses.

Does an increase in mass have a greater effect on life expectancy for a larger star or a smaller star? A graph of versus can help you answer this question. ANS: The effect is greater for a smaller star. PTS: 1

DIF: medium

ESSAY 1. The concentration of cholesterol, in milligrams per deciliter, in the blood exercise program was discontinued is given by

months after a diet and

A. Make a graph of cholesterol levels over the first 36 months since the diet and exercise program ended. B. Is the graph concave up or concave down? Explain in practical terms what the concavity means. C. What is the maximum concentration that will be achieved? ANS: A

C 270 240 210 180 150 120 90 60 30 –5 –30

–60

B. It is concave down. The cholesterol concentration increases rapidly at first, but later the rate of increase decreases. C. 235 milligrams per deciliter PTS: 3

DIF: hard

2. Suppose a puppy (of medium-sized breed) weighs 7 pounds at age t, in weeks. The adult weight in pounds, can be estimated using

A. Make a graph of adult weight estimates W for puppy ages t up to 15 weeks. B. Does a weight of 7 pounds at an early age indicate a larger or smaller adult weight than a weight of 7 pounds at a later age? ANS: A. W 200 180 160 140 120 100 80 60 40 20 –2

B. A weight of 7 pounds at an early age indicates a larger adult weight.

PTS: 2

DIF: hard

3. For a person who is 100 centimeters tall, the body surface area from the weight measured in kilograms. The relation is

, in square meters, can be estimated

A. Make a graph of body surface area versus weight. Include weights up to 100 kilograms. B. Is the graph concave up or concave down? C. Among people who are 100 centimeters tall, would weight gain have a greater effect on surface area for a lighter person or for a heavier person? ANS: A. B 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

90 100

B. Concave down. C. Weight gain would have a greater effect on surface area for a lighter person. PTS: 3

DIF: hard

4. If a person drives 201 miles, his average speed in hours, spent driving. The relationship is

, in miles per hour, for the trip depends on the time ,

A. Make a graph of average speed versus time spent driving. Include driving times of up to 5 hours. B. Is the graph concave up or concave down?

C. For a 201-mile trip, would an increase in speed make more difference in average velocity for a slow driver or for a fast driver? ANS: A. S 200 180 160 140 120 100 80 60 40 20 –1

B. Concave up. C. A change in speed has a greater effect on average velocity for a fast driver. PTS: 3

DIF: hard

5. A potato is placed in a preheated oven to bake. Its temperature baking is

, in degrees, after

minutes of

A. Make a graph of the temperature of the potato over the first 4 hours of baking. B. Did the potato’s temperature rise more during the first 30 minutes of baking or the second 30 minutes of baking? C. What is the temperature of the oven? ANS: A.

P 450 400 350 300 250 200 150 100 50 –60 –30 –50

90 120 150 180 210 240

B. During the first 30 minutes of baking C. About 399 degrees PTS: 3

DIF: hard

6. The amount , in pounds, of food consumed in a day by a sheep depends on the amount per acre, of vegetation present. The relationship is

, in pounds

A. Make a graph of

versus

. Include vegetation levels of up to 1000 pounds per acre.

B. Does a change in vegetation make a greater difference in the amount a sheep will consume when there is little vegetation present or when there is a lot of vegetation present? C. What is the most a sheep will eat no matter how much vegetation is available? ANS: A. C 3

100 200 300 400 500 600 700 800 900 1000

–1

B. When there is little vegetation present. C. About 2.9 pounds. (Any answer near 2.9 should be acceptable.) PTS: 3

DIF: hard

7. You want to invest in an account that pays an annual interest rate of , as a decimal. The amount , in dollars, you need to invest so that $5403 per year can be withdrawn each year for 10 years is given by .

A. Make a graph of

versus . Include interest rates from 0.1 to 0.9.

B. Is the graph concave up or concave down? C. Explain in practical terms the meaning of the concavity of the graph. ANS: A. M 35000 30000 25000 20000 15000 10000 5000 –0.1 –5000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

–10000 –15000

B. It is concave up C. The needed investment decreases rapidly when interest rates are low but less rapidly for higher interest rates. PTS: 3

DIF: hard

8. The running speed , in meters per second, for certain dinosaurs can be determined from the hip height , in meters. The relationship is .

A. Make a graph of speed versus hip height. Include hip heights from 0.5 to 5 meters. B. What happens to speed as hip height increases? C. Among these dinosaurs, does speed change more rapidly for smaller hip heights or for larger hip heights? ANS: A. s 10 9 8 7 6 5 4 3 2 1

0.5

1.5

2.5

3.5

4.5

B. Speed decreases. C. There is more rapid change for smaller hip heights. PTS: 3

DIF: hard

9. It is the rotation of a space station that provides Earth-level gravity. The number of rotations per minute needed to provide Earth-level gravity depends on the radius , in meters, of the spinning space station. The relationship is .

A. Make a graph of the number of rotations per minute versus the radius. Include radii from 10 to 100 meters. B. What happens to the required number of rotations as the radius increases? C. What number of rotations per minute is required if the radius is 35 meters? Report your answer correct to 2 decimal places. ANS: A.

N 10 9 8 7 6 5 4 3 2 1

90 100

B. The required number of rotations decreases as the radius increases. C. 5.05 revolutions per minute. PTS: 3

DIF: hard

10. The fraction is given by

of the surface of Earth that is visible from an altitude of

kilometers above the surface

A. Make a graph of

versus . Include altitudes up to 100,000 kilometers.

B. What fraction of Earth’s surface is visible from 580 kilometers above Earth’s surface? Report your answer both as a decimal (rounded to two places) and as a percentage of Earth’s surface. C. Use your graph to determine the largest fraction of Earth’s surface that is visible no matter what the altitude. ANS: A. F 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

20000

40000

60000

80000 100000

B. The fraction is 0.04. This is equivalent to 4% of Earth’s surface. C. 0.5 or 50%. PTS: 3

DIF: hard

Section 2.3 Solving Linear Equations TRUE/FALSE 1. Linear equations are always difficult to solve. ANS: F

PTS: 1

DIF:

easy

2. To say that an equation is linear means that it is a straight line. ANS: F

PTS: 1

DIF:

easy

3. Logarithms are sometimes needed to solve linear equations. ANS: F

PTS: 1

DIF:

easy

4. Techniques for solving linear equations can sometimes be used to reverse the roles of variables in equations. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. An iPad costs $491 at your local computer store. The cable company charges $24 per month for a DSL Internet connection. Write a formula that gives the cost , in dollars, of buying an iPad and paying for months of Internet service. a. c. b. d. ANS: A 2. The life expectancy

PTS: 1

DIF:

, in years, for a child born

easy years after 2000 is given by .

In what birth year would a child have a life expectancy of 77.34 years? a. 2005 b. 2004 ANS: B

c. 2002 d. That life expectancy never occurs. PTS: 1

DIF:

easy

3. A rental car charges $72.87 for insurance and other fees plus a flat rate of $24.95 per day. Write a formula that gives the cost , in dollars, of renting a car for days. a. c. b. d. ANS: A

PTS: 1

DIF:

easy

4. One rental company charges $52.71 in fees for a television plus $22.77 per week. A second rental company charges $31.75 in fees plus $25.61 per week. How many weeks would make the total cost of renting a television the same for both companies? You can be charged for part of a week, so you should report your answer correct to two decimal places.

a. 7.38 weeks b. 52.71 weeks ANS: A 5. The amount given by

c. 10.28 weeks d. It never happens PTS: 1

DIF: medium

, in billions of dollars, collected by the Internal Revenue Service

years after 2003 is

. According to this model, in what year would IRS collections reach 3554 billion dollars? a. 2015 b. 2012 ANS: B

c. 2010 d. It never happens PTS: 1

DIF: medium

6. You got a birthday gift of $45.64, which you choose to spend on music.You pay $12.75 for an iPad app that lets you download songs for $1.43 each. How many songs can you download before your birthday money is used up? a. 26 songs b. 22 songs ANS: C 7. The long jump record formula

c. 23 songs d. 25 songs PTS: 1

DIF:

medium

, in meters, for the th Olympics after 1900 is given approximately by the

. When did the long jump record reach 9.02 meters? a. In the 13th Olympics after 1900 b. In the 12th Olympics after 1900 ANS: A

PTS: 1

c. In the 16th Olympics after 1900 d. The record never reached 9.02 meters. DIF:

medium

8. If you wish to have an item you ordered delivered quickly, you must pay extra for shipping, and the amount you pay depends on the weight , in pounds, of the item. The shipping cost , in dollars, is given by . You have $33.78 to spend on shipping costs. How large an item can you have shipped? a. 13.16 pounds b. 22.39 pounds ANS: D

c. 13 pounds d. 11.39 pounds PTS: 1

9. The cost C, in dollars, of downloading

DIF: medium tunes from an Internet music service is given by

. Here a and b are constants. Solve this equation for download as a function of the cost C.

in order to express the number of tunes you can

d. There is no solution.

ANS: B

PTS: 1

DIF:

medium

10. The speed , in meters per second, of sound in the ocean depends on the depth , in kilometers. (It depends on other quantities as well, but those will be ignored for this problem.) The relationship is . At what depth is the speed of sound 1633.18 meters per second? a. 6.14 kilometers b. 8.04 kilometers ANS: D

c. 29461.38 kilometers d. 5.8 kilometers PTS: 1

DIF: medium

11. The length , in feet, of a baby blue whale depends on its age

, in months. The relationship is

. How old is a baby wale that is 44 feet long? a. 5.37 months b. 17.14 months ANS: A

c. 6.79 months d. A baby whale is never that long. PTS: 1

DIF:

medium

12. You start making regular deposits to your piggy bank. The amount after weeks is given by . The price of the computer you want to buy is decreasing. The price started saving money is given by

, in dollars, in your piggy bank

, in dollars,

weeks after you

. When will there be enough money in your piggy bank to buy the computer? Report your answer as a whole number. a. After 8 weeks b. After 7 weeks ANS: A

c. After 10 weeks d. There will never be enough. PTS: 1

DIF: medium

13. For connection to the system and water usage

, in hundreds of gallons, water company A charges

. For connection to the system and water usage

, in hundreds of gallons, water company B charges .

Both and are in dollars. How many hundreds of gallons of water will result in the same charge from water company A and water company B? a. The charge will never be the same. b. 394 hundred gallons ANS: D

PTS: 1

c. 190.43 hundred gallons d. 394.12 hundred gallons DIF:

14. For connection to the system and water usage

medium

, in hundreds of gallons, a water company charges dollars.

Solve this equation for

to express water consumption as a function of the charges.

a. There is no solution.

ANS: C

PTS: 1

15. As you drive up a hill, your elevation relationship is

DIF:

medium

, in feet, increases with each mile

. Express the miles you drive as a function of elevation. a.

ANS: A 16.

PTS: 1

DIF:

Solve the equation

a. There is no solution. b. ANS: D

PTS: 1

17. Solve the equation

c. d. DIF:

medium

a. There is no solution. b. ANS: D

medium

PTS: 1

c. d. DIF:

medium

that you drive. The

18. Solve the equation

a. There is no solution. b. ANS: B

c. d.

PTS: 1

19. Solve the equation

DIF:

medium

for .

b. ANS: C

d. PTS: 1

20. Solve the equation

PTS: 1

21. Solve the equation

c. d. DIF:

d. PTS: 1

22. Solve the equation a. There is no solution. b. ANS: B

PTS: 1

DIF:

medium for .

c. d. DIF:

23. Solve the equation

medium for y.

a. There is no solution. b. ANS: D

medium

for .

a. There is no solution.

ANS: B

medium

a. There is no solution. b. ANS: D

DIF:

PTS: 1

24. Solve the equation a. There is no solution.

c. d. DIF: for c.

medium .

ANS: D 25.

PTS: 1

DIF:

Solve the equation

for .

a. There is no solution.

ANS: B 26.

PTS: 1

DIF:

Solve the equation

c. d.

PTS: 1

27. Solve the equation

DIF:

PTS: 1

28. Solve the equation

DIF:

d. PTS: 1

29. Solve the equation

DIF:

30. Solve the equation

medium

for z.

a. There is no solution.

ANS: C

medium

for .

a. There is no solution.

ANS: C

medium

for .

a. There is no solution.

ANS: B

medium

for .

a. There is no solution.

ANS: B

medium

PTS: 1

DIF: for z.

medium

a. There is no solution.

ANS: C

PTS: 1

31. The value V, in dollars, of an automobile

DIF:

medium

years after 2010 is given by the formula .

Solve this equation for

to express the time in years since 2010 as a function of the value of the car.

a. There is no solution.

ANS: C

PTS: 1

DIF:

medium

32. Solve the following equation for .

a. There is no solution.

ANS: D

PTS: 1

DIF:

medium

ESSAY 1. Music source A charges a monthly fee of $6.17 plus 99 cents per song. Music source B charges a monthly fee of $10.17 plus 79 cents per song. A: Find a formula that gives the monthly cost , in dollars, of buying songs from Company A as a function of the number of songs you buy in a month. B: Find a formula that gives the monthly cost , in dollars, of buying songs from Company B as a function of the number of songs you buy in a month. C: What number of songs purchased will give the same monthly cost for Company A and Company B? ANS: A: B: C: The cost is the same if you buy 20 songs per month.

PTS: 3

DIF: medium

2. An electric company charges a connection fee of $54 plus $0.19 for each kilowatt-hour of electricity used. A: Write a formula that gives the cost electricity.

, in dollars, of connecting and using

kilowatt-hours of

B: Solve the equation you found in part A to express the kilowatt-hours used as a function of the cost E. C: How many kilowatt-hours are used if the total charge (connection plus charge for electricity usage) is $152.67? ANS: A: B: C: 519.32 kilowatt-hours PTS: 3

DIF: medium

3. The number , in hundreds, of soup cans you can sell in a week depends on the price each can of soup. The relationship is

, in dollars, for

. A: How many cans of soup can you sell in a week if the price is $1.49 per can? B: Solve for

in the equation above to obtain a formula expressing

as a function of

C: What is the price per can if you are selling 23.83 hundred cans per week? ANS: A: 23.83 hundred cans of soup. B: C: $1.49 per can. PTS: 3

DIF: medium

4. You have 552 pairs of shoes in stock. You sell 35 pairs of shoes each week. A: Find a formula that gives the number

of pairs of shoes in stock after

weeks of sales.

B: You need to reorder when stock supplies are reduced to 167 pairs of shoes. After how many weeks will you need to reorder? ANS:

A: B: After 11 weeks. PTS: 2

DIF: medium

5. The chicken sandwiches you are ordering contain a total of 13 grams of trans fat. Fry orders have 10 grams of fat per order. A: Find a formula that gives the number fries to your sandwich order.

of grams of fat in your order if you add

orders of

B: How many orders of fries can you make if you wish to limit trans fat in the total order to 43 grams? ANS: A: B: 3 orders of fries to accompany your sandwich order. PTS: 2

DIF: medium

6. The amount , in billions of bushels, that a country is willing to supply depends on the price dollars per bushel, of wheat. The relationship is

, in

The demand , in billions of bushels, is the quantity of wheat that customers will buy at a price of dollars per bushel. The relationship is .

A: Solve the latter equation for D and so express the demand as a function of the price P.

B: The equilibrium price is the price at which supply and demand are the same. Find the equilibrium price. ANS: A: B: 3.9 dollars per bushel PTS: 2

DIF: medium

7. You are considering jobs that offer similar pay scales. Pay scale 1: You get a base yearly salary of $10463 plus a commission of 9 percent of yearly sales in dollars.

Pay scale 2: You get a base yearly salary of $13368 plus a commission of 7 percent of yearly sales in dollars.

A: Find a formula for your yearly pay pay scale 1.

, in dollars, as a function of yearly sales

, in dollars, under

B: Find a formula for your yearly pay pay scale 2.

, in dollars, as a function of yearly sales

, in dollars, under

C: What amount of yearly sales will result in the same pay under both scales? ANS: A: B: C: Annual sales of 145,250 dollars will result in the same pay. PTS: 3

DIF: medium

8. The relationship between Fahrenheit temperature

and Celsius temperature

is given by

. The relationship between the Kelvin temperature scale and the Celsius scale is . A: Find a formula that expresses

as a function of

Find a formula that expresses

as a function of

C: Find a formula that expresses

as a function of

ANS: A: B: C: PTS: 3

DIF: medium

9. The running speed , in centimeters per second, for certain ants depends on the temperature degrees Celsius. The relationship is

. A: Find a formula that expresses the temperature as a function of the running speed. B:

What is the temperature when the running speed is 6.27 centimeters per second?

ANS: A: B: 44.85 degrees Celsius. PTS: 2

DIF: medium

10. For a certain species of cricket, the temperature , in degrees Fahrenheit, can be estimated from the number of cricket chirps per minute. The relationship is . A: Find a formula that expresses the number of chirps per minute as a function of the temperature. B: How many chirps per minute can be expected if the temperature is 56 degrees Fahrenheit? ANS: A: B: 68 chirps per minute PTS: 2

DIF: medium

11. Between the ages of 7 and 11 years, the weight

, in pounds, of a certain girl is given by the formula ,

where

is her age in years.

During the same age range the relationship between her age t and her height

, in inches, is given by

. A: Find a formula that expresses the girl’s height as a function of her weight. Write your answer in the form , and round the coefficient a of w to two decimal places. B: Use your answer to part A to determine her height when her weight is 7 pounds. C: Use your answer to part A to find a formula that expresses the girl’s weight as a function of her height. ANS: A: B: 41.54 inches tall

C: PTS: 3

DIF: hard

12. The weight of an adult blue whale is related to its length by the formula , where

is the length, in feet, and

A: Find a formula that expresses B:

is the weight, in tons. as a function of

How long would the whale need to be to weigh 101 tons?

C: A blue whale is spotted, and its length is estimated to be 84 feet. Use this length to find the whale’s weight. ANS: A: B: 82.67 feet C: 105.68 tons PTS: 3

DIF: medium

13. The weight of a baby blue whale over the first 7 months of its life can be calculated using , where

is the weight, in tons, and

A: Find a formula that expresses B:

is the age, in months. as a function of

The length , in feet, is given by .

Use your answer from part A to find a formula that gives the length as a function of the weight. C: How long is a baby blue whale if its weight is 19.2 tons? ANS: A:

B: C: 43.58 feet

PTS: 3

DIF: medium

14. An automobile is worth $28857 in 2010. Each year after 2010 the value decreases by $1842. A: Find a formula that gives the value B:

, in dollars,

years after 2010.

When will the value of the car be $14121?

ANS: A: B: In 2018 PTS: 2 15. The running speed relationship is

DIF: medium , in feet per second, of certain animals depends on their length

, in inches. The

. A: Solve this equation for

to get an expression that gives the length in terms of the running speed.

B: If an animal 26.32 is inches long, how fast does it run? C: How long is an animal that runs 26.41 feet per second? ANS: A: B: 60.15 feet per second C: 10.02 inches PTS: 3

DIF: medium

Section 2.4 Solving Nonlinear Equations TRUE/FALSE 1. The crossing-graphs method can be used to solve linear equations. ANS: T

PTS: 1

2. The solution(s) of the equation crosses the horizontal axis. ANS: T

PTS: 1

DIF:

easy

can be found by locating the point(s) where the graph of

DIF:

easy

3. The solution(s) of the equation can be found by locating the point(s) where the graph of crosses the horizontal line determined by the equation . ANS: T

PTS: 1

4. The solution(s) of the equation and cross. ANS: T

PTS: 1

DIF:

easy

can be found by locating the point(s) where the graphs of

DIF:

easy

5. Linear equations cannot be solved using the crossing-graphs method. ANS: F 6. The equation ANS: T 7. The equation ANS: F

PTS: 1

DIF:

easy

has the same solution(s) as the equation PTS: 1

DIF:

easy

has the same solution(s) as the equation PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Solve the equation

. Restrict your attention to the horizontal span of 0 to 5.

a. 1.32 b. 1.5 ANS: A

c. 2.44 d. 2.52 PTS: 1

2. Solve the equation

. Restrict your attention to the horizontal span of 0 to 15.

a. 7.81 b. 112 ANS: A

DIF: medium

c. 8.91 d. 113.11 PTS: 1

DIF: medium

3. Solve the equation

. Restrict your attention to the horizontal span of 0 to 5.

a. 1.64 b. 0.25 ANS: C

c. 0.53 d. 2.77 PTS: 1

4. Solve the equation

. Restrict your attention to the horizontal span of 0 to 20.

a. 5.63 b. 2.37 ANS: A 5. Solve the equation

c. 6.93 d. 4.14 PTS: 1

c. 3.82 d. 2.56 PTS: 1

6. Solve the equation

c. 24.79 d. 23.31 PTS: 1

7. Solve the equation

c. 6.25 d. 1.05 PTS: 1

8. Solve the equation

c. 0.19 d. 4.46 PTS: 1

9. Solve the equation

10. Solve the equation

DIF: medium . Restrict your attention to the horizontal span of 0 to 10.

a. 5.29 b. 3.51 ANS: D

DIF: medium . Restrict your attention to the horizontal span of 0 to 5.

a. 1.41 b. 2.73 ANS: A

DIF: medium . Restrict your attention to the horizontal span of 0 to 10.

a. 2.76 b. 5.01 ANS: B

DIF: medium . Restrict your attention to the horizontal span of 0 to 75.

a. 17.87 b. 16 ANS: D

DIF: medium

. Restrict your attention to the horizontal span of 0 to 5.

a. 14.87 b. 13 ANS: D

DIF: medium

c. 5.77 d. 4.2 PTS: 1

DIF: medium . Restrict your attention to the horizontal span of 0 to 10.

a. 6.2 b. 4.36 ANS: D

c. 6.58 d. 8.13 PTS: 1

11. Solve the equation 10.

c. 1.94 d. 3.37 PTS: 1

12. Solve the equation

13. Solve the equation

PTS: 1

DIF: medium

. Restrict your attention to the horizontal span of 0 to 5. c. 2.69 d. 4.38 PTS: 1

14. Solve the equation

DIF: medium . Restrict your attention to the horizontal span of 0 to 5.

a. 4.57 b. 3.14 ANS: C

c. 1.74 d. 3.38 PTS: 1

DIF:

15. Solve the equation to 5.

c. 4.2 d. 6.01 PTS: 1

16. Solve the equation

17. Solve the equation a. 4.47

DIF:

medium

. Restrict your attention to the horizontal span of 0 to 5.

a. 2.79 b. 0.75 ANS: B

medium . Restrict your attention to the horizontal span of 0

a. 5.65 b. 4.11 ANS: B

medium

c. 9.99 d. 8.13

a. 4.77 b. 3.14 ANS: C

DIF:

. Restrict your attention to the horizontal span of 0 to 10.

a. –0.63 b. –2.36 ANS: D

medium

. Restrict your attention to the horizontal span of 0 to

a. 3.36 b. 1.42 ANS: D

DIF:

c. 1.33 d. 1.88 PTS: 1

DIF: medium . Restrict your attention to the horizontal span of 0 to 5. c. 4.97

b. 6.24

d. 5.9

ANS: A

PTS: 1

DIF: medium

18. There are 3 solutions of

. Find all three solutions.

a. 3.14, 2.72, –3.14 b. 0, 4.85, –1.85 ANS: B

c. 0, 5.61, –2.33 d. 0, 1, –1 PTS: 1

DIF:

medium

19. There is one positive solution of . Find it.a.

0.42

3.16 b. 1.2

c. d. –1

ANS: B

PTS: 1

DIF:

medium

20. There is one negative solution of . Find it.a. 1.55 b. –1

–

c. d. –0.83

ANS: D 21. The amount

0.42

PTS: 1

DIF:

medium

, in grams, of one radioactive substance remaining after

years is given by

. The amount

, in grams, of a second radioactive substance remaining after

years is given by

. When will the amounts remaining of each be the same? a. After 2.4 years b. After 11.57 years ANS: B 22. The temperature

c. After 0.53 years d. After 14.32 years PTS: 1

DIF:

medium

, in degrees Fahrenheit, of a cup of coffee

minutes after it is poured is given by .

When does the temperature of the coffee reach 154 degrees? a. After 13.71 minutes b. After 71.51 minutes ANS: A 23. The temperature

PTS: 1

c. After 68.75 minutes d. After 16.09 minutes DIF:

medium

, in degrees Fahrenheit, of a roast

minutes after it is placed in the oven is given by .

When does the temperature of the roast reach 144 degrees? a. After 128.57 minutes b. After 94.14 minutes ANS: B

PTS: 1

c. After 124.91 minutes d. After 95.63 minutes DIF:

medium

24. You are running an election campaign. The number vote for your candidate after days is given by

, in thousands, of people who have decided to

. You expect your candidate to win if she gets 50 thousand votes. Will you ever expect your candidate to win? If so when will the required number of votes be reached? Round your answer to the nearest whole number of days. a. b. c. d.

The required number of votes will never be reached. The required number of votes is reached after 73 days. The required number of votes is reached after 76 days. The required number of votes is reached after 79 days.

ANS: C

PTS: 1

DIF:

medium

25. You sell heating oil. Your yearly storage cost , in dollars, depends on the amount , in thousands of gallons, of heating oil you order at the beginning of the season. The relationship is . There are two heating oil order sizes that will result in a yearly storage cost of 154 dollars. Find both of them. a. 4.2 thousand gallons and 10.33 thousand gallons b. 0.34 thousand gallons and 7.09 thousand gallons c. 25.3 thousand gallons and 0.36 thousand gallons d. 30.25 thousand gallons and 30.8 thousand gallons ANS: C 26. The number per day by

PTS: 1

DIF:

medium

of customers patronizing a restaurant

days after the grand opening is given

. There are two times when 600 customers per day patronize the restaurant. Find both times. Round your answers to the nearest whole number of days.

a. There are never 600 customers per day. b. and ANS: D

PTS: 1

c. d.

and and

DIF:

medium

27. The amount

, in milligrams, of a drug in the bloodstream

minutes after injection is given by .

There are two times when there are 7 milligrams of the drug in the bloodstream. Find both times.

a. There are never 7 milligrams of the drug in the blood. b. and c. and d. and ANS: C

PTS: 1

DIF:

medium

28. The profit , in thousands of dollars, a manufacturer makes if produced is given by

, measured in thousands, items are

. The formula is valid for up to 10 thousand items produced. What number of items should be produced to yield a profit of 7 thousand dollars? There are two solutions. Find both of them.

a. That profit level cannot be achieved. b. 0.36 thousand and 9.31 thousand ANS: B 29. The weight

PTS: 1

c. 1.44 thousand and 7.63 thousand d. 7.33 thousand and 8.10 thousand DIF:

, in pounds, of a certain animal at age

medium years is given by .

The formula is valid for ages up to 10 years. What age gives a weight of 11 pounds? a. That weight is never achieved. b. 4.21 years ANS: D

PTS: 1

c. 4.12 years d. 2.76 years DIF:

30. Water is being pumped from a tank. The amount hours is given by

medium , in gallons, of water remaining in the tank after

. How long will it be before there are 1370 gallons of water left in the tank? a. After 8.19 hours b. After 7.11 hours ANS: C 31. Solve the equation

c. After 3.16 hours d. After 4.31 hours PTS: 1

DIF:

medium

. Restrict your attention to the horizontal span of 0 to 10.

a. 5.37 b. 1.79

c. 2 d. 1.54

ANS: D

DIF: medium

PTS: 1

ESSAY 1. The number of deer present on a reserve after

years is given by .

A: Make a graph of

versus

that shows the population over the first 30 years.

B: We want to know when the deer population reaches 89 deer. The answer to that question is the solution of an equation. Which equation? C: Solve the equation you found in part B. ANS: A: N 360 320 280 240 200 160 120 80 40

–40

B: C: After 4.21 years PTS: 3

DIF: hard

2. For a certain automobile engine, the bore inches, by the formula

, in inches, is related to the displacement

. A: Make a graph of

versus

. Include bore sizes up to 5 inches.

, in cubic

B: What displacement corresponds to a bore of 3.41 inches? C: What bore corresponds to a displacement of 184 cubic inches? ANS:

A: D 200 180 160 140 120 100 80 60 40 20 –2

–1

B: 112.68 cubic inches C: 4.36 inches PTS: 3

DIF: hard

3. For certain ships the diameter , in feet, of the propeller depends on the maximum revolutions per minute the propeller turns. The relation is given by

. A: Make a graph of

versus . Include up to 200 maximum revolutions per minute.

B: What diameter corresponds to a maximum revolutions per minute of 114? C: What maximum revolutions per minute corresponds to a diameter of 16 feet? ANS:

d 40 35 30 25 20 15 10 5 –70 –35 –5

70 105 140 175 210

–10

B: 16.68 feet C: 122.2 revolutions per minute PTS: 3 4. The number

DIF: hard of geese nesting in a protected area after

years is given by

. A: Make a graph of

versus

over the first 20 years.

B: What was the initial population of geese in the area? Round your answer to the nearest whole number. C: When did the population reach 115 geese? ANS:

N 180 160 140 120 100 80 60 40 20 –2 –20

8 10 12 14 16 18 20

B: 26 geese C: After 9.78 years PTS: 3

DIF: hard

5. When a skydiver jumps from an airplane, his downward velocity the fall is given by

, in feet per second,

seconds into

. A: Make a graph of

versus

over the first 20 seconds of the fall.

B: The terminal velocity of the skydiver is 179 feet per second. What is 90% of terminal velocity? C: When does the skydiver reach 90% of terminal velocity? ANS:

A: V 180 160 140 120 100 80 60 40 20 –2 –20

8 10 12 14 16 18 20

B: 161.1 feet per second C: After 13.21 seconds PTS: 2

DIF: hard

6. If an average-sized man falls with a parachute, he will fall A: Make a graph of

versus

feet in

seconds.

over the first minute of the fall.

B: How far will the man fall in 51 seconds? C: How long does it take for the man to fall 750 feet? ANS:

A: D 1000 900 800 700 600 500 400 300 200 100 –20 –10

B: 1007.5 feet C: 38.13 seconds PTS: 3 7. The amount by

DIF: hard , in grams, of a certain isotope of uranium remaining after

. A: Make a graph of

versus

over one hour.

B: How much of the isotope is present after 13 minutes?

minutes of decay is given

C: What is the half-life of this isotope? That is, how long will it take for half of the substance to decay? ANS:

A: A 100 90 80 70 60 50 40 30 20 10

B: 34.86 grams C: 9.5 minutes PTS: 3

DIF: hard

8. If you borrow money at an APR of payment will be

as a decimal, then under certain conditions your monthly

. A: Make a graph of M versus . Include interest rates of up to 10% (corresponding to

B: Your budget allows a monthly payment of $150. What is the largest interest rate you can accept? ANS:

M 200 180 160 140 120 100 80 60 40 20 –0.02

0.02 0.04 0.06 0.08

0.1

or 5%

PTS: 2

DIF: hard

9. The amount , in pounds, of grass a kangaroo will eat in a day depends on the amount per acre, of vegetation present. The relationship is given by

, in pounds

. A: Make a graph of

versus

. Include vegetation levels up to 2000 pounds per acre.

B: What vegetation level will cause a kangaroo to consume 2.1 pounds in a day? C: Is the graph concave up or concave down? Explain in practical terms what this means about how the kangaroo feeds. D: What is the most a kangaroo will eat in a day no matter how much vegetation is present? ANS:

G 3 2.5 2 1.5 1 0.5 –1000 –500 –0.5

500 1000 1500 2000

–1 –1.5 –2

B: 621.23 pounds per acre C: The graph is concave down. As vegetation levels increase the amount the kangaroo eats increases, but at a decreasing rate. Additional availability of vegetation makes a greater difference in the amount the kangaroo eats when there is less vegetation present. D: About 2.5 pounds PTS: 4

DIF: hard

10. The number of cocoons a certain wasp will parasitize in a day depends on the number per square inch that are present. The relationship is given by

of cocoons

. A: Make a graph of

versus . Include values of

up to 4 cocoons per square inch.

B: What number of cocoons per square inch will cause the wasp to parasitize 7 of them? C: What is the largest number of cocoons the wasp will parasitize no matter how many cocoons are present? ANS:

10 8 6 4 2

–2 –4

B: 1.61 cocoons per square inch C: About 8.8 cocoons PTS: 3 11. The circulation

DIF: hard , in thousands, of a certain magazine

years after 2010 is given by .

This formula is valid over the period from 2010 to 2015 only. A: Make a graph of

versus .

B: There are two values of

that have a circulation of 9 thousand. Find both of them.

C: Explain in practical terms what is happening to magazine sales over the period from 2010 to 2015. ANS:

24 20 16 12 8 4

–1

–4

and

C: The circulation increases until early 2013 and decreases after that. PTS: 3

DIF: hard

12. The growth rate , in hundreds of cases per day, in the spread of an epidemic depends on the number , in hundreds, of currently sick individuals. The relationship is given by . A: Make a graph of

versus . Include up to 4000 currently sick people (corresponding to

B: There are two numbers of currently sick individuals that give a growth rate of 500 new cases per day (corresponding to ). Find both of them. C: When there are no sick individuals, the growth rate is 0. There is a second level where the growth rate is 0. Find that number of currently sick people. ANS:

G 14 12 10 8 6 4 2

–2

B: 5.73 hundred and 36.54 hundred C: 39.60 hundred PTS: 3

DIF: hard

Section 2.5 Inequalities TRUE/FALSE 1. Graphs can be used to solve equations but not inequalities. ANS: F

PTS: 1

DIF:

2. The solution to the inequality . ANS: T

occurs where the graph of

PTS: 1

DIF:

3. Complicated inequalities such as ANS: F

easy lies above the graph of

easy cannot be solved using graphs.

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Solve the inequality

a. b.

c. d.

ANS: A

PTS: 1

2. Solve the inequality a. b.

c. d. PTS: 1

3. Solve the inequality

DIF:

PTS: 1

DIF:

a. b.

and medium

a. b.

5. Solve the inequality

medium

c. d.

4. Solve the inequality

ANS: A

and

. Restrict your attention to positive values of

and

ANS: D

medium

and

ANS: D

a. b.

DIF:

c. d. PTS: 1

DIF: . c. d.

medium

ANS: A

PTS: 1

6. Solve the inequality

DIF:

. Restrict your attention to positive values of

a. b. ANS: A

PTS: 1

DIF:

c. d. PTS: 1

DIF:

8. Solve the inequality

c. d. PTS: 1

9. Solve the inequality

DIF:

c. d. PTS: 1

DIF:

10. Solve the inequality

c. d. The inequality is never true.

PTS: 1

DIF:

11. Solve the inequality c. d. PTS: 1

12. Solve the inequality

DIF:

medium

a. b.

13. Solve the inequality

medium .

a. b.

ANS: B

medium

. Restrict your attention to positive values of .

a. b. The inequality is always true.

ANS: A

medium

. Restrict your attention to positive values of .

a. b.

ANS: B

medium

. Restrict your attention to positive values of

a. b.

ANS: B

medium

. Restrict your attention to positive values of .

a. b.

ANS: B

c. d.

7. Solve the inequality

ANS: D

medium

c. d. PTS: 1

DIF: .

medium

a. b. ANS: B

c. d. PTS: 1

14. Solve the inequality

DIF:

. Restrict your attention to positive values of .

a. b. ANS: B

c. d. PTS: 1

15. Solve the inequality

DIF:

c. d. There is no solution. PTS: 1

16. Solve the inequality

DIF:

17. Solve the inequality a. b. ANS: D

c. d. PTS: 1

DIF:

c. d. PTS: 1

DIF:

c. d. PTS: 1

DIF:

medium

. Be careful. The graphs cross twice.

a. b.

c. d. PTS: 1

20. Solve the inequality

DIF:

medium

a. b. ANS: C

medium

19. Solve the inequality

ANS: D

medium

18. Solve the inequality a. b. ANS: D

hard

a. b. ANS: A

medium

a. b. ANS: D

medium

c. d. PTS: 1

DIF:

medium

21. Solve the inequality

a. b.

c. d.

ANS: C

PTS: 1

22. Solve the inequality a. b.

DIF: .

and and

ANS: A

c. d. PTS: 1

DIF:

23. Solve the inequality

c. d.

ANS: A

PTS: 1

DIF:

24. Solve the inequality

c. d. PTS: 1

25. Solve the inequality

c. d. PTS: 1

26. Solve the inequality

medium

DIF:

and medium

. Restrict your attention to positive values of

a. b. ANS: A

DIF:

and

ANS: B

medium

and

ANS: B

a. b.

medium

a. b.

medium

c. d. PTS: 1

DIF:

medium

27. The body surface area , in square meters, of a man who is 180 centimeters tall can be estimated from his weight , in kilograms. The relationship is . What weight range results in a body surface area of more than 2.26 meters? a. b.

kilograms kilograms

c. d.

kilograms kilograms

ANS: A 28. The distance

PTS: 1

DIF:

medium

, in miles, from a moving train to a home is given by .

Here is the time in hours since the train left the station. For what values of miles from the home? a. b.

hours

c. d.

hours

ANS: B

PTS: 1

DIF:

is the train less than 2

hours hours medium

29. A metal bar is heated and then placed on a workbench for bending. The temperature Fahrenheit, minutes after heat is applied is given by

, in degrees

. The bar can be worked so long as its temperature is higher than 440 degrees. For what values of the bar suitable for working? a. b.

minutes minutes

ANS: B

PTS: 1

c. d. DIF:

minutes minutes medium

30. City water, which is slightly chlorinated, is being used to flush a tank of heavily chlorinated water. The concentration , in milligrams per gallon, of chlorine in the tank hours after flushing begins is given by . The water in the tank is safe for use when chlorine levels are less than 2.7 milligrams per gallon. For what values of is the water in the tank safe for use? a. b.

hours hours

ANS: D

PTS: 1 DIF:

31. A tubeworm of length

c. d.

meters is

hours hours

medium years old, where .

In a certain location tubeworms are known to be between the ages of 107 years to 187 years. the range of lengths of the tubeworms in this location?

a. From 1.44 meters to 1.46 meters b. From 11.2 meters to 13.74 meters ANS: C

PTS: 1

c. From 1.58 meters to 1.92 meters d. From 1.86 meters to 3.09 meters DIF:

medium

What is

32. Plutonium-239 is a primary component of modern nuclear weapons. If there is initially 9 kilograms, then after thousand years, the amount remaining is given by . Over what time range will there be between 1 and 2 kilograms of plutonium-239 remaining? a. From 52.51 to 76.71 thousand years b. From 33.72 to 38.52 thousand years ANS: A

PTS: 1

c. Up to 76.71 thousand years d. Up to 33.72 thousand years DIF:

medium

33. A roast is baking in the oven. The internal temperature minutes after it is put in the oven is given by

, in degrees Fahrenheit, of the roast

. The roast should be removed from the oven when its internal temperature is between 160 and 165 degrees. During what time period should the roast be removed? a. Between b. Between c. Between d. Between ANS: C

115.93 101.62 101.93 106.75

and 118.43 minutes after it is placed in the oven and 103.62 minutes after it is placed in the oven and 107.9 minutes after it is placed in the oven and 111.37 minutes after it is placed in the oven PTS: 1

DIF:

medium

34. The length , in centimeters, of a certain fish depends on the age , in years. The relationship is . Fish between lengths of 34.5 and 40.5 centimeters long are suitable for harvesting. Over what age range is the fish suitable for harvesting? a. 5.14 years or older b. 12.25 years or older ANS: C

PTS: 1

c. Between 3.49 and 5.14 years old d. Between 7.68 and 12.25 years old. DIF:

medium

35. If an average-sized man falls with a parachute, then his velocity , in feet per second, the fall is given by

seconds into

. During which period will his velocity be between 9.79 and 12.79 feet per second? a. From 0.66 seconds on b. From 9.02 seconds on ANS: C

PTS: 1

c. From 0.43 to 0.66 seconds into the fall d. From 4.67 to 9.02 seconds into the fall DIF:

36. Buffalo are introduced into a protected area. After by

medium years the number of buffalo in the herd is given

. During what period are there between 36 and 62 buffalo in the herd?

a. b. c. d.

From 5.36 years on From 15.12 years on From 2.57 to 5.36 years since the herd was introduced to the area From 7.26 to 15.12 years since the herd was introduced to the area

ANS: C

PTS: 1

DIF:

medium

ESSAY 1. The concentration , in milligrams per deciliter, of a drug in the bloodstream injection is given by

hours after an

. A: Make a graph of

versus

over the first 10 hours since the injection.

B: The drug is effective as long as the concentration is 50 milligrams per deciliter or higher. Add to your graph the horizontal line . C: During what time period is the drug effective? ANS: A: C 90 80 70 60 50 40 30 20 10

–10 –20

C 90 80 70 60 50 40 30 20 10

–10

–20

C: From 1.03 hours to 5.94 hours after the injection PTS: 3

DIF: hard

2. Beginning in 2001 the price, in dollars per ounce, of gold was modeled approximately by , where

is the time in years since the beginning of 2001.

A: Make a graph of

versus

that shows estimated gold prices for 2001 to 2010.

B: A broker recommended purchase of gold as long as the price was between $400 and $700 per ounce. Add the horizontal lines and to your graph. C: During what time period was the purchase of gold advised? Give your answer in terms of correct to two decimal places. ANS: A: G 1200 1100 1000 900 800 700 600 500 400 300 200 100 –2 –100 –200

G 1200 1100 1000 900 800 700 600 500 400 300 200 100 –2 –100

–200

C: From

years

PTS: 3

DIF: hard

3. The income , in dollars per month, for a fish farm depends on the fish population hundreds of fish. The relationship is

, measured in

A: Make a graph of

versus

for fish populations up to 15 hundred fish.

B: The expense , in dollars per month, also depends on the population (again measured in hundreds). The relationship is . Add the graph of

versus

to the graph you made in part A.

C: The fish farm is profitable if income exceeds expenses. What range of fish population results in a profitable fish farm? ANS: A:

I 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 –500

–1000

B: I 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 –500

–1000

C: From 0.81 hundred fish to 11.79 hundred fish. PTS: 3

DIF: hard

4. When a car makes an emergency stop on dry pavement, it leaves skid marks on the pavement. The length , in feet, of the skid marks is related to the speed , in miles per hour, when the brakes were applied. The relationship is .

A: Make a graph of

against

for skid mark lengths up to 200 feet.

B: A man involved in an emergency stop claimed his speed was between 30 and 40 miles per hour. Add the horizontal lines represented by and to the graph you made in part A. C: What range of skid mark length would support the driver’s claim?

ANS: A: S 90 80 70 60 50 40 30 20 10 –40 –20 –10

80 100 120 140 160 180 200

B: S 90 80 70 60 50 40 30 20 10 –40 –20 –10

80 100 120 140 160 180 200

C: From 29.32 to 52.13 feet PTS: 3 5. The function hour.

A: Make a graph of

DIF: hard is used to estimate wave height , in feet, from wind speed

versus

, in miles per

for wind speeds up to 25 miles per hour.

B: A small boat can sail safely provided wave heights are no more than 4 feet. Add the horizontal line represented by to the graph you made in part A. C: What range of wind speeds will result in safe sailing for small boats? ANS: A:

14 12 10 8 6 4 2

B: h

14 12 10 8 6 4 2

C: From 0 to 14.14 miles per hour PTS: 3

DIF: hard

6. A certain drug has a concentration of C milligrams per liter in the bloodstream administered. The relationship is

hours after it is

. A: Make a graph of

versus

for up to 10 hours after the drug is administered.

B: The drug is most effective when the concentration is at least 1.5 milligrams per liter. Add the graph of to the graph you made in part A. C: During what time period is the drug most effective? ANS: A:

C 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 –2 –1 –0.2

B: C 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 –2 –1 –0.2

C: From 0.56 to 2.81 hours since the drug was administered PTS: 3

DIF: hard

7. A cohort of fish is a group of fish of the same age. For a certain fish in the North Sea, the total biomass , in pounds, of a cohort of age years is given by . A: Make a graph of

versus

for fish up to 30 years old.

B: Fishing of a cohort is allowed if the biomass of that cohort is at least 2000 pounds. Add the graph of to the graph you made in part A. C: What age cohorts are allowed to be fished? ANS: A:

B 4000 3500 3000 2500 2000 1500 1000 500 –10

–5 –500

–1000

B: B 4000 3500 3000 2500 2000 1500 1000 500 –10

–5 –500

–1000

C: Fishing is allowed for cohorts of ages 6.39 years through 24.84 years. PTS: 3

DIF: hard

8. A pyrotechnic display is fired upward with an initial velocity of 57 feet per second. Its height feet, after seconds (until it hits the ground) is given by

, in

. A: Make a graph of

versus .

B: The device is visible when it is 30 feet high or higher. Add the graph of made in part A. C: During what time period will the display be visible? ANS: A:

to the graph you

H 80 70 60 50 40 30 20 10 –0.5 –10

0.5

1.5

2.5

3.5

0.5

1.5

2.5

3.5

–20

B: H 80 70 60 50 40 30 20 10 –0.5 –10

–20

C: From

seconds to

PTS: 3

seconds

DIF: hard

9. For a certain population the growth rate , measured in thousands per year, depends on the population , measured in thousands. The relationship is . A: Make a graph of B:

versus . Include populations of up to 120,000 (corresponding to

Explain what is happening to the population at

C: Over what population ranges is

thousand and at

thousand.

negative?

D: Explain in practical terms what is happening to the population over the population ranges you found in part C. ANS:

A: G 60 50 40 30 20 10 –20–10

10 20 30 40 50 60 70 80 90 100110120

–20 –30 –40 –50 –60 –70 –80 –90

B: The growth rate is zero, so the population is not changing. C: From 0 to 23 thousand and above 107 thousand D. The population is decreasing over these ranges. PTS: 4

DIF: hard

10. For a certain population the growth rate , measured in thousands per year, depends on the population , measured in thousands. The relationship is . A: Make a graph of

versus . Include populations of up to 200,000 (corresponding to

B: Ecologists believe the population will be self-sustaining as long as the growth rate is 50 thousand per year or higher. Add the horizontal line corresponding to a growth rate of 50 thousand per year to the graph you made in part A. C: Over what population range is the growth rate 50 thousand per year or more? ANS: A:

G 220 200 180 160 140 120 100 80 60 40 20 –40 ––2200

20 40

80 100 120 140 160 180 200

B: G 220 200 180 160 140 120 100 80 60 40 20 –40 ––2200

20 40

C: From 52.35 thousand to 192.47 thousand PTS: 3

DIF: hard

11. You put a drink in the freezer for a while. You take the drink out, but being forgetful, you leave the drink sitting on the kitchen table. The temperature , in degrees, of the drink minutes after it is put in the freezer is given by . A: Make a graph of

versus

for up to one hour after the drink is placed in the freezer.

B: During what period was the temperature of the drink less than 32 degrees? ANS: A:

D 70 60 50 40 30 20 10

–5 –10

B: From PTS: 3

10 15 20 25 30 35 40 45 50 55

minutes DIF: hard

Section 2.6 Optimization TRUE/FALSE 1. Graphs can be used to find maxima and minima for functions. ANS: T

PTS: 1

DIF:

easy

2. Maxima for functions can be found at peaks or at endpoints. ANS: T

PTS: 1

DIF:

easy

DIF:

easy

3. Minima always occur at endpoints. ANS: F

PTS: 1

4. No function can have both a maximum and a minimum value. ANS: F

PTS: 1

DIF:

easy

5. Assume that the function is increasing on the horizontal span of 2 to 7. Then the maximum value of on this horizontal span is . ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Find the minimum value of

. Restrict your attention to the horizontal span of 0 to 10.

a. 5.43 b. 0.71 ANS: A

c. 1.93 d. 9.38 PTS: 1

2. Find the maximum value of

DIF: medium . Restrict your attention to the horizontal span of 0 to 10.

a. 2.35 b. 0.56 ANS: B 3. What value of of 0 to 10.

c. 3.63 d. 4.48 PTS: 1

gives the maximum value of

a. 2.5 b. 0.72 ANS: A

? Restrict your attention to the horizontal span c. 3.57 d. 3.99

PTS: 1

4. What is the minimum value of 1.

DIF: medium

DIF: medium ? Restrict your attention to the horizontal span of 0 to

a. 0.13 b. 1.48 ANS: B

c. 1.6 d. 4.52 PTS: 1

5. Find the maximum value of

a. 0.5 b. 7.5 ANS: B

PTS: 1

a. 2.39 b. 20.04 PTS: 1

a. 7 b. 11.08 PTS: 1

a. 2.54 b. 1.72

9. For what value of span of 0 to 7.

10. For which value of span of 0 to 9.

a. 13

reach a maximum? Restrict your attention to the

DIF: medium ? Restrict your attention to the horizontal span of 0 to 7.

DIF: medium ? Restrict your attention to the horizontal span of 0 to

c. 0.86 d. 5.12 PTS: 1 does

a. 2.59 b. 14.09 ANS: C

medium

c. 1.17 d. 14.81

8. What is the minimum value of 11.

ANS: B

DIF:

c. 0.92 d. 23.67

7. What is the maximum value of

ANS: B

. Restrict your attention to the horizontal span of 0 to 6.

c. 1.71 d. 10.77

6. For which value of does horizontal span of 0 to 11.

ANS: C

DIF: medium

DIF: medium reach a minimum? Restrict your attention to the horizontal

c. 0.62 d. 17.99 PTS: 1

DIF: medium

does

reach a minimum? Restrict your attention to the horizontal

c. 1.5

b. 6.25 ANS: C

d. 0 PTS: 1

11. For which value of does horizontal span of 0 to 12.

DIF: medium reach a maximum? Restrict your attention to the

a. 1.66 b. 3.77 ANS: C

c. 0.05 d. 7.46 PTS: 1

DIF: medium

12. What is the maximum value of 14.

a. 1.68 b. 4.84 ANS: B

Restrict your attention to the horizontal span of 0 to

c. 0.11 d. 8.33 PTS: 1

DIF:

medium

13. What value of gives the minimum value of horizontal span of 0 to 16.

a. 27.73 b. 3.9 ANS: D

c. 31.5 d. 2.13 PTS: 1

14. What is the minimum value of to 18.

a. 22.44 b. 3.95 ANS: A

PTS: 1

a. –1.51 b. 4.78

medium

? Restrict your attention to the horizontal span of 0

DIF:

medium

? Restrict your attention to the horizontal span of 1 to

c. 87 d. 3.17 PTS: 1

16. What is the maximum value of 20.

a. 38.23

DIF:

c. 25.68 d. 2.4

15. What is the maximum value of 5.

ANS: C

? Restrict your attention to the

DIF:

medium

? Restrict your attention to the horizontal span of 0 to

c. 42.04

b. 12.7 ANS: A

d. 11.24 PTS: 1

DIF: medium

17. What is the minimum value of of 0 to 10.

? Restrict your attention to the horizontal span

a. 29.26 b. 5.34 ANS: A

c. 32.78 d. 3.95 PTS: 1

DIF: medium

18. What value of gives the minimum value of horizontal span of 0 to 10. a. 32.48 b. 5.13 ANS: C

? Restrict your attention to the

c. 4.03 d. 35.61 PTS: 1

DIF: medium

19. What value of gives a maximum for span of 0 to 40.

a. 31.49 b. 9.81 ANS: D

? Restrict your attention to the horizontal

c. 35.15 d. 8.38 PTS: 1

DIF:

medium

20. What value of gives a minimum for ? Restrict your attention to the horizontal span of 5 to 11. Be careful. You are asked about a minimum, not a maximum.

a. 32.08 b. 11 ANS: B

c. 23.31 d. 8.42 PTS: 1

DIF:

medium

21. What is the minimum value for ? Restrict your attention to the horizontal span of 5 to 11. Be careful. You are asked about a minimum, not a maximum. a. 32.42 b. 11 ANS: C

c. 23.75 d. 8.44 PTS: 1

DIF: medium

22. One model for expenditures at high schools gives the cost number of students enrolled. The relationship is

, in dollars per pupil, as a function of the

Find the enrollment and cost that correspond to the minimum cost per pupil. Round the enrollment to the nearest whole number.

a. b. c. d.

An enrollment of 1464 gives a minimum cost of $443.82 per pupil. An enrollment of 443.82 gives a minimum cost of $1464 per pupil. An enrollment of 1466 gives a minimum cost of $447.06 per pupil. An enrollment of 447.06 gives a minimum cost of $1466 per pupil.

ANS: A

PTS: 1

DIF:

medium

23. For a certain epidemic the number of new cases expected in a day depends on the number days since the outbreak began. The relationship is

. What is the greatest number of new cases that occur on a day, and on what day does that happen? Round both answers to the nearest whole number.

a. b. c. d.

A maximum of A maximum of A maximum of A maximum of

ANS: D

42 new cases occurs on day 23. 21 new cases occurs on day 38. 23 new cases occurs on day 42. 38 new cases occurs on day 21. PTS: 1

DIF:

medium

24. You put a drink in the freezer to cool. When it is chilled, you take the drink out but, being forgetful, leave it sitting on the kitchen counter. The temperature , in degrees, of the drink minutes after it is put in the freezer is given by . When is the drink the coldest, and what is its coldest temperature? a. b. c. d.

It reaches its coldest temperature of 9.76 degrees after 25.76 minutes. It reaches its coldest temperature of 25.76 degrees after 9.76 minutes. It reaches its coldest temperature of 11.97 degrees after 29.15 minutes. It reaches its coldest temperature of 29.15 degrees after 11.97 minutes.

ANS: B

PTS: 1

DIF:

medium

25. You put a drink in the freezer to cool. When it is chilled, you take the drink out but, being forgetful, leave it sitting on the kitchen counter. The temperature , in degrees, of the drink minutes after it is put in the freezer is given by . When during the first 20 minutes is the drink the warmest, and what is its warmest temperature? Be careful. You are asked for the warmest temperature, not the coldest. a. It reaches its warmest temperature of 72 degrees when it is first put in the freezer.

b. It reaches its warmest temperature of 0 degrees when it is first put in the freezer. c. It reaches its warmest temperature of 20 degrees after 54.93 minutes. d. It reaches its warmest temperature of 54.93 degrees after 20 minutes. ANS: A

PTS: 1

DIF:

medium

26. You put a drink in the freezer to cool. When it is chilled, you take the drink out but, being forgetful, leave it sitting on the kitchen counter. The temperature , in degrees, of the drink minutes after it is put in the freezer is given by . During the period from 5 minutes after it is put in the freezer until 20 minutes after, when is the drink the warmest, and what is its warmest temperature? Be careful. You are asked for the warmest temperature, not the coldest. a. b. c. d.

It reaches its warmest temperature of 45.13 degrees after 5 minutes. It reaches its warmest temperature of 5 degrees after 45.13 minutes. It reaches its warmest temperature of 20 degrees after 54.93 minutes. It reaches its warmest temperature of 54.93 degrees after 20 minutes.

ANS: D

PTS: 1

DIF:

medium

27. One model for the growth rate , in thousands per year, of a certain species depends on the population , in thousands. The relationship is

What population corresponds to the maximum growth rate, and what is that growth rate? a. The maximum growth rate of 5.75 thousand per year corresponds to a population of 3.5 thousand. b. The maximum growth rate of 3.5 thousand per year corresponds to a population of 5.75 thousand. c. The maximum growth rate of 4.51 thousand per year corresponds to a population of 8.32 thousand. d. The maximum growth rate of 8.32 thousand per year corresponds to a population of 4.51 thousand. ANS: A

PTS: 1

DIF:

medium

28. You are laying phone cable between two cities and must cross a lake. You will lay miles on land. The cost , in dollars, depends on how much cable is on land and how much is in water. The relationship is . How much cable do you lay on land to achieve a minimum cost, and what is that minimum cost? a. b. c. d.

The minimum cost of $9.32 is achieved if you lay 3022.53 miles of cable on land. The minimum cost of $3022.53 is achieved if you lay 9.32 miles of cable on land. The minimum cost of $7.4 is achieved if you lay 3020.17 miles of cable on land. The minimum cost of $3020.17 is achieved if you lay 7.4 miles of cable on land.

ANS: D

PTS: 1

DIF:

medium

29. A can is made from 6 square inches of aluminum. We can calculate both the height the volume , in cubic inches, from the radius , in inches, of the can:

, in inches, and

, . Find the radius and height of the can of maximum volume. What is the maximum volume? a. The radius is 1.13 inches, and the height is 0.56 inches. b. The radius is 3.79 inches, and the height is 2.39 inches. c. The radius is 2.39 inches, and the height is 3.79 inches. d. The radius is 0.56 inches, and the height is 1.13 inches. ANS: D

PTS: 1

DIF:

medium

30. The number , measured in millions of students, enrolled in high school approximately by

years after 1965 is given

According to this model, in what year (as a date) did high school enroll reach a maximum, and what was the maximum enrollment? a. b. c. d.

The maximum enrollment of 12.55 million occurred in 1976. The maximum enrollment of 11 million occurred in 1977. The maximum enrollment of 11.58 million occurred in 1980. There is no maximum value.

ANS: A

PTS: 1

DIF:

medium

31. The circulation , in thousands, of a magazine depends on the time , in years, since initial publication. The following relationship is valid over the first 7 years: .

When did magazine circulation reach a maximum, and what was the maximum circulation? a. b. c. d.

The maximum circulation of 4.67 thousand occurred 54.81 years after initial publication. The maximum circulation of 54.81 thousand occurred 4.67 years after initial publication. The maximum circulation of 3.31 thousand occurred 57.77 years after initial publication. The maximum circulation of 57.77 thousand occurred 3.31 years after initial publication.

ANS: B

PTS: 1

DIF:

medium

ESSAY 1. A cannonball fired from the origin follows the graph of

. Here denotes the distance downrange, and measured in feet.

denotes the height of the cannonball. Both are

A: Make a graph of versus . In choosing a viewing window, bear in mind that the cannonball follows the graph only until it strikes the ground. B: How far downrange does the cannonball strike the ground? C: How far downrange does the cannonball reach its maximum height? D: What is the maximum height the cannonball reaches? ANS: A: y 500 450 400 350 300 250 200 150 100 50

250

500

750 1000 1250 1500 1750

B: 1860.5 feet downrange C: 930.25 feet downrange D: 465.13 feet PTS: 4

DIF: hard

2. The growth rate , in cubic feet per acre per year, of a certain forest stand depends on its age years. The relationship is

A: Make a graph of

versus

. Include ages up to 60 years.

B: Explain in practical terms what the graph tells you about the growth rate of the forest stand over the first 60 years.

C: At what age does the maximum growth rate occur? D: What is the maximum growth rate? ANS: A: G 500 450 400 350 300 250 200 150 100 50 –10

B: The growth rate increases when the forest is young but declines when the forest is older. C: At age 16 years D: The maximum growth rate is 372.62 cubic feet per acre per year. PTS: 4

DIF: hard

3. Suppose a wire that is 103 feet long is cut into one piece of length feet and another of length feet. The first piece is bent into a square, and the second is bent into a circle. The total area enclosed is given by

square feet. A: Make a graph of A versus x. Include values of B: What -value gives the minimum area? C: What is the minimum area? ANS: A:

up to 103 feet.

A 1000 900 800 700 600 500 400 300 200 100 –10

feet

C: 371.38 square feet PTS: 3

DIF: hard

4. Suppose a wire that is 99 feet long is cut into one piece of length feet and another of length feet. The first piece is bent into a square, and the second is bent into a circle. The total area enclosed is given by

square feet.

A: Make a graph of A versus x. In choosing your viewing window, bear in mind that the formula makes sense only when is between 0 and 99. B: What is the area at the left-hand endpoint? What physical situation does that represent? C: What is the area at the right-hand endpoint? What physical situation does that represent? D: How should the wire be used so that the maximum area is enclosed? ANS: A:

A 1000 900 800 700 600 500 400 300 200 100 –10

B: The area is 779.94 square feet. All the wire is used for the circle. C: The area is 612.56 square feet. All the wire is used for the square. D: The maximum area is obtained if all the wire is used for the circle. PTS: 3

DIF: hard

5. The concentration , in milligrams per deciliter, of a drug in the bloodstream injection is given by .

A: Make a graph of

versus

over the first 5 hours after the injection.

B: When is the drug concentration at a maximum? C: What is the maximum concentration? ANS: A: C 2 1.75 1.5 1.25 1 0.75 0.5 0.25

–0.25 –0.5

hours after an

B: After 1.28 hours. C: 1.71 milligrams per deciliter PTS: 3

DIF: hard

6. The growth rate , in hundreds per year, of a certain bird species depends on the bird population thousands. The relationship is .

A: Make a graph of

versus n. Include populations of up to 1.5 thousand birds.

B: What bird population yields a maximum growth rate? C: What is the maximum growth rate? ANS: A: G 0.5

0.4

0.3

0.2

0.1

0.2

0.4

0.6

0.8

1.2

1.4

–0.1

B: The maximum growth rate occurs when there are 0.9 thousand birds. C: The maximum growth rate is 0.5 hundred birds per year. PTS: 3

DIF: hard

7. You use 181 feet of wire to make three sides of a rectangular pen. The fourth side is the wall of a barn. If you use feet for the width, the area you enclose is given by square feet.

A: Make a graph of versus . In choosing a viewing window, you should bear in mind that the width will certainly be no more than half the total amount of fence. B: What width gives the maximum area?

, in

C: What is the maximum area? ANS: A: A 4500 4000 3500 3000 2500 2000 1500 1000 500 –10 –500

B: 45.25 feet C: The maximum area is 4095.13 square feet. PTS: 3

DIF: hard

8. For a certain company, the growth rate G in sales, in thousands of dollars per month, depends on current sales , in thousands of dollars. The relationship is .

A: Make a graph of

versus s. Include sales of up to 10 thousand dollars.

B: What sales give the maximum growth rate in sales? C: What is the maximum growth rate? ANS: A:

G 2.1 1.8 1.5 1.2 0.9 0.6 0.3

–2

–0.3

B: The maximum growth rate occurs when sales are 5 thousand dollars. C: The maximum growth rate is 1.25 thousand dollars per month. PTS: 3

DIF: hard

9. The rate of growth , in pounds per year, of a certain animal depends on the weight the animal. The relationship is .

A: Make a graph of

versus

. Include weights up to 40 pounds.

B: At what weight is the animal growing at the fastest rate? C: What is the maximum growth rate? ANS: A: G 7 6 5 4 3 2 1

5 –1

, in pounds, of

B: The animal is growing at the fastest rate when it weighs 21.95 pounds.. C: The maximum growth rate is 5.49 pounds per year. PTS: 3

DIF: hard

10. In a certain laboratory experiment the amount minutes is given by

, in grams, of a radioactive substance remaining after

A: Make a graph of A versus

over the first 60 minutes of the experiment.

B: When is there a maximum amount of material present? C: What is the maximum amount of material? ANS: A: A 0.7 0.6 0.5 0.4 0.3 0.2 0.1

–10 –0.1

B: After 17.33 minutes. C:

0.75 grams

PTS: 3

DIF: hard

11. A can has a volume of 25 cubic inches. We can calculate both the height area , in square inches, from the radius , in inches, of the can: ,

. A: Find the radius of the can of minimum surface area.

, in inches, and the surface

B: What is the height of the can of minimum surface area? C: What is the minimum surface area? ANS: A: 1.58 inches B: 3.17 inches C: 47.33 square inches PTS: 3

DIF: hard

12. A can is made from 28 square inches of aluminum. We can calculate both the height the volume , in cubic inches, from the radius , in inches, of the can:

, in inches, and

. A: Find the radius of the can of maximum volume. B: What is the height of the can of maximum volume? C: What is the maximum volume? ANS: A: 1.22 inches B: 2.44 inches C: 11.38 cubic inches PTS: 3

DIF: hard

13. A box with a square base, feet by feet, and no lid is made from 33 square feet of cardboard. We can calculate both the height , in feet, and the volume , in cubic feet, from the length : ,

. A: Find the value of

that makes the box of maximum volume.

B: What is the height of the box of maximum volume? C: What is the maximum volume? ANS:

feet

B: 1.66 feet C: 18.24 cubic feet PTS: 3

DIF: hard

14. A box with a square base, feet by feet, and no lid is to have a volume of 32 cubic feet. We can calculate both the height , in feet, and the amount , in square feet, of cardboard needed to make the box from the length :

. A: Find the value of

for the box using the minimum amount of cardboard.

B: What is the height of the box you found in part A? C: How much cardboard is used for the box you found in part A? ANS: A:

feet

B: 2 feet C: 48 square feet PTS: 3

DIF: hard

Section 3.1 The Geometry of Lines TRUE/FALSE 1. If a line is represented on coordinate axes, then the horizontal intercept is the point where the line meets the horizontal axis. ANS: T

PTS: 1

DIF:

easy

2. The slope of a line tells the rate at which it rises or falls. ANS: T

PTS: 1

DIF:

easy

3. The slope of a line varies from point to point on the line. ANS: F

PTS: 1

DIF:

easy

4. The slope of a line tells how much the line rises for each unit of run. ANS: T

PTS: 1

DIF:

easy

5. Among lines with positive slope, a larger slope corresponds to a steeper line. ANS: T

PTS: 1

DIF:

easy

DIF:

easy

DIF:

easy

6. A vertical line has a slope of 0. ANS: F

PTS: 1

7. The slope of a line is the run over the rise. ANS: F

PTS: 1

8. The rise of a line can be found by multiplying the run by the slope. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Consider a line with slope 1.3. What rise corresponds to a run of 4 feet? a. 5.2 feet c. 3.08 feet b. 0.33 feet d. 5.44 feet ANS: A

PTS: 1

DIF:

medium

2. Consider a line with slope 1.7. What run corresponds to rise of 5 feet? a. 8.5 feet c. 2.94 feet b. 0.34 feet d. 8.86 feet ANS: C

PTS: 1

DIF:

medium

3. For a certain line, a run of 3 feet corresponds to a rise of 4 feet. What is the slope of this line?

a. 1.33 feet b. 0.75 feet ANS: A

c. 1.71 feet d. 1.58 feet PTS: 1

DIF: medium

4. At the outside wall the roof is 9 feet high. The slope of the roof is 1.39 feet per foot. What is the height of the roof 8 horizontal feet from the outside wall toward the interior of the room? a. 20.8 feet b. 14.76 feet ANS: D

c. 15.74 feet d. 20.12 feet PTS: 1

DIF:

medium

5. At the outside wall the roof is 9 feet high. The slope of the roof is 1.53 feet per foot. If the roof is extended beyond the outside wall, what is the height of the roof 5 horizontal feet from the outside wall toward the exterior of the room? a. 17.56 feet b. 1.35 feet ANS: B

c. 2.06 feet d. 16.65 feet PTS: 1

DIF: medium

6. At the outside wall the roof is 10 feet high. Also, 3 horizontal feet from the outside wall toward the interior of the room, the roof is 13 feet tall. What is the height of the roof 8 horizontal feet from the outside wall toward the interior of the room? a. 18.85 feet b. 11.85 feet ANS: D

c. 12.7 feet d. 18 feet PTS: 1

DIF: medium

7. At the outside wall the roof is 9 feet high. Also, 5 horizontal feet from the outside wall toward the interior of the room, the roof is 10 feet tall. If the roof is extended beyond the outside wall, what is the height of the roof 4 horizontal feet from the outside wall toward the exterior of the room? a. 9.15 feet b. 7 feet ANS: C

c. 8.2 feet d. 7.14 feet PTS: 1

DIF:

medium

8. The bottom of a ladder sits on the ground 3 horizontal feet from a wall. The top of the ladder rests atop a wall that is 11 feet high. What is the slope of the line made by the ladder? (Assume that the positive direction points from the base of the ladder toward the wall.) a. 4.24 feet per foot b. 0.27 feet per foot ANS: C

c. 3.67 feet per foot d. 0.28 feet per foot PTS: 1

DIF:

medium

9. The bottom of a ladder sits on the ground 4 horizontal feet from a wall against which its top rests. The slope of the line made by the ladder is 3.87 feet per foot. What is the vertical height of the top of the ladder? (Assume that the positive direction points from the base of the ladder toward the wall.) a. 8.03 feet

c. 7.87 feet

b. 15.48 feet ANS: B

d. 16.25 feet PTS: 1

DIF:

medium

10. A ladder leans against a wall so that its slope is 3.63 feet per foot. The top of the ladder is 13 vertical feet above the ground. What is the horizontal distance from the base of the ladder to the wall? (Assume that the positive direction points from the base of the ladder toward the wall.) a. 3.58 feet b. 0.28 feet ANS: A

c. 4.01 feet d. 0.78 feet PTS: 1

DIF: medium

11. Take west to be the positive direction. You are driving in a westerly direction up a straight road. You pass a sign that says Elevation 1036 feet. Also, 5 horizontal miles further you see a sign that says Elevation 1833 feet. What is the slope of the road you are driving on? a. 159.4 feet per mile c. 160.04 feet per mile b. 366.6 feet per mile d. 366.79 feet per mile ANS: A

PTS: 1

DIF:

medium

12. You are driving up a straight road. You pass a sign that says Elevation 1765 feet. Also, 3 horizontal miles further you see a sign that says Elevation 2422 feet. The peak of the mountain is 8 horizontal miles from the first sign. What is the elevation of the mountain? a. 8223.67 feet b. 3517 feet ANS: B

c. 8223.75 feet d. 3517.44 feet PTS: 1

DIF: medium

13. Take west to be the positive direction. The height of a sloped roof above the place you are standing is 13 feet. If you move 3 feet to the west, the height is 8 feet. What is the slope of the roof line? a. –1.67 feet b. 1.67 feet ANS: A

c. –0.94 feet d. 2.06 feet PTS: 1

DIF: medium

14. What is the slope of the line through the points (12, 5) and (8,10)? a. 0.42 b. –0.8 ANS: D

c. 1.25 d. –1.25 PTS: 1

DIF: medium

15. What is the slope of the line through the points (11, 5) and (17,10)? a. 0.45 b. –0.83 ANS: D

c. 0.59 d. 0.83 PTS: 1

DIF:

medium

16. A line through (6, 5) has slope 1.19. What is the horizontal intercept? a. 2.41

c. –1.96

b. –2.14 ANS: D

d. 1.8 PTS: 1

DIF:

medium

17. A line through (5, 8) has slope –1.66. What is the horizontal intercept? a. 10.3 b. 16.3 ANS: D

c. 16.72 d. 9.82 PTS: 1

DIF:

medium

18. A line through (5, 6) has slope –1.93. What is the vertical intercept? a. 8.29 b. 15.65 ANS: B

c. 16.48 d. 8.11 PTS: 1

DIF:

medium

19. A line through (5, 6) has slope 0.47. What is the vertical intercept? a. –7.58 b. 3.65 ANS: B

c. 3.71 d. –7.77 PTS: 1

DIF:

medium

20. A line passes through the points (4, 7) and (8,9). For what line? a. b.

12.46 16

ANS: D

PTS: 1

c. d.

16.08 11.5

DIF:

medium

21. A line passes through the points (5, 1) and (11,8). For what line? a. b.

16.43 19.57

ANS: B

PTS: 1

c. d.

19.61 16.17

DIF:

medium

value does the point (13, ) lie on the

value does the point ( ,18) lie on the

22. A line passes through the point (5, 2) and has slope –2.13. For what lie on the line? a. b.

4.47 0.74

ANS: D

PTS: 1

c. d.

1.67 4.06

DIF:

medium

23. A line passes through the point (3, 5) and has slope 1.43. For what on the line? a. b.

12.29 12.15

ANS: B

PTS: 1

c. d.

8.23 7.29

DIF:

medium

value does the point ( ,4)

value does the point (8, ) lie

24. A pipeline is to span a horizontal distance of 108 feet. But the pipeline is sloped downward to promote drainage. The pipeline drops by a total of 13 feet over that span. How much lower is the pipe at the end of each 12-foot horizontal stretch? a. 2.38 feet b. 0.12 feet ANS: D

c. 0.86 feet d. 1.44 feet PTS: 1

DIF:

medium

25. You are driving up a straight mountain road which has a slope of 75 feet per mile. If you are initially at an altitude of 623 feet, what is your altitude after you drive 6 horizontal miles? a. 1073.75 feet b. 704 feet ANS: C

c. 1073 feet d. 704.04 feet PTS: 1

DIF: medium

26. You are driving up a straight mountain road which has a slope of 71 feet per mile. If you are initially at an altitude of 930 feet and later at an altitude of 1141 feet, how many horizontal miles have you driven? a. 3.86 miles b. 29.17 miles ANS: C

c. 2.97 miles d. 29.35 miles PTS: 1

DIF: medium

27. The peak of a roof is 19 feet tall. At the outside wall, 15 horizontal feet away, the roof is 8 feet tall. If the roof line were extended until it struck the ground, how many horizontal feet from the outside wall would that occur? a. 11.83 feet b. 10.91 feet ANS: B

c. 15 feet d. 15.2 feet PTS: 1

DIF:

medium

28. A vertical 8-foot pole casts a shadow that is 9 feet long. How long a shadow will a vertical 11-foot pole cast at the same time? a. 12.38 feet b. 12.94 feet ANS: A

c. 9.78 feet d. 9.8 feet PTS: 1

DIF:

medium

29. A vertical 7-foot pole casts a shadow that is 8 feet long. How tall is a vertical pole that casts a shadow that is 11 feet long at the same time? a. 12.57 feet b. 13.01 feet ANS: C

c. 9.63 feet d. 9.64 feet PTS: 1

DIF:

medium

30. The floor of a lake slopes downward in a straight line. The depth 26 feet from shore is 14 feet. What is the depth 28 feet from shore?

a. 52 feet b. 15.09 feet ANS: C

c. 15.08 feet d. 52.3 feet PTS: 1

DIF: medium

31. The floor of a lake slopes downward in a straight line. The depth 26 feet from shore is 12 feet. At what distance from shore is the depth 30 feet? a. 65 feet b. 14.24 feet ANS: A

c. 13.85 feet d. 65.79 feet PTS: 1

DIF:

medium

32. The floor of a lake slopes downward in a straight line. The depth 21 feet from shore is 16 feet. At what distance from shore is the depth 41 feet? a. 53.81 feet b. 31.81 feet ANS: A

c. 31.24 feet d. 54.32 feet PTS: 1

DIF:

medium

33. At a distance 6 miles west of town, oil-bearing deposits are found 3027 feet deep. Also, 10 miles west of town, the oil is found 3221 feet down. Assuming the oil-bearing stratum lies on a straight line, how deep would you expect to drill 11 miles west of town in order to find oil? a. 3075.5 feet b. 3075.59 feet ANS: D

c. 3269.85 feet d. 3269.5 feet PTS: 1

DIF:

medium

34. At a distance 6 miles west of town, oil-bearing deposits are found 3176 feet deep. Also, 8 miles west of town, the oil is found 3328 feet down. Assuming the oil-bearing stratum lies on a straight line, how far west of town would you expect to drill 3700 feet deep in order to find oil? a. 12.89 miles b. 13.79 miles ANS: A

c. 11.79 miles d. 10.89 miles PTS: 1

DIF: medium

35. The figure below shows a simplified pattern for a wrap skirt that is 20 inches long and has a top hem of 57 inches. The bottom hem has a length of 63 inches. Suppose you decide to alter the pattern to make a wrap skirt with the same top hem length but a length of 25 inches. How long should be the bottom hem?

a. 64.94 inches b. 63.75 inches ANS: D

c. 63.43 inches d. 64.5 inches PTS: 1

DIF: medium

36. Your office window is 35 feet high. Looking out your window, you find that the top of a statue lines up exactly with the bottom of a building that is 600 horizontal feet from your office, as is shown in the figure below. The statue is 125 horizontal feet from the building. How tall is the statue?

a. 6.48 feet b. 9.71 feet ANS: C

c. 7.29 feet d. 10 feet PTS: 1

DIF: medium

37. A surveyor whose eye is 6 feet above the ground views a mountain peak which is 2 horizontal miles distant. See the figure below. Directly in his line of sight is the top of a surveying pole which is 10 horizontal feet distant and 8 feet high. How tall is the mountain peak? Note: 1 mile is 5280 feet.

a. 5143 feet b. 2118 feet ANS: B

c. 2546 feet d. 4798 feet PTS: 1

DIF:

medium

38. An ice cream cone is 4 inches deep and 2 inches across the top, as shown in the figure below. If we wanted to make a king-size cone which has the same shape but is 2.5 inches across the top, how deep would the cone be?

a. 5 inches deep b. 4.5 inches deep ANS: A

c. 6 inches deep d. 5.5 inches deep PTS: 1

DIF:

medium

ESSAY 1. The base of a ramp sits on the ground. Its slope is 0.3 feet per foot. It extends to the top of the front steps of a building 19 horizontal feet away. A: How high is the ramp 1 horizontal foot toward the building from the base of the ramp? B: How high is the top of the steps relative to the ground? ANS: A: 0.3 feet high

B: 5.7 feet high PTS: 2

DIF: medium

2. A cathedral ceiling (shown below) is 8 feet high at the west wall of a room. As you go from the west wall toward the east wall, the ceiling slants upward. Three feet from the west wall, the ceiling is 10.5 feet high.

A: What is the slope of the ceiling? B: The width of the room is 19 feet. Use your answer to part A to determine how high the ceiling is at the east wall. C: If you stand on a ladder, you can reach 14 feet high. Use your answer to part A to determine the farthest distance from the west wall at which you can stand on the ladder and reach the ceiling. ANS: A: 0.83 feet per foot B: 23.77 feet high

C: 7.23 feet from the west wall PTS: 3

DIF: medium

3. You are driving on a straight road that is inclined upward toward the peak of a mountain. You pass a sign that reads “Elevation 1700 feet”. Three horizontal miles further you pass a sign that reads “Elevation 2240 feet”.

A: Think of the direction in which you are driving as the positive direction. What is the slope of the road? (Report your answer in feet per mile.) B: What is the elevation of the road 5 horizontal miles from the first sign? C: The peak of the mountain is 3343 feet. How far in horizontal distance is the first sign from the peak of the mountain? ANS: A: 180 feet per mile B: 2600 feet C: 9.13 miles PTS: 3

DIF: medium

Section 3.2 Linear Functions TRUE/FALSE 1. A linear function is a function with a constant rate of change. ANS: T

PTS: 1

DIF:

easy

2. The graph of a linear function is a straight line. ANS: T

PTS: 1

DIF:

easy

3. The slope of the graph of a linear function is the same as the rate of change of the linear function. ANS: T

PTS: 1

DIF:

easy

4. The rate of change of a linear function depends on the variable of the function. ANS: F

PTS: 1

DIF:

easy

5. You can get the rate of change of a linear function by dividing the change in the variable by the change in the function. ANS: F

PTS: 1

DIF:

easy

6. You can get the change in a linear function from the formula Change in function = Slope ANS: T

PTS: 1

DIF:

Change in variable.

easy

7. If I add $5 to a cookie jar each week, then the amount of money in the cookie jar is a linear function of time with slope 5 dollars per week. ANS: T

PTS: 1

DIF:

easy

8. If a population doubles in size each day, then the population is a linear function of time with slope 2. ANS: F

PTS: 1

DIF:

easy

9. If snow is falling at a rate of 0.5 inch each hour, then the depth of snow on the ground is a linear function of time as long as the snow continues to fall. ANS: T

PTS: 1

DIF:

easy

10. The height of a staircase is a linear function of the number of steps. ANS: T

PTS: 1

DIF:

easy

11. You pay $10 to register at a web site. After registering you pay 59 cents for each song you download. Then the total amount you pay for music at this site is a linear function of the number of songs you download.

ANS: T

PTS: 1

DIF:

easy

12. It is a fact that if you drop a rock, it will fall 16 feet the first second of the fall. The next second it will fall 48 feet. In view of this fact, I can conclude that the distance a rock falls is a linear function of the time since the rock was dropped. ANS: F

PTS: 1

DIF:

easy

13. It is a fact that if you drop a rock, its velocity increases by 32 feet per second for each additional second it falls. We can conclude that the velocity of the rock is a linear function of time since the rock was dropped. ANS: T

PTS: 1

DIF:

easy

14. If is increased by 5 units from x=0, then is increased by 9 units. If is increased by an additional 5 units, then is increased by 11 units. We can conclude that a linear function of . ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. My business pays tax on income, and the tax paid is a linear function of income. If my income is $12511 then the tax owed is $939.77. If my income is $13236 then the tax owed is $990.52. How much tax do I owe if my income is $13610? a. $1016.7 b. $1016.82 ANS: A

c. $952.7 d. $953.41 PTS: 1

DIF: medium

2. The money I pay for music is a linear function of the number of songs I download. If I download 7 songs, then I pay $15.66. If I download 13 songs, then I pay $23.94. How much do I pay if I download 19 songs? a. $27.01 b. $32.51 ANS: D

c. $26.22 d. $32.22 PTS: 1

DIF: medium

3. If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity per second, after seconds is given by . By how much does velocity increase over a 6-second period?

a. 32.14 feet per second b. 192.51 feet per second

c. 32 feet per second d. 192 feet per second

, in feet

ANS: D

PTS: 1

DIF:

medium

4. When you order merchandise from a web site you pay shipping costs , in dollars. The shipping costs depend on the weight , in pounds, of the merchandise. The relationship is . . If my order weighs 4 pounds more than yours, how do shipping costs compare?

a. My costs are $2.77 more than yours. b. My costs are $9.39 more than yours. ANS: C

PTS: 1

c. My costs are $9.36 more than yours. d. My costs are $2.34 more than yours. DIF:

medium

5. The distance , in miles, I can drive without stopping for gas depends on the amount fuel in the tank. The relationship is

, in gallons, of

. If I add 6 gallons of fuel to my tank, how much farther can I drive? a. 198 miles b. 198.92 miles ANS: A 6. The sales tax

c. 33.11 miles d. 33 miles PTS: 1

DIF: medium

, in dollars, I have to pay when I buy

dollars worth of merchandise is given by

. If I decide to buy $35 more in merchandise, how much additional sales tax will I have to pay? a. $2.8 b. $3.04 ANS: A

c. $0.6 d. $0.08 PTS: 1

DIF: medium

7. I am saving money in a cookie jar. The amount of money given by

, in dollars, in the jar after

weeks is

. By how much do my savings grow over a 5-week period? a. $105.86 b. $105 ANS: B

c. $21.34 d. $21 PTS: 1

DIF: medium

8. For a certain group of animals, the running speed length , in inches.The relationship is

, in feet per second, is a linear function of the

If one animal is 5 inches longer than another, how much faster does it run? a. 12.34 feet per second b. 11.8 feet per second ANS: B

PTS: 1

9. The life expectancy

c. 2.44 feet per second d. 2.36 feet per second DIF:

medium

, measured in years, of a child born

years after 2003 is given by .

If one child is born 7 years after another, how much greater is the life expectancy? Assume that both children are born after 2003, so the formula above applies. a. 2.12 years b. 1.26 years ANS: B

c. 7.17 years d. 0.18 years PTS: 1

10. Suppose

DIF:

medium

is a linear function with

a. 0.67 b. 3.19

and

. What is the slope of

c. 0.4 d. 2.48

ANS: D

PTS: 1

11. Suppose ?

DIF:

is a linear function with

a. 34.75 b. 38.68

. If the slope of

is 2.89, what is the value of

c. 34.68 d. 38.72

ANS: B

PTS: 1

12. Suppose

medium

is a linear function with

DIF: medium . If the slope of

is 2.4, what value of

gives

. Find an equation for

? a. b.

36.46 2.08

ANS: B 13. Suppose

PTS: 1

35.8 2.98

DIF:

medium

is a linear function with

a. b. ANS: C

c. d.

and c. d.

PTS: 1

DIF:

medium

14. Suppose is a linear function with slope 2.63. Suppose further that formula to express as a function of . a. b.

c. d.

when

. Use a

ANS: A 15. Suppose in ?

PTS: 1

DIF:

medium

is a linear function of . The slope is 2.44. An increase of 6 units in

a. A decrease of 14.64 units b. An increase of 14.64 units ANS: B

PTS: 1

c. An increase of 2.46 units d. A decrease of 2.46 units DIF:

medium

16. Suppose is a linear function of . The slope is 2.59. An increase of 6 units in change in ? a. A decrease of 15.54 units b. An increase of 15.54 units ANS: C 17. Suppose in ?

PTS: 1

ANS: A 18. Suppose in x?

PTS: 1

DIF:

medium

ANS: D

PTS: 1

causes what change

c. An increase of 2.99 units d. A decrease of 2.99 units DIF:

medium

is a linear function of . The slope is 2.04. A decrease of 7 units in

a. A decrease of 14.28 units b. An increase of 14.28 units

causes what

c. An increase of 2.32 units d. A decrease of 2.32 units

is a linear function of . The slope is 2.01. A decrease of 6 units in

a. A decrease of 12.06 units b. An increase of 12.06 units

causes what change

c. An increase of 3.43 units d. A decrease of 3.43 units DIF:

medium

SHORT ANSWER 1. If a rock is thrown downward with an initial velocity of 13 feet per second, then its velocity per second, after seconds is given by

, in feet

. Identify the slope, and explain in practical terms its meaning. ANS: The slope is 32 feet per second per second. Each second of the fall the velocity increases by 32 feet per second. PTS: 1

DIF: medium

2. When you order merchandise from a web site you pay shipping costs , in dollars. The shipping costs depend on the weight , in pounds, of the merchandise. The relationship is .

Identify the slope, and explain in practical terms its meaning. ANS: The slope is 1.81 dollars per pound. Shipping costs increase by $1.81 for each additional pound of merchandise. PTS: 1

DIF: medium

3. The distance , in miles, I can drive without stopping for gas depends on the amount fuel in the tank. The relationship is

, in gallons, of

. Identify the slope, and explain in practical terms its meaning. ANS: The slope is 30 miles per gallon. This is the gas mileage that my car gets--the number of miles I can drive on 1 gallon. PTS: 1 4. The sales tax

DIF: medium , in dollars, I have to pay when I buy

dollars worth of merchandise is given by .

Identify the slope, and explain in practical terms its meaning in terms of percentages. ANS: The slope is 0.07 dollars per dollar. The sales tax is 7 percent. PTS: 1

DIF: medium

5. I am saving money in a cookie jar. The amount of money given by

, in dollars, in the jar after

weeks is

. Identify the slope, and explain in practical terms what it means. ANS: The slope is 19 dollars per week. Each week $19 is added to the cookie jar. PTS: 1

DIF: medium

6. For a certain group of animals, the running speed length , in inches.The relationship is

, in feet per second, is a linear function of the

. Identify the slope, and explain in practical terms what it means. ANS:

The slope is 2.56 feet per second per inch. Each additional inch in length adds 2.56 feet per second to the running speed. PTS: 1 7. The life expectancy

DIF: medium , measured in years, of a child born

years after 2003 is given by .

Identify the slope, and explain in practical terms what it means. ANS: The slope is 0.18 year per year. For children born after 2003, each 1-year increase in birth date adds 0.18 year to life expectancy. PTS: 1

DIF: medium

ESSAY 1. Your workout this morning burned 658 calories. After your workout, you take a walk that burns 252 calories per hour. A: Find a formula that gives the total calories walking.

you burn in your morning workout plus

hours of

B: How long do you need to walk to burn a total of 1119 calories? ANS: A: B: 1.83 hours PTS: 2

DIF: hard

2. The temperature , in degrees Fahrenheit, is a linear function of the number of cricket chirps per minute. Now 23 cricket chirps per minute corresponds to a temperature of 42 degrees. Each additional chirp per minute corresponds to an increase of 0.27 degree. A: Find a formula that gives

as a linear function of

B: What temperature is indicated by 104 cricket chirps per hour? C: How many cricket chirps per minute would give a temperature of 60.36 degrees? ANS: A: B: 63.87 degrees C: 91 cricket chirps per minute PTS: 3

DIF: hard

3. Water freezes at 0 degrees Celsius, which is the same as 32 degrees Fahrenheit. Water boils at 100 degrees Celsius, which is the same as 212 degrees Fahrenheit. A: Find a formula that gives Celsius temperature

as a linear function of Fahrenheit temperature

B: What is the slope of the formula you found in part A? Be sure to include proper units. C: Explain in practical terms the meaning of the slope you found in part B. ANS: A: B: The slope is 0.56 degree Celsius per degree Fahrenheit. C: Each 1-degree increase in Fahrenheit temperature corresponds to a 0.56-degree increase in Celsius temperature. PTS: 3

DIF: hard

4. A school is taking a busload of students to a music camp. It costs $179 to operate the bus, and the school pays $27 per student for use of camp facilities. A: Find a formula that gives total cost

, in dollars, of taking

students to the camp.

B: What is the cost of taking 22 students to music camp? C: Solve the equation

for , and explain what the solution represents.

ANS: A: B: $773 C:

27 is the solution. For $908 the school can take 27 students to the camp.

PTS: 3

DIF: hard

5. Hall rental for a wedding reception costs $190. The caterers charge $28 for each guest. A: Find a formula that gives total cost

, in dollars, of inviting

B: Solve the equation you found in part A for

guests to the reception.

C: Explain in practical terms what the equation from part B tells you. ANS: A: B: C:

or This formula tells you how many guests you can invite for a given amount of money.

PTS: 3

DIF: hard

6. The speed of sound in air changes with temperature. When the temperature is 39 degrees Fahrenheit, the speed of sound is 1095.2 feet per second. For each degree increase in temperature, the speed of sound increases by 1.1 feet per second. A: Explain why the speed Fahrenheit.

, in feet per second, is a linear function of the temperature

B: Use a formula to express

as a linear function of

, in degrees

C: What temperature corresponds to a speed of sound of 1143.2 feet per second? ANS: A: Now is a linear function of because each one degree change in temperature always results in the same change, 1.1 feet per second, in the speed of sound. B: C:

82.64 degrees Fahrenheit

PTS: 3

DIF: hard

7. Your lean body weight , in pounds, is the amount you would weight if all the fat in your body were to disappear. The following formula gives an estimate for lean body weight in young adult males: . Here

is total body weight in pounds and

is abdominal circumference in inches.

A: If a young adult male’s abdominal circumference remains the same, but his total weight increases by 5 pounds, how is lean body weight affected? B: If a young adult male’s total weight remains the same, but abdominal circumference increases by 3 inches, how is lean body weight affected? C: A young adult male has a lean body weight of 148 pounds. Over a period of time he gains 15 pounds and his abdominal circumference increases by 2 inches. What is his lean body weight now? ANS: A: Lean body weight increases by 5.3 pounds. B: Lean body weight decreases by 14.46 pounds. C: The new lean body weight is 154.26 pounds. PTS: 3

DIF: hard

8. There are initially 5 inches of snow on the ground. It begins snowing, and snow falls at a steady rate of 3 inches per hour.

A: Explain why the depth , in inches, of snow on the ground is a linear function of the number of hours since the snowfall began.

B: Find a formula that gives the depth of snow on the ground after C: Solve the equation you found in part B for tells you.

hours of snowfall.

and explain in practical terms what this equation

ANS: A: The depth increases by the same amount each hour. B: C:

. This equation tells you how long it takes for the snow to reach a

given depth. PTS: 3

DIF: hard

9. Each month, a salesman earns a base salary of $767 plus 5% commission on sales. Use monthly compensation and for total monthly sales, both in dollars.

for total

A: Find a formula that gives total compensation as a linear function of total monthly sales. B: What monthly sales lead to a total monthly compensation of $4111? C:

How is total monthly compensation affected if sales are increased by $6002?

ANS: A: B: $66880 C: Compensation is increased by $300.1. PTS: 3

DIF: hard

10. City A charges a $56 connection fee plus $0.17 per kilowatt hour for electricity. City B charges a $72 connection fee plus $0.13 per kilowatt hour. A: Use for the total charges, in dollars, for connecting and using from City A. Find an equation that gives as a linear function of

kilowatt hours of electricity

B: Use for the total charges, in dollars, for connecting and using from city B. Find an equation that gives as a linear function of

kilowatt hours of electricity

What number of kilowatt hours will result in the same charges for electricity from each city?

ANS: A: B: C: 400 kilowatt hours

PTS: 3

DIF: hard

11. Company A charges a flat fee of $59 plus $34 per day for car rental. City B charges a flat fee of $119 plus $29 per day for car rental. A: Use for the total charges, in dollars, for renting a car for equation that gives as a linear function of .

days from company A. Find an

B: Use for the total charges, in dollars, for renting a car for equation that gives as a linear function of .

days from company B. Find an

C: What number of days of car rental will result in the same charges for car rental from each company? ANS: A: B: C: 12 days PTS: 3

DIF: hard

Section 3.3 Modeling Data with Linear Functions TRUE/FALSE 1. A data table for with evenly spaced values for shows a constant change in . ANS: T 2. If data for ANS: F 3. If data for ANS: T

PTS: 1

DIF:

can be modeled with a linear function if it

easy

are linear, then the slope of the linear function is the change in PTS: 1

DIF:

easy

are linear, then the plot of the data falls on a straight line. PTS: 1

DIF:

easy

4. The graph shown below can be appropriately modeled by a linear function.

ANS: T

PTS: 1

DIF:

easy

5. The graph shown below can be appropriately modeled by a linear function.

ANS: F 6. Suppose ANS: T 7. Suppose ANS: F 8. Suppose

PTS: 1 is a linear function and that PTS: 1 is a linear function and that PTS: 1 is a linear function and that

DIF:

easy

, and DIF:

is positive.

. Then the slope of

is positive.

easy

, and DIF:

. Then the slope of

easy . Then the slope of

is negative.

ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Find a linear model for the following data table. 1 6

2 11

3 16

a. b.

4 21

c. d.

ANS: A

PTS: 1

DIF:

medium

2. The following table shows net income of an art dealer when he pays for supplies and sells Find a linear model for net income as a function of number of items sold. 2 7

3 35

4 63

a. b.

items.

5 91

c. d.

ANS: A

PTS: 1

DIF:

medium

3. Find a linear model for the following data table. Round the slope to three decimal places. 3 6

4.82 12.56

6.64 19.12

a. b.

8.46 25.68 c.

ANS: A

PTS: 1

DIF:

medium

4. The following table shows the total cost of joining a music club and downloading songs. Find a linear model for the total cost as a function of songs downloaded. Round the slope to three decimal places. 3 12.38

8 16.94

13 21.5

a. b.

18 26.06

c. d.

ANS: A

PTS: 1

DIF:

medium

5. Find a linear model for the following data table. 3 17

4 12

5 7

6 2

a. b.

c. d.

ANS: D

PTS: 1

DIF:

6. The following table shows the money Find a linear model for M in terms of W. 3 136

medium

left in your cookie jar after

4 121

5 106

a. b.

withdrawals from the jar.

6 91

c. d.

ANS: D

PTS: 1

DIF:

medium

7. Find a linear model for the following data table. 1 17

2.45 12.94

3.9 8.88

a. b.

5.35 4.82 c.

ANS: D

PTS: 1

DIF:

medium

8. The following table shows the miles from home after hours of driving. Find a linear model for miles from home as a function of hours driven. Round the slope to three decimal places. 4 663

5.07 578.59

a. b.

6.14 494.18

7.21 409.77

c. d.

ANS: D

PTS: 1

DIF:

medium

9. Find a linear model for the following data table. 3 6

6 12

a. b. ANS: A

9 18

12 24

c. d. PTS: 1

DIF:

medium

10. The following table shows the cost of renting a lakeside cabin and taking your family there on a vacation that is days long. Find a linear model for the cost as a function of the days spent on vacation. 1 443

3 552

5 661

7 770

a. b.

c. d.

ANS: A

PTS: 1

DIF:

medium

11. Find a linear model for the following data table. 1 26

3 20

a. b.

5 14

7 8

c. d.

ANS: C

PTS: 1

DIF:

medium

12. The following table shows the depth , in inches, of water in a pond after Find a linear model for depth as a function of days. 4 779

6 777

a. b.

8 775

days of evaporation.

10 773

c. d.

ANS: C

PTS: 1

DIF:

medium

13. Which of the tables below show linear data? Table 1 2 4 y 35.51 33.51

6 31.51

8 29.51

Table 2 y

2 36.21

4 33.64

6 32.01

a. Table 1 only b. Table 2 only ANS: A

8 29.7

c. Both Table 1 and Table 2 d. Neither table is linear. PTS: 1

DIF: medium

14. Which of the tables below show linear data? Table 1 2 4 y 36.07 39.39

6 43.45

8 46.89

Table 2 y

a. Table 1 only b. Table 2 only

2 35.82

4 39.38

6 42.94

8 46.5

c. Both Table 1 and Table 2 d. Neither table is linear.

ANS: B

PTS: 1

DIF:

medium

15. Which of the tables below show linear data? Table 1 y

4 38.25

7 43.83

10 49.41

13 54.99

Table 2 4 35.86

7 41.44

a. Table 1 only b. Table 2 only ANS: C

10 47.02

13 52.6

c. Both Table 1 and Table 2 d. Neither table is linear. PTS: 1

DIF: medium

16. Which of the tables below show linear data? Table 1 y

1 38.21

7 40.87

13 43.45

19 45.19

Table 2 y

1 35.99

7 38.65

a. Table 1 only b. Table 2 only ANS: D

13 40.81

19 42.97

c. Both Table 1 and Table 2 d. Neither table is linear. PTS: 1

DIF: medium

SHORT ANSWER 1. For the following data table, complete the table of changes, and determine if the data can be modeled exactly by a linear function. 2 9.21 Change in Change in ANS: Change in Change in

6 20.29

10 31.37

14 42.45

From 2 to 6

From 6 to 10

From 10 to 14

From 2 to 6 11.08

From 6 to 10 11.08

From 10 to 14 11.08

The data can be modeled with a linear function. PTS: 1

DIF: medium

2. Plot the data from the following table and then add the graph of 2 7.39

5 13.81

8 20.23

to the plot. 11 26.65

ANS:

The horizontal span is 0 to 12, and the vertical span is 0 to 30. PTS: 1

DIF: medium

3. Plot the data from the following table and then add the graph of 3 8.25

6 1.83

9 –4.59

to the plot. 12 –11.01

ANS:

The horizontal span is 0 to 13, and the vertical span is -15 to 15. PTS: 1

DIF: medium

4. The following table is generated from a linear function

. Fill in the missing data points.

3 16.39

6 26.83

ANS:

16.39 PTS: 1

26.83

37.27

47.71

DIF: medium

5. The following table is generated from a linear function

. Fill in the missing data points.

3 15.25

12 45.94

3 15.25

6 25.48

9 35.71

12 45.94

ANS:

PTS: 1

DIF: medium

6. The following table is generated from a linear function careful: The data are not evenly spaced.

. Fill in the missing data points. Be

3 14.78

22 73.3

3 14.78

10 36.34

14 48.66

22 73.3

ANS:

PTS: 1

DIF: medium

7. Complete the following table, which is generated from the linear function

3 17.77

8 41.07

13 64.37

18 87.67

ANS:

PTS: 1

DIF: medium

8. Complete the following table, which is generated from the linear function

9.97 ANS:

11.18

11.47

12.62

3.11 9.97 PTS: 1

3.69 11.18

3.83 11.47

4.38 12.62

DIF: medium

9. Complete the following table, which is generated from the linear function entries to three decimal places. 3.03

. Round the

3.585 –0.43

0.76 ANS:

3.03 1.22 PTS: 1

3.385 0.76

3.585 0.5

4.3 –0.43

DIF: medium

10. The following table is generated from a linear function

. Fill in the missing data points.

3 14.95

9 34.27

3 14.95

6 24.61

9 34.27

12 43.93

ANS:

PTS: 1

DIF: medium

11. The following table is generated from a linear function careful: The data are not evenly spaced.

. Fill in the missing data points. Be

3 6.12

6 –3.54

3 6.12

6 –3.54

14 –29.3

20 –48.62

ANS:

PTS: 1

DIF: medium

12. For the following data table, complete the table of changes, and determine if the data can be modeled exactly by a linear function.

4 21.31 Change in Change in ANS: Change in Change in

8 10.91

12 0.51

16 –9.89

From 4 to 8

From 8 to 12

From 12 to 16

From 4 to 8 –10.4

From 8 to 12 –10.4

From 12 to 16 –10.4

The data can be modeled exactly by a linear function. PTS: 1

DIF: medium

13. For the following data table, complete the table of changes, and determine if the data can be modeled exactly by a linear function. 3 16.06 Change in Change in ANS: Change in Change in

4 13.83

5 11.6

6 9.37

From 3 to 4

From 4 to 5

From 5 to 6

From 3 to 4 –2.23

From 4 to 5 –2.23

From 5 to 6 –2.23

The data can be modeled exactly by a linear function. PTS: 1

DIF: medium

14. For the following data table, complete the table of changes, and determine if the data can be modeled exactly by a linear function. 2 8.23 Change in Change in ANS: Change in Change in

3 11.24

4 14.25

From 2 to 3

From 3 to 4

From 4 to 5

From 2 to 3 3.01

From 3 to 4 3.01

From 4 to 5 3.01

The data can be modeled exactly by a linear function. PTS: 1 ESSAY

5 17.26

DIF: medium

1. The following table shows enrollment 0 7021

at your university 1 7300

years after 2010.

2 7579

3 7858

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: If this trend persists, what enrollment do you anticipate in 2017? ANS: A Change in Change in

From 0 to 1 279

From 1 to 2 279

From 2 to 3 279

B: C: 8974 PTS: 3

DIF: hard

2. The following table shows the price initial offering. 0 707

, in dollars, of a popular electronics item

1 685

2 663

months after its

3 641

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: Explain in practical terms the meaning of the slope of the linear function you found in part B. ANS: A Change in Change in

From 0 to 1 –22

From 1 to 2 –22

From 2 to 3 –22

B: C: The cost of the item is decreasing by $22 each month. PTS: 3

DIF: hard

3. The following table shows the value 0 17476

, in dollars, of an automobile 2 16681

4 15886

years after it is purchased. 6 15091

From 0 to 2 –795

From 2 to 4 –795

From 4 to 6 –795

B: C: The value of the car is decreasing by $397.5 each year. PTS: 3

DIF: hard

4. The following table shows average tuition 3 23401

, in dollars, at a regional university

5 23635

7 23869

years after 2010.

9 24103

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: If this trend persists, when will tuition reach $24805? ANS: A Change in Change in

From 3 to 5 234

From 5 to 7 234

From 7 to 9 234

B: C: In 2025 PTS: 3

DIF: hard

5. The following table shows the velocity downward on the planet Mercury.

0 15

, in meters per second, of a rock

1 18.61

2 22.22

seconds after it is thrown

3 25.83

ANS: A Change in Change in

From 0 to 1 3.61

From 1 to 2 3.61

From 2 to 3 3.61

B: C: For each second that passes, the velocity increases by 3.61 meters per second. Therefore, the acceleration due to gravity on Mercury is 3.61 meters per second per second. PTS: 3

DIF: hard

6. The following table shows the temperature temperature of

, in degrees Fahrenheit, of an object that has a

on the Kelvin temperature scale.

200 –99.67

221 –61.87

242 –24.07

263 13.73

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: If the temperature of an object increases by 19 degrees on the Kelvin scale, by how much does its temperature increase on the Fahrenheit scale? ANS: A Change in Change in

From 200 to 221 37.8

From 221 to 242 37.8

From 242 to 263 37.8

B: C: It increases by 34.2 degrees Fahrenheit. PTS: 3

DIF: hard

7. A gas-filled balloon is released from the top of a tall building, and it rises. The following table shows the altitude , in meters, of a balloon seconds after release.

4 65

6 73.8

8 82.6

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data.

10 91.4

C: How tall is the building from which the balloon was released? D: Plot the data along with the linear model you found in part B. ANS: A Change in Change in

From 4 to 6 8.8

From 6 to 8 8.8

From 8 to 10 8.8

B: C: It is 47.4 meters high. D: The horizontal span is 0 to 10, and the vertical span is 60 to 100.

PTS: 4

DIF: hard

8. A child sells lemonade. She must pay for the ingredients but then earns an income on each glass she sells. The following table shows her net income , in dollars, from selling glasses of lemonade. 4 1.65

6 5.19

8 8.73

10 12.27

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: How much did she pay for ingredients? D: Plot the data along with the linear model you found in part B. ANS: A Change in Change in

From 4 to 6 3.54

From 6 to 8 3.54

B: C: Ingredients cost $5.43. D: The horizontal span is 0 to 11, and the vertical span is 0 to 15.

From 8 to 10 3.54

PTS: 4

DIF: hard

9. The following table shows the velocity downward near the surface of Venus. 4 51

, in meters per second, of a rock

7 77.49

10 103.98

seconds after it is thrown

13 130.47

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: Explain in practical terms the meaning of the initial value of the linear function you found in part B. ANS: A Change in Change in

From 4 to 7 26.49

From 7 to 10 26.49

From 10 to 13 26.49

B: C: The rock was thrown downward with an initial velocity of 15.68 meters per second. PTS: 3

DIF: hard

10. A sports car is moving at a constant speed and then starts to accelerate. The following table shows the velocity , in miles per hour, of a sports car seconds after it begins accelerating.

1 45

1.5 50.67

2 56.34

2.5 62.01

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: By how much does the velocity increase over a 4-second period? ANS: A Change in Change in

From 1 to 1.5 5.67

From 1.5 to 2 5.67

From 2 to 2.5 5.67

B: C: The velocity increases by 45.36 miles per hour. PTS: 3

DIF: hard

11. The speed of sound , in meters per second, at a fixed depth and temperature in the ocean depends on the salinity , in parts per thousand. The following table shows selected data. 35 1515.36

35.6 1516.08

36.2 1516.8

36.8 1517.52

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: Explain in practical terms the meaning of the slope for the linear function in part B. ANS: A Change in Change in

From 35 to 35.6 0.72

From 35.6 to 36.2 0.72

From 36.2 to 36.8 0.72

B: C: Each increase of one part per thousand in salinity corresponds to an increase of 1.2 meters per second in the speed of sound. PTS: 3

DIF: hard

12. A salesman earns a base salary plus a percentage of sales. The following table shows the income in dollars, of a salesman who has sales of dollars. $2500 $1515.36

$3003 $1555.6

$3506 $1595.84

$4009 $1636.08

A: Complete a table of changes to show that the data are linear. B: Find a linear model for the data. C: Explain in practical terms the meaning of the linear function in part B. Your answer should be stated in terms of base salary plus percentage of sales. ANS: A Change in Change in B:

From 2500 to 3003 40.24

From 3003 to 3506 40.24

From 3506 to 4009 40.24

C: The salesman earns a base salary of $1315.36 plus 8% of sales. PTS: 3

DIF: hard

Section 3.4 Linear Regression TRUE/FALSE 1. The regression line always gives an exact model for data. ANS: F

PTS: 1

DIF:

easy

2. It may be appropriate to model data using linear regression if the data points nearly fall on a straight line. ANS: T

PTS: 1

DIF:

easy

3. The slope of the regression line gives an estimate of the rate of change of the data being modeled. ANS: T

PTS: 1

DIF:

easy

4. If the regression line does not fit data points exactly, then it should not be used as a model. ANS: F

PTS: 1

DIF:

easy

5. Linear regression always gives a linear function with positive slope. ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. The following table shows the gross national product , in trillions of dollars, the regression line for the data to estimate the gross national product in 2013. 0 12.77

1 13.51

a. 15.86 trillion dollars b. 16.46 trillion dollars ANS: A

3 14.25

DIF:

medium

2. The following table shows the gross national product linear regression to model as a function of .

a. b. ANS: B

5 14.61

c. 17.66 trillion dollars d. 18.46 trillion dollars

PTS: 1

0 12.72

years after 2005. Use

1 13.52

, in trillions of dollars,

3 14.27

years after 2005. Use

5 14.65

c. d. PTS: 1

DIF:

medium

3. The following table shows the gross national product , in trillions of dollars, Explain in practical terms the meaning of the slope of the regression line.

years after 2005.

0 12.74 a. b. c. d.

1 13.52

3 14.25

5 14.62

The gross national product is increasing by 0.36 trillion dollars per year. The gross national product is increasing by 12.97 trillion dollars per year. The gross national product is increasing by 0.78 trillion dollars per year. The gross national product is increasing by 1.37 trillion dollars per year.

ANS: A

PTS: 1

DIF:

medium

4. The following table shows the gross national product , in trillions of dollars, years after 2005. Based on the regression equation, what increase would you expect to see in the gross national product over a 5-year period? 0 12.74

1 13.59

a. 2.51 trillion dollars b. 1.85 trillion dollars ANS: B

3 14.25

5 14.69

c. 1.95 trillion dollars d. 2.2 trillion dollars

PTS: 1

DIF:

medium

5. The following table shows the number , in millions, of cars sold in the U.S. the regression line for the data to estimate the number of cars sold in 2013. 0 5.54

1 5.43

a. 3.37 million b. 4.51 million ANS: D

3 5.28

years after 2005. Use

5 4.54

c. 3.16 million d. 4.11 million DIF: medium

PTS: 1

6. The following table shows the number , in millions, of cars sold in the U.S. years after 2005. Use linear regression to model as a function of . 0 1 3 5 5.59 5.41 5.24 4.54 a. b.

c. d.

ANS: D

PTS: 1

DIF:

medium

7. The following table shows the number , in millions, of cars sold in the U.S. Explain in practical terms the meaning of the slope of the regression line. 0 5.58 a. b. c. d.

1 5.47

Sales are increasing by 0.2 million cars per year. Sales are decreasing by 0.2 million cars per year. Sales are increasing by 0.11 million cars per year. Sales are decreasing by 0.11 million cars per year.

3 5.26

years after 2005.

5 4.57

ANS: B

PTS: 1

DIF:

medium

8. The following table shows the number , in millions, of cars sold in the U.S. years after 2005. Based on the regression equation, how would you expect sales to be affected over a 5-year period? 0 5.56 a. b. c. d.

1 5.46

3 5.24

5 4.57

Sales would decrease by 0.19 million cars. Sales would increase by 0.95 million cars. Sales would increase by 0.19 million cars. Sales would decrease by 0.95 million cars.

ANS: D

PTS: 1

DIF:

medium

9. The following table shows the running speed , in feet per second, of animals of length , in inches. Use the regression line for the data to estimate the running speed of animals that are 14 inches long. 3.5 8.22

6.3 15.78

a. 17.64 feet per second b. 36.33 feet per second ANS: D

9.4 24.02

9.8 24.93

c. 17.36 feet per second d. 36.19 feet per second

PTS: 1

DIF:

medium

10. The following table shows the running speed , in feet per second, of animals of length Use linear regression to model as a function of . 3.5 8.27

6.3 15.74

a. b.

9.4 24.01

, in inches.

9.8 24.99

c. d.

ANS: D

PTS: 1

DIF:

medium

11. The following table shows the running speed , in feet per second, of animals of length , in inches. If one animal is 5 inches longer than another, then based on the regression equation, how would their running speeds compare? 3.5 8.24 a. b. c. d.

6.3 15.78

9.4 24.09

9.8 24.91

The longer animal would run 2.66 times faster than the shorter animal. The longer animal would run 13.3 times faster than the shorter animal. The longer animal would run 13.3 feet per second faster than the shorter animal. The longer animal would run 2.66 feet per second faster than the shorter animal.

ANS: C

PTS: 1

DIF:

medium

12. The following table shows the running speed , in feet per second, of animals of length Explain in practical terms the meaning of the slope of the regression equation.

, in inches.

3.5 8.28

6.3 15.77

9.4 24.08

9.8 24.91

a. An increase of 1 inch in length corresponds to an increase of 2.68 feet per second in running speed. b. An increase of 1 inch in length corresponds to an increase of 2.65 feet per second in running speed. c. An increase of 1 inch in length corresponds to an animal that runs 2.65 times faster. d. An increase of 1 inch in length corresponds to an animal that runs 2.68 times faster. ANS: B

PTS: 1

DIF:

medium

13. The following table shows the running speed , in centimeters per second, of ants as a function of the temperature in degrees Celsius. Explain in practical terms the meaning of the slope of the regression equation. T

25.6 2.33

30.3 3.63

32.3 4.33

34.8 5.27

a. An increase of 1 degree in temperature results in an increase of 0.32 centimeters per second in running speed. b. An increase of 1 degree in temperature results in an increase of 1.08 centimeters per second in running speed. c. An increase of 1 degree in temperature makes the running speed 0.32 times faster. d. An increase of 1 degree in temperature makes the running speed 1.08 times faster. ANS: A

PTS: 1

DIF:

medium

14. The following table shows the running speed , in centimeters per second, of ants as a function of the temperature in degrees Celsius. Based on the regression equation, how fast would an ant run when the temperature is 30.9 degrees Celsius? T

25.6 2.46

30.3 3.96

a. 4.4 centimeters per second b. 4.98 centimeters per second ANS: D

32.3 4.14

34.8 4.75

c. 4.3 centimeters per second d. 3.99 centimeters per second

PTS: 1

DIF:

medium

15. The following table shows the running speed , in centimeters per second, of ants as a function of the temperature in degrees Celsius. Based on the regression equation, how would an increase of 5 degrees Celsius affect running speed? T

a. b. c. d.

25.6 2.01

30.3 3.63

32.3 4.74

The ants run 2 centimeters per second faster. The ants run 2 times as fast. The ants run 0.4 centimeters per second faster. The ants run 0.4 times as fast.

ANS: A

PTS: 1

DIF:

medium

34.8 5.6

16. The following table shows the running speed , in centimeters per second, of ants as a function of the temperature in degrees Celsius. Use linear regression to model running speed as a function of temperature. T

25.6 2.45

30.3 3.34

a. b. ANS: C

32.3 4.57

34.8 4.9

c. d. PTS: 1

DIF:

medium

17. The following table shows the running speed , in centimeters per second, of ants as a function of the temperature in degrees Celsius. Use linear regression to model running speed as a function of temperature. T

25.6 2.84

30.3 3.24

a. b. ANS: C

32.3 4.14

34.8 5.08

c. d. PTS: 1

DIF:

medium

18. The following table shows the average rice yield , in tons per hectare, on Asian farms years after 1980. Use linear regression to model average rice yield as a function of years since 1980. t Y

5 3.31

10 3.66

a. b. ANS: C

20 3.96

25 4.16

c. d. PTS: 1

DIF:

medium

19. The following table shows the average rice yield , in tons per hectare, on Asian farms years after 1980. Based on the regression equation, how much increase in yield is expected over a 5-year period? t Y

5 3.34

10 3.62

a. 0.2 tons per hectare b. 0.04 tons per hectare ANS: A

20 3.98

25 4.11

c. 1.11 tons per hectare d. 3.19 tons per hectare

PTS: 1

DIF:

medium

20. The following table shows the average rice yield , in tons per hectare, on Asian farms 1980. Explain in practical terms the meaning of the slope of the regression equation. t Y

5 3.35

10 3.61

20 3.95

a. Each year yield decreases by an average of 0.04 tons per hectare. b. Each year yield increases by an average of 0.04 tons per hectare.

25 4.16

years after

c. Each year yield increases by an average of 0.91 tons per hectare. d. Each year yield decreases by an average of 0.91 tons per hectare. ANS: B

PTS: 1

DIF:

medium

21. The following table shows the average rice yield , in tons per hectare, on Asian farms years after 1980. Based on the regression equation, what would you expect to be the average rice yield in 2011? t Y

5 3.37

10 3.68

a. 5.31 tons per hectare b. 5.29 tons per hectare ANS: D

20 3.98

25 4.18

c. 5.41 tons per hectare d. 4.47 tons per hectare

PTS: 1

DIF:

medium

SHORT ANSWER 1. Find the linear regression equation for the following table of values. 1.83 12.35

3.77 20.61

4.89 24.66

6.06 29.5

ANS: PTS: 1

DIF: medium

2. Use the regression formula for the data below to estimate the value of 1.21 11.03

3.49 17.56

4.48 21.07

. 6.97 28.76

ANS: PTS: 1

DIF: medium

3. Use the regression formula for the data below to estimate the value of 1.1 43.94

3.9 35.6

4.19 34.9

. 6.34 28.46

ANS: PTS: 1

DIF: medium

4. Find the linear regression equation for the following table of values. 1.96 19.9

3.51 15.35

4.97 10.45

6.19 7.35

ANS: PTS: 1

DIF: medium

5. Find the slope of the linear regression equation for the following table of values. 1.6 21.6

3.58 15.93

4.41 13.01

6.88 6.35

ANS: –2.9 PTS: 1

DIF: medium

6. Find the slope of the linear regression equation for the following table of values. 1.32 7.33

3.06 10.55

4.45 13.54

6.22 16.87

ANS: 1.96 PTS: 1

DIF: medium

7. Plot the graph of the regression line for the following table of values. 1.76 6.23

3.32 8.88

4.33 10.86

6.78 16.3

ANS: y 20 18 16 14 12 10 8 6 4 2

PTS: 1

DIF: medium

8. Plot the graph of the regression line for the following table of values. 1.25

3.8

4.8

6.09

16.45

9.52

5.85

2.69

ANS: y 20 18 16 14 12 10 8 6 4 2

PTS: 1

DIF: medium

9. Plot the graph of the data along with the regression line for the following table of values. 1.44 15.12

3.17 8.98

4.19 8.19

6.78 –1.57

ANS: The horizontal span is 0 to 8, and the vertical span is -5 to 18.

PTS: 1

DIF: medium

10. The following table shows total sales , in dollars, months after the first of the year. Use linear regression to model sales as a function of months since the first of the year. 2 7732.36 ANS: PTS: 1

DIF: medium

3 11654.49

6 22863.76

7 26751.81

11. The following table shows the population of a small town years after 2000. Use linear regression to model the population as a function of years since 2000. 2 3117

3 3362

6 4081

7 4296

ANS: PTS: 1

DIF: medium

12. The following table shows the enrollment in my college model population as a function of years since 2000. 0 8148

3 8541

years after 2000. Use linear regression to

7 9032

9 9263

ANS: PTS: 1

DIF: medium

13. Plot the graph of the data along with the regression line for the following table of values. 1.44 6.32

3.17 8.83

4.19 13.14

6.78 16.33

ANS: Horizontal span 0 to 8. Vertical span from 0 to 20

PTS: 1

DIF: medium

ESSAY 1. The following table shows federal expenditures 2004. 0 269.72

2 298.98

A: Find the equation of the regression line.

, in billions of dollars, on Medicare

3 329.44

4 375

years after

B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression equation. D: What expenditure does the regression line predict for 2012? ANS: A: B: E 380 360 340 320 300 280 260 240 220 200

–1

C: During the period from 2004 to 2007, federal Medicare expenditures increased by about 25.34 billion dollars each year. D: 463.99 billion dollars PTS: 4

DIF: hard

2. The following table shows federal expenditures M, in billions of dollars, on national defense after 2004. 0 455.88

2 495.22

3 521.49

4 551.63

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression equation. D: What expenditure does the regression line predict for 2009? ANS: A: B:

years

M 600 580 560 540 520 500 480 460 440 420 –1

C: During the period from 2004 to 2007, national defense expenditures increased by about 23.65 billion dollars each year. D: 571.09 billion dollars PTS: 4

DIF: hard

3. The following table shows the average price 2000. 0 1.59

, in dollars, of a gallon of regular gas

2 1.41

4 1.93

7 2.81

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression equation. D: What price does the regression line predict for 2010? ANS: A: B:

years after

P 4.5 4 3.5 3 2.5 2 1.5 1 0.5 –1 –0.5

–1

C: During the period from 2000 to 2004, the price of a gallon of regular gas increased by about 0.19 dollar each year. D: $3.22 PTS: 4

DIF: hard

4. The following table shows the number 0 10.92

, in millions, of traffic accidents 1 10.71

3 10.61

years after 2004. 4 10.21

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression equation. D: What number of accidents does the regression line predict for 2010? ANS: A: B: A 12 11 10 9 8 7 6 5

–1

C: During the period from 2000 to 2004, the number of traffic accidents decreased by about 0.15 million each year. D: 10.02 million PTS: 4

DIF: hard

5. The following table shows the number after 2006. 0 5.82

, in thousands, of federal methamphetamine arrests

1 5.52

2 4.72

years

3 4.64

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression equation. D: What number of arrests does the regression line predict for 2011? ANS: A: B: A 7 6 5 4 3 2 1

–1

C: During the period from 2000 to 2004, the number of federal methamphetamine arrests decreased by about 0.43 thousand each year. D: 3.68 thousand PTS: 4

DIF: hard

6. The following table shows the number States years after 2005. 0 49.21

, in millions, of international tourists visiting the United

1 51.63

2 55.34

3 57.85

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression line. D: What number of tourists does the regression line predict for 2013? ANS: A: B: T

60 50 40 30 20 10

–10

C: During the period from 2005 to 2008, the number of international tourists increased by approximately 2.96 million each year. D: 72.74 million PTS: 4

DIF: hard

7. The following table shows the amount after 2000. 5 114.85

, in billions of dollars, spent on cell phone service

6 125.02

7 139.25

years

8 142.96

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Based on the regression equation, how much would be expected to be spent on cellular service over a 5-year period? D: What amount does the regression line predict for 2013? ANS: A:

B: y 180 160 140 120 100 80 60 40 20 –4

–2 –20

C: About 49.3 billion dollars D: 194.64 billion dollars PTS: 4

DIF: hard

8. The following table shows the number D , in millions, of DirecTV subscribers 0 2.35

4 7.61

9 13.54

years after 1995.

14 19.69

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Based on the regression equation, how many new DirecTV subscribers would be expected over a 5-year period? D: What number of subscribers does the regression line predict for 2012? ANS: A: B:

D 35 30 25 20 15 10 5 –6

–4

–2 –5

16 t

–10 –15

C: About 6.15 million D: 23.39 million PTS: 4

DIF: hard

9. The following table shows the number 4 9

of Crayola colors available 59 67

91 82

years after 1900. 104 122

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Based on the regression equation, how many new colors are made available over a 5-year period? Round your answer to the nearest whole number. D: What number of colors does the regression line predict for 1991? Round your answer to the nearest whole number. ANS: A: B:

C 150 135 120 105 90 75 60 45 30 15 ––1105

10 20 30 40 50 60 70 80 90 100 110

–30 –45

C: About 5 D: 97 PTS: 4

DIF: hard

10. The following table shows the speed of a car, in miles per hour, as a function of the time , in seconds, since observation of the car began. 15 60

30 65

45 67

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression line. D: How fast would you expect this car to be traveling after 73 seconds? ANS: A: B: S 90 80 70 60 50 40 30 20 10 –10 –10

60 72

C: The speed increases by about 0.25 mile per hour for each second that passes, so the car is accelerating at a rate of 0.25 mile per hour per second. D: About 74.75 miles per hour PTS: 4

DIF: hard

11. The following table shows the length after 1900. 0 7.19

, in meters, of the winning long jump in the Olympics

4 7.34

8 7.48

years

12 7.6

A: Find the equation of the regression line. B: Plot the data along with the regression line. C: Explain in practical terms the meaning of the slope of the regression line. D: Would you expect this model to apply over an indefinite period of time? Explain the reasoning behind your answer. ANS: A: B: L 9 8 7 6 5 4 3 2 1

–1

C: The length of the winning long jump is increasing by about 0.03 meter each year. D: The model would not be expected to apply over an indefinite period because there is a physical limit to the distance any human can ever jump. PTS: 4

DIF: hard

Section 3.5 Systems of Equations TRUE/FALSE 1. A system of two linear equations in two unknowns can be solved graphically. ANS: T

PTS: 1

DIF:

easy

2. There is no solution of a system of two linear equations in two unknowns if the graphs of the equations are parallel lines. ANS: T

PTS: 1

DIF:

easy

3. If the lines determined by two linear equations meet, then the crossing point corresponds to the solution of the system of two linear equations in two unknowns. ANS: T

PTS: 1

DIF:

easy

4. It is never possible to solve a system of three equations in three unknowns. ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Solve the equation a. b.

for . c.

d. None of the above.

ANS: A

PTS: 1

2. Solve the equation a. b.

DIF:

PTS: 1

3. Solve the equation

b. ANS: A 4. Solve the equation

d. None of the above.

medium

for .

ANS: A

DIF:

medium

for . or or PTS: 1

d. None of the above. DIF: for .

medium

d. None of the above.

ANS: A

PTS: 1

5. Solve the equation a.

for

PTS: 1

DIF:

PTS: 1

7. Solve the equation

ANS: C

DIF: for

d. None of the above.

medium

for .

ANS: C

d. None of the above.

6. Solve the equation

medium

ANS: C

DIF:

PTS: 1

medium

. c.

d. None of the above. DIF:

medium

8. Solve the system of equations

a. b.

and and

ANS: A

PTS: 1 DIF:

9. Solve the system of equations

c. d. medium

and and

a. b.

and and

c. d.

ANS: A

PTS: 1 DIF:

and and

medium

10. Solve the system of equations

a. b. ANS: C

and and

c. d.

PTS: 1

DIF:

and and medium

11. We have $69.52 to spend on drinks and chips. Drinks cost $0.78 each and chips cost $2.42 per bag. There are 5 times as many drinks as bags of chips. How many of each are bought?

a. 65 drinks and 13 bags of chips b. 11 drinks and 55 bags of chips ANS: C

PTS: 1

c. 55 drinks and 11 bags of chips d. None of the above DIF:

medium

12. We have $33.26 to spend on citrus. Lemons cost $0.74 each, and limes cost $0.62 each. We need a total of 49 pieces of citrus. How many of each are bought?

a. 23 lemons and 26 limes b. 25 lemons and 24 limes ANS: C

PTS: 1

c. 24 lemons and 25 limes d. 26 lemons and 23 limes DIF:

medium

13. We have $18.33 to spend on vegetable plants for our garden. Tomato plants cost $0.87 each, and pepper plants cost $0.33 each. There is space in the garden for 31 plants.

How many of each are bought?

a. 13 tomatoes and 18 peppers b. 16 tomatoes and 15 peppers ANS: D

PTS: 1

c. 18 tomatoes and 13 peppers d. 15 tomatoes and 16 peppers DIF:

medium

14. After returning from a trip to Mexico you find that you have several dollars and several pesos in your pocket. Each peso has a value of $0.18. There are a total of 80 items, and they have a total value of $19.32.

How many dollars and how many pesos are in your pocket?

a. 5 dollars and 75 pesos b. 74 dollars and 6 pesos ANS: D

PTS: 1

c. 75 dollars and 5 pesos d. 6 dollars and 74 pesos DIF:

medium

15. A group of 70 parents and children plan to go to the zoo. For safety, there must be one parent for each 6 children.

How many parents and how many children go to the zoo?

a. 12 parents and 72 children b. 10 parents and 60 children ANS: B

PTS: 1

c. 72 parents and 12 children d. 60 parents and 10 children DIF:

medium

16. A business pays 9% tax on income from New York and 6% tax on income from New Jersey. The total tax paid is $856.2, and the total income is $10980.

How much income came from each state?

a. b. c. d.

$6583 from New York and $4397 from New Jersey $6580 from New York and $4400 from New Jersey $4397 from New York and $6583 from New Jersey $4400 from New York and $6580 from New Jersey

ANS: B

PTS: 1

DIF:

medium

17. Type A grain containing 29% fiber is mixed with type B grain containing 6% fiber to make 536 pounds of feed. The resulting mixture is to contain 80.69 pounds of fiber.

How much of each type of grain goes in the mixture?

a. b. c. d.

228 pounds of type A and 308 pounds of type B 211 pounds of type A and 325 pounds of type B 308 pounds of type A and 228 pounds of type B 325 pounds of type A and 211 pounds of type B

ANS: B

PTS: 1

18. Solve the equation a.

DIF:

medium

for . or

b. ANS: C

or PTS: 1

19. Solve the equation

DIF:

medium

for

a. b. ANS: C

d. None of the above.

c. d. None of the above. PTS: 1

DIF: medium

SHORT ANSWER 1. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

2. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

3. Solve the following system of equations by graphing.

ANS:

y 3.5 3 2.5 2 1.5 1 0.5

–0.5 –0.5

0.5

1.5

PTS: 1

2.5

3.5

DIF: medium

4. Solve the following system of equations by graphing.

ANS: y 3.5 3 2.5 2 1.5 1 0.5

–0.5 –0.5

0.5

1.5

2.5

3.5

PTS: 1

DIF: medium

5. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

6. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

7. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

8. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

9. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

10. Solve the following system of equations by graphing.

ANS: y 3.5 3 2.5 2 1.5 1 0.5

–0.5 –0.5

PTS: 1

DIF: medium

11. Solve the following system of equations.

ANS:

0.5

1.5

2.5

3.5

PTS: 1

DIF: medium

12. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

13. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

14. Solve the following system of equations.

ANS:

PTS: 1

DIF: medium

15. Solve the following system of equations by graphing.

ANS:

y 3.5 3 2.5 2 1.5 1 0.5

–0.5 –0.5

0.5

1.5

PTS: 1

2.5

3.5

DIF: medium

16. Solve the following system of equations by graphing.

ANS: y 3.5 3 2.5 2 1.5 1 0.5

–0.5 –0.5

PTS: 1

0.5

1.5

2.5

3.5

DIF: medium

17. Solve the following system of equations by graphing.

ANS: y 3.5 3 2.5 2 1.5 1 0.5

–0.5 –0.5

PTS: 1

0.5

1.5

2.5

3.5

DIF: medium

ESSAY 1. We have $44.58 to spend on chips and drinks for a party. Bags of chips cost $4.11 each, and drinks cost $ 0.83 each. We need 4 times as many drinks as bags of chips. A: Using for number of bags of chips and equations that determine our party purchase.

for number of drinks, write a system of two

B: How many bags of chips and how many drinks are bought for the party? ANS: A:

B: We buy 6 bags of chips and 24 drinks. PTS: 2

DIF: hard

2. We have $22.09 to spend on fruit. Apples cost $0.79 each, and oranges cost $0.63 each. We need a total of 31 pieces of fruit. A: Using for number of apples and determine our fruit purchase.

for number of oranges, write a system of two equations that

B: How many apples and how many oranges do we buy? ANS:

B: We buy 16 apples and 15 oranges. PTS: 2

DIF: hard

3. We have $101.4 to spend on flowering bulbs. Canna bulbs cost $3.15 each, and lily bulbs cost $1.51 each. There is space in the garden for 40 bulbs. A: Using for number of canna bulbs and L for number of lily bulbs, write a system of two equations that determine our flower purchase.

B: How many canna bulbs and how many lily bulbs do we buy? ANS: A:

B: We buy 25 canna bulbs and 15 lily bulbs. PTS: 2

DIF: hard

4. After returning from a trip to Spain, you find that you have several dollars and several Euros in your purse. Each Euro has a value of $1.21. There are a total of 29 items, and they have a total value of $32.15. A: Using for the number of dollar bills and the for number of Euros, write a system of two equations that describes the money in your purse.

B: How many dollar bills and how many Euros do you have? ANS: A:

B: We have 14 dollar bills and 15 Euros. PTS: 2

DIF: hard

5. A group of 30 parents and children plan to go to the zoo. For safety, there must be one parent for each 4 children. A: Using for parents and traveling to the zoo.

for children, write a system of two equations that describes the group

B: How many parents and how many children make the trip? ANS:

B: 24 children and 6 parents travel to the zoo. PTS: 2

DIF: hard

6. Animal type A and animal Type B compete for the same resources. Suppose there are animals of type A and animals of type B. Both and are measured in thousands. Then the growth rates of the two types of animals are given by Growth rate for Type A: Growth rate for Type B:

A: Equilibrium occurs when both growth rates are 0. Find two expressions of equilibrium occurs. B: Plot the graphs of versus shows the crossing point.

in terms of

when

for both the expressions you found in part A. Be sure your graph

B: How many of each type of animal is present when equilibrium occurs? ANS: A:

b 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 –0.2 –0.1 –0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

–0.2

B: C: 0.33 thousand type A animals and 0.67 thousand type B animals are present. PTS: 3

DIF: hard

7. Type A liquid containing 25% chlorine is mixed with type B liquid containing 15% chlorine to fill a 399 gallon container. The resulting liquid is to contain 19% chlorine.

A: If we use gallons of type A liquid and gallons of type B liquid, write a system of two equations that determine the amounts of each that go into the mixture. B: How much of each liquid type goes into the mixture? ANS: A:

B: 159.6 gallons of type A liquid and 239.4 gallons of type B liquid PTS: 2

DIF: hard

8. Type A grain containing 13% protein is mixed with type B grain containing 26% protein to make 1342 pounds of feed. The resulting mixture is to contain 22% protein.

A: If we use pounds of type A grain and pounds of type B grain, write a system of two equations that determine the amounts of each that go into the mixture. B: How much grain of each type goes into the mixture? ANS: A:

B: 412.92 pounds of type A grain and 929.08 pounds of type B grain PTS: 2

DIF: hard

9. A business pays 7% tax on income from Oklahoma and 12% tax on income from Arkansas. The total tax paid is $2,268, and the total income is $25,201.

A: Using O for Oklahoma income and for Arkansas income, write a system of two equations that determine the income from of each of the two states. B: How much income was earned in each state? ANS: A:

B: $15,122.4 from Oklahoma and $10,078.6 from Arkansas

PTS: 2

DIF: hard

10. We have $944 to spend on gravel for a drive. Fine gravel costs $28 per ton, while coarse gravel costs $16 per ton. We need a total of 44 tons of gravel.

A: Using fine gravel and for coarse gravel, both in tons, write a system of two equations that determine the amount of each to be purchased. B: How much of each type of gravel was purchased? ANS: A:

B: 20 tons of fine gravel and 24 tons of coarse gravel. PTS: 2

DIF: hard

Section 4.1 Exponential Growth and Decay TRUE/FALSE 1. Exponential functions are functions with a constant proportional change. ANS: T

PTS: 1

DIF:

easy

2. For a function that shows exponential growth, an increase of 1 unit in the variable causes the function to be multiplied by the growth factor. ANS: T

PTS: 1

DIF:

easy

3. For a function that shows exponential decay, an increase of 1 unit in the variable causes the function to be divided by the decay factor. ANS: F

PTS: 1

DIF:

easy

4. A graph showing exponential growth is concave up. ANS: T

PTS: 1

DIF:

easy

5. A graph showing exponential decay is concave down. ANS: F

PTS: 1

DIF:

easy

6. The graph of exponential growth is decreasing. ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. A population grows so that we can always get next year’s population from this year’s population by multiplying by 1.53. The initial population is 540. The population after years is given by a. b. ANS: A

c. d. PTS: 1

DIF:

medium

2. The growth factor of an exponential function formula for is a. b. ANS: D

is 2.49. The initial value is 3.75. Then a

c. d. PTS: 1

DIF:

3. The decay factor of an exponential function for is a.

medium is 0.61. The initial value is 1.48. Then a formula

b. ANS: D

d. PTS: 1

DIF:

medium

4. The formula for an exponential function is a. 131 b. 2.4 ANS: B

. Then the growth factor is: c. d.

PTS: 1

DIF: medium

5. The formula for an exponential function is a. 639 b. ANS: D

. Then the decay factor is: c. d. 0.62

PTS: 1

DIF: medium

6. The formula for an exponential function is a. 29 b. 138.91 ANS: A

. Then the initial value is: c. d. 4.79

PTS: 1

DIF: medium

7. If there are grams of a certain radioactive substance present today, then in one year there will be grams remaining. There are initially 34 grams present. Find a formula that gives the amount , in grams, remaining after years. a. b. ANS: B

c. d. PTS: 1

DIF:

medium

8. If the population of a certain town today is , then in one year the population will be initial population is 2320. Find a formula that gives the population after years. a. b. ANS: C

. The

c. d. PTS: 1

DIF:

medium

9. If there are grams of a certain radioactive substance present today, then in one year there will be grams remaining. There are initially 22 grams present. How much remains after 10 years? a. 17.25 grams b. 11.85 grams ANS: B

c. 12.6 grams d. None of the above PTS: 1

DIF:

medium

10. The population of a certain town is growing exponentially with yearly growth factor 1.06. The initial population is 1732. What will be the population after 10 years? Round your answer to the nearest whole number. a. 2418

c. 1721

b. 3102 ANS: B

d. None of the above PTS: 1

DIF:

medium

11. A certain function is an exponential function of time. The yearly growth factor is 1.61. What is the decade growth factor? a. 0.16 b. 1.05 ANS: C

c. 117.02 d. 16.1 PTS: 1

DIF:

medium

12. A certain function is an exponential function of time. The weekly growth factor is 1.41. What is the monthly growth factor? (Assume there are four weeks in a month.) a. 0.35 b. 1.09 ANS: C

c. 3.95 d. 5.64 PTS: 1

DIF: medium

13. A certain function is an exponential function of time. The monthly growth factor is 1.54. What is the weekly growth factor? (Assume there are four weeks in a month.) a. 0.39 b. 1.11 ANS: B

c. 5.62 d. 6.16 PTS: 1

DIF: medium

14. A radioactive substance is decaying exponentially with yearly decay factor 0.93. What is the decade decay factor? a. 0.09 b. 0.99 ANS: D

c. 9.3 d. 0.48 PTS: 1

DIF: medium

15. A radioactive substance is decaying exponentially with yearly decay factor 0.94. What is the monthly decay factor? a. 0.08 b. 0.99 ANS: B

c. 11.28 d. 0.48 PTS: 1

DIF: medium

16. A population is growing exponentially with a weekly growth factor of growth factor? a. 0.02 b. 1 ANS: D

1.03. What is the yearly

c. 53.56 d. 4.65 PTS: 1

DIF: medium

17. A population is growing exponentially with a yearly growth factor of 1.23. What is the monthly growth factor? a. 0.1

c. 14.76

b. 1.02 ANS: B

d. 11.99 PTS: 1

DIF:

medium

18. The half-life of a certain radioactive substance is 9 years. What is the yearly decay factor? a. 4.5 b. 0.99 ANS: C

c. 0.93 d. None of the above PTS: 1

DIF:

medium

19. Over a certain time period the price of gold doubled every 6 years. What was the yearly growth factor for gold during this period? a. 3 b. 1.12 ANS: B

c. 2.01 d. None of the above PTS: 1

DIF:

medium

20. For a certain substance, we get next week’s amount by dividing by 5. That means the amount is showing exponential decay. What is the weekly decay factor? a. 0.2 b. 5 ANS: A

c. 0.74 d. None of the above PTS: 1

DIF: medium

21. Next year’s value of quantity 1 is obtained by multiplying this year’s value of quantity 1 by 1.12. Next year’s value of quantity 2 is obtained by adding 1.12 to this year’s value of quantity 1. Which of the two quantities are growing exponentially? a. Only quantity 1 b. Only quantity 2 ANS: A

c. Both quantity 1 and quantity 2 d. Neither quantity 1 nor quantity 2 PTS: 1

DIF:

medium

22. Next year’s value of quantity 1 is obtained by multiplying this year’s value of quantity 1 by 0.77. Next year’s value of quantity 2 is obtained by adding 0.77 to this year’s value of quantity 1. Which of the two quantities are decaying exponentially? a. Only quantity 1 b. Only quantity 2 ANS: A

c. Both quantity 1 and quantity 2 d. Neither quantity 1 nor quantity 2 PTS: 1

DIF:

medium

23. Next year’s value of a certain quantity is obtained by multiplying this year’s value of the quantity by 0.74. Which of the following can you conclude? a. The quantity grows exponentially. b. The quantity decays exponentially. ANS: B

PTS: 1

c. The quantity grows linearly. d. The quantity decays linearly. DIF:

medium

24. Next year’s value of a certain quantity is obtained by multiplying this year’s value of the quantity by 1.71. Which of the following can you conclude?

a. The quantity grows exponentially. b. The quantity decays exponentially. ANS: A

PTS: 1

c. The quantity grows linearly. d. The quantity decays linearly. DIF:

medium

25. The initial value of one exponential function is 46, and the growth factor is 1.02. The initial value of a second exponential function is 96, and the decay factor is 0.95. If is the independent variable for both functions, for what value of are the two functions equal? a. b.

c. d.

ANS: D

PTS: 1

DIF:

medium

26. The initial value of an exponential function is 12, and the growth factor is 1.01. If variable, for what value of does the function have the value 42? a. b.

is the independent

c. d.

ANS: D

PTS: 1

DIF:

27. Select the graph of

medium

–1

10 11

–1

–5

10 11

–1

–5 y

10 11

–1 –5

10 11

ANS: B

PTS: 1

DIF:

28. Select the graph of

medium

–1

10 11

–1

–5

10 11

–1

–5 y

10 11

–1

–5

ANS: A

–5

PTS: 1

DIF:

29. Solve the exponential equation

a. b. ANS: C

c. d. PTS: 1

DIF:

30. Solve the exponential equation

31. Solve the equation

medium .

a. b. ANS: C

medium

c. d. PTS: 1

DIF: .

medium

10 11

a. b. ANS: C

c. d. PTS: 1

DIF:

32. Solve the equation

a. b. ANS: C

c. d. PTS: 1

DIF:

33. Solve the inequality

c. d. PTS: 1

DIF:

34. Solve the inequality

medium .

a. b. ANS: B

medium

a. b. ANS: D

medium

c. d. PTS: 1

DIF:

medium

SHORT ANSWER 1. Solve the inequality

ANS: PTS: 1

DIF: medium

ESSAY 1. A certain radioactive substance decays exponentially with yearly decay factor of 0.93. There are initially 107 grams present. A: Write a formula that gives the amount

, in grams, of the substance remaining after

B: How much is left after 4 years? C: Plot the graph of

versus

over the first 20 years.

D: How long will it take until only 36 grams are left?

years.

ANS: A: B: 80.04 grams. C: A 110 100 90 80 70 60 50 40 30 20 10 –4 –10

10 12 14 16 18 20 22

D: 15.01 years PTS: 4

DIF: hard

2. A certain population grows exponentially with a yearly growth factor of 1.06. The population is initially 508 individuals. A: Write a formula that gives the population

after

years.

B: What is the population after 5 years? Round your answer to the nearest whole number. C: Plot the graph of

versus

over the first 20 years.

D: How long will it take for the population to reach 913 individuals? ANS: A: B: 680 individuals. C:

N 1800 1600 1400 1200 1000 800 600 400 200 –4 –2– 0 20

8 10 12 14 16 18 20 22

–400 –600 –800

D: 10.06 years PTS: 4

DIF: hard

3. There are initially 8 milligrams per deciliter of a drug in the bloodstream of a patient. The concentration of the drug decays exponentially with hourly decay factor 0.91. A: Using for the concentration, in milligrams per deciliter, of the drug after formula that gives the value of after hours.

hours, write a

B: What is the concentration after 4 hours? C: Plot the graph of

versus

over the first 20 hours.

D: What is the half-life of the drug? That is, how long will it take for the concentration to reach half of its initial level? ANS: A: B: 5.49 milligrams per deciliter. C: C 10 9 8 7 6 5 4 3 2 1 –4 –2

D: 7.35 hours

10 12 14 16 18 20 22

PTS: 4

DIF: hard

4. There are initially 6.61 thousand people in a city. The population grows exponentially with yearly growth factor 1.06. A: Using for the population in thousands after after years.

years, write a formula that gives the value of

B: What is the population after 4 years? C: Plot the graph of

versus

over the first 20 years.

D: What is the doubling time for the population? That is, how long will it take for the population to double its initial level? ANS: A: B: 8.34 thousand C: N 25

–4 –2

10 12 14 16 18 20 22

–5

D: 11.9 years PTS: 4

DIF: hard

5. On a certain island new species are being introduced at a rate of species per year. But conditions cause species extinction at a rate of species per year. Here t is the time in years. Equilibrium occurs when the rate of extinction is the same as the rate new species are being introduced. A: The time when equilibrium occurs is the solution of an equation. Which equation?

B: Plot the graphs of both

and

on the same coordinate axes over the first 10 years of activity.

C: When does equilibrium occur? D: What is extinction rate when equilibrium occurs? ANS: A:

–2 –1

–1

C: After 4.75 years D: 3.35 species per year PTS: 4

DIF: hard

6. A certain bacteria population doubles each 20 minutes. A: Explain why the bacteria population is an exponential function of time. B: What is the hourly growth factor for the bacteria population? C: If there are initially

bacteria present, write a formula that gives the population

ANS: A: The population grows exponentially because it grows by constant multiples. B: 8 C: PTS: 4

DIF: hard

7. The amount of data passing through mobile phone networks doubles each year.

after

hours.

A: Explain why the amount of data passing through mobile phone networks is an exponential function of time. B: Use for the initial amount of data, and write a formula that gives the amount through mobile phone networks after years.

of data passing

C: If this trend continues, how long will it be before the amount of data is 138 times its initial value? ANS: A: The amount of data grows exponentially because it grows by constant multiples. B: C: 7.11 years PTS: 3

DIF: hard

8. Between 1995 and 2005, each year the number of Internet domain hosts was 1.46 times the number of hosts in the preceding year. In 1995 the number of hosts was 8.2 million. A: Explain why the number of domain hosts was an exponential function of time over the period from 1995 to 2005.. B: Write a formula that gives the number

, in millions, of domain hosts

years after 1995.

C: Make a graph that shows the growth of domain hosts from 1995 to 2005, that is, for

D: According to this model, when did the number of domain hosts reach 21 million? ANS: A: The number of domain hosts grew exponentially because it changed by constant multiples. B: D 400 350 300 250 200 150 100 50 –2 –1 –50 –100

C: D: 2.48 years after 1995

PTS: 3

DIF: hard

9. City 1 has an initial population of 88.9 thousand. Each year we get next year’s population by multiplying by 0.92. City 2 has an initial population of 54.95 thousand. Each year we get next year’s population by multiplying by 1.04.

A: Write a formula that gives the population

, in thousands, of City 1 after years.

B: Write a formula that gives the population

, in thousands, of City 2 after years.

C: Make a graph that shows the populations of both cities over the first 20 years. D: When do the two cities have the same population? ANS: A: B: C: 120 105 90 75 60 45 30 15 –5

–15 –30 –45

D: After 3.92 years PTS: 3

DIF: hard

Section 4.2 Constant Percentage Change TRUE/FALSE 1. An exponential function is a function that shows constant percentage change. ANS: T

PTS: 1

DIF:

easy

2. The graph of a function that shows constant percentage growth is concave up. ANS: T

PTS: 1

DIF:

easy

3. The graph of a function that shows constant percentage decay is concave up. ANS: T

PTS: 1

DIF:

easy

4. If a function has constant percentage increase, then its graph shows exponential growth. ANS: T

PTS: 1

DIF:

easy

5. If a function has constant percentage decrease, then its graph show exponential growth. ANS: F

PTS: 1

DIF:

easy

6. If I get a 5% raise each year for four consecutive years, that is the same as a 20% raise. ANS: F

PTS: 1

DIF:

easy

7. If an investment earns interest that is compounded, then the balance will show exponential growth. ANS: T

PTS: 1

DIF:

easy

8. If the compound interest rate on an investment is small, then the value of the investment will never grow very fast. ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. If a function increases by 7% each year, then the yearly growth factor is a. 1.07 b. 7 ANS: A 2. A function formula for a. b.

c. 0.07 d. 0.93 PTS: 1

DIF: medium

, where t is measured in years, increases by 11% each year. Its initial value is 16. A is c. d.

ANS: B

PTS: 1

DIF:

medium

3. If a function decreases by 5% each year, then the yearly decay factor is a. 1.05 b. 5 ANS: D 4. If

c. 0.05 d. 0.95 PTS: 1

is increased by 1, then

a. 97% b. 0.03% ANS: D 5. If

is increased by what percentage? c. 1.03% d. 3%

PTS: 1

is increased by 1, then

a. 85% b. 15% ANS: B

DIF: medium

DIF:

medium

is decreased by what percentage? c. 0.85% d. 0.15%

PTS: 1

DIF:

medium

6. The monthly percentage growth rate for a certain exponential function is 3%. By what percentage does the function grow in a year? a. 42.88% b. 42.58% ANS: B

c. 35.3% d. 36% PTS: 1

DIF: medium

7. The monthly percentage decay rate for a certain exponential function is 8%. By what percentage does the function decay in a week? (Assume there are four weeks in a month.) a. 2.06% b. 2.28% ANS: A

c. 1.88% d. 2% PTS: 1

DIF: medium

8. I got a salary increase of 6% each year for 5 years. What is the total percentage increase in my salary over the 5-year period? a. 29.69% c. 33.82% b. 34.06% d. 30% ANS: C

PTS: 1

DIF: medium

9. A quantity increases by 6% each month for 9 months. What is the total percentage increase over the9month period? a. 53.11% b. 69.05% ANS: C

c. 68.95% d. 54% PTS: 1

DIF: medium

10. A quantity decreases by 8% each week for 7 weeks. What is the total percentage decrease over the 7-week period?

a. 55.52% b. 44.25% ANS: D

c. 56% d. 44.22% PTS: 1

DIF: medium

11. A population declines by 8% each year. The initial population is 5617. The population after years is given by a. b. ANS: A

c. d. PTS: 1

DIF:

medium

12. A population grows by 6% each year. The initial population is 7957. The population after years is given by a. b. ANS: A

c. d. PTS: 1

DIF:

medium

13. The formula for an exponential function is monthly percentage growth rate for is

a. 1.08% b. 8% ANS: B

, where

c. 0.08% d. 8.02% PTS: 1

DIF: medium

14. The formula for an exponential function is monthly percentage decay rate for is

a. 0.93% b. 7.02% ANS: D

is measured in months. Then the

, where

is measured in months. Then the

c. 0.07% d. 7% PTS: 1

DIF: medium

15. A population is growing at a constant yearly percentage rate of 8%. The initial population is 623. What will be the population after 14 years? Round your answer to the nearest whole number. a. 1325 b. 1831 ANS: C

c. 1830 d. 1321 PTS: 1

DIF: medium

16. A radioactive substance is decaying at a constant yearly percentage rate of 6%. The initial amount is 435 grams. How much will remain after 12 years? a. 123.77 grams b. 207.03 grams

c. 209.47 grams d. 121.8 grams

ANS: B

PTS: 1

DIF:

medium

17. Over a certain period of time, a precious metal doubled in price every 7 years. What was the yearly percentage growth rate for the price of this metal during this period? a. 29.46% b. 10.41% ANS: B

c. 11.22% d. 28.57% PTS: 1

DIF: medium

18. The half-life of a certain radioactive substance is 7 years. What is the yearly percentage decay rate for this substance? a. 7.35% b. 9.43% ANS: B

c. 10.32% d. 7.14% PTS: 1

DIF: medium

19. Next year’s value of a certain quantity is 6% of last year’s value. Which of the following can you conclude? a. The quantity grows exponentially. b. The quantity grows linearly. ANS: C

PTS: 1

c. The quantity decays exponentially. d. The quantity decays linearly. DIF:

medium

20. Next year’s value of a certain quantity is 108% of last year’s value. Which of the following can you conclude? a. The quantity grows exponentially. b. The quantity grows linearly. ANS: A

PTS: 1

c. The quantity decays exponentially. d. The quantity decays linearly. DIF:

medium

21. The initial value of one function is 32. It has a constant percentage growth rate of 10%. The initial value of a second function is 129. It has a constant percentage decay rate of 8%. If is the independent variable for both functions, for what value of are the two functions equal? a. b. ANS: B

c. d. PTS: 1

DIF:

medium

22. A function shows constant percentage growth. Which of the following may be the graph of this function?

–1

10 11

–1

ANS: A

10 11

PTS: 1

–1

DIF:

10 11

–1

medium

23. A function shows constant percentage decay. Which of the following may be the graph of this function? y

–1

10 11

–1

10 11

–1

ANS: B

10 11

PTS: 1

DIF:

–1

10 11

medium

SHORT ANSWER 1. Enrollment at my college is increasing at a rate of 5% per year. What will be the total percentage increase after 8 years? ANS: 47.75% PTS: 1

DIF: medium

2. Enrollment at my college is increasing at a rate of 49% per decade. What is the yearly percentage increase? ANS: 4.07% PTS: 1

DIF: medium

3. Enrollment at my college is declining at rate of 48% per decade. What will be the yearly percentage decrease? ANS: 6.33% PTS: 1

DIF: medium

4. Enrollment at my college is declining at rate of 5% per year. What is the total percentage decrease after 8 years? ANS: 33.66% PTS: 1

DIF: medium

5. Bank A pays 5% per year compounded monthly on investments. Bank B pays 5% per year compounded weekly on investments. In which bank should I invest my money?

ANS: Bank B. PTS: 1

DIF: medium

ESSAY 1. An initial investment of $5742 earns interest of 6% compounded yearly. A: Write a formula that gives the value

, in dollars, of the investment after

years.

B: What is the balance after 9 years? C: Make a graph that shows the value of the investment over the first 20 years. D: When will the value of the investment reach $7096? ANS: A: B: $9700.99 C: V 21000 18000 15000 12000 9000 6000 3000

–4 –2 –3000

10 12 14 16 18 20 22

D: After 3.63 years PTS: 4

DIF: hard

2. A cleaning process removes 7% of a contaminant from a water source each week. There are initially 586 pounds of the contaminant in the water source. A: Write a formula that gives the amount weeks.

, in pounds, of the contaminant in the water source after

B: How much contaminant remains after 7 weeks?

C: Make a graph that shows the amount of contaminant over the first 20 weeks. D: When will the contaminant level reach 198.33 pounds? ANS: A: B: 352.6 pounds C: C

600 500 400 300 200 100

–4 –2 –100

10 12 14 16 18 20 22

D: After 14.93 weeks PTS: 4

DIF: hard

3. An endangered population is declining by 5% per year. There are initially 862 animals present. A: Write a formula that gives the number

of animals remaining after

B: What is the percentage decrease over any 10-year period? C: Make a graph that shows the population level over the first 20 years. D: How long will it take for the population to decline by 15%? ANS: A: B: 40.13% C:

years.

1200 1000 800 600 400 200

–4 –2 –200

10 12 14 16 18 20 22

D: 3.17 years PTS: 4

DIF: hard

4. There are initially 37 milligrams per deciliter of a drug in the bloodstream of a patient. The concentration of the drug shows a constant percentage decay of 6% per hour. A: Using for the concentration, in milligrams per deciliter, of the drug after formula for . B: What is the percentage decrease in concentration over any 4-hour period? C: Make a graph that shows the drug concentration over the first 20 hours. D: How long will it take for the concentration to decline by 24%? ANS: A: B: 21.93% C: C 35 30 25 20 15 10 5

–4 –2 –5

10 12 14 16 18 20 22

hours, write a

D: 4.44 hours PTS: 4

DIF: hard

5. In 1995 there were 56.22 thousand performances in the U.S. by nonprofit professional theaters. From 1995 through 2007, the number increased by 11% each year. A: Find a formula that gives the number

, in thousands, of theater performances

years after 1995.

B: What is the percentage increase over any 4-year period during the time from 1995 to 2007? C: Make a graph that shows the number of theater performances from 1995 to 2012. D: How long did it take for the number of performances to double? ANS: A: B: 51.81% C: T 225 200 175 150 125 100 75 50 25 –1 –25

9 10 11 12 13

–50

D: 6.64 years PTS: 4

DIF: hard

6. At age 25 you start to work and are offered two retirement options. Retirement option 1: When you retire, you receive a lump sum of $25000 for each year of service. Retirement option 2: When you start to work, the company will deposit $10000 into an account that pays a yearly compound interest rate of 13%. A: Write a formula that gives your retirement payment service under retirement option 1.

, in dollars, if you retire after

years of

B: Write a formula that gives your retirement payment service under retirement option 2.

, in dollars, if you retire after

years of

C: Make a graph that shows the retirement payment for each option over the 40 years of service. D: How many years of service will result in the same payment from either option? (You will work more than 1 year.) ANS: A: B:

1000000 900000 800000 700000 600000 500000 400000 300000 200000 100000 –10 –5

D: 37.05 years PTS: 4

DIF: hard

7. Inflation in a certain country remains constant at 6% per year. That means prices increase by 6% each year. A: Explain why prices are an exponential function of time. B: Using

for initial price, write a formula that gives the price

after

C: How long will it take prices to double? ANS: A: Prices change by a constant percentage, so they increase exponentially. B: C: 11.9 years PTS: 3

DIF: hard

years.

8. In a certain lake the intensity of light from the surface decreases by 59% for each additional meter of depth. A: Explain why light intensity is an exponential function of depth. B: Using for intensity at the surface of the lake, write a formula that gives the intensity light meters below the surface.

C: At what depth will light intensity be one tenth of surface intensity? ANS: A: Intensity changes by the same percentage for each additional meter of depth. B: C: 2.58 meters PTS: 3

DIF: hard

9. An overweight man goes on a diet. He initially weighs 277 pounds. He has set a target weight of 200 pounds. Each month the difference , in pounds, between his current weight and his target weight decreases by 5%. A: Find a formula that gives his weight difference

after

months on the diet.

B: If , in pounds, is his weight after months, then his weight difference is formula that gives his weight after months on the diet.

. Find a

C: When will he weigh 220 pounds? ANS: A: B: C: After 26.28 months PTS: 3

DIF: hard

10. The maximum length of a certain fish is 21 inches. Each year the difference , in inches, between its maximum length and its current length , in inches, decreases by 10%. The initial difference is 19 inches. A: Find a formula that gives the length difference B: The length difference is

years.

. Find a formula that gives its length after

C: When will the fish reach 17 inches long? ANS: A:

after

months.

B: C: After 14.79 years PTS: 3

DIF: hard

11. A bad investment loses 4% of its value each year. The initial investment was $2703. A: Find a formula that gives the value

, in dollars, of the investment after

years.

B: What is the value of the investment after 7 years? C: When will the investment decline to half its original value? ANS: A: B: $2031.16 C: After 16.98 years PTS: 3

DIF: hard

12. City A has an initial population of 2.9 thousand. The population grows by 5% each year. City B has an initial population of 6.3 thousand. The population declines by 7% per year.

A: Find a formula that gives the population

, in thousands, of city A after

years.

B: Find a formula that gives the population

, in thousands, of city B after

years.

C: Make a graph that shows both populations over the first 20 years. D: When will the populations of the two cities be the same, and what is that common population? ANS: A: B: C:

10 9 8 7 6 5 4 3 2 1 –4 –2

10 12 14 16 18 20 22

D: The populations are the same after 6.39 years. The common population is 3.96 thousand. PTS: 3

DIF: hard

13. On a certain island there are initially 4.4 new species per year introduced. This number decreases by 5% each year. But conditions cause some species to become extinct. Initially 1.7 species per year become extinct. The number increases by 14% each year. A: Using for new species per year, find a formula that gives the number of new species per year introduced after years. B: Using

for extinction rate, find a formula that gives the rate of extinction

C: Make a graph that shows both

and

years.

over the first 10 years.

D: When will the rate of extinction be the same as the rate new species are being introduced? ANS: A: B: C: 10 9 8 7 6 5 4 3 2 1

–4 –2

10 12 14 16 18 20 22

D: The rates are the same after 5.22 years. PTS: 3

DIF: hard

14. A certain bacteria population increases by 76% each day. A: Explain why the bacteria population is an exponential function of time. B: What is the hourly growth factor for the bacteria population? Report your answer correct to 3 decimal places. C: If there are initially

bacteria present, write a formula that gives the population

D: How long does it take for the bacteria population to double? ANS: A: The population is exponential because it shows constant percentage change. B: 1.024 C: D: The population doubles after 29.23 hours. PTS: 3

DIF: hard

after

hours.

Section 4.3 Modeling Exponential Data TRUE/FALSE 1. Evenly spaced data can be modeled by an exponential function if successive quotients are all the same. ANS: T

PTS: 1

DIF:

easy

2. Data that show a constant percentage change can be modeled by an exponential function. ANS: T

PTS: 1

DIF:

easy

3. Data that decrease at an increasing rate are not exponential data. ANS: T

PTS: 1

DIF:

easy

4. If data are exponential, then only the first data point can be used to determine the initial value. ANS: F

PTS: 1

DIF:

easy

5. If data are exponential, but the data are not measured in 1-unit increments, it is necessary to adjust the successive quotients in order to get the growth/decay factor. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Find an exponential model for the following data set. 0 3.06

1 5.51

a. b.

2 9.92

3 17.86

c. d.

ANS: A

PTS: 1

DIF:

medium

2. Find an exponential model for the following data set. 0 8.09

1 5.66

a. b.

2 3.96

3 2.77

c. d.

ANS: A

PTS: 1

DIF:

medium

3. Find an exponential model for the following data set. 0

3.07

3.38

a. b. ANS: B

3.72

4.09

c. d. PTS: 1

DIF:

medium

4. Find an exponential model for the following data set. 0 13.04

1 7.82

2 4.69

ANS: B

PTS: 1

DIF:

3 2.81

medium

5. Find an exponential model for the following data set. 0 3.06

1 4.59

a. b. ANS: C

2 6.89

3 10.34 c.

d. PTS: 1

DIF:

medium

6. Find an exponential model for the following data set. 0 36.02

1 21.61

a. b. ANS: C

2 12.97

3 7.78 c.

d. PTS: 1

DIF:

medium

7. Find an exponential model for the following data set. 0 3.07

2 6.02

a. b. ANS: A

4 11.8

6 23.13

c. d. PTS: 1

DIF:

medium

8. Find an exponential model for the following data set. 0 19.06

2 12.2

4 7.81

6 5

a. b. ANS: A

c. d. PTS: 1

DIF:

medium

9. Find an exponential model for the following data set. 0 3.04

2 5.14

4 8.69

a. b. ANS: D

6 14.69 c.

d. PTS: 1

DIF:

medium

10. Find an exponential model for the following data set. 0 27.06

2 18.64

4 12.84

a. b. ANS: D

6 8.85 c.

d. PTS: 1

DIF:

medium

11. Find an exponential model for the following data set. 0 30.05

3 40

6 53.24

a. b. ANS: A

9 70.86

c. d. PTS: 1

DIF:

medium

12. Find an exponential model for the following data set. 0 535.09

3 183.54

a. b. ANS: A

6 62.95

9 21.59

c. d. PTS: 1

DIF:

medium

13. Find an exponential model for the following data set. 0 3.05

3 8.37

6 22.97

9 63.03

a. b. ANS: B

c. d. PTS: 1

DIF:

medium

14. Find an exponential model for the following data set. 0 52.09

3 17.87

6 6.13

ANS: B

PTS: 1

DIF:

9 2.1

medium

15. Find an exponential model for the following data set. 0 3.09

3 12.66

6 51.86

ANS: A

PTS: 1

DIF:

9 212.42

medium

16. Find an exponential model for the following data set. 0 77.05

3 39.45

a. b. ANS: A

6 20.2

9 10.34

c. d. PTS: 1

DIF:

medium

17. Find an exponential model for the following data set. 0 61.07

2.13 48.79

a. b. ANS: A

4.26 38.98

6.39 31.14

c. d. PTS: 1

DIF:

medium

18. Find an exponential model for the following data set. 0 6.03 a.

2.13 8.58

4.26 12.21 c.

6.39 17.37

b. ANS: A

d. PTS: 1

DIF:

medium

19. The value , in dollars, of an investment years after 2010 is given in the following table. Use an exponential model for these data to determine when the value of the investment will be $19904.

0 $6622

1 $7085.54

a. 4.13 years after 2005 b. 3.39 years after 2005 ANS: D

2 $7581.53

3 $8112.24

c. 17.23 years after 2005 d. 16.27 years after 2005

PTS: 1

DIF:

medium

20. An animal population after years of growth is given in the following table. Use an exponential model for these data to determine the percentage growth of the population over any 4-year period. 0 1264

1 1352.48

a. 28.29% b. 31.08% ANS: B

2 1447.15

3 1548.45

c. 31.71% d. 28% PTS: 1

DIF: medium

21. The concentration , in milligrams per deciliter, of a drug in the bloodstream hours after injection is given by the following table. Use an exponential model for these data to determine the initial concentration of the drug in the bloodstream. 1 34.95

2 31.11

3 27.69

a. 34.97 milligrams per deciliter b. 39.27 milligrams per deciliter ANS: B

PTS: 1

4 24.64

c. 39.7 milligrams per deciliter d. 34.95 milligrams per deciliter DIF:

medium

22. The amount , in grams, of a radioactive substance remaining after years is given by the following table. Use an exponential model for these data to determine the half-life of the radioactive substance. That is, determine how long it takes for half of the substance to decay.

3 157.72

a. 25.18 years b. 35.04 years ANS: C

4 154.57

5 151.48

6 148.45

c. 34.31 years d. 25 years PTS: 1

DIF: medium

23. The average purchase price , in dollars, of a new home in a certain town years after 2010 is given by the following table. Use an exponential model for these data to determine the yearly percentage increase in home prices each year since 2010.

3 $158,685.02

4 $168,206.12

a. 106.8% b. 8%

5 $178,298.49

6 $188,996.4

c. 6% d. 106%

ANS: C

PTS: 1

DIF: medium

24. The average enrollment at a nearby school years after 2010 is given by the following table. Use an exponential model for these data to determine the yearly percentage decrease in enrollment each year since 2010. 0 802

1 705.76

a. 12% b. 15%

2 621.07

3 546.54

c. 88.88% d. 88% PTS: 1

ANS: A

DIF: medium

25. The average enrollment at a nearby school years after 2010 is given by the following table. Use an exponential model for these data to determine the yearly percentage decrease in enrollment each year since 2010. 0 1,172

1 1,043.08

a. 11% b. 13%

2 928.34

3 826.22

c. 89.76% d. 89%

ANS: A

PTS: 1

DIF: medium

26. Which of the following data plots may be modeled by an exponential function? y

ANS: C

PTS: 1

DIF:

medium

27. Which of the following data plots may be modeled by an exponential function? y

ANS: C

PTS: 1

DIF:

medium

SHORT ANSWER 1. The following table is generated using an exponential function. Fill in the missing values.

3.3 16.04

4.3 25.66

5.3

6.3

3.3 16.04

4.3 25.66

5.3 41.06

6.3 65.7

ANS:

PTS: 1

DIF: medium

2. The following table is generated using an exponential function. Fill in the missing values. 0 2.39

2 9.28

0 2.39

1 4.71

2 9.28

3 18.28

ANS:

PTS: 1

DIF: medium

3. The following table is generated using an exponential function. Fill in the missing values. 3.35 18.81

4.35

5.35

6.35 6.45

3.35 18.81

4.35 13.17

5.35 9.22

6.35 6.45

ANS:

PTS: 1

DIF: medium

4. The following table is generated using an exponential function. Fill in the missing values. 3.3 43.97

4.3

3.3 43.97

4.3 39.57

32.05

6.3 32.05

5.3 35.61

6.3 32.05

ANS:

PTS: 1

DIF: medium

5. Determine whether the following table can be modeled exactly by an exponential function. 3.62 17.29

4.62 12.1

5.62 8.47

ANS: The data can be modeled exactly by an exponential function.

6.62 5.93

PTS: 1

DIF: medium

6. Determine whether the following table can be modeled exactly by an exponential function. 3.69 68.15

4.69 122.67

5.69 220.81

6.69 397.46

ANS: The data can be modeled exactly by an exponential function. PTS: 1

DIF: medium

7. Determine whether the following table can be modeled exactly by an exponential function. 3.19 36.69

4.19 33.02

5.19 30.15

6.19 27.13

ANS: The data cannot be modeled exactly by an exponential function. PTS: 1

DIF: medium

8. Determine whether the following table can be modeled exactly by an exponential function. 3.72 56.18

4.72 106.74

5.72 204

6.72 387.59

ANS: The data cannot be modeled exactly by an exponential function. PTS: 1

DIF: medium

ESSAY 1. The following table shows the value 0 $3357

, in dollars, of an investment after

years.

1 $3524.85

3 $3886.14

2 $3701.09

A: Make a table of successive ratios to show that the value of the investment can be modeled by an exponential function. B: Find an exponential model for the data. C: Plot the graph of the data points along with the exponential model. Include up to 10 years. D: According to the model, when will the value of the account reach $8393? ANS: A: Time increment

From

Ratios of

B: C: V

6000 5000 4000 3000 2000 1000

–2 –1 –1000

D: After 18.78 years PTS: 4

DIF: hard

2. The following table shows the value 0 $301

, in dollars, of a stamp collection after 1 $313.04

2 $325.56

years.

3 $338.58

A: Make a table of successive ratios to show that the value of the stamp collection can be modeled by an exponential function. B: Find an exponential model for the data. C: Plot the graph of the data points along with the exponential model. Include up to 10 years. D: According to the model, what is the percentage increase in the value of the collection over a 6-year period? ANS: A: Time increment Ratios of

B: C:

From

V 675 600 525 450 375 300 225 150 75 –2 –75

–150 –225 –300

26.53%

PTS: 4

DIF: hard

3. The amount remaining , in grams, of a certain radioactive is measured at monthly intervals and recorded in the table below. 0 74

1 59.2

2 47.36

3 37.89

A: Make a table of successive ratios to show that the amount remaining of the radioactive substance can be modeled by an exponential function. B: Find an exponential model for the data. C: Plot the graph of the data points along with the exponential model. Include up to 10 months. D: According to the model, what is the half-life of this substance? That is, how long will it take for half the substance to decay? ANS: A: Time increment Ratios of

B: C:

From

A 90 80 70 60 50 40 30 20 10 –2 –1 –10

3.11 months

PTS: 4

DIF: hard

4. The concentration , in milligrams per deciliter, of a certain drug in the bloodstream injection is given in the following table. 0 40

1 32

2 25.6

hours after

3 20.48

A: Make a table of successive ratios to show that the concentration of the drug can be modeled by an exponential function. B: Find an exponential model for the data. C: Plot the graph of the data points along with the exponential model. Include up to 10 hours. D: According to the model, how long will it take for the drug concentration to decay to 25% of the initial amount? ANS: A: Time increment Ratios of

B: C:

From

C 45 40 35 30 25 20 15 10 5 –2 –1 –5

6.21 hours

PTS: 4

DIF: hard

5. The following table shows the hourly wage 0 $15.3

, in dollars, of a certain worker

2 $16.55

4 $17.9

years after 2000.

6 $19.36

A: Make a table of successive ratios to show that the wage can be modeled by an exponential function. B: Find an exponential model for the data. Be careful. The data are presented in 2-year increments. C: Plot the graph of the data points along with the exponential model. Include up through 2010. D: According to the model, the worker got a pay raise each year. What percentage raise did the worker get each year? ANS: A: Time increment Ratios of W

B: C:

From

W 30 25 20 15 10 5

–2 –1 –5

–10

PTS: 4

DIF: hard

6. As a certain animal grows, its length increases toward a maximum of 18 inches. At age years its length is measured in inches. The difference between the maximum length and the current length is shown in the table below. 0 13

1 11.57

2 10.3

A: Make a table of successive ratios to show that time.

3 9.17

can be modeled by an exponential function of

B: Find an exponential model for the data. C: Plot the graph of the data points along with the exponential model. Include up to 10 years. D: Find a formula that gives the length ANS: A: Time increment Ratios of

B: C:

From

after

years.

From

D 22 20 18 16 14 12 10 8 6 4 2 –2 –1 –2

D: PTS: 4

DIF: hard

7. The salinity , in parts per thousand of dissolved solids, of a tank is being raised to a target value of 84 parts per thousand. The table below shows the difference between the target salinity and the current salinity after hours. 0 59

2 39.67

4 26.67

6 17.93

A: Find an exponential model for the data. Be careful. The data are given in steps of 2 hours. B: Find a formula that gives the salinity C: Make a graph of target value.

versus

after

hours.

over the first 10 hours. Include the horizontal line representing the

E: According to the model, when does salinity reach 64 parts per thousand? ANS: A: B: C:

S 90 80 70 60 50 40 30 20 10 –2 –1 –10

D: After 5.45 hours PTS: 4

DIF: hard

8. When a skydiver jumps from an airplane his velocity , in feet per second, increases toward a maximum of 178 feet per second. (This is terminal velocity.) The table below shows the difference between the terminal velocity and the velocity after seconds. 0 178

5 99.4

10 55.51

15 31

A: Find an exponential model for the data. Be careful. The data are given in steps of 5 seconds. B: Find a formula that gives the velocity C: Make a graph of V versus terminal velocity.

after

seconds.

over the first 10 seconds. Include the horizontal line representing the

E: According to the model, how long does it take for the skydiver to reach 88% of terminal velocity? ANS: A: B: C:

V 200 180 160 140 120 100 80 60 40 20 –2 –1 –20

D: 18.19 seconds PTS: 4

DIF: hard

9. A water-filled ice tray is placed in a freezer that maintains a constant temperature of 5 degrees Fahrenheit. The table below shows the difference between the temperature T of the water in the ice tray, in degrees Fahrenheit, and the temperature the freezer.

0 74

5 54.31

10 39.86

15 29.25

A: Find an exponential model for the data. Be careful. The data are given in steps of 5 seconds. B: Find a formula that gives the temperature

after

minutes.

C: Make a graph of T versus over the first 30 minutes. Include the horizontal line representing the freezing temperature of water, 32 degrees. E: According to the model, how long does it take for the water to reach 32 degrees? ANS: A: B: C:

T 90 80 70 60 50 40 30 20 10

–10

–20

16.29 minutes

PTS: 4

DIF: hard

10. A lizard is growing toward a maximum length of 7 inches. The table below shows the difference between the maximum length and the length , in inches, at age years.

1 4.6

2 4.23

3 3.89

4 3.58

A: Find an exponential model for the data. B: Find a formula that gives the length C: Make a graph of maximum length.

versus

after

years.

over the first 15 years. Include the horizontal line representing the

E: According to the model, how long was the lizard at birth? ANS: A: B: C:

L 9 8 7 6 5 4 3 2 1

–1

2 inches

PTS: 4

DIF: hard

Section 4.4 Modeling Nearly Exponential Data TRUE/FALSE 1. If linear regression does not provide the proper fit for data, then exponential regression is always appropriate. ANS: F

PTS: 1

DIF:

easy

2. Exponential regression always provides an exact fit for data. ANS: F

PTS: 1

DIF:

easy

3. For data that are approximately exponential, exponential regression can be used to find the approximate growth rate. ANS: T

PTS: 1

DIF:

easy

4. Data that are decreasing at an increasing rate should not be modeled by an exponential function. ANS: T

PTS: 1

DIF:

easy

5. Exponential regression should never be used if data do not exhibit constant percentage change. ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Use exponential regression to provide an approximate model for the following data. 1.97 3.54

4.89 8.01

a. b.

7.62 16.24

10.06 30.7

12.24 54.95

9.11 23.61

11.13 40.62

c. d.

ANS: A

PTS: 1

DIF:

medium

2. Use exponential regression to estimate the value of 1.41 3.15

3.79 6.19

a. 144.28 b. 144.81

6.43 11.69

c. 23.83 d. 24.69

ANS: A

PTS: 1

DIF: medium

3. Use exponential regression to determine what happens to 1.08

3.85

6.8

when t is increased by 1 unit. 9.46

11.84

2.95

6.42

a. is multiplied by 2.18. b. 2.18 is added to . ANS: D

13.48

28.37

54.43

c. 1.31 is added to . d. is multiplied by 1.31.

PTS: 1

DIF:

medium

4. Use exponential regression to provide an approximate model for the following data. 1.61 17.49

3.92 10.89

6.68 6.75

ANS: B

PTS: 1

DIF:

3.38 16.45

a. 9.13 b. 10

8.88 7.8

11.47 6.09

6.14 11.82 c. 11 d. 12.52

ANS: B

PTS: 1

DIF:

medium

6. Use exponential regression to determine what happens to 1.1 21.83 a. b.

11.38 3.01

medium

5. Use exponential regression to provide an estimate for 1.21 20.59

9 4.27

3.64 15.97

is multiplied by 0.89. is multiplied by 0.73.

ANS: A

when 1 is added to t.

5.66 12.56

8.21 9.06

11.13 6.54

c. 0.73 is added to . d. 0.89 is added to .

PTS: 1

DIF:

medium

7. Use exponential regression to provide an approximate model for the following data. 1.33 2.85

4.71 7.03

a. b.

7.74 15.82

11.05 36.47

14.92 100.62

12.13 19.18

15.33 32.8

c. d.

ANS: B

PTS: 1

DIF:

medium

8. Use exponential regression to estimate the value of 1.75 3.44

5.36 5.35

8.78 10.79

a. 4.43 b. 4.4

c. 3.67 d. 4.83

ANS: C

PTS: 1

DIF: medium

9. Use exponential regression to determine what happens to 1.96 7.37

5.91 89.26

a. 1.89 is added to b. 12.11 is added to ANS: C

9.61 954.71 c. d.

PTS: 1

when 1 is added to t.

DIF:

12.71 6982.54

16.03 58809.97

is multiplied by 1.89 is multiplied by 12.11 medium

10. The value , in dollars, of a certain investment after years is given in the table below. Use exponential regression to determine the yearly percentage growth rate for the investment. 2 2249.28

4 2480.59

a. 105.63% b. 105% ANS: D

6 2734.31

8 3014.15

10 3323.64

c. 6.47% d. 5% PTS: 1

DIF: medium

11. The value , in dollars, of a certain investment after years is given in the table below. Use exponential regression to determine the percentage increase in the investment over any 4-year period. 2 3540.85

4 5710.95

a. 109.43% b. 108% ANS: D

6 9210.67

8 14854.83

10 23959.74

c. 161.75% d. 160.14% PTS: 1

DIF:

medium

12. The value , in dollars, of a certain investment after years is given in the table below. Use exponential regression to model the value as a function of time. 2 2404.02

4 2712.19

a. b. ANS: C

6 3043.89

8 3412.91

10 3796.08

c. d. PTS: 1

DIF:

medium

13. The value , in dollars, of a certain investment after years is given in the table below. Use exponential regression to determine the yearly percentage decay rate for the investment. 1 2022.16

4 1157.06

7 661.92

10 378.02

13 216.12

a. 84.71% b. 83%

c. 17% d. 17.79%

ANS: C

DIF: medium

PTS: 1

14. The value , in dollars, of a certain investment after years is given in the table below. Use exponential regression to determine the percentage decrease in the value of the investment over any 4-year period. 1 1968.51

4 1085.4

a. 73.25% b. 72%

7 599.25

10 330.72

13 182.41

c. 54.79% d. 57.16%

ANS: C

PTS: 1

DIF: medium

15. The size , in thousands, of a certain animal population after years is given in the table below. By what percentage can the population be expected to grow over any 9-year period? 0 5.42

1 5.07

a. 54.62% b. 54%

2 5.12

3 6.17

4 6.47

c. 68.95% d. 70.46%

ANS: C

PTS: 1

DIF: medium

16. The size , in thousands, of a certain animal population after years is given in the table below. Use exponential regression to find the yearly percentage growth rate of the population. 0 3.76

1 4.4

a. 10.01% b. 9.54%

2 4.35

3 4.7

4 5.5

c. 9% d. 7.06%

ANS: C

PTS: 1

DIF: medium

17. The size , in thousands, of a certain animal population after years is given in the table below. Use exponential regression to model the population size as a function of time. 0 6.56

1 8.17

a. b. ANS: A

2 10.62

3 14.63

4 19.53

c. d. PTS: 1

DIF:

medium

18. The population , in thousands, of a small town after years is given in the table below. Use exponential regression to model the population as a function of time.

2 8.34

4 5.52

a. b.

6 4.17

8 3.41

10 2.24

c. d.

ANS: D

PTS: 1

DIF:

medium

19. The population , in thousands, of a small town after years is given in the table below. Use exponential regression to find the yearly percentage decay rate of the population. 2 8.46

4 5.71

a. 15% b. 13.46%

6 4.7

8 3.62

10 2.13

c. 16.18% d. 18.08%

ANS: A

PTS: 1

DIF: medium

20. The population , in thousands, of a small town after years is given in the table below. Use exponential regression to find the percentage decrease over any 5-year period. 2 8.57

4 6.97

a. 50.16% b. 49.1%

6 4.83

8 4.13

10 2.92

c. 65% d. 65.73%

ANS: A

PTS: 1

DIF: medium

21. The following table shows the number , in thousands, of newspaper subscriptions years after initial publication. Use exponential regression to determine when the number of subscriptions can be expected to reach 125.93 thousand. 0 5.44

3 10.65

a. 12.17 years after initial publication b. 14.8 years after initial publication ANS: A

PTS: 1

6 24.32

9 53.51

12 120.07

c. 16.08 years after initial publication d. 14.05 years after initial publication DIF:

medium

22. The following table shows the number , in thousands, of newspaper subscriptions years after initial publication. Use exponential regression to model the number of subscriptions as a function of time. 0 5.36

3 11.17

a. b. ANS: B

6 26.78 c. d.

PTS: 1

DIF:

medium

9 61.81

12 144.79

23. The following table shows the number , in thousands, of newspaper subscriptions years after initial publication. Use exponential regression to find the yearly percentage growth in the number of subscriptions. 0 5.3

3 11.07

a. 28.36% b. 30%

6 23.88

9 54.17

12 121.03

c. 31.36% d. 31.27%

ANS: B

PTS: 1

DIF: medium

24. The following table shows the number , in thousands, of newspaper subscriptions years after initial publication. Use exponential regression to find the percentage growth rate in the number of subscriptions over any 4-year period. 0 6.06

3 13.53

a. 193.87% b. 194.5%

6 30.4

9 69.36

12 159.84

c. 124.76% d. 124%

ANS: B

PTS: 1

DIF: medium

25. The following table shows the number , in thousands, of magazine subscriptions years after initial publication. Use exponential regression to determine when the number of subscriptions can be expected to reach 0.5825 thousand. 1.67 9.47

5.43 5.43

a. 28.72 years after initial publication b. 27.19 years after initial publication ANS: B

PTS: 1

9.01 3.65

12.78 2.17

16.17 2.13

c. 28.95 years after initial publication d. 27.63 years after initial publication DIF:

medium

26. The following table shows the number , in thousands, of magazine subscriptions years after initial publication. Use exponential regression to model the number of subscriptions as a function of time. 1.84 8.97

5.53 5.58

a. b.

9.35 3.47

12.9 2.31

16.12 2.16

c. d.

ANS: C

PTS: 1

DIF:

medium

27. The following table shows the number , in thousands, of magazine subscriptions years after initial publication. Use exponential regression to find the yearly percentage decay rate for the number of subscriptions. 1.82 8.7

5.35 5.57

8.57 3.58

11.72 3.11

14.75 1.78

a. 12.96% b. 13.29%

c. 11% d. 9.17%

ANS: C

DIF: medium

PTS: 1

28. The following table shows the number , in thousands, of magazine subscriptions years after initial publication. Use exponential regression to find the percentage decline in subscriptions over any 5-year period. 1.54 6.39

4.56 4.24

a. 65% b. 66.94%

7.66 2.2

11.6 0.96

15.18 1.28

c. 50.16% d. 50.09%

ANS: C

PTS: 1

DIF: medium

29. A cleaning process is removing salt from a pool. The amount , in pounds, of salt remaining after hours is given in the following table. Use exponential regression to determine the amount of salt remaining after 19.44 hours. 1.35 21.93

4.99 12.89

a. 3.61 pounds b. 1.44 pounds ANS: B

8.48 7.62

12.02 4.81

15.44 2.65

c. 5.22 pounds d. 3.3 pounds PTS: 1

DIF:

medium

30. A cleaning process is removing salt from a pool. The amount , in pounds, of salt remaining after hours is given in the following table. Use exponential regression to find the hourly percentage decay factor. 1.77 21.78

5.47 11.84

a. 16.43% b. 16%

9.19 6.03

13.1 3.26

16.92 1.61

c. 17.4% d. 14.95% PTS: 1

ANS: B

DIF: medium

31. A cleaning process is removing salt from a pool. The amount , in pounds, of salt remaining after hours is given in the following table. Use exponential regression to find the percentage decrease in salt over any 5-year period. 1.23 23.16

4.88 14.36

a. 60% b. 47.23% ANS: B

8.2 9.34 c. 60.82% d. 45.63%

PTS: 1

DIF: medium

11.59 6.05

15.13 3.83

32. A cleaning process is removing salt from a pool. The amount , in pounds, of salt remaining after hours is given in the following table. Use exponential regression to model the amount of salt remaining as a function of time. 1.73 22.21

4.89 15.18

8.34 10.4

a. b.

11.95 6.11

15.17 4.04

c. d.

ANS: A

PTS: 1

DIF:

medium

33. The population , in thousands, of a small town after years is given in the following table. Use exponential regression to determine the population after 17.98 years. 1.08 22.93

4.84 13.42

8.79 7.46

a. 5.88 thousand b. 3.96 thousand ANS: D

12.61 4.45

15.98 3.1

c. 6.49 thousand d. 2.11 thousand PTS: 1

DIF:

medium

SHORT ANSWER 1. Use exponential regression to get an approximate model for the following data points. Then plot the data points along with the exponential model. 0 2.33

2.52 3.2

4.89 7.42

7.34 10.15

10.01 13.04

ANS:

y 45 40 35 30 25 20 15 10 5 –4

–2

–5

–10

PTS: 1

DIF: medium

2. Use exponential regression to get an approximate model for the following data. Then plot the data points along with the exponential model. 0 43.67

2.33 28.23

4.42 24.08

6.45 19.49

9.03 11.8

ANS:

y 45 40 35 30 25 20 15 10 5 –4

–2

–5

–10

PTS: 1

DIF: medium

ESSAY 1. The population following table.

, in millions, of insects in a certain area after

3 9.87

5 7.04

7 11.41

years of observation is given in the

9 11.31

11 13.73

A: Use exponential regression to model the insect population as a function of time. B: Plot the exponential model along with the data points. C: According to the exponential model, when is the insect population expected to reach 18.98 million? ANS: A: B:

N 24 21 18 15 12 9 6 3 –4

–2

–3

–6

C: After 17.09 years PTS: 3

DIF: hard

2. The table below shows the number 0 212.9

, in millions, of cell phone subscribers

1 242.17

2 252.41

3 277.33

years after 2005. 4 289.64

A: Use exponential regression to model the number of subscribers as a function of time. B: Plot the exponential model along with the data points. C: According to the exponential model, what is the yearly percentage increase in cell phone subscribers? ANS: A: B: C 400 350 300 250 200 150 100 50

–1

C: 8%

PTS: 3

DIF: hard

3. The table below shows the cost 19 2

, in cents, of domestic first-class postage 58 4

78 15

A: Use exponential regression to model

years after 1900.

85 22

102 37

as a function of t.

B: Plot the exponential model along with the data points. C: According to the exponential model, what is the percentage increase in postal rates each decade? ANS: A: B: N 60

–40

–20

100

120

–15

–30

C: 48.02% PTS: 3

DIF: hard

4. The table below shows the value 0 33

, in dollars, of a certain rare coin

30 410

A: Use exponential regression to model

45 652

years since 1950.

54 1894

as a function of t.

B: Plot the exponential model along with the data points. C: According to the exponential model, what is the value of the coin in 2015? ANS:

60 2383

A: B: N

2400 2000 1600 1200 800 400

–10 –400

C: $2943.7 PTS: 3

DIF: hard

5. The table below shows nation health care costs 0 84

10 270

, in billions of dollars, 20 728

30 1375

years after 1970. 40 2633

A: Use exponential regression to model H as a function of t. B: Plot the exponential model along with the data points. C: According to the exponential model, when will national health care costs reach 4332 billion dollars? Report your answer as a year rounded to the nearest whole number. ANS: A: B:

I 5000 4000 3000 2000 1000

–5 –1000

–2000 –3000

C: 2013 PTS: 3

DIF: hard

6. The table below shows the atmospheric pressure kilometers. h P

5 587

10 316

, in grams per square centimeter, at an altitude of

15 179

20 105

25 58

A: Use exponential regression to model atmospheric pressure as a function of altitude. B: Plot the exponential model along with the data points. C: According to the exponential model, what is the percentage decrease in atmospheric pressure for each one-kilometer increase in altitude? ANS: A: B: P 675 600 525 450 375 300 225 150 75 –5 –75

C: 11% PTS: 3

DIF: hard

7. The table below shows the pressure , in dynes per square centimeter, exerted on the ear by a sound with loudness measured in decibels. 65 0.36

85 5.6

90 8.4

105 31

110 53

A: Use exponential regression to model pressure as a function of loudness. Round the initial value of the model to four decimal places. B: Plot the exponential model along with the data points. C: According to the exponential model, how is pressure on the ear affected when loudness increases by 1 decibel? Report your answer as a percentage. ANS: A: B: P

60 50 40 30 20 10

–20 –10

100

120

C: When loudness increases by 1 decibel, pressure increases by 11%. PTS: 3

DIF: hard

8. The table below shows the number of persons injured per 100 accident-involved vehicles when the accident speed was miles per hour. 20 26

30 33

A: Use exponential regression to model

40 38

as a function of

50 46

60 70

B: Plot the exponential model along with the data points. C: Do the data show that accidents at speeds of 60 miles per hour are more dangerous or less dangerous than predicted by the model? ANS: A: B: N 90 75 60 45 30 15

–10 –15

–30

C: They are more dangerous. PTS: 3

DIF: hard

9. A small dog will weigh 23 pounds at adulthood. The following table shows the weight , in pounds, months after birth. The difference between his maximum weight and his current weight is given in the third row of the table.

3 8 15

5 12.23 10.77

7 14.4 8.6

A: Use exponential regression to model the difference

9 17.89 5.11

11 17.66 5.34

as a function of t.

B: Use the model you found in part A to find a formula that gives the approximate weight months. C: Plot the graph of

versus

along with the data points for weight.

D: At what age does the dog weigh 19.9 pounds? ANS: A:

at age

B: C: W 24 21 18 15 12 9 6 3 –4

–2

–3

–6

D: 14.08 months PTS: 3

DIF: hard

10. The maximum length of a certain snake is 24 inches. The following table shows the length , in inches, of the snake years after birth. The difference between its maximum length and its current length is given in the third row of the table.

2 6.23 17.77

3 10.22 13.78

4 11.25 12.75

A: Use exponential regression to model the difference

5 12.87 11.13

6 15.27 8.73

as a function of age.

B: Use the model you found in part A to find a formula that gives the approximate length years. C: Plot the graph of L versus

along with the data points for length.

D: What is the age of a snake that is 13.91 inches long? ANS: A: B: C:

at age

L 24 21 18 15 12 9 6 3

–3 –6

D: 5.33 years PTS: 3

DIF: hard

Section 4.5 Logarithmic Functions TRUE/FALSE 1. The Richter scale is a logarithmic scale. ANS: T

PTS: 1

DIF:

easy

2. The graph of the logarithm is concave down. ANS: T

PTS: 1

DIF:

easy

3. The logarithm function is always less than 100. ANS: F

PTS: 1

DIF:

easy

DIF:

easy

4. The decibel scale is a logarithmic scale. ANS: T

PTS: 1

5. The logarithm is the inverse of an exponential function. ANS: T

PTS: 1

DIF:

easy

6. The inverse of a logarithm is an exponential function. ANS: T 7. Multiplying

PTS: 1 by

ANS: T 8.

is the power of ANS: T

adds

PTS: 1

DIF:

easy

DIF:

easy

DIF:

easy

that gives . PTS: 1

9. The graph of the logarithm has a point of inflection. ANS: F

PTS: 1

DIF:

easy

10. The crossing graphs method is not useful for solving equations involving logarithms. ANS: F

PTS: 1

DIF:

easy

11. The magnitude scale for brightness of stars is a logarithmic scale. ANS: T MULTIPLE CHOICE

PTS: 1

DIF:

easy

1. The magnitude of earthquake 1 is 6.3. If earthquake 2 is 1000 times as intense as earthquake 1, then the magnitude of earthquake 2 is a. 9.3 b. 3.3 ANS: A

c. 1006.3 d. 6300 PTS: 1

DIF:

medium

2. The magnitude of earthquake 1 is 5.6. If earthquake 2 is 81.28 times as intense as earthquake 1, find the magnitude of earthquake 2. Round your answer to 1 decimal place. a. 7.5 b. 3.7 ANS: A

c. 86.9 d. 8.2 PTS: 1

DIF: medium

3. The magnitude of earthquake 1 is 4.4, and the magnitude of earthquake 2 is 6.4. How do their intensities compare? a. b. c. d.

Earthquake 1 is 100 times as intense as earthquake 2. Earthquake 1 is 2 times as intense as earthquake 2. Earthquake 2 is 2 times as intense as earthquake 1. Earthquake 2 is 100 times as intense as earthquake 1.

ANS: D

PTS: 1

DIF:

medium

4. The magnitude of earthquake 1 is 3.7, and the magnitude of earthquake 2 is 5.6. How do their intensities compare? a. b. c. d.

Earthquake 1 is 79.43 times as intense as earthquake 2. Earthquake 1 is 1.9 times as intense as earthquake 2. Earthquake 2 is 1.9 times as intense as earthquake 1. Earthquake 2 is 79.43 times as intense as earthquake 1.

ANS: D

PTS: 1

DIF:

medium

5. The magnitude of earthquake 1 is 8.4. Earthquake 2 is half as intense as earthquake 1. What is the magnitude of earthquake 2? Round your answer to 1 decimal place. a. 7.9 b. 8.1 ANS: B 6. The magnitude relationship is

c. 4.2 d. 4.7 PTS: 1

DIF: medium

of an earthquake depends on the relative intensity

. Express the relative intensity as a function of the magnitude. a. b. ANS: A

c. d. PTS: 1

DIF:

medium

of the quake. The

7. If

is 3.14, then the value of

a. 6.28 b. 7.05

c. 5.14 d. 5.75

ANS: C 8. If

PTS: 1

DIF:

is 8.83, then the value of

a. 0.34 b. 5.83 PTS: 1

10. If

is 7.52, then the value of

medium

is c. 12.61 d. 9.88

PTS: 1

DIF:

is 2.23, then the value of

a. 8.61 b. 7.08 ANS: D

DIF:

a. 12.03 b. 9.12 ANS: B

medium

c. 1.32 d. 5.85

ANS: B 9. If

medium

is c. 9.55 d. 6.09

PTS: 1

DIF:

medium

11. The decibel, a measure of loudness, is related to the relative intensity of sound by the formula Decibels

Relative intensity).

If relative intensity is multiplied by 100, how are decibels affected? a. Decibels are increased by 2. b. Decibels are decreased by 2. ANS: C

c. Decibels are increased by 20. d. Decibels are decreased by 20.

PTS: 1

DIF:

medium

12. The decibel, a measure of loudness, is related to the relative intensity of sound by the formula Decibels

Relative intensity).

If relative intensity is divided by 100, how are decibels affected? a. Decibels are increased by 2. b. Decibels are decreased by 2. ANS: D

c. Decibels are increased by 20. d. Decibels are decreased by 20.

PTS: 1

DIF:

medium

13. The decibel, a measure of loudness, is related to the relative intensity of sound by the formula Decibels

Relative intensity).

If relative intensity is multiplied by 190.55, how are decibels affected? a. Decibels are increased by 22.8. b. Decibels are decreased by 2.28. ANS: A

PTS: 1

c. Decibels are increased by 2.28. d. Decibels are decreased by 22.8. DIF:

medium

14. The decibel, a measure of loudness, is related to the relative intensity of sound by the formula Decibels

Relative intensity).

A whisper in a library is about 31 decibels. What decibel level is produced by 12 people whispering in the library? Round your answer to the nearest whole number. a. 372 decibels b. 42 decibels ANS: B 15. The decibel level formula

c. 374 decibels d. 44 decibels PTS: 1

DIF:

medium

, a measure of loudness, is related to the relative intensity

of sound by the

. Express relative intensity as a function of decibels. a.

b. ANS: D

d. PTS: 1

DIF:

medium

16. The magnitude of a star is a measure of its brightness, and it is related to the relative intensity of light reaching Earth from the star. The relationship is Magnitude

Relative intensity).

If relative intensity is multiplied by 100, how is magnitude affected? a. Magnitude is decreased by 2. b. Magnitude is increased by 5. ANS: B

PTS: 1

c. Magnitude is increased by 2. d. Magnitude is decreased by 5. DIF:

medium

17. The magnitude of a star is a measure of its brightness, and it is related to the relative intensity of light reaching Earth from the star. The relationship is Magnitude

Relative intensity).

If relative intensity is divided by 100, how is magnitude affected? a. Magnitude is decreased by 2. b. Magnitude is increased by 5.

c. Magnitude is increased by 2. d. Magnitude is decreased by 5.

ANS: D

PTS: 1

DIF:

medium

18. The magnitude of a star is a measure of its brightness, and it is related to the relative intensity of light reaching Earth from the star. The relationship is Magnitude

Relative intensity).

If relative intensity is multiplied by 275.42. how is magnitude affected? a. Magnitude is decreased by 2.44. b. Magnitude is increased by 6.1. ANS: B 19.

PTS: 1

c. Magnitude is increased by 2.44. d. Magnitude is decreased by 6.1. DIF:

medium

The magnitude of a star is a measure of its brightness, and it is related to the relative intensity of light reaching Earth from the star. The relationship is . Express relative intensity as a function of magnitude. a.

ANS: C

PTS: 1

20. The solution for

of the equation

DIF:

medium

is: a.

ANS: B

PTS: 1

DIF:

medium

21. We use ln to denote the natural logarithm. The solution for is: a.

ANS: B

PTS: 1

22. The solution for

of the equation

DIF:

medium

of the equation

is: a.

ANS: B

PTS: 1

DIF:

medium

23. We use ln to denote the natural logarithm. The solution for

of the equation

is: a.

ANS: B

PTS: 1

DIF:

medium

24. We use ln to denote the natural logarithm. What is the value of a. 5 b. 2.17 ANS: A

c. 148.41 d. 2.85 PTS: 1

DIF:

medium

25. We use ln to denote the natural logarithm. What is the value of a. 3.72 b. 1.62 ANS: A

c. 41.26 d. 1.75 PTS: 1

DIF: medium

26. The graph of the logarithm is a. Decreasing and concave down. b. Decreasing and concave up. ANS: D 27. The graph of

PTS: 1 is

c. Increasing and concave up. d. Increasing and concave down. DIF:

medium

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

–5 –0.2

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

ANS: D

PTS: 1

28. The inverse of

PTS: 1

–5 –0.2

DIF:

medium

DIF:

medium

a. b.

c. d.

ANS: C

PTS: 1

DIF:

medium

a. b.

c. d.

ANS: D 31. The inverse of

c. d.

ANS: A

30. The inverse of

a. b.

29. The inverse of

PTS: 1 is

DIF:

1.8

–5 –0.2

medium

a. b. ANS: B

c. d. PTS: 1

DIF:

medium

ESSAY 1. Quake 1 has a reading of 4.5 on the Richter scale. A: Quake 2 has a magnitude of 6.1. How do the relative intensities of quakes 1 and 2 compare? B: Quake 3 is 398.11 times as intense as quake 1. What is the magnitude of quake 3? Round your answer to one decimal place. C: An aftershock following quake 1 was one tenth as intense as quake 1. What was the magnitude of the aftershock? ANS: A: Quake 2 is 39.81 times as intense as quake 1. B: The magnitude of quake 3 is 7.1. C: The magnitude of the aftershock is 3.5. PTS: 3

DIF: hard

2. The magnitude of an earthquake is related to the energy The relationship is given by the following equation:

, in terajoules, released by the quake.

A: What is the magnitude of a quake that releases 6.37 terajoules of energy? Round your answer to one decimal place. B: Make a graph of magnitude versus energy. Include up to 10 terajoules. C: What is the energy released by an earthquake of magnitude 4.66? ANS: A: 5.5 B:

6 5 4 3 2 1

–2 –1 –1

C: 0.31 terajoules PTS: 3

DIF: hard

3. The gross tonnage is a measure of a ship’s capacity. It has no units. It is calculated in terms of the volume , in cubic meters, of the ship. The relationship is . A: What is the gross tonnage of a ship with volume 10,722 cubic meters? B: Make a graph of gross tonnage versus volume. Include volumes up to 20,000 cubic meters. C: What is the volume of a ship that has a gross tonnage of 4000? ANS: A: 3008.65 B: G

6000 5000 4000 3000 2000 1000

–5000 –1000

5000

10000

C: 14,134.01 cubic meters

15000

20000

PTS: 3

DIF: hard

4. An animal grows according to the formula . Here

is the length of the animal in feet, and

is the age in years.

A: What is the length of a 6.31-year-old animal? B: Make a graph of length versus age. Include ages up to 20 years. C: Explain in practical terms what the concavity of the graph from part B means. ANS: A: 1.22 feet B: L

1.5

0.5

–4 –2

10 12 14 16 18 20

C: Length increases at a decreasing rate. That is, younger animals grow faster than older animals. PTS: 3

DIF: hard

5. The spectroscopic parallax of a star is the difference between the apparent magnitude, which is the brightness of light from the star reaching Earth, and the absolute magnitude, which is the intrinsic brightness. The spectroscopic parallax can be used to calculate the distance , in parsecs, from Earth to the star. The relationship is .

A: Find the spectroscopic parallax of a star that is 10.08 parsecs from Earth. B: Find the distance from Earth of a star that has a spectroscopic parallax of 4. C: Star A is 5 times as far from Earth as star B. How do their spectroscopic parallaxes compare?

ANS: A: 0.02 B: 63.1 parsecs C: The spectroscopic parallax of star A is 3.49 more than that of star B. PTS: 3

DIF: hard

6. The decibel level , a measure of loudness of sound, can be calculated from the relative intensity of sound. The relationship is .

A: If one sound is 1,000 times as intense as another, how do their decibel ratings compare? B: Is the graph of practical terms.

versus

concave up or concave down? Explain what the concavity means in

C: Solve the equation above to express relative intensity as a function of decibels. ANS: A: The louder sound is 30 decibels more than the quieter sound. B: The graph is concave down. Decibels increase at a decreasing rate, which means that a small change in intensity makes a greater contribution to the decibel rating for quieter sounds than for louder sounds. C: PTS: 3

DIF: hard

7. The magnitude of a star is related to the relative intensity The relationship is

of light reaching Earth from the star.

A: Star A has a relative intensity 9.02 times that of star B. How do their magnitudes compare? B: Make a graph of

versus . Include relative intensities up to 100.

C: If the magnitude of a star is 4, what is the relative intensity of light reaching Earth from the star? ANS: A: The magnitude of star A is 2.39 more than that of star B.

M 5

–20 –10

10 20 30 40 50 60 70 80 90 100

–1

C: 39.81 PTS: 3

DIF: hard

8. For certain decisions, the time it takes to respond is a function of the number of choices faced. The relationship is

where

is reaction time in seconds, and

is the number of choices.

A: Express using functional notation the reaction time if there are 6 choices. Then calculate that value. B: Make a graph of

versus

. Include up to 10 choices.

C: Is the graph concave up or concave down? Explain in practical terms what the concavity of the graph means. ANS: A: In functional notation the reaction time is

B: R 0.7 0.6 0.5 0.4 0.3 0.2 0.1

–1 –0.1

10 11

. The value is 0.5 seconds.

C: The graph is concave down. Additional choices make more difference in reaction time when there are fewer choices available. PTS: 3

DIF: hard

Section 5.1 Logistic Functions TRUE/FALSE 1. A logistic function always has a limiting value. ANS: T

PTS: 1

DIF:

easy

2. A logistic population model shows the most rapid population growth at the inflection point. ANS: T

PTS: 1

DIF:

easy

3. For a logistic function, the point of inflection and the limiting value are the same. ANS: F

PTS: 1

DIF:

easy

4. The theory of maximum sustainable yield says that a renewable resource that grows logistically should be harvested at the carrying capacity. ANS: F

PTS: 1

DIF:

easy

5. For a logistic function, the function shows a maximum growth rate at half the limiting value. ANS: T

PTS: 1

DIF:

easy

6. Logistic regression is appropriate for all data sets that are neither exponential nor linear. ANS: F

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. A population is modeled by the function . The limiting value of the population is a. 2821 b. 1410.5 ANS: A

c. 564.2 d. 4 PTS: 1

2. Enrollment at a certain university

DIF:

medium

years after 2000 is given by .

According to the model, what is the maximum enrollment that can be attained? a. 11488 b. 22976

c. 3282.29 d. 23125

ANS: B 3. The number function

PTS: 1

DIF:

medium

, in millions, of Americans with Internet access

years after 1990 is modeled by the

. According to this model, what is the maximum number of Americans who will ever have Internet access? a. 195.1 million b. 97.55 million ANS: A

c. 2.08 million d. 93 million PTS: 1

4. Magazine circulation after

DIF:

medium

years is modeled by the function .

The largest magazine circulation that can be expected is a. 11.5 thousand b. 23 thousand ANS: B

c. 3.83 thousand d. 5 thousand PTS: 1

5. The cumulative number

DIF:

medium

of flu cases reported after

weeks is modeled by the function .

The maximum number of people who can be expected to get the flu is a. 1419.5 b. 2839 ANS: B

c. 101.39 d. 27 PTS: 1

DIF: medium

6. A population is modeled by the function . For what value of a. b.

is the population growing at the fastest rate?

2800 1400

ANS: B

PTS: 1

7. Enrollment at a certain university

c. d.

400 6

DIF:

medium

years after 2000 is given by

. At what enrollment level is enrollment growing at the fastest rate? a. b.

7744 3872

ANS: B 8. The number function

PTS: 1

c. d.

1936 3901

DIF:

medium

, in millions, of Americans with Internet access

years after 1990 is modeled by the

. For what value of a. b.

is Internet access growing at the fastest rate?

188 94

ANS: B

PTS: 1

9. The cumulative number

c. d.

1.92 97

DIF:

medium

of flu cases reported after

weeks is modeled by the function .

For what cumulative number of flu cases is the disease spreading at the fastest rate? a. b.

2069 1034.5

ANS: B

c. d. PTS: 1

DIF:

66.74 30 medium

10. A population is modeled by the function . What is the initial value of the population? a. b.

5933 2966.5

ANS: C

c. d. PTS: 1

11. Enrollment at a certain university

DIF:

847.57 6 medium

years after 2000 is given by .

What is the enrollment in 2000?

a. b.

5470 2735

ANS: C 12. The number function

PTS: 1

c. d.

1094 0

DIF:

medium

, in millions, of Americans with Internet access

years after 1990 is modeled by the

. According to the model, how many Americans had Internet access in 1990? a. 187 million b. 93.5 million ANS: D

c. 98 million d. 1.89 million PTS: 1

13. The cumulative number

DIF: medium

of flu cases reported after

weeks is modeled by the function

. How many people had the flu at the start of the epidemic? Round your answer to the nearest whole number. a. b.

5736 2868

ANS: C

PTS: 1

14. The cumulative number

c. d.

143 39

DIF:

medium

of flu cases reported after

weeks is modeled by the function .

For what value of

will the cumulative number of flu cases be 1254?

a. b. ANS: B

c. d. PTS: 1

DIF:

medium

15. A population is modeled by the function . For what value of a. b.

will the population reach 3900? c. d.

ANS: B

PTS: 1

16. Enrollment at a certain university

DIF:

medium

years after calendar year 2000 is given by .

According to the model, for which calendar year will enrollment reach 25512? Round your answer to the nearest whole number. a. b.

c. d.

ANS: B

PTS: 1

DIF:

medium

17. The number , in millions, of Americans with Internet access modeled by the function

years after calendar year 1990 is

. For what calendar year will the number of Americans with Internet access reach 132.3 million? Round your answer to the nearest whole number. a. b.

c. d.

ANS: B

PTS: 1

DIF:

medium

18. A population is modeled by the function . What value of a. b.

corresponds to the point of inflection?

2.36 1749

ANS: D

c. d. PTS: 1

19. Enrollment at a certain university

DIF:

1749.08 1.96 medium

years after calendar year 2000 is given by .

According to the model, for which calendar year is enrollment growing at the fastest rate? Round your answer to the nearest whole number. a. 2006 b. 2009 ANS: D

c. 2011 d. 2003 PTS: 1

DIF: medium

20. The number , in millions, of Americans with Internet access modeled by the function

years after calendar year 1990 is

. According to the model, for which calendar year is Internet access growing at the fastest rate? Round your answer to the nearest whole number. a. 2001 b. 2003

c. 1998 d. 2004

ANS: A

PTS: 1

21. The cumulative number

DIF: medium

of flu cases reported after

weeks is modeled by the function .

For what value of a. b.

is the epidemic spreading most rapidly?

6.67 1592

ANS: D

c. d. PTS: 1

22. Make a logistic model conditions:

DIF:

medium

, with t in years, for a population that satisfies all of the following

A The carrying capacity is

B: The initial value is

C: In the absence of limiting factors,

would grow by 17% per year.

ANS: A

1592.42 6.3

PTS: 1

23. Make a logistic model following information.

DIF:

medium

for enrollment at a university

years after 2000 making use of the

A University facilities can accommodate at most 5753 students. B: Enrollment in 2000 is 2179. C: In the absence of limiting factors, enrollment would grow by 15% per year.

ANS: C

PTS: 1

24. Make a logistic model weeks.

DIF:

medium

for the cumulative number of disease cases in a cattle herd reported after

A There are a total of 304 animals susceptible to the disease. B: There are initially 7 diseased animals. C: In the absence of limiting factors, the cumulative number of diseased animals would grow by 25% per week.

ANS: A

PTS: 1

DIF:

medium

25. Show the graph of a logistic function.

200

180

160

140

120

100

–5

200

180

160

140

120

100

–5

ANS: D

PTS: 1

–5

DIF:

medium

26. Use logistic regression to find a logistic model for the following data. 5 23.77

10 65.13

15 96.24

ANS: B

PTS: 1

DIF:

20 155.61

25 180.92

medium

27. The following table shows the population of an ant colony months after it is established. Use logistic regression to find a logistic model for the ant population. 5 233

10 664

15 988

ANS: B

PTS: 1

DIF:

8 502

25 1933

medium

28. The following table shows newspaper circulation logistic model for newspaper circulation. 4 121

20 1721

9 701

after

months. Use logistic regression to find a

13 874

17 963

ANS: D

PTS: 1

DIF:

medium

29. Use logistic regression to find a logistic model for the following data. 2.13 22.44

5.11 51.34

7.93 88.87

ANS: C

PTS: 1

30. Magazine circulation after

DIF:

10.41 141.21

medium

years is modeled by the function .

At what circulation level is magazine circulation growing at the fastest rate? a. 11.5 thousand b. 23 thousand ANS: A

c. 5.75 thousand d. 3 thousand PTS: 1

31. Magazine circulation after

DIF:

medium

years is modeled by the function .

What was the initial magazine circulation? a. 11.5 thousand b. 23 thousand ANS: C

c. 5.75 thousand d. 3 thousand PTS: 1

32. Magazine circulation after

DIF:

medium

years is modeled by the function

12.61 204.35

. When did magazine circulation reach 8.33 thousand? a. After 6.95 years b. After 1.58 years ANS: D

c. After 5.5 years d. After 1.37 years PTS: 1

33. Magazine circulation after

DIF:

medium

years is modeled by the function .

When was the circulation increasing at the fastest rate? a. After 3.85 years b. After 3.35 years ANS: B

c. After 3.83 years d. After 4.63 years PTS: 1

DIF:

medium

SHORT ANSWER 1. Plot the following data points along with a graph of the model obtained using logistic regression. 5 28.77

10 54.12

15 102.11

20 161.44

25 185.21

30 190.45

ANS: y 200 180 160 140 120 100 80 60 40 20 –5

PTS: 1

DIF: medium

2. Plot the following data points along with a graph of the model obtained using logistic regression. 2.13 8.25

6.32 30.76

9.98 91.24

13.13 144.62

17.01 163.08

19.78 171.59

ANS: y 200 180 160 140 120 100 80 60 40 20 –3

PTS: 1

DIF: medium

ESSAY 1. The following data show the number 1990. 7 46

10 113

A: Use logistic regression to model

, in millions, of Americans with Internet access

13 166

16 177

years after

19 196

as a function of .

B: Plot the graph of the data along with the model you found in part A. C: According to the model, what is the maximum number of people who will ever have Internet access? D: When did Internet access reach 144.8 million? ANS: A: B:

N 200 180 160 140 120 100 80 60 40 20 –2

C: 191.85 million D: 11.52 years after 1990 PTS: 4

DIF: hard

2. The following data show the number years after 1990.

0 18.93

5 29.33

A: Use logistic regression to model

, in thousands, of concerts presented by symphony orchestras

10 33.15

12 37.12

17 37.17

as a function of .

B: Plot the graph of the data along with the model you found in part A. C: According to the model, what is the maximum number of concerts that will ever be performed in a year? D: When did the number of concerts reach 30.09 thousand? ANS: A: B:

C 40 35 30 25 20 15 10 5

–2

C: 38.1 thousand D: 5.75 years after 1990 PTS: 1

DIF: hard

3. The following data show the number

0 3170

2 5531

A: Use logistic regression to model

of elk in a national park 4 7230

years after 2000.

5 9941

as a function of .

B: Plot the graph of the data along with the model you found in part A. C: According to the model, what is the carrying capacity for elk in this park? D: When did the elk population reach 89% of carrying capacity? ANS: A: B:

7 10430

E 12000 10500 9000 7500 6000 4500 3000 1500 –1 –1500

–3000

C: 12224 D: 7.36 years after 2000 PTS: 4

DIF: hard

4. The following data show the world population

0 2.56

20 3.71

, in billions,

years after 1950.

30 4.45

50 6.09

60 6.85

A: Use logistic regression to model W as a function of . Round the r value to three decimal places. B: Plot the graph of the data along with the model you found in part A. Your graph should show the world population up to the year 2100. C: According to the model, when was the world population growing at its fastest rate? ANS: A: B:

12 10 8 6 4 2

–40 –20 –2

80 100 120 140 160

C: 54.36 years after 1950 PTS: 1

DIF: hard

5. Initially there are 213 newspaper subscriptions. In the absence of limiting factors, subscriptions would grow by 26% per week. There are only 2244 households in town, and that is the maximum possible number of subscriptions.

A: Make a logistic model of newspaper subscriptions

after

weeks.

B: Plot the graph of the model you found in part A over the first 20 weeks. C: According to the model, at what subscription level are subscriptions growing at the fastest rate? Round your answer to the nearest whole number. D: At what time did subscriptions reach the value you found in part C? Round your answer to the nearest whole number. ANS: A: B:

2500

2000

1500

1000

500

–2

C: 1122 subscriptions D: After 10 weeks PTS: 4

DIF: hard

6. In a certain region there are 3267 bee hives that are threatened by African bees. Initially 347 of the hives have been taken over by African bees. In the absence of limiting factors, the number of hives taken over by African bees would increase by 27% per year.

A: Make a logistic model of the number

of hives taken over by African bees after

weeks.

B: Plot the graph of the model you found in part A over the first 20 years. C: We want to know then there are 2087 hives taken over. Add the graph of that target value to the graph you made in part B. D: When did the number of hives taken over reach 2087? ANS: A: B:

H 4000 3500 3000 2500 2000 1500 1000 500

–2

C: H 4000 3500 3000 2500 2000 1500 1000 500

–2

D: After 11.25 years PTS: 4

DIF: hard

7. A child is 20 inches long at birth and grows to an adult height of 72 inches.

A: Make a logistic model of the height

, in inches, at age

given sufficient information to determine the value of

years. Note: Thus far you are not

in the expression

. Just leave

that term as a variable. B: In the model from part A we think of restricting the age to 15. That corresponds to putting in the formula, so H is a function of r. Make a graph of versus with . You may assume that r is no larger than 1. C: Use your graph in part B to determine for what value of height at age 15.

the person will reach 90% of adult

D: Use the information you found in part C to make a logistic model of height versus age.

ANS: A: B: H

70 60 50 40 30 20 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

per year

D: PTS: 4

DIF: hard

8. A fish population grows according to the logistic model . Here

is measured in years and

is measured in thousands of tons.

A: Make a graph of the fish population over the first 20 years. B: According to the model, what is the carrying capacity for this fish population in this environment? C: According to the model, what is the optimum yield level for harvesting the fish? D: How long does it take for the fish population to grow from the optimum yield level to 90% of carrying capacity? ANS: A:

N 100 90 80 70 60 50 40 30 20 10 –1.–510

1.5

4.5

7.5

10.5

–20 –30

B: 85 thousand tons C: 42.5 thousand tons D: 3.67 years PTS: 4

DIF: hard

9. In a city of 7 thousand, there are initially 0.13 thousand flu cases. In the absence of limiting factors, the cumulative number flu cases is expected to grow by 19% per week.

A: Make a logistic model of the cumulative number

, in thousands, of flu cases after

weeks.

B: Plot the graph of the model you found in part A over the first 35 weeks. C: At what cumulative number of flu cases will the disease be spreading at the fastest rate? D: When will the cumulative number of flu cases reach the level you found in part C? Round your answer to the nearest whole number. ANS: A: B:

F 9 8 7 6 5 4 3 2 1 –15 –10 –5

–1

C: 3.5 thousand

D: After 23 weeks PTS: 4

DIF: hard

Section 5.2 Power Functions TRUE/FALSE 1. The homogeneity property applies to both power functions and exponential functions. ANS: F

PTS: 1

2. For the power function ANS: F 3.

, if

DIF:

is negative, then the power function PTS: 1

easy

is multiplied by 2, then

PTS: 1

ANS: T 4. If

DIF:

PTS: 1

is a decreasing function for positive values of DIF:

easy is concave up.

DIF:

easy

MULTIPLE CHOICE 1. Let

. If

a. b.

is multiplied by 64 is multiplied by 81

ANS: A 2. Let a. b.

. If

3. Let

is increased by 5%, how is

affected?

PTS: 1

4. Let

. If

a. b.

is multiplied by 33.33 is multiplied by 36.94

ANS: A

c. d. DIF:

is decreased by 4%, how is

is decreased by 18.67% is decreased by 18.23%

ANS: D

c. 64 is added to d. 81 is added to medium

PTS: 1 . If

affected?

DIF:

is increased by 1.54% is increased by 1.22%

ANS: D

a. b.

is multiplied by 4, how is

PTS: 1

c. d. DIF:

is multiplied by 3.53, how is

PTS: 1

easy

is not zero, the graph of the power function

ANS: T

is multiplied by

is increased by 22.52% is increased by 21.55% medium affected? is decreased by 16.82% is decreased by 16% medium affected?

c. 33.33 is added to d. 36.94 is added to DIF:

medium

5. Let

. If

a. b.

is multiplied by 2.48 is multiplied by 0.4

ANS: B

is multiplied by 1.59, how is

PTS: 1

6. Let

. If

a. b.

is decreased by 1.18% is increased by 8.71 %

ANS: A 7. Let

. If

a. b.

is decreased by 3.54% is increased by 3.05 %

8. Let

DIF:

9. Let

DIF:

10. Let

c. d.

11. Let percent. a. b.

12. Let percent.

medium

affected? c. 81 is added to d. is multiplied by 1.32

DIF:

medium affected?

c. 47.67 is added to d. is multiplied by 1.29 DIF:

is increased by 32%, how is

PTS: 1 . If

DIF:

is multiplied by 2.68, how is

is decreased by 9.51% is increased by 9.51%

ANS: D

c. is multiplied by 0.06 d. 1.23 is added to

is multiplied by 3, how is

PTS: 1 . If

is increased by 3.54% is decreased by 3.05%

affected?

a. 1.29 is added to b. is multiplied by 47.67 ANS: D

affected?

is divided by 3.69, how is

PTS: 1 . If

medium

a. 1.32 is added to b. is multiplied by 81 ANS: D

is increased by 1.18% is decreased by 8.71%

DIF:

PTS: 1 . If

affected?

c. d.

a. 0.06 is added to b. is multiplied by 1.23 ANS: C

medium

is decreased by 1.98%, how is

PTS: 1 . If

c. 0.4 is added to d. 2.48 is added to

is increased by 1.14%, how is

PTS: 1

ANS: C

affected?

c. d. DIF:

is decreased by 34%, how is

medium affected? Express your answer in terms of a

is decreased by 7.46% is increased by 7.46% medium affected? Express your answer in terms of a

a. b.

is decreased by 10.21% is increased by 10.21%

ANS: A 13. Let percent. a. b.

PTS: 1 . If

14. Let percent.

c. d.

PTS: 1 . If

15. Let

c. d.

PTS: 1 . If

when

a. b. PTS: 1 . If

when

a. b. ANS: B

DIF:

is decreased by 7.88% is increased by 7.88% medium affected? Express your answer in terms of a

is decreased by 47.1% is increased by 47.1% medium affected? Express your answer in terms of a

is decreased by 69.97% is increased by 69.97% medium

, what is the value of ? c. d.

ANS: B 16. Let

DIF:

is increased by 33%, how is

is decreased by 52.52% is increased by 52.52%

ANS: D

DIF:

is decreased by 26%, how is

is decreased by 39.52% is increased by 39.52%

ANS: A

a. b.

c. d.

DIF:

medium

, what is the value of ? c. d.

PTS: 1

DIF:

medium

17. The length of skid marks when a car skids to a stop is a power function of the speed the car is traveling when the brakes are applied. It is observed that if the speed is doubled, the skid mark length increases by a factor of 4. What is the value of the power for this power function? a. b. ANS: D

c. d. PTS: 1

DIF:

medium

18. The life expectancy of a star is a power function of its mass. It is observed that if the mass is doubled, the life expectancy is multiplied by 0.18. What is the value of the power for this power function? a. b. ANS: D

c. d. PTS: 1

DIF:

medium

19. The weight of a certain type of fish is a power function of its length. It is observed that if the length is doubled, the weight increases by a factor of 2.77. What is the value of the power for this power function?

a. b. ANS: D

c. d. PTS: 1

DIF:

medium

20. The speed for certain animals is related to the stride length by a power function. It is observed that if the stride length is doubled, the speed increases by a factor of 2.47. What is the value of the power for this power function? a. b. ANS: A

c. d. PTS: 1

DIF:

medium

21. The speed for certain animals is related to the stride length by a power function. The value of the power in this function is . If the stride length is doubled, how is the speed affected? a. Speed is multiplied by 1.12. b. Speed is multiplied by 2.08. ANS: B

PTS: 1

c. Speed is multiplied by 2.59. d. Speed is multiplied by 1.39. DIF:

medium

22. The weight of a certain type of fish is a power function of its length. The value of the power in this function is . If the length is doubled, how is the weight affected? a. Weight is multiplied by 1.17. b. Weight is multiplied by 2.11. ANS: B

PTS: 1

c. Weight is multiplied by 2.99. d. Weight is multiplied by 1.29. DIF:

medium

23. The length of skid marks when a car skids to a stop is a power function of the speed the car is traveling when brakes are applied. The value of the power in this function is . If the skid length is doubled, how is the speed affected? a. Speed is multiplied by 4. b. Speed is multiplied by 2.29 ANS: C

PTS: 1

c. Speed is multiplied by 1.41. d. Speed is multiplied by 4.12. DIF:

medium

24. The life expectancy of a star is a power function of its mass. The value of the power in this function is . If the life expectancy is doubled, how is the mass affected? a. Life expectancy is multiplied by 6.25. b. Life expectancy is multiplied by 1.44 ANS: C

PTS: 1

c. Life expectancy is multiplied by 0.76. d. Life expectancy is multiplied by 6.77. DIF:

medium

25. The life expectancy of a star is a power function of its mass. The value of the power in this function is . If the mass is increased by 36%, how is the life expectancy affected? a. Life expectancy is increased by 46.36% b. Life expectancy is decreased by 46.36% ANS: C

PTS: 1

c. Life expectancy is decreased by 53.64%. d. Life expectancy is increased by 53.64%. DIF:

medium

26. The life expectancy of a star is a power function of its mass. The value of the power in this function is . If the mass is decreased by 11%, how is the life expectancy affected? a. Life expectancy is increased by 77.04% b. Life expectancy is decreased by 77.04% ANS: D

PTS: 1

c. Life expectancy is decreased by 33.82%. d. Life expectancy is increased by 33.82%. DIF:

medium

27. The speed for certain animals is related to the stride length by a power function. The value of the power in this function is . If the stride length is multiplied by 1.7, how is the speed affected? a. Speed is multiplied by 1.14. b. Speed is multiplied by 1.77. ANS: B

PTS: 1

c. Speed is multiplied by 1.87. d. Speed is multiplied by 1.21. DIF:

medium

28. The weight of a certain type of fish is a power function of its length. The value of the power in this function is . If the length is multiplied by 2.02, how the is weight affected? a. Weight is multiplied by 1.06. b. Weight is multiplied by 2.06. ANS: B

PTS: 1

c. Weight is multiplied by 2.49. d. Weight is multiplied by 1.92. DIF:

medium

29. The length of skid marks when a car skids to a stop is a power function of the speed the car is traveling when the brakes are applied. The value of the power in this function is . If the speed is multiplied by 2.11, how is the skid mark length affected? a. Speed is multiplied by 4.32. b. Speed is multiplied by 4.45. ANS: B

PTS: 1

c. Speed is multiplied by 4.79. d. Speed is multiplied by 4.34. DIF:

medium

30. The life expectancy of a star is a power function of its mass. The value of the power in this function is . If the life expectancy is multiplied by 2.28, how is the mass affected? a. Mass is multiplied by 0.72. b. Mass is multiplied by 0.13. ANS: A 31. The body surface area relationship is

PTS: 1

c. Mass is multiplied by 0.91. d. Mass is multiplied by 1.19. DIF:

medium

of an 80-kilogram man depends on his height

measured in meters. The

. If the height is multiplied by 1.06, how is the body surface area affected? a. Body surface area is multiplied by 0.48. b. Body surface area is multiplied by 1.2. ANS: C

PTS: 1

c. Body surface area is multiplied by 1.03. d. Body surface area is multiplied by 1.44. DIF:

medium

32. The speed for certain animals is related to the stride length by a power function. The value of the power in this function is . If the speed is multiplied by 1.63, how is the stride length affected?

a. Stride length is multiplied by 1.65. b. Stride length is multiplied by 2.03 ANS: C

c. Stride length is multiplied by 1.61. d. Stride length is multiplied by 1.92.

PTS: 1

DIF:

medium

33. The length of skid marks when a car skids to a stop is a power function of the speed the car is traveling when brakes are applied. The value of the power in this function is . If the skid mark length is multiplied by 2.58, how is the speed affected? a. Stride length is multiplied by 6.66. b. Stride length is multiplied by 1.94 ANS: C

c. Stride length is multiplied by 1.61. d. Stride length is multiplied by 6.83.

PTS: 1

DIF:

medium

34. The body surface area of an 80-kilogram man depends on his height measured in meters. The value of the power in this function is . If the body surface area is multiplied by 1.01, how is the height affected? a. Height is multiplied by 1. b. Height is multiplied by 1.24 ANS: D

c. Height is multiplied by 1.37. d. Height is multiplied by 1.02.

PTS: 1

DIF:

medium

35. The weight of a certain type of fish is a power function of its length. The value of the power in this function is . If the weight is multiplied by 2.42, how is the length affected? a. Length is multiplied by 5.86. b. Length is multiplied by 2.08 ANS: C

c. Length is multiplied by 1.56. d. Length is multiplied by 6.28.

PTS: 1

36. Show the graph of

DIF:

when

is a positive number larger than 1.

medium

–3 –6

x –3 –6

b. 24

–3

–6

PTS: 1

37. Show the graph of

DIF:

when

24 21

–3

–6

–6 y

–6

ANS: B

–3

medium

is a negative number.

–6

ANS: C

ESSAY

–3 –6

PTS: 1

DIF:

medium

1. When a rock falls near the surface of Earth, it will fall

feet in

seconds.

A: Make a graph of distance fallen versus time for up to 10 seconds. B: How long will it take the rock to fall 69 feet? C: If the time falling is doubled, how is distance fallen affected? ANS: A: D 1800 1600 1400 1200 1000 800 600 400 200 –2 –200

B: 2.08 seconds C: Distance fallen is multiplied by 4. PTS: 3

DIF: hard

2. The volume , in cubic inches, of a spherical balloon depends on its radius , measured in inches. The relationship is . A: Make a graph of the volume of the balloon versus the radius. Include radii up to 10 inches. B: What is the radius of the balloon if its volume is 744 cubic inches? C: If one balloon has twice the radius of another balloon, how do their volumes compare? ANS: A:

V 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 ––2500

–1000

B: 5.62 inches C: The volume of the larger balloon is 8 times that of the smaller balloon. PTS: 3

DIF: hard

3. The speed , in feet per second, of certain animals depends on their stride length relationship is

in inches. The

. A: If one animal has a stride length 2.52 times that of another, how do their running speeds compare? B: If one animal has double the speed of anther, how do their stride lengths compare? C: If an animal with stride length 25 inches runs 18 feet per second, find the value of . ANS: A: The animal with the longer stride length runs 4.35 times as fast. B: The faster animal has 1.55 times the stride length of the slower animal. C: PTS: 3

DIF: hard

4. The body surface area , in square meters, depends on the height , in centimeters, and the weight in kilograms. The relationship is . A: What is the body surface area of a man who is 182 centimeters tall and weighs 85 kilograms? B: If a man’s height increases by a factor of 1.08, but his weight remains the same, how is his body surface area affected? C: If a man’s height does not change, but he loses 5% of his weight, how is his body surface area affected? Express your answer in terms of a percent.

ANS: A: 2.07 square meters B: Body surface area is multiplied by 1.04. C: Body surface area is decreased by 2.53%. PTS: 3

DIF: hard

5. When a car skids to a stop, the length , in feet, of the skid marks is related to the speed per hour, when the brakes were applied. The formula is

, in miles

. A: If the skid marks are 223 feet long, how fast was the car going when the brakes were applied? B: If your speed doubles, how is the skid mark length affected? C: You are driving at 67 miles per hour. You wish to slow to a speed that will cut the length of potential skid marks in half (thus reducing the emergency stopping distance). What should be your new speed? ANS: A: 74.67 miles per hour B: Skid mark length is multiplied by 4. C: 47.38 miles per hour PTS: 3 6. The life expectancy formula

DIF: hard , in solar lifetimes, of a star is related to the mass

, in solar masses, by the

. A: Make a graph of the life expectancy versus the mass. Include up to 20 solar masses. B: Does a larger star have a longer or shorter life expectancy? C: If one star is double the mass of another, how do their life expectancies compare? Express your answer as a percent. ANS: A:

E 4.5 4 3.5 3 2.5 2 1.5 1 0.5 –2

–0.5

B: The larger star has a shorter life expectancy. C: The life expectancy of the larger star is 82.32% less than that of the smaller star. PTS: 3

DIF: hard

7. The number

of species of birds on an island is given approximately by

, where

is the area, in square kilometers, of the island that is usable for birds.

A: If a conservation program is able to increase the usable area by 23%, how is the number of species affected? Express your answer in terms of a percent. B: If a natural disaster reduces the usable area of the island by 23%, how is the number of species affected? Express your answer in terms of a percent. C: If the number of bird species is 56, and the usable area is 242 square kilometers, find the value of . ANS: A: The number of species is increased by 4.88%.

B: The number of species is decreased by 5.83%. C: PTS: 3

DIF: hard

8. The period , in years, of a planet is the time it takes to make a single orbit of the sun. The period is related to the distance from the sun by the formula .

A: Neptune is 30 times as far from the sun as Earth. How long does it take Neptune to make a single orbit around the sun? B: The period of Mercury is 88 days. Express the period of mercury in terms of years. C: The Earth is 93 million miles from the sun. Use your answer to part B to determine how far Mercury is from the sun. ANS: A: 164.32 years B: The period of Mercury is 0.24 year. C: Mercury is 35.92 million miles from the sun. PTS: 3

DIF: hard

9. If a rock falls near the surface of any planet, the distance fallen , in feet, is proportional to the square of the time t, in seconds, since the rock was dropped. This fact can be expressed by the formula .

A: From time

to time

, by what factor has time increased?

B: By what factor does distance increase from time C: By what percentage does distance increase from time ANS: A: Time increases by a factor of 5. B: Distance increases by a factor of 25. C: Distance increases by 2400%. PTS: 3

DIF: hard

to time to time

? ?

Section 5.3 Modeling Data with Power Functions TRUE/FALSE 1. Power regression always applies to data that are not exponential. ANS: F

PTS: 1

DIF:

easy

2. Power regression does not apply to data that should be fit by a decreasing function. ANS: F

PTS: 1

DIF:

easy

3. The homogeneity property applies to power functions but not to exponential functions. ANS: T

PTS: 1

DIF:

easy

DIF:

medium

4. Some linear functions are power functions. ANS: T

PTS: 1

MULTIPLE CHOICE 1. Use power regression to model the following data. 1 3.71

3 14.58

a. b.

4 16.79

7 34.63

8 41.06

c. d.

ANS: A

PTS: 1

DIF:

medium

2. The following table shows the length , in centimeters, and the weight , in grams, of a certain type of fish. Use power regression to model the weight as a function of the length. L W

28.13 216

32.7 312

a. b.

38.98 516

42.16 690

46.04 902

c. d.

ANS: A

PTS: 1

DIF:

medium

3. The following table shows the stopping distance , in feet, for a car that is traveling at a speed of miles per hour. Use power regression to model the stopping distance as a function of the speed. 25 87 a. b.

35 132

40 168 c. d.

60 301

75 433

ANS: B

PTS: 1

DIF:

4. The following table shows the stopping distance miles per hour. 25 82

35 132

medium , in feet, for a car that is traveling at a speed of

40 167

60 302

75 434

Use power regression to determine the stopping distance of a car that is traveling at a speed of 67 miles per hour. a. 363.71 feet c. 363.88 feet b. 24.63 feet d. 19.76 feet PTS: 1

ANS: C

DIF: medium

5. The following table shows the stopping distance miles per hour. 25 81

35 132

, in feet, for a car that is traveling at a speed of

40 162

60 309

75 436

Use power regression to determine the speed of a car that requires 151 feet to stop. a. 35.03 miles per hour b. 44.44 miles per hour ANS: C

c. 37.88 miles per hour d. 42.35 miles per hour

PTS: 1

DIF:

6. The following table shows the stopping distance miles per hour. 25 83

35 135

medium , in feet, for a car that is traveling at a speed of

40 164

60 302

75 431

Use power regression to answer the following question. If the speed is cut in half, how is the stopping distance affected? a. b. c. d.

The stopping distance is multiplied by 0.35. The stopping distance is multiplied by 0.5. The stopping distance is multiplied by 2.26. The stopping distance is multiplied by 1.22.

ANS: A

PTS: 1

DIF:

7. The following table shows the stopping distance miles per hour. 25 82

35 138

medium , in feet, for a car that is traveling at a speed of

40 169

60 307

75 431

Use power regression to answer the following question. If the speed increases by 29%, how is the stopping distance affected?

a. Stopping distance increases by 46.22%. b. Stopping distance increases by 46.52%. ANS: B

PTS: 1

c. Stopping distance increases by 42.35%. d. Stopping distance increases by 40.17%. DIF:

medium

8. The following table shows the length , in inches, and the flying speed , in feet per second, of certain flying animals. Use power regression to model the flying speed as a function of the length. L

0.57 21.35

3.14 36.18

a. b.

47.03 61.86

62.74 74.47

c. d.

ANS: D

PTS: 1

DIF:

9. The following table shows the length certain flying animals. L

4.19 39.26

0.51 21.71

3.16 36.3

medium

, in inches, and the flying speed

4.72 39.99

, in feet per second , of

47.07 61.4

62.37 74.32

Use power regression to estimate the flying speed of an animal that is 11 inches long. a. 50.7 feet per second b. 47.19 feet per second ANS: B

c. 49.73 feet per second d. 51.24 feet per second

PTS: 1

DIF:

10. The following table shows the length certain flying animals. L

0.54 21.19

3.33 36.66

medium

, in inches, and the flying speed

4.63 39.93

, in feet per second , of

47.48 61.58

62.59 74.35

Use power regression to estimate the length of an animal that has a flying speed of 46 feet per second. a. 9.78 inches b. 67.74 inches ANS: A

c. 69.79 inches d. 10.15 inches PTS: 1

DIF:

11. The following table shows the length certain flying animals. L

0.55 21.45

3.27 36.33

medium

, in inches, and the flying speed

4.51 39.83

, in feet per second , of

47.17 61.27

62.86 74.73

Use power regression to answer the following question. If the length is doubled, how is the flying speed affected? a. It is multiplied by 0.06. b. It is multiplied by 2.09.

c. It is multiplied by 2. d. It is multiplied by 1.18.

ANS: D

PTS: 1

DIF:

12. The following table shows the length certain flying animals. L

0.51 21.97

medium

, in inches, and the flying speed

3.25 36.23

4.66 39.34

, in feet per second , of

47.28 61.25

62.97 74.7

Use power regression to answer the following question. If the length is halved, how is the flying speed affected? a. It is multiplied by 0.49. b. It is multiplied by 1.16. ANS: D

c. It is multiplied by 0.5. d. It is multiplied by 0.85.

PTS: 1

DIF:

13. The following table shows the length certain flying animals. L

0.55 21.81

medium

, in inches, and the flying speed

3.62 36.58

4.22 39.37

, in feet per second , of

47.68 61.1

62.22 74.1

Use power regression to answer the following question. If the flying speed is doubled, how is the length affected? a. It is multiplied by 1.18. b. It is multiplied by 2.31. ANS: C

c. It is multiplied by 17.96. d. It is multiplied by 20.14.

PTS: 1

DIF:

medium

14. The following table shows the area , in square miles, and the number of species on islands in a certain island chain. Use power regression to find a formula expressing the number of species as a function of the area. 44113 74

29108 68

a. b. ANS: B

4277 41

2531 41

41 9

c. d. PTS: 1

DIF:

15. The following table shows the area certain island chain. 44188 72

medium

, in square miles, and the number

29190 64

4268 36

2580 43

of species on islands in a

41 11

Use power regression to estimate the number of species on an island in the chain that is 3175 square miles in area. Round your answer to the nearest whole number. a. 38

c. 35

b. 39

d. None of the above

ANS: A

PTS: 1

DIF:

16. The following table shows the area certain island chain. 44193 80

medium

, in square miles, and the number

29159 64

4295 40

of species on islands in a

2591 42

44 10

Use power regression to estimate the area of an island in the chain on which there are 54 species. a. 12161.38 square miles b. 12158.87 square miles ANS: B

c. 12149.44 square miles d. None of the above

PTS: 1

DIF:

medium

17. The following table shows the length , in centimeters, and the weight , in grams, of a certain type of fish. Use power regression to estimate the weight of a fish that is 53.36 centimeters long. L W

30.2 256

34.1 362

40.09 579

a. 3.01 grams b. 980.97 grams ANS: B

44.9 806

48.5 1005

c. 982.82 grams d. 2.43 grams PTS: 1

DIF:

medium

18. Use a power regression model for the following data to answer the following question. If multiplied by 5, what is the effect on ? 1 4.2 a. b.

3 20.44

4 33.38

is multiplied by 12.23. is multiplied by 14.23.

ANS: B

c. d.

PTS: 1

DIF:

19. The following table shows the length of fish. L W

28.6 215

7 99.52

8 131.66

is multiplied by 15.66. is multiplied by 11.43. medium

, in centimeters, and the weight

32.9 309

38.7 505

42.8 671

, in grams, of a certain type

46.02 918

Use a power model to answer the following. If the length is increased by a factor of 3, how is the weight affected? a. b. c. d.

Weight is increased by a factor of 27. Weight is increased by a factor of 29.46. Weight is increased by a factor of 28.32. Weight is increased by a factor of 27.

ANS: D

PTS: 1

DIF:

medium

20. The following table shows the length of fish. L W

28.5 214

32.5 310

, in centimeters, and the weight

38.8 510

, in grams, of a certain type

42.9 675

46.05 903

Use a power model to answer the following. If one fish is half the length of another, how do their weights compare? a. The shorter fish is 86.88% lighter. b. The shorter fish is 88.34% lighter. ANS: A

PTS: 1

c. The shorter fish is 14.92% lighter. d. The shorter fish is 13.12% lighter. DIF:

medium

21. Use a power regression model for the following data to answer the following question. If increased by 27%, what is the effect on ? 1 4.05 a. b.

3 12.93

is increased by 19.72%. is decreased by 34.5%.

ANS: C

c. d.

PTS: 1

DIF:

22. The following table shows the length of fish. L W

1 4.63

4 18.83

3 19.16

7 44.2

8 51.94

is increased by 34.5%. is decreased by 19.72%. medium

, in centimeters, and the weight

4 28.45

, in grams, of a certain type

7 74.9

8 94.11

Use a power regression model to answer the following question. If the length is increased by 34%, how is the weight affected? a. Weight is increased by 20.92%. b. Weight is increased by 52.86%. ANS: B

PTS: 1

c. Weight is increased by 21.11%. d. Weight is increased by 53.55%. DIF:

medium

23. Use power regression to model the following data. 1 6.54

3 1.59

a. b.

4 3.69

7 1.42

8 2.36

c. d.

ANS: A

PTS: 1

DIF:

medium

24. Use a power regression model for the following data to estimate 1

. 7

4.57

0.67

a. –1.01 b. –0.15

2.86

0.83

4.07

c. 3.31 d. 1.42

ANS: D

PTS: 1

DIF: medium

25. Use a power regression model for the following data to estimate 1 4.19

3 13.71

a. 9.23 b. 9.76

4 15.89

. 7 34.17

8 38.19

c. 38.32 d. 37.84

ANS: D

PTS: 1

DIF: medium

26. Use a power regression model for the following data to answer the following question. If multiplied by 5, what is the effect on ? 1 13.59 a. b.

3 3.88

is multiplied by 2.61. is multiplied by 2.94.

ANS: D

PTS: 1

4 3.54

7 2.1

c. d.

is multiplied by 1.99. is unchanged.

DIF:

8 1.48

medium

27. Use a power regression model for the following data to answer the following question. If half, what is the effect on ? 1 12.63 a. b.

3 3.07

is multiplied by 2.33. is multiplied by 1.1.

ANS: A

PTS: 1

4 1.57

7 0.95

c. d.

is multiplied by 2.38. is multiplied by 1.14.

DIF:

a. b.

3 2.17

is increased by 35.65%. is increased by 4.32%.

ANS: D

PTS: 1

medium

4 1.05

7 0.76

c. d.

is decreased by 4.32%. is decreased by 35.65%.

DIF:

8 1.11

medium

29. Use power regression to model the following data. 2.47

3.38

is cut in

8 1.23

28. Use a power regression model for the following data to answer the following question. If increased by 40%, what is the effect on ? 1 13.18

3.55

6.3

7.79

16.88

29.45

a. b.

88.37

132.73

c. d.

ANS: C

PTS: 1

DIF:

medium

30. Use a power regression model for the following data to estimate 2.26 10.64

3.41 12.61

a. 54851.49 b. 36.57

5.75 21.43

. 7.32 27.86

9.06 34.71

3.61 0.87

5.64 2.52

c. 54852.16 d. 37.48

ANS: B

PTS: 1

DIF: medium

31. Use power regression to model the following data. 1.41 3.67

1.51 3.08

a. b.

3.41 3.33

c. d.

ANS: C

PTS: 1

DIF:

medium

32. Use a power regression model for the following data to answer the following question. If multiplied by 5, what is the effect on ? 1.16 20.53 a. b.

2.72 5.38

is multiplied by 0.13. is multiplied by 0.14.

ANS: A

PTS: 1

2.76 4.53

5.41 2.73

c. d.

is multiplied by 0.28. is multiplied by 0.31.

DIF:

6.19 2.4

medium

33. Use a power regression model for the following data to answer the following question. If what is the effect on ? 1.11 21.36 a. b.

2.9 8.75

is decreased by 95.88%. is increased by 95.88%.

ANS: B

PTS: 1

5.47 4.22 c. d. DIF:

8.25 2.85

11.16 2.43

is decreased by 195.88%. is increased by 195.88%. medium

34. Use a power regression model for the following data to estimate

is halved,

2.47 6.54

4 4.46

4.13 5.24

a. 1.8 b. 2.2

4.83 1.04

7.37 2.78

c. 2.18 d. 2.88

ANS: A

PTS: 1

DIF: medium

SHORT ANSWER 1. Use regression to find a power model for the following data table. Then plot the data along with the graph of the power regression model. 1.89 7.24

2.39 10.33

2.5 10.86

4.49 30.45

4.93 36.73

ANS: Power model: y

120 100 80 60 40 20

–1 –20

PTS: 1

DIF: medium

2. Plot the graph of the following data and add the graph of the power regression model. 1.22 21.55 ANS: –1.02

1.92 13.35

2.22 12.82

3.19 8.27

3.44 8.43

120 100 80 60 40 20

–1 –20

PTS: 1

DIF: medium

ESSAY 1. The following data show the distance , in millions of miles, and the period planets. (The period is the time required for one revolution about the sun.) Planet

Mercury 36 0.24

Venus 67.1 0.62

Earth 92.9 1

Mars 141.7 1.88

, in years, of five of the

Jupiter 483.4 11.87

A: Use power regression to model the period as a function of the distance. Use four digits of accuracy for the coefficient and two digits for the power. B: Plot the data along with the power model you found in part A. C: If one planet is twice as far away from the sun as another, how do their periods compare? ANS: A: B:

12 10 8 6 4 2

–50 –2

50 100 150 200 250 300 350 400 450

C: The more distant planet has a period 2.83 times as long as that of the nearer planet. PTS: 3

DIF: hard

2. The following data show the distance , in millions of miles, and the period planets. (The period is the time required for one revolution about the sun.) Planet

Mercury 0.24 36

Venus 0.62 67.1

Earth 1 92.9

Mars 1.88 141.7

, in years, of five of the

Jupiter 11.87 483.4

A: Use power regression to model the distance as a function of the period. B: Plot the data along with the power model you found in part A. C: If one planet takes twice as long to orbit the sun as another, how do their distances from the sun compare? ANS: A: B: D 450 400 350 300 250 200 150 100 50 –2 –50

C: The planet with the longer period is 1.59 times as far away from the sun as the planet with the shorter period. PTS: 3

DIF: hard

3. The generating time for an animal is the time it takes to reach reproductive maturity. The following data show the length , in feet, and the generation time , in years, for some animals. Animal

Deermouse 0.3 0.12

Salamander 0.67 1

Beaver 2.23 2.8

Grizzly bear 5.91 4

Elephant 11.5 12.3

A: Use power regression to model the generation time as a function of the length. B: Plot the data along with the power model you found in part A. C: According to the power model, if one animal is 3 times as long as another, how do their generation times compare? D: Does the grizzly bear have a longer or shorter generation time than would be predicted by the power model? ANS: A: B: G

12 10 8 6 4 2

–2

C: The longer animal has a generation time 3.39 times as long as that of the smaller animal. D: It has a shorter generation time. PTS: 4

DIF: hard

4. The generating time for an animal is the time it takes to reach reproductive maturity. The following data show the generation time , in years, and the length , in feet, for some animals.

Animal G L

Deermouse 0.3 0.19

Salamander 0.67 1

Beaver 2.23 2.8

Grizzly bear 5.91 4

Elephant 11.5 12.3

A: Use power regression to model the length as a function of the generation time. B: Plot the data along with the power model you found in part A. C: According to the power model, if the generating time of one animal is 4 times that of another, how do their lengths compare? D: Is the grizzly bear longer or shorter than would be predicted by the power model? ANS: A: B: G

12 10 8 6 4 2

–2

C: The animal with the longer generation time is 3.63 times as long the animal with the shorter generation time. D: Longer PTS: 4

DIF: hard

5. The following data show the speed stopping distance , in feet. 15 44.06

, in miles per hour, of a car when the brakes are applied and the

25 85.94

35 136.52

60 304.55

75 433.23

A: Use power regression to model the stopping distance as a function of the speed. B: Plot the data along with the power model you found in part A.

C: According to the power model, if the speed is multiplied by 5, how is the stopping distance affected? ANS: A: B: D

350 300 250 200 150 100 50

–20 –10

C: The stopping distance is multiplied by 9.83. PTS: 3

DIF: hard

6. The following data show the height , in feet, above the surface of the ocean and the distance miles, to the visible horizon. 6 3.3

8 3.6

12 4.7

16 5.4

, in

19 5.9

A: Use power regression to model the distance to the horizon as a function of the height above the ocean. B: Plot the data along with the power model you found in part A. C: According to the power model, if the height is increased by 20%, how is the distance to the horizon affected? Express your answer in terms of a percentage. ANS: A: B:

D 6 5 4 3 2 1

–2 –1

–2

C: The distance to the horizon is increased by 9.94%. PTS: 3

DIF: hard

7. The following data show the population of residents traveling to work. City

Los Angeles 6490 16.2

Pittsburgh 1869 12.9

, in thousands, and the average driving time

Washington 1829 14.7

Nashville 369 10.6

, in minutes,

Tallahassee 48 11

A: Use power regression to model the driving time as a function of the population. B: Plot the data along with the power model you found in part A. C: According to the power model, if the population is increased by 17%, how is the driving time affected? Express your answer in terms of a percentage. ANS: A: B:

D 18 16 14 12 10 8 6 4 2 –1000 –2

1000 2000 3000 4000 5000 6000

–4

C: The driving time is increased by 1.26%. PTS: 3

DIF: hard

8. The following data show the mass , in solar masses, and the relative luminosity star’s luminosity to that of the sun) of various stars. Star

Spica 7.3 1050

Vega 3.1 55

Altair 1 1.1

Sun 1 1

(the ratio of a

61 Cygni A 0.17 0.002

A: Use power regression to model the luminosity as a function of the mass. B: What is the mass of a star with a relative luminosity of 5.82? C: If one star has a mass of 25 times that of another, how do their luminosities compare? ANS: A: B: 1.64 solar masses

C: The relative luminosity of the larger star is 78125 times that of the other star. PTS: 3

DIF: hard

9. The following data show the speed in flight selected flying animals. Animal

, in feet per second, and the length

House fly

Hummingbird

0.51 21.06

3.26 37.84

Willow warbler 4.32 39.03

, in inches, of

Flying fish

White pelican

13.52 53.26

63.97 72.38

A: Use power regression to model the flying speed as a function of the length. B: According to the model, what is the length of a flying animal that flies 6.7 feet per second? C: If one flying animal is 13 times as long as another, how do their flying speeds compare? ANS: A: B: 0.01 inches C: The longer animal flies 1.95 times as fast. PTS: 3

DIF: hard

10. The following data show the diameter Tree

and the height

Cottonwood

Hackberry

2.8 80.03

5.02 113.48

Weeping willow 6.08 95.29

, both in feet, of certain mature trees. Ponderosa pine 8.03 165.53

Douglas fir 14.04 222.84

A: Use power regression to model the height as a function of the diameter. B: According to the model, what is the diameter of a mature tree that is 6.02 feet tall? C: If one mature tree has 5 times the diameter of another, how do their heights compare? ANS: A: B: 0.06 feet C: The larger tree is 2.89 times as tall as the smaller tree. PTS: 3

DIF: hard

Section 5.4 Combining and Decomposing Functions TRUE/FALSE 1. Function composition is the same as the product of functions. ANS: F

PTS: 1

DIF:

easy

2. Piecewise-defined functions occur in such applications as the cost of postage. ANS: T

PTS: 1

DIF:

easy

3. A piecewise-defined function cannot have a limiting value. ANS: F

PTS: 1

4. The maximum value of ANS: F

DIF:

easy

is always the same as the maximum value of

PTS: 1

DIF:

easy

5. If we have a function giving the weight of a man in terms of his height and a function giving the height of this man in terms of his age, we can compose the two functions to get the man’s weight in terms of his age. ANS: T 6. If

PTS: 1

is positive and

ANS: T

DIF:

easy

, then the limiting value of PTS: 1

DIF:

is 0.

easy

7. When two functions are composed to make a new function, it makes no difference in which order the functions are used. ANS: F

PTS: 1

8. The limiting value of ANS: F

DIF:

easy

is always the same as the limiting value of PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Use a formula to express

as a function of

a. b. ANS: A

and

. c.

d. PTS: 1

2. Use a formula to express

as a function of

DIF: if

medium and

ANS: B

PTS: 1

3. Use a formula to express

DIF: as a function of

medium if

a. b.

d. PTS: 1

4. Use a formula to express

DIF:

as a function of

medium

and

a. d.

ANS: C

PTS: 1

5. Use a formula to express

DIF:

as a function of

and

ANS: A

PTS: 1 and

medium

DIF:

c. d.

ANS: A

PTS: 1 and

DIF:

medium

, find a formula for

a. b.

8. If

medium

, find a formula for

a. b.

ANS: B

7. If

ANS: D

6. If

, and

c. d. PTS: 1 and

DIF:

medium

, find a formula for

a. 0

. Assume that

ANS: C 9. If

PTS: 1 and

DIF:

medium

, find a formula for

ANS: A 10. If

PTS: 1 and

DIF:

, find a formula for

a. b.

c. d.

ANS: D 11. If

PTS: 1 , find a formula for

DIF:

medium

a. b.

c. d.

ANS: B 12. If

medium

PTS: 1 , find a formula for

DIF:

medium

a. b.

c. d.

ANS: B

PTS: 1

DIF:

medium

13. If is the piecewise defined function value of .

a. 3 b. 19

c. 9 d. –1

ANS: C 14. If

PTS: 1 , find a formula for

DIF:

c. d.

ANS: B

a. 9 b. 60

medium

a. b.

15. If

and

PTS: 1 and

DIF:

, what is the value of

medium when

c. 22 d. 12

, find the

ANS: A

PTS: 1

16. If

and

DIF:

medium

, what is the value of

when

a. 3.16 b. 56.92

c. 4.9 d. 12

ANS: C 17. If

PTS: 1 and

DIF:

, what is the value of

a. 33.49 b. 54.92 PTS: 1 and

, what is the value of

PTS: 1

19. Suppose )?

and

PTS: 1

20. Suppose if

. Let

. What is the value of

medium

. Let be the piecewise-defined function . What is the value of )? c. 360 d. 42

ANS: D

PTS: 1 and

DIF:

medium

, what is the value of

a. 489.37 b. 40.85

c. 75.57 d. None of the above PTS: 1 and

a. 7.48 b. 183.34 ANS: A

medium if

DIF:

a. 720 b. 78

c. 126 d. 46

ANS: D

22.

when

DIF:

a. 168 b. 52

ANS: A

c. 29.3 d. 27.13

ANS: D

when

DIF: medium

a. –71.02 b. 286.21

21.

medium

c. 5.16 d. 62.35

ANS: D 18. If

DIF: ,

medium

what is the value of

c. 14 d. None of the above PTS: 1

DIF:

medium

, and

23.

and

what is the value of

a. 21.59 b. 0.83

c. 15.71 d. None of the above

ANS: B 24.

PTS: 1 , what is the value of

DIF:

ANS: B If

c. 700 d. None of the above PTS: 1

DIF:

, what is the value of

26. Suppose

c. 23.6 d. None of the above PTS: 1

DIF:

. If the limiting value of

a. 10 b. 0 ANS: A 27. Suppose

28. Suppose

PTS: 1

is 0, what is the limiting value of

DIF: medium

. If the limiting value of

is 4, what is the limiting value of

c. 10 d. None of the above PTS: 1

DIF: medium

. If the limiting value of

a. 20.3 b. 0 ANS: A

medium

c. 13 d. None of the above

a. 14 b. 0 ANS: A

medium ?

a. 17.29 b. 18.11 ANS: B

medium

a. 4900 b. 490

25.

is 2.46, what is the limiting value of

c. 8 d. None of the above PTS: 1

DIF: medium

SHORT ANSWER 1. Graph the piecewise-defined function . ANS:

, and

for

y 45 40 35 30 25 20 15 10 5 –2 –5

PTS: 1

20 x

DIF: medium

2. Graph the piecewise-defined function .

, and

for

, and

for

ANS: y 45 40 35 30 25 20 15 10 5 –2 –5

PTS: 1

20 x

DIF: medium

3. Graph the piecewise-defined function . ANS:

y 45 40 35 30 25 20 15 10 5 –2 –5

PTS: 1

20 x

DIF: medium

4. Graph the piecewise-defined function

, and

for

ANS: y 45 40 35 30 25 20 15 10 5 –2 –5

PTS: 1

20 x

DIF: medium

5. Graph the piecewise-defined function ANS:

, and

for

y 60 50 40 30 20 10

–1 –10

–20

PTS: 1

DIF: medium

ESSAY 1. The cross-sectional area

, in square feet, of a river is given by the formula ,

where

is the width of the river and

The flow rate

is the depth. Both of these variables are measured in feet.

, in cubic feet per minute, of the river is given by ,

where

is the speed of the river measured in feet per minute.

A: Find a formula that gives the flow rate of a river in terms of the variables ,

, and

B: Use the formula you found in part A to find the flow rate of a river with width 22 feet, a depth of 3 feet, and a speed of 30 feet per minute. ANS: A: B: 1980 cubic feet per minute PTS: 2

DIF: medium

2. If a skydiver jumps from an airplane, his velocity , in feet per second, starts at 0 and increases toward a terminal velocity of feet per second. The difference is an exponential function of time. A: What is the initial value of

B: For each second that passes, C: Find an exponential formula for

decreases by 17%. What is the one-second decay factor for . Use

for the number of seconds since the skydiver jumped.

D: Find a formula for . ANS: A: 176 feet per second B: 0.83 C: D: PTS: 4

DIF: hard

3. A certain type of lizard grows to a maximum length of length of the lizard after years, then the difference

centimeters. If , in inches, is the is an exponential function of time.

A: A newborn lizard is 4 centimeters long. What is the initial value of

B: A 2-year-old lizard is 8 centimeters long. Find the yearly decay factor for D, and use your answer to find an exponential formula for . Use for the age, in years, of the lizard. C: Find a formula for

ANS: A: 16 B: 0.71; C: PTS: 3

DIF: hard

4. A cake is placed in a 350-degree oven to bake. The difference between the temperature of the oven and the temperature of the cake is an exponential function of time. All temperatures are measured in degrees. A: The cake is initially at room temperature of 75 degrees. What is the initial value of B: The temperature difference decreases by 3% each minute. Find a formula for C: Find a formula for the temperature

of the cake after

minutes..

ANS: A: 275 B: C: PTS: 3

DIF: hard

5. The length , in inches, of a certain type of fish at age

years is given by

after

? minutes.

. A: The weight

, in pounds, of the fish is given by .

Find a formula for the weight as a function of the age. B: A cohort of fish is a group of fish born at the same time. After years, there are fish remaining in the cohort. The biomass , in pounds, is the total weight of the cohort. It is the number of fish times the weight of each fish. Find a formula that gives the biomass of the cohort after years. C: What is the biomass of the cohort after 5 years? ANS: A: B: C: 3553.18 pounds PTS: 3

DIF: hard

6. The average delay time

, in seconds, for a car waiting at a certain stop sign is given by

, where

is the number of cars per second passing the stop sign.

A: What is the average delay time if there are 0.15 cars per second passing the sign? B: The service rate , for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay time by . Find a formula for

in terms of .

C: What flow rate will permit a service rate of 0.85 cars per second? ANS: A: 2.45 seconds B:

C: 0.08 cars per second PTS: 3

DIF: hard

7. For a certain species, the probability extinct within years is given by

that a colony founded by a single individual will become

A: What is the probability that this population will become extinct within 3 years? B: The probability that the population will eventually become extinct is the limiting value of is the probability that this population will eventually become extinct? C: If the colony is founded by years is given by

. What

individuals, then the probability that it will survive for at least

. Find a formula for

in terms of

and

C: What is the probability that a population founded by Your answer will involve the variable .

individuals will survive indefinitely? Note:

ANS: A: 0.24 B: 0.29 C:

D: PTS: 4

or DIF: hard

8. The outside wall of a house is 12 feet high. From the western wall, the roof rises 6 feet over a horizontal span of 24 feet and then descends at the same rate over another horizontal span of 24 feet to the eastern wall of the house. For this problem, take the origin to be at the base of the west wall, and take west-to-east to be the positive direction.

A: Find a formula that describes the ascending part of the roof. Use of the west wall and for the height above the ground, both in feet.

for the horizontal distance east

B: Find a formula that describes the descending part of the roof. C: Find a formula for a piecewise-defined function that describes the entire roof line. ANS:

A: B: C:

PTS: 4

, and

DIF: hard

9. Initially a city has a population of 101 thousand. Over the first 10 years the population grows exponentially at a rate of 13% per year. Over the next 10 years the population grows linearly at a rate of 39 thousand individuals per year. A: Find a formula that gives the population

, in thousands, over the first 10 years.

B: Find a formula that describes the population from year 10 to year 20. Be careful: Year 10 corresponds to , not to . C: Find a formula for a piecewise-defined function that describes the population over the 20-year period. D: Make a graph of the population over the 20-year period. ANS: A: B: C:

for

, and

, for

D: N 900 800 700 600 500 400 300 200 100 –2 –100

PTS: 4

DIF: hard

Section 5.5 Polynomials and Rational Functions TRUE/FALSE 1. A quadratic function has the form ANS: T

PTS: 1

. DIF:

easy

2. A quadratic function always has a limiting value. ANS: F

PTS: 1

DIF:

easy

3. The graph of a quadratic function is a parabola. ANS: T

PTS: 1

DIF:

easy

4. A quadratic function is also a polynomial function. ANS: T

PTS: 1

DIF:

easy

5. A cubic function may have both a maximum and a minimum. ANS: T

PTS: 1

DIF:

easy

6. A rational function is one that always produces rational values. ANS: F

PTS: 1

DIF:

easy

7. The limiting value of a rational function can be used to find a horizontal asymptote. ANS: T

PTS: 1

DIF:

easy

8. A rational function is a function for which the horizontal and vertical asymptotes are the same. ANS: F

PTS: 1

DIF:

easy

9. A quartic function may have more than one maximum. ANS: T

PTS: 1

10. The horizontal asymptote for ANS: T

PTS: 1

DIF:

easy

is given by the equation DIF:

easy

11. High-degree polynomials may have many zeros. ANS: T

PTS: 1

DIF:

easy

12. A rational function is a quotient of polynomial functions. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Use quadratic regression to model the following data. 0 4

1 2

3 10

ANS: A

PTS: 1

DIF:

5 30

6 45

23 11

31 3

6.87 6.14

8.15 8.77

6.18 7.32

8.22 3.19

medium

2. Use quadratic regression to model the following data. 0 4

10 12

12 18

a. b.

c. d.

ANS: D

PTS: 1

DIF:

medium

3. Use quadratic regression to model the following data. 1.44 7.93

3.11 3.65

5.43 4.41

a. b.

c. d.

ANS: B

PTS: 1

DIF:

medium

4. Use quadratic regression to model the following data. 2.13 3.64

3.45 5.82

4.76 9.16

a. b.

c. d.

ANS: C

PTS: 1

DIF:

medium

5. Use cubic regression to model the following data. 1 0 a. b.

2 2

3 5

4 1

5 2 c. d.

6 6

7 9

8 10

ANS: D

PTS: 1

DIF:

medium

6. Use cubic regression to model the following data. 1 7

3 4

4 1

6 2

7 6

a. b.

9 9

10 5

11 2

6.71 3.74

7.33 4.06

8.65 6.98

9.53 5.74

10.71 3.72

13.84 1.23

c. d.

ANS: A

PTS: 1

DIF:

medium

7. Use cubic regression to model the following data. 1.32 1.24

2.17 3.75

3.11 4.16

4.68 2.73

a. b.

5.27 2.49 c. d.

ANS: A

PTS: 1

DIF:

medium

8. Use cubic regression to model the following data. 1.52 8.03

3.69 5.22

a. b. ANS: B

4.68 4.18

6.04 3.14

8.22 5.66 c. d.

PTS: 1

DIF:

medium

9. What value of x gives the maximum value of the quadratic function a. 0.3 b. 5.45 ANS: A

c. 0.93 d. 5.91 PTS: 1

DIF: medium

10. What value of x gives the minimum value of the quadratic function a. 5.82 b. 5.33 ANS: D 11. For

c. 0.68 d. 0.33 PTS: 1

DIF: medium

, what is the minimum value of the cubic function

a. 0.58 b. 2.69 ANS: B

c. 0 d. 5 PTS: 1

DIF: medium

12. For

, what value of

gives the minimum value of the cubic function

a. 0.58 b. 4.08 ANS: A 13. What value of

c. 0 d. 6 PTS: 1

DIF:

medium

gives the minimum value of the quadratic function

a. 0.25 b. 0.85 ANS: A

PTS: 1

DIF:

a. b.

medium ? c.

d. PTS: 1

DIF:

15. What is the vertical asymptote of the function a. b. ANS: D

c. 2.63 d. 3.43

14. What is the horizontal asymptote of the function

ANS: A

medium ? c.

d. PTS: 1

DIF:

medium

16. The amount , in pounds, of food consumed in a day by a certain grazing animal depends on the amount , in pounds per acre, of vegetation available. The relationship is given by the rational function . What is the most the animal will consume in a day no matter how much vegetation is available? a. 5 pounds b. 53 pounds ANS: A

c. 0.09 pounds d. 10.6 pounds PTS: 1

DIF:

medium

17. The average number of prey that certain predators will consume depends on the density number per square foot, of the prey. The relationship is

, in

. What is the largest average number of prey that predators will consume no matter what the density of the prey is?

a. 2 b. 6 ANS: C

c. 3 d. None of the above PTS: 1

DIF:

medium

18. A ball is tossed upward from the ground. The height by

, in feet, of the ball after

seconds is given

. What is the maximum height reached by the ball? a. 4 feet b. 156.38 feet ANS: D

c. 3.13 feet d. 156.25 feet PTS: 1

DIF:

medium

19. A ball is tossed upward from the ground. The height by

, in feet, of the ball after

seconds is given

. How long after the ball is tossed will it reach its maximum height? a. 1.83 seconds b. 40.85 seconds ANS: C

c. 1.59 seconds d. 40.64 seconds PTS: 1

DIF:

medium

20. The growth rate , in millions per year, of a certain population depends on the number of individuals present. The relationship is

, in millions,

. For what population size is the population growing at the fastest rate? a. 0.98 million b. 0.26 million ANS: C

c. 0.8 million d. 0.14 million PTS: 1

DIF: medium

21. The growth rate , in millions per year, of a certain population depends on the number of individuals present. The relationship is . What is the maximum growth rate of the population? a. 1.79 million per year b. 1.13 million per year ANS: D

PTS: 1

c. 0.93 million per year d. 0.18 million per year DIF:

medium

22. Use the quadratic formula to write the solution of the quadratic equation

, in millions,

. a.

ANS: A

PTS: 1

DIF:

medium

23. Use the quadratic formula to write the solution of the quadratic equation . a.

ANS: A

PTS: 1

DIF:

medium

SHORT ANSWER 1. Plot the following data along with data from the quadratic regression model. 0 4

1 2

3 10

5 30

6 45

ANS: y 50 45 40 35 30 25 20 15 10 5

PTS: 1

DIF: medium

2. Plot the following data along with the quadratic regression model. 1.44 7.93

3.11 3.65

5.43 4.41

6.87 6.14

8.15 8.77

ANS: y 10 9 8 7 6 5 4 3 2 1

PTS: 1

DIF: medium

3. Plot the following data along with the quadratic regression model. 2.13 3.64

3.45 5.82

4.76 9.16

6.18 7.32

8.22 3.19

ANS: y 10 9 8 7 6 5 4 3 2 1

PTS: 1

DIF: medium

4. Plot the following data along with the cubic regression model. 1 0 ANS:

2 2

3 5

4 1

5 2

6 6

7 9

8 10

y 10 9 8 7 6 5 4 3 2 1

PTS: 1

DIF: medium

5. Plot the following data along with the cubic regression model. 1.32 1.24

2.17 3.75

3.11 4.16

4.68 2.73

5.27 2.49

6.71 3.74

7.33 4.06

8.65 6.98

ANS: y 10 9 8 7 6 5 4 3 2 1

PTS: 1

DIF: medium

6. Plot the following data along with the cubic regression model. For this problem use 3 decimal places of accuracy for the cubic regression model. 1.52 8.03 ANS:

3.69 5.22

4.68 4.18

6.04 3.14

8.22 5.66

9.53 5.74

10.71 3.72

13.84 1.23

y 10 9 8 7 6 5 4 3 2 1

PTS: 1

DIF: medium

7. Plot the graph of

. Include a horizontal span of 0 to 100.

ANS: y 45 40 35 30 25 20 15 10 5

–5

PTS: 1

90 100

DIF: medium

ESSAY 1. A ball is tossed upward from the top of a building and allowed to fall to the ground. Its distance feet, above the ground seconds after the toss is given by . A: How tall is the building? B: What is the maximum height reached by the ball, and when does this occur? C: How long does it take for the ball to hit the ground? ANS:

, in

A: 54 feet B: The ball reaches a maximum height of 91.52 feet after 1.53 seconds. C: 3.92 seconds PTS: 3

DIF: hard

2. The growth rate , in thousands per year, of a certain population depends on the number thousands, of individuals present. The relationship is

, in

. A: Make a graph of growth rate versus population size. B: What population level results in a maximum growth rate? What is that maximum growth rate? C: When population demands exceed available resources, a population decline can be expected. What population levels would result in a declining population? ANS: A: y 0.25

0.2

0.15

0.1

0.05

–0.4 –0.2

0.2 0.4 0.6 0.8

1.2 1.4 1.6

–0.05

B: There is a maximum growth rate of 0.11 thousand per year when there are 0.67 thousand individuals present. C: A population decline can be expected if there are more than 1.33 thousand individuals present. PTS: 3

DIF: hard

3. The average number of prey that certain predators will consume depends on the density number per square foot, of the prey. The relationship is . A: Make a graph of

versus

. Include densities up to 30 per square foot.

, in

B: Does a small increase in the density of prey have a greater effect on the number consumed if there are many or few prey present? C: What is the greatest average number of prey this predator will consume no matter what the density of prey? ANS: A: N 10 9 8 7 6 5 4 3 2 1

B: A small increase in the density of prey has a greater effect on the number consumed if there are few prey present. C: 9 PTS: 3

DIF: hard

4. The time , in minutes, required to drive 1 mile in a certain congested city depends on the traffic flow , in vehicles per hour. The relationship is . A: Make a graph of

versus . Include traffic flows of up to 3000 vehicles per hour.

B: What traffic flow value corresponds to the pole for this rational function? What happens when traffic flow nears this level? C: What traffic flow will result in a 1-mile driving time of 20 minutes? ANS: A:

q 35 30 25 20 15 10 5

500

1000

1500

2000

2500

3000

–5

3387.5 vehicles per hour. When traffic flow nears this value, traffic is nearly stopped.

3006.88 vehicles per hour

PTS: 3

DIF: hard

5. The following table shows the number 0 732

1 820

2 896

, in thousands, of condos sold

years after 2003.

3 801

6 590

4 713

5 563

7 599

. A: Use cubic regression to model condos sold as a function of years since 2003. B: Plot the data along with the graph of the model you found in part A. C: During the period from 2003 to 2010, what is the minimum number of condos sold? D: What number of sales does the model predict for 2012? ANS: A: B:

y 1000 900 800 700 600 500 400 300 200 100 –1 –100

C: 551.92 thousand D: 1148.57 thousand PTS: 4

DIF: hard

6. The following table shows the rate , in accidents per hundred million miles, of accidents versus the speed , in miles per hour. The data refers to urban streets at night. s

20 1600

25 700

30 250

35 300

40 700

45 1300

. A: Use quadratic regression to model the accident rate as a function of the speed. B: Plot the data along with the graph of the model you found in part A. C: According to the model you found, what is the safest speed to drive on urban streets at night? ANS: A: B:

R 1600 1400 1200 1000 800 600 400 200 –5 –200

–400

C: 33.01 miles per hour PTS: 3

DIF: hard

7. Suppose a lake is stocked with fish of the same age. The total number fishing over the life span of the fish is given by

of fish caught by

. Here is the percentage of the fish population caught annually, and is the percentage of fish that die each year. Suppose the lake is initially stocked with 1400 fish that have a mortality rate of 12% per year, so . A: Write a formula that gives C as a function F. B: What percentage of fish caught annually will result in a total catch of 630 fish? C: What is the horizontal asymptote for this function? ANS: A: B: 9.82% C: 1400 PTS: 3

DIF: hard

8. The amount , in pounds, of food consumed in a day by a sheep depends on the amount per acre, of vegetation available. The relationship is given by the rational function .

, in pounds

A: What is the horizontal asymptote for this rational function? B: Make a graph of the amount of food consumed in a day versus the amount of vegetation available, together with the graph of the horizontal asymptote. Include vegetation levels up to 1000 pounds per acre. C: Explain in practical terms the meaning of the horizontal asymptote. ANS: A: The horizontal asymptote is the line

B: C 4 3.5 3 2.5 2 1.5 1 0.5

–0.5

100 200 300 400 500 600 700 800 900 1000

–1

C: The most that a sheep will consume in a day, no matter how much vegetation is available, is 3 pounds. PTS: 3

DIF: hard

Section 6.1 Velocity TRUE/FALSE 1. Velocity may be negative. ANS: T

PTS: 1

DIF:

easy

DIF:

easy

2. Speed may be negative. ANS: F

PTS: 1

3. Velocity is the rate of change in directed distance. ANS: T

PTS: 1

DIF:

easy

4. When directed distance is increasing, velocity is negative. ANS: F

PTS: 1

DIF:

easy

5. When a rock is tossed upward, its velocity at the peak of its flight is 0. ANS: T

PTS: 1

DIF:

easy

6. When velocity is constant, directed distance is a linear function of time. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. When directed distance is decreasing, what can be said about velocity? a. Velocity is positive. b. Velocity is negative. ANS: B

PTS: 1

c. Velocity is 0. d. Velocity is increasing. DIF:

medium

2. When directed distance is increasing, what can be said about velocity? a. Velocity is positive. b. Velocity is negative. ANS: A

PTS: 1

c. Velocity is 0. d. Velocity is increasing. DIF:

medium

3. When directed distance is not changing, what can be said about velocity? a. Velocity is positive. b. Velocity is negative. ANS: C

PTS: 1

c. Velocity is 0. d. Velocity is increasing. DIF:

medium

4. When the graph of directed distance is increasing, what can be said about the graph of velocity?

a. It lies above the horizontal axis. b. It lies below the horizontal axis. ANS: A

PTS: 1

c. It lies on the horizontal axis. d. The graph of velocity is increasing. DIF:

medium

5. When the graph of directed distance is decreasing, what can be said about the graph of velocity? a. It lies above the horizontal axis. b. It lies below the horizontal axis. ANS: B

PTS: 1

c. It lies on the horizontal axis. d. The graph of velocity is increasing. DIF:

medium

6. When the graph of directed distance is a horizontal line, what can be said about the graph of velocity? a. It lies above the horizontal axis. b. It lies below the horizontal axis. ANS: C

PTS: 1

c. It lies on the horizontal axis. d. The graph of velocity is increasing. DIF:

medium

7. When velocity is constant and negative, what can be said about directed distance? a. b. c. d.

Directed distance is a linear function with negative slope. Directed distance is a linear function with positive slope. Directed distance is constant and negative. Directed distance is constant and positive.

ANS: A

PTS: 1

DIF:

medium

8. When velocity is 0, what can be said about directed distance? a. b. c. d.

Directed distance is a linear function with negative slope. Directed distance is a linear function with positive slope. Directed distance is constant. Directed distance is 0.

ANS: C

PTS: 1

DIF:

medium

9. When the graph of directed distance is a straight line with positive slope, what can be said about the graph of velocity? a. b. c. d.

The graph of velocity is a horizontal line above the horizontal axis. The graph of velocity is a horizontal line below the horizontal axis. The graph of velocity is a straight line with positive slope. The graph of velocity is a straight line with negative slope.

ANS: A

PTS: 1

DIF:

medium

10. When the graph of directed distance is a straight line with negative slope, what can be said about the graph of velocity? a. b. c. d.

ANS: B

PTS: 1

DIF:

medium

11. When the graph of directed distance is a horizontal line above the horizontal axis, what can be said about the graph of velocity? a. b. c. d.

ANS: D

PTS: 1

DIF:

medium

12. When the graph of directed distance is a horizontal line below the horizontal axis, what can be said about the graph of velocity? a. b. c. d.

The graph of velocity is a horizontal line above the horizontal axis. The graph of velocity is a horizontal line below the horizontal axis. The graph of velocity is a straight line with negative slope. The graph of velocity lies on the horizontal axis.

ANS: D

PTS: 1

DIF:

medium

13. When the graph of directed distance reaches a maximum (at a peak), what can be said about velocity at this point? a. Velocity is a linear function. b. Velocity reaches a minimum. ANS: D

PTS: 1

c. Velocity reaches a maximum. d. Velocity is 0. DIF:

medium

14. When the graph of directed distance reaches a minimum (at a valley), what can be said about velocity at this point? a. Velocity is a linear function. b. Velocity reaches a minimum. ANS: D

PTS: 1

c. Velocity reaches a maximum. d. Velocity is 0. DIF:

medium

15. A car is driving north at a constant velocity of 56 miles per hour. Since velocity is constant, distance north, in miles, is a linear function. What is the slope of this linear function? a. 56 miles per hour b. –56 miles per hour ANS: A

PTS: 1

c. 0 miles per hour d. None of the above. DIF:

medium

16. A car is driving north at a constant velocity of 64 miles per hour. Since velocity is constant, distance south, in miles, is a linear function. What is the slope of this linear function? a. 64 miles per hour b. –64 miles per hour ANS: B

PTS: 1

c. 0 miles per hour d. None of the above. DIF:

medium

17. We locate the position of a car driving on an east-west highway as distance east. If the car is driving east at a speed of 65 miles per hour, what can be said about velocity?

a. Velocity is 65 miles per hour. b. Velocity is –65 miles per hour. ANS: A

PTS: 1

c. Velocity is 0 miles per hour. d. None of the above. DIF:

medium

18. We locate the position of a car driving on an east-west highway as distance east. If the car is driving west at a speed of 60 miles per hour, what can be said about velocity? a. Velocity is 60 miles per hour. b. Velocity is –60 miles per hour. ANS: B

PTS: 1

c. Velocity is 0 miles per hour. d. None of the above. DIF:

medium

19. We locate the position of a car driving on an east-west highway as distance east. If the car stopped at a light, what can be said about velocity? a. Velocity is positive. b. Velocity is negative. ANS: C

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

20. The graph of velocity lies on the horizontal axis. What can be said about the graph of directed distance? a. It is a horizontal line. b. It is a vertical line. ANS: A

PTS: 1

c. It is increasing. d. It is decreasing. DIF:

medium

21. The graph of velocity lies above the horizontal axis. What can be said about the graph of directed distance? a. It is above the horizontal axis. b. It is below the horizontal axis. ANS: C

PTS: 1

c. It is increasing. d. It is decreasing. DIF:

medium

22. The graph of velocity lies below the horizontal axis. What can be said about the graph of directed distance? a. It is above the horizontal axis. b. It is below the horizontal axis. ANS: D

PTS: 1

c. It is increasing. d. It is decreasing. DIF:

medium

23. A basketball is tossed onto the hardwood, where it bounces several times. We measure its location as distance up from the floor. When the ball is at the peak of its bounce, what can be said about its velocity? a. Velocity is positive. b. Velocity is negative. ANS: C

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

24. A basketball is tossed onto the hardwood, where it bounces several times. We measure its location as distance down from the ceiling. When the ball is at the peak of its bounce, what can be said about its velocity? a. Velocity is positive. b. Velocity is negative. ANS: C

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

25. A basketball is tossed onto the hardwood, where it bounces several times. We measure its location as distance up from the floor. When the ball is moving upward, what can be said about its velocity? a. Velocity is positive. b. Velocity is negative. ANS: A

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

26. A basketball is tossed onto the hardwood, where it bounces several times. We measure its location as distance down from the ceiling. When the ball is moving upward, what can be said about its velocity? a. Velocity is positive. b. Velocity is negative. ANS: B

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

27. A basketball is tossed onto the hardwood, where it bounces several times. We measure its location as distance up from the floor. When the ball is moving downward, what can be said about its velocity? a. Velocity is positive. b. Velocity is negative. ANS: B

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

28. A basketball is tossed onto the hardwood, where it bounces several times. We measure its location as distance down from the ceiling. When the ball is moving downward, what can be said about its velocity? a. Velocity is positive. b. Velocity is negative. ANS: A

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

29. A man hikes up a mountain to the peak, where he rests for a few minutes and then continues down the other side of the mountain. We measure his location as distance up from the base of the mountain. What can be said about the man’s velocity when he is hiking up the mountain? a. Velocity is positive. b. Velocity is negative. ANS: A

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

30. A man hikes up a mountain to the peak, where he rests for a few minutes and then continues down the other side of the mountain. We measure his location as distance down from the peak of the mountain. What can be said about the man’s velocity when he is hiking up the mountain? a. Velocity is positive.

c. Velocity is 0.

b. Velocity is negative. ANS: B

PTS: 1

d. None of the above. DIF:

medium

31. A man hikes up a mountain to the peak, where he rests for a few minutes and then continues down the other side of the mountain. We measure his location as distance up from the base of the mountain. What can be said about the man’s velocity when he is resting at the peak? a. Velocity is positive. b. Velocity is negative. ANS: C

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

32. A man hikes up a mountain to the peak, where he rests for a few minutes and then continues down the other side of the mountain. We measure his location as distance up from the base of the mountain. What can be said about the man’s velocity when he is hiking down the mountain? a. Velocity is positive. b. Velocity is negative. ANS: B

PTS: 1

c. Velocity is 0. d. None of the above. DIF:

medium

SHORT ANSWER 1. A rock is tossed upward from the ground. It rises and then falls back to the ground. Sketch a graph of distance up from the ground. Be sure to label clearly your graph. ANS:

PTS: 1

DIF: medium

2. A rock is tossed upward from the ground. The graph of distance up is shown. Make a graph of velocity.

. ANS:

PTS: 1

DIF: medium

3. A car accelerates in a westerly direction from a yield sign to a constant speed on the freeway. It then exits the freeway and parks. Measure the location of the car as its distance west of the yield sign. Make a graph of velocity. Be sure to label you graph appropriately.

. ANS:

PTS: 1

DIF: medium

4. A car accelerates in a westerly direction from a yield sign to a constant speed on the freeway. It then exits the freeway and parks. The graph of velocity is shown below. Make a graph of distance west of the yield sign. Be sure to label you graph appropriately.

. ANS:

PTS: 1

DIF: medium

5. An airplane leaves New York and flies south at a constant speed to Miami, where it lands and takes on new passengers. It then makes the return flight at a constant speed. Make a graph that shows distance south of New York over the period of the round-trip flight. Be sure to label your graph appropriately.

. ANS:

PTS: 1

DIF: medium

6. An airplane leaves New York and flies south at a constant speed to Miami, where it lands and takes on new passengers. It then makes the return flight at a constant speed. Make a graph that shows distance north of Miami over the period of the round-trip flight. Be sure to label your graph appropriately.

. ANS:

PTS: 1

DIF: medium

7. An airplane leaves New York and flies south at a constant speed to Miami, where it lands and takes on new passengers. It then makes the return flight at a constant speed. The graph of distance south of New York is shown. Make a graph of velocity for the round-trip. Be sure to label your graph appropriately.

. ANS:

PTS: 1

DIF: medium

8. An airplane leaves New York and flies south at a constant speed to Miami, where it lands and takes on new passengers. It then makes the return flight at a constant speed. The graph of distance north of Miami is shown. Make a graph of velocity for the round-trip. Be sure to label your graph appropriately.

. ANS:

PTS: 1

DIF: medium

9. A man leaves home driving west on a straight road. We locate the position of the car as the distance west from home. Suppose the man begins his trip and sets his cruise control, but after a few minutes he notices that he has forgotten his wallet. He returns home, gets his wallet, and then resumes his journey. He figures that he may be late, so he sets the cruise control a little higher than before. Make a graph of distance west from home. . ANS:

PTS: 1

DIF: medium

10. You leave your apartment at 10:00 am and take a leisurely stroll to get lunch at a pizza place that is 2500 yards west of your home. You have lunch there from 12:00 to 1:00 pm. You return home, arriving at 2:00 pm, and take a one-hour nap. Then you go to a friend's house, which is east of your home. You stay there from 4:00 pm on. Make a graph of distance west from your home versus time. Be sure to label your graph appropriately. ANS:

PTS: 1

DIF: medium

11. A man leaves home driving west on a straight road. We locate the position of the car as the distance west from home. Suppose the man begins his trip and sets his cruise control, but after a few minutes he notices that he has forgotten his wallet. He returns home, gets his wallet, and then resumes his journey. He figures that he may be late, so he sets the cruise control a little higher than before. The graph of distance west from home is shown. Make a graph velocity for the trip.

ANS:

PTS: 1

DIF: medium

ESSAY 1. From atop a building 30 feet high, a rock is tossed upward with an initial velocity of 18 feet per second. The distance , in feet, above the ground after seconds is given by . A: Make a graph of distance up from the ground versus time. B: Make a graph of the velocity of the rock from the time it is tossed until it hits the ground. Be sure to label your graph appropriately.

ANS: A: D

–0.2

0.2 0.4 0.6 0.8

1.2 1.4 1.6 1.8

PTS: 2

DIF: hard

Section 6.2 Rates of Change for Other Functions TRUE/FALSE 1. The rate of change in velocity is acceleration. ANS: T 2.

PTS: 1

DIF:

represents the rate of change of the function ANS: T

3. If

PTS: 1

ANS: F

ANS: T

PTS: 1

easy

is negative. DIF:

easy

DIF:

easy

is 0.

denotes the tax owed on an income of

ANS: T

easy

is decreasing. DIF:

PTS: 1

6. At a minimum value of

7. If

DIF:

PTS: 1

5. At a maximum value of

easy

is certain to be positive.

is negative then the function

ANS: T

with respect to the variable .

DIF:

is positive then the function

ANS: F 4. If

PTS: 1

easy

dollars, then

DIF:

represents the marginal tax rate.

easy

8. The marginal tax rate is the additional tax owed on each additional dollar of income. ANS: T 9. If

PTS: 1

DIF:

easy

denotes the profit earned from an investment of

dollars, then

represents the marginal

profit. ANS: T

PTS: 1

DIF:

easy

10. The marginal profit is the additional profit earned on each additional dollar of investment. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. Consider a function

. When

is increasing, what can be said about

c. is positive.

is zero.

d. is negative.

ANS: A 2. Consider a function

is increasing. PTS: 1 . When

DIF:

medium

is decreasing, what can be said about

c. is positive.

is zero.

d. is negative.

ANS: B 3. Consider a function a.

is increasing. PTS: 1 . If the graph of

DIF:

medium

is decreasing, what can be said about the graph of

The graph of

is above the horizontal axis.

The graph of

is below the horizontal axis.

The graph of

lies on the horizontal axis.

The graph of

is decreasing.

b. c. d.

ANS: B 4. Consider a function a.

PTS: 1 . If the graph of

DIF:

medium

is increasing, what can be said about the graph of

The graph of

is above the horizontal axis.

The graph of

is below the horizontal axis.

The graph of

lies on the horizontal axis.

The graph of

is increasing.

b. c. d.

ANS: A

PTS: 1

DIF:

medium

5. Consider a function

. If the graph of

be said about the graph of

is constant, neither increasing nor decreasing, what can

a. The graph of

is above the horizontal axis.

The graph of

is below the horizontal axis.

The graph of

lies on the horizontal axis.

The graph of

is increasing.

b. c. d.

ANS: C

PTS: 1

6. Consider a function

. If the graph of

be said about the graph of a. b. c. d.

The graph of The graph of The graph of The graph of

ANS: D

graph of a. b. c. d.

ANS: C

is above the horizontal axis. is below the horizontal axis. lies on the horizontal axis. is a straight line.

. If the graph of

DIF:

medium

is above the horizontal axis, what can be said about the

is above the horizontal axis. is below the horizontal axis. is increasing. is a straight line. PTS: 1

8. Consider a function

a. b. c. d.

is constant, neither increasing nor decreasing, what can

The graph of The graph of The graph of The graph of

graph of

medium

PTS: 1

7. Consider a function

DIF:

. If the graph of

DIF:

medium

is below the horizontal axis, what can be said about the

The graph of The graph of The graph of The graph of

ANS: C 9. Consider a function

is above the horizontal axis. is below the horizontal axis. is decreasing. is a straight line. PTS: 1 . Suppose

DIF:

medium

. What can be said about the function

a. is a linear function with slope –9. b. is an exponential function with growth factor 9. c. is a linear function with slope 9. d. None of the above. ANS: C 10. Consider a function

PTS: 1

DIF:

. When

medium

is not changing, what can be said about

c. is positive.

is zero.

d. is negative.

ANS: B

is increasing. PTS: 1

DIF:

medium

11. You are driving with your cruise control set. What can be said regarding your acceleration? a. Acceleration is 0. b. Acceleration is positive. ANS: A

c. Acceleration is negative. d. Acceleration is increasing.

PTS: 1

DIF:

medium

12. You are driving and press on the accelerator to increase your velocity. What can be said regarding your acceleration? a. Acceleration is 0. b. Acceleration is positive. ANS: B

c. Acceleration is negative. d. Acceleration is increasing.

PTS: 1

DIF:

medium

13. You are driving and press on the brake to decrease your velocity. What can be said regarding your acceleration? a. Acceleration is 0. b. Acceleration is positive. ANS: C

c. Acceleration is negative. d. Acceleration is increasing.

PTS: 1

DIF:

medium

14. You have the opportunity to earn some extra income by working at $23 per hour. If your marginal tax rate is 33%, how much of the hourly income will you take home after taxes? a. $23 b. $33 ANS: C 15. Consider a function a. b.

c. $15.41 d. $7.59 PTS: 1 If

DIF: medium is constant and positive, what can be said about the function

is linear with positive slope. is linear with negative slope.

c. d.

is constant and positive. is constant and negative.

ANS: A

PTS: 1

16. Consider a function a. b.

DIF:

is constant and negative, what can be said about the function

is linear with positive slope. is linear with negative slope.

ANS: B

PTS: 1

medium

c. d. DIF:

is constant and positive. is constant and negative. medium

17. The CEO of a tire company has kept records for the profit change

when

tires are produced. The rate of

is the marginal profit. The graph of marginal profit is shown. According to the graph, how

many tires per day should the company be producing in order to maximize profit?

a. 150 b. 270 ANS: A

c. 30 d. None of the above PTS: 1

DIF:

medium

18. The CEO of a tire company has kept records for the profit change

when

tires are produced. The rate of

is the marginal profit. The graph of marginal profit is shown. If the current rate of

production is 180 tires, what effect on profit will an increase in production have?

a. Profits will be increased. b. Profits will remain the same. ANS: C

PTS: 1

c. Profits will be reduced. d. None of the above DIF:

medium

19. You are considering buying three stocks, Stock 1, Stock 2, and Stock 3, and their prices at time are given by

, and

, respectively. You observe that

is positive,

is near 0, and

is negative. Which stock is likely to provide the best growth in the short term?

a. Stock 1 b. Stock 2 ANS: A 20. Let

c. Stock 3 d. None of the above. PTS: 1

denote sales when

DIF: medium dollars are spent on advertising. If

is positive, what will be the

effect on sales if advertising is increased?

a. Sales will increase. b. Sales will decrease. ANS: A

PTS: 1

c. Sales will remain the same. d. None of the above DIF:

medium

21. Let

denote sales when

dollars are spent on advertising. If

is negative, what will be the

effect on sales if advertising is increased?

a. Sales will increase. b. Sales will decrease. ANS: B 22. Let

c. Sales will remain the same. d. None of the above

PTS: 1 denote sales when

DIF:

dollars are spent on advertising. We find that

maximum value. What is the value of

is positive.

is negative.

ANS: C 23. Let

medium

c. d. PTS: 1

DIF:

is 0. The value of medium

to negative at

What can be concluded about the graph of

a. b. c. d.

reaches a maximum at reaches a minimum at is increasing at is decreasing at

ANS: A 24. Let

PTS: 1

at the point

medium is positive at

What

is negative at

What

The price of groceries reaches a maximum at The price of groceries reaches a minimum at The price of groceries is increasing at The price of groceries is decreasing at

ANS: C 25. Let

DIF:

changes sign from positive

at the point

denote the price of a bag of groceries at time . It is found that

can be concluded about the graph of

a. b. c. d.

cannot be determined.

denote the price of a bag of groceries at time . It is found that

The graph of The graph of The graph of The graph of

has reached its

PTS: 1

DIF:

medium

denote the price of a bag of groceries at time . It is found that

can be concluded about the graph of

at the point

a. b. c. d.

The price of groceries reaches a maximum at The price of groceries reaches a minimum at The price of groceries is increasing at The price of groceries is decreasing at

ANS: D 26. Let

PTS: 1

medium

denote the price of a bag of groceries at time . It is found that

negative to positive at

a. b. c. d.

DIF:

The graph of The graph of The graph of The graph of

ANS: B

What can be concluded about the graph of

changes sign from at the point

reaches a maximum at reaches a minimum at is increasing at is decreasing at PTS: 1

DIF:

medium

SHORT ANSWER 1. You are driving with your cruise control set when you encounter a slowly moving truck. You speed up to pass the truck. When you have overtaken the truck, you slow down and resume your previous speed. If represents your velocity at time , make a graph of . Be sure to label your graph appropriately. ANS:

PTS: 1

DIF: medium

2. You are driving with your cruise control set when you encounter a slowly moving truck. You speed up to pass the truck. When you have overtaken the truck, you slow down and resume your previous speed. The graph of velocity,

, is shown. Make a graph of

, the acceleration. Be sure to label

your graph appropriately.

ANS:

PTS: 1

DIF: medium

3. A balloon is initially full of air, but then it springs a leak. Air leaks at a constant rate from the balloon. After a while the leak is patched and a pump is attached to the balloon. It is re-inflated to its original volume: Air is pumped at a constant rate that is larger than the rate of the leak. Let denote the volume, in cubic inches, of air in the balloon after minutes. Make a graph of versus from the time the balloon springs a leak until it is fully re-inflated. ANS:

PTS: 1

DIF: medium

4. A balloon is initially full of air, but then it springs a leak. Air leaks at a constant rate from the balloon. After a while the leak is patched and a pump is attached to the balloon. It is re-inflated to its original volume: Air is pumped at a constant rate that is larger than the rate of the leak. Let denote the volume, in cubic inches, of air in the balloon after minutes. The graph of versus from the time the balloon springs a leak until it is fully re-inflated is shown. Make the graph of

ANS:

PTS: 1

DIF: medium

5. The acceleration due to gravity near the surface of Earth has a constant value of 32 feet per second per second. Explain what this fact tells us about the downward velocity of a falling object near the surface of Earth. ANS: Velocity is a linear function with slope 32 feet per second per second. PTS: 1

DIF: medium

ESSAY 1. The CEO of a tire company has kept records for the profit change

when

tires are produced. The rate of

is the marginal profit.

A: Explain in practical terms the meaning of the marginal profit. B: What action should be taken if the marginal profit is positive? C: What action should be taken if the marginal profit is negative? ANS: A: The marginal profit is the additional profit expected from the production of one additional tire. B: Production should be increased. C: Production should be decreased. PTS: 3

DIF: hard

2. A balloon is initially full of air, but then it springs a leak. It leaks at a constant rate of 4 cubic inches per minute. After a while the leak is patched, a pump is attached, and the balloon is re-inflated to its original volume. The pumping rate is constant at 6 cubic inches per minute. Let denote the volume, in cubic inches, of air in the balloon after minutes. A: Explain in practical terms the meaning of

B: During the time the balloon is leaking air, what is the value of

C: During the time air is being pumped into the balloon, what is the value of ANS: A:

is the change in volume we expect over one minute.

cubic inches per minute.

PTS: 3

DIF: hard

3. Consider the function A: Make a graph of B: At

, is

. . Use a horizontal span of 0 to 5.

positive or negative?

C: For what value of

ANS: A: f 70 60 50 40 30 20 10

–1

1 –10

is positive.

when

PTS: 3

. DIF: hard

4. You are hiking in a hilly region, and

is your elevation at time .

A: Explain in practical terms the meaning of

B: Where might you be when C: Where might you be when D: Where might you be when

is a large positive number? ? is a large negative number?

ANS: A:

is the change in elevation we expect over one unit of time.

B: You might be ascending a hill. C: You might be at the peak of a hill, at the bottom of a valley, or stopped. D: You might be descending a hill. PTS: 4

DIF: hard

5. The gas mileage

that your car gets depends on its age .

A: Explain in practical terms the meaning of

B: As your care ages, gas mileage decreases. Would

be positive or negative?

C: Your mechanic tells you that if you use his super gasoline additive, your gas mileage will improve over time. Assuming the additive works, would

be positive or negative once you start using the

additive? ANS: A:

is the change in your gas mileage we expect over one unit of time.

is negative.

would be positive.

PTS: 3

DIF: hard

6. Your company produces at most 15 items per week. Its profit items produced in a week. The marginal profit is given by

is a function of , the number of

A: Make a graph of

versus .

B: For what values of

is the profit increasing?

C: How many items per week should be produced in order to maximize profit? ANS: A: dP/dN 100 80 60 40 20 –6

–3 –20

–40 –60 –80 –100

B: C: 12 items PTS: 3

DIF: hard

7. Let denote the balance of an account rate of change of B:

years after the initial investment. Here is a formula for the

A: Make a graph of

versus

over the first 10 years of the investment.

B: During what time period is the value of the investment decreasing? C: When does the investment reach a minimum value? ANS: A: dB/dt 300

200

100

–1

–100

–200

–300

B: The value is decreasing over the first 5.16 years of the investment. C: After 5.16 years PTS: 3 8. Let

DIF: hard

denote the number of raccoons present in a certain area at time .

A: From 1990 to 2000 favorable conditions allow the population to increase at a constant rate of 58 animals per year. What can be said about

during this time period?

B: From 2000 to 2005 drought caused the food supply to diminish, and the population size decreased. What can be said about

during this time period?

C: From 2005 on the racoon population remained stable. It neither increased nor decreased. can be said about

during this time period?

ANS: A:

animals per year.

is negative during this period.

What

C: PTS: 3

DIF: hard

9. The following graph shows a man’s height

A: Explain in practical terms the meaning of

B: During what period is

, in inches, at age

years.

positive?

ANS: A:

is the change in the man’s height we expect over one year.

positive up to age 20.

PTS: 2

DIF: hard

Section 6.3 Estimating Rates of Change TRUE/FALSE 1. The value of

can be approximated using the average rate of change for

ANS: T

PTS: 1

DIF:

2. To get a good estimate of the value of average rate of change for ANS: T 3. The value of

easy

we should use a very short interval to calculate the

PTS: 1

DIF:

easy

can be calculated exactly from a table of values.

ANS: F

PTS: 1

DIF:

easy

4. When a function reaches a maximum value (at a peak), the rate of change is 0. ANS: T

PTS: 1

DIF:

easy

MULTIPLE CHOICE 1. If

is the linear function

then

is:

a. 3

b. 6

d. 0

ANS: A 2. If

PTS: 1

is the linear function

ANS: C

PTS: 1 and

is:

DIF:

medium

. Use this information to estimate the value of

a. 0 b. –10 ANS: D

then

medium

c. –5 d. 0

a. 5 b. –4

3. Suppose

DIF:

c. 2 d. 10 PTS: 1

DIF:

medium

when

4. Suppose

and

. Use this information to estimate the value of

a. 0 b. 150 PTS: 1 and

DIF:

PTS: 1 and

. Use this information to estimate the value of

PTS: 1

DIF: medium

a. 0 b. 4.92

PTS: 1

DIF: medium

a. 0.08 b. 4.02

c. 0 d. None of the above PTS: 1

DIF: medium

9. Use an increment of 0.3 to estimate the value of a. –0.13 b. –0.04

c. 0 d. None of the above PTS: 1

DIF: medium

10. Use an increment of 0.03 to estimate the value of a. 0.03 b. 1.02

c. 24.6 d. None of the above

8. Use an increment of 0.02 to estimate the value of

ANS: A

when

c. –100 d. –2

7. Use an increment of 0.2 to estimate the value of

ANS: B

DIF: medium

a. 0 b. 100

ANS: C

when

c. –1 d. –5

ANS: D

ANS: C

medium

. Use this information to estimate the value of

a. 0 b. 5

6. Suppose

c. 3 d. –150

ANS: B 5. Suppose

when

c. 0 d. None of the above

ANS: B

PTS: 1

DIF:

11. Use an increment of 0.3 to estimate the value of a. 0.44 b. 1.47

medium at

c. 0 d. None of the above

ANS: B

PTS: 1

DIF: medium

12. Use the following table of values to estimate the value of 1 5

3 7

a. 4 b. 1.33

6 11

9 13

c. 0 d. None of the above PTS: 1

ANS: B

DIF: medium

13. Use the following table of values to estimate the value of 1 5

3 7

a. 1.33 b. 4

6 11

9 13

c. 0 d. None of the above

ANS: A

PTS: 1

DIF: medium

14. Use the following table of values to estimate the value of 1 8

3 3

a. –2 b. –0.67

6 1

9 5

c. 0 d. None of the above

ANS: B

PTS: 1

DIF: medium

15. Use the following table of values to estimate the value of 1 8

3 3

a. –0.67 b. –2 ANS: A

6 1

9 5

c. 0 d. None of the above. PTS: 1

DIF:

medium

16. Use the following table of values to estimate the value of 1.6

3.7

at 7.2

. 11.8

6.6

12.4

21.5

a. 2.6 b. 9.1 ANS: A

c. 0 d. None of the above PTS: 1

DIF: medium

17. Use the following table of values to estimate the value of 1.6 6.6

3.7 12.4

PTS: 1

2.7 6.6

DIF: medium

5.5 5.7

8.6 1.4

c. 0 d. None of the above PTS: 1

DIF:

medium

19. Use the following table of values to estimate the value of 2.7 6.6

5.5 5.7

at 7.2 2.1

a. –2.12 b. –3.6

. 8.6 1.4

c. 0 d. None of the above PTS: 1

DIF:

20. A balloon is floating upward. Its height in feet We know that

7.2 2.1

a. –2.12 b. –3.6

ANS: A

11.8 32.6

c. 0 d. None of the above

18. Use the following table of values to estimate the value of

ANS: A

7.2 21.5

a. 2.6 b. 9.1 ANS: A

32.6

and

is 4 when

medium seconds after release is given by the function

is 3. Use this information to estimate the height of the

balloon 8 seconds after release. a. 25 feet b. 17 feet ANS: C

c. 33 feet d. None of the above PTS: 1

DIF: medium

21. Water is draining from a pond. The amount of water, in gallons, remaining in the pond after draining begins is given by the function

. We know that

and

minutes is –599

when is 9. Use this information to estimate the amount of water in the pond 16 minutes after draining begins.

a. 2387 gallons b. 1195 gallons ANS: B

c. 4789 gallons d. None of the above PTS: 1

DIF:

medium

22. A ball is tossed upward and then falls to the ground. Its height after function a. b.

. During the period the rock is rising, what can be said about c.

is positive.

is zero.

PTS: 1

DIF:

medium

23. A ball is tossed upward and then falls to the ground. Its height after function

is positive.

d. None of the above

is negative.

function

is zero.

PTS: 1

DIF:

medium

24. A ball is tossed upward and then falls to the ground. Its height after

seconds is given by the

. During the period the rock is falling, what can be said about

ANS: B

d. None of the above

is negative.

ANS: A

seconds is given by the

. At the time the rock reaches its peak, what can be said about c.

is positive.

is zero.

d. None of the above

is negative.

ANS: C

PTS: 1

25. The profit earned when

DIF:

medium

items are produced is given by the function

will result in an increase in profits, what can be said about marginal profit

is positive.

is negative.

ANS: A

c. d. PTS: 1

26. The profit earned when

DIF:

. If increasing production ?

is zero. is linear. medium

items are produced is given by the function

will result in a decrease in profits, what can be said about marginal profit

. If increasing production ?

is positive.

is negative.

ANS: B

d. PTS: 1

DIF:

is zero. is linear. medium

27. The profit earned when items are produced is given by the function . If either increasing production or decreasing production will result in a decrease in profits, what can be said about marginal profit

is positive.

is negative.

ANS: C

c. d. PTS: 1

DIF:

28. A population changes with time. Its level after

is zero. is linear. medium years is given by the function

. Over a certain

period, the population is growing. What can be said about over this period? c. a. is positive. is zero. b.

is negative.

ANS: A

PTS: 1

DIF:

29. A population changes with time. Its level after

is linear. medium years is given by the function

. Over a certain

period, the population is declining. What can be said about over this period? c. a. is positive. is zero. b.

is negative.

ANS: B

PTS: 1

DIF:

30. A population changes with time. Its level after

is linear. medium years is given by the function

. After a period

of population increase, the population reaches a maximum value. What can be said about time? a.

is positive.

is negative.

ANS: C

c. d. PTS: 1

DIF:

is zero. is linear. medium

at this

31. A population changes with time. Its level after

years is given by the function

. After a period

of population decrease, the population reaches a minimum value. What can be said about

at this

time? a.

is positive.

is negative.

ANS: C

is zero.

d. PTS: 1

DIF:

is linear. medium

SHORT ANSWER 1. If

, use the calculator to get the value of

Suggestion: Make a graph of f

using a horizontal span of 0 to 5. ANS: 0.12 PTS: 1 2. If

DIF: medium , use the calculator to get the value of

Suggestion: Make a graph of

f using a horizontal span of 0 to 5. ANS: 0.58 PTS: 1 3. If

DIF: medium , use the calculator to get the value of

Suggestion: Make a graph of f

using a horizontal span of 0 to 3. ANS: 6.07 PTS: 1 4. If

DIF: medium , use the calculator to get the value of

at . Suggestion: Make a graph of f using a

horizontal span of 0 to 5. ANS: –0.89 PTS: 1 5. If

DIF: medium , use the calculator to get the value of

using a horizontal span of 0 to 5.

. Suggestion: Make a graph of f

ANS: –0.76 PTS: 1

DIF: medium

ESSAY 1. The following table shows the cubic inches leaking started. 0 814

of air remaining in a leaky balloon

3 704

6 605

A: Explain in practical terms the meaning of

B: Use the average rate of change from

seconds after the

9 422

to estimate

C: Use your answer from part B to estimate the amount of air remaining in the balloon after 5 seconds. ANS: A: B: At

is the change in the volume of the balloon we expect over one second. ,

is approximately –33 cubic inches per second.

C: 638 cubic inches PTS: 3

DIF: hard

2. The following table shows the population 0 41

of reindeer on an island

5 169

10 657

A: Explain in practical terms the meaning of

B: Use the average rate of change from

years after 1945. 15 2612

to estimate

C: Use your answer from part B to estimate the number of reindeer present in 1953. Round your answer to the nearest whole number. ANS: A:

is the change in the reindeer population expected over one year.

is approximately 97.6 reindeer per year.

C: 462 reindeer PTS: 3

DIF: hard

3. The following table shows, for males ages 55 to 64, the number of heart disease deaths at time years after 2000. 0 375

3 335

4 317

per 100,000

7 288

A: Use the average rate of change from 2004 to 2007 to estimate the value of

for 2004.

B: Explain in practical terms the meaning of the number you found in part A. C: Use your answer from part A to estimate the number heart disease deaths per 100,000 males in 2006. Round your result to the nearest whole number. ANS: A:

is approximately –9.67 deaths per 100,000 per year.

B: In 2004 the number of heart disease deaths among males was declining by 9.67 per 100,000 over the year. C: 298 deaths per 100,000 PTS: 3

DIF: hard

4. The following table shows, for females ages 55 to 64, the number of heart disease deaths 100,000 at time years after 2000. 0 152

3 146

4 138

per

7 117

A: Use the average rate of change from 2004 to 2007 to estimate the value of

for 2004.

B: Explain in practical terms the meaning of the number you found in part A. C: Use your answer from part A to estimate the number heart disease deaths per 100,000 females in 2006. Round your result to the nearest whole number. ANS: A:

is approximately –7 deaths per 100,000 per year.

B: In 2004 the number of heart disease deaths among males was declining by 7 per 100,000 over the year.

C: 124 deaths per 100,000 PTS: 3

DIF: hard

5. The height , in feet, of a cannonball

feet downrange is given by

. A: Plot the graph of the flight of the cannonball. Use a horizontal span of 0 to 2000. B: Use the calculator to estimate the value of

at 1595 feet downrange.

C: Explain in practical terms what the number you found in part B tells us. ANS: A:

feet per foot

C: At 1595 feet downrange, the cannonball is dropping 0.63 vertical foot for each horizontal foot. PTS: 3

DIF: hard

6. When a man with a parachute jumps from an airplane, he will fall seconds.

feet in

A: Plot the graph of the graph of distance fallen versus time. Use a horizontal span of 0 to 10 seconds. B: Use the calculator to estimate the value of

at 5 seconds after he jumps from the plane.

C: Explain in practical terms what the number you found in part B tells us. ANS: A:

B: 19.99 feet per second C: After 5 seconds the man is falling 19.99 feet for each second. This is his velocity after 5 seconds. PTS: 3

DIF: hard

7. Taking into account air resistance, a man will fall

feet in

seconds. The relationship is .

A: Plot the graph of the graph of distance fallen versus time. Use a horizontal span of 0 to 10 seconds. B: Use the calculator to estimate the value of

at 5 seconds into the fall.

C: Explain in practical terms what the number you found in part B tells us. ANS: A:

B: 105.16 feet per second C: After 5 seconds the man is falling 105.15 feet for each second. This is his velocity after 5 seconds. PTS: 3 8. The temperature

DIF: hard of a potato

minutes after being placed in the oven is given by .

A: Plot the graph of the graph of temperature versus time over the first hour of baking. B: Use the calculator to estimate the value of

after 15 minutes of baking.

C: Use the calculator to estimate the value of

after 30 minutes of baking.

D: Explain in practical terms what the answers from parts B and C tell you about how the potato bakes over time. ANS: A:

B: 4.82 degrees per minute C: 3.57 degrees per minute D: The temperature of the potato increases more rapidly early in the baking process than it does later. PTS: 4

DIF: hard

9. The profit , in thousands of dollars, for a certain company depends on the number produced. The relationship is given by .

of items

A: Plot the graph of the profits versus number of items produced. B: What number of items produced will result in maximum profit? C: The marginal profit is the rate of change you found in part B? ANS: A:

. What is the value of

at the production level

B: 13 C: 0 PTS: 3

DIF: hard

10. Consider the function

A: Estimate the value of

by calculating the average rate of change from

B: Estimate the value of

by calculating the average rate of change from

C: Which of the numbers you calculated in parts A and B is likely to give a better estimate for the true value of

ANS: A: 7 B: 6.5 C: The answer in part B. PTS: 3

DIF: hard

Section 6.4 Equations of Change: Linear and Exponential Functions TRUE/FALSE 1. If slope

satisfies the equation of change

PTS: 1

ANS: T

is a linear function with

is a constant, then

is an exponential

easy

where

PTS: 1

is a linear function with slope

ANS: F 4. If

DIF:

satisfies the equation of change

function with growth factor

3. If

is a constant, then

ANS: T 2. If

where

DIF: , then

PTS: 1

easy

satisfies the equation of change DIF:

easy

is an exponential function with growth factor

, then

satisfies the equation of change

. ANS: T

PTS: 1

DIF:

easy

5. Both an equation of change and an initial value are required to determine a function. ANS: T

PTS: 1

DIF:

easy

6. Exponential population growth satisfies an equation of change of the form ANS: T 7. The difference

PTS: 1

DIF:

easy

between the temperature of an oven and the temperature of an object baking in the

oven satisfies an equation of change of the form ANS: T 8. The difference

PTS: 1

DIF:

MULTIPLE CHOICE

. easy

between the temperature of a hot but cooling cup of tea and room temperature

satisfies an equation of change of the form ANS: T

PTS: 1

. DIF:

easy

1. If a. b.

satisfies the equation of change is linear with slope .

2. If a. b.

a. b.

PTS: 1

DIF:

is exponential with decay factor

c. d.

PTS: 1

DIF:

is exponential with growth factor

medium

is linear with slope is linear with slope

c. d.

PTS: 1

DIF:

is exponential with decay factor is exponential with decay factor

, then what can be said about the function

is linear with slope . is linear with slope .

ANS: A

d. DIF:

medium , and the initial value of

c. d. DIF:

is exponential with growth factor . is exponential with growth factor .

PTS: 1

medium

satisfies the equation of change

a. b.

? .

, then what can be said about the function

for the function

is exponential with growth factor

satisfies the equation of change

medium

. .

PTS: 1

is exponential with decay factor

is linear with slope is linear with slope

, then what can be said about the function

ANS: C 6. If

is exponential with growth factor medium

c. d.

is exponential with growth factor .

satisfies the equation of change

ANS: D 5. If

DIF:

is linear with slope is linear with slope

ANS: B 4. If

PTS: 1

satisfies the equation of change

ANS: B 3. If

c. d.

is linear with slope

ANS: A

, then what can be said about the function

medium

is 5.93, find a formula

7. If

satisfies the equation of change

function

c. d.

ANS: A

PTS: 1

satisfies the equation of change

the function

, and the initial value of

is 2.7, find a formula for

d. PTS: 1

satisfies the equation of change

function

DIF:

medium

, and the initial value of

is 8, find a formula for the

a. b.

c. d.

ANS: D

PTS: 1

satisfies the equation of change

the function

DIF:

medium

, and the initial value of

is 2.1, find a formula for

a. b.

c. d.

ANS: D 11. If

medium

ANS: C

10. If

DIF:

a. b.

9. If

is 8, find a formula for the

a. b.

8. If

, and the initial value of

PTS: 1

satisfies the equation of change

the function

medium

, and the initial value of

is 9.38, find a formula for

a. b. ANS: D

DIF:

c. d. PTS: 1

DIF:

medium

12. Suppose

is a linear function with slope 10. Find the equation of change for

b. ANS: A

d. PTS: 1

DIF:

medium

13. Suppose

is a linear function with slope 8.01. Find the equation of change for

ANS: A

PTS: 1

DIF:

medium

14. Suppose

is a linear function with slope –6.2. Find the equation of change for

ANS: C 15. Suppose

PTS: 1

DIF:

ANS: A 16. Suppose

PTS: 1

DIF:

ANS: D 17. Suppose

PTS: 1

DIF:

ANS: D

PTS: 1

DIF:

. Find the equation of change for

medium

is an exponential function with growth factor

. Find the equation of change for

medium

is an exponential function with growth factor

medium

is an exponential function with growth factor

. Find the equation of change for

medium

18. A balloon is being inflated at a constant rate of 4 liters of air per minute. If is the volume, in liters, of the balloon after minutes, write an equation of change that is satisfied by .

b. ANS: A

d. PTS: 1

DIF:

medium

19. A balloon is leaking air at a constant rate of 6 liters of air per minute. If is the volume, in liters, of the balloon after minutes, write an equation of change that is satisfied by . a.

ANS: B

PTS: 1

20. The rate of growth of a population of satisfied by .

DIF:

medium

individuals is

. Write an equation of change that is

ANS: C

PTS: 1

DIF:

medium

21. The rate of growth in price of an item that costs that is satisfied by .

dollars is

. Write an equation of change

ANS: D

PTS: 1

DIF:

medium

22. If a rock falls near the surface of Earth, it has a constant downward acceleration of 9.8 meters per second per second. If is the downward velocity, in meters per second, of the rock after seconds, then satisfies which equation of change? a.

ANS: D

PTS: 1

DIF:

medium

23. If a rock falls near the surface of Mars, it has a constant downward acceleration of 3.77 meters per second per second. If is the downward velocity, in meters per second, of the rock after seconds, then satisfies which equation of change?

ANS: D

PTS: 1

DIF:

medium

24. If a rock is tossed upward near the surface of Earth, then it has a constant downward acceleration of –9.8 meters per second per second. If is the upward velocity, in meters per second, of the rock after seconds, then satisfies which equation of change? a.

ANS: D

PTS: 1

DIF:

medium

25. If a rock is tossed upward near the surface of Mars, then it has a constant downward acceleration of –3.77 meters per second per second. If is the upward velocity, in meters per second, of the rock after seconds, then satisfies which equation of change? a.

ANS: D

PTS: 1

DIF:

medium

26. The water level in a tank rises 6 inches every minute. Write an equation of change that describes the height , in inches, of the water at time minutes. a.

ANS: C

PTS: 1

DIF:

medium

27. Which of the following are described by an equation of change of the form a. b. c. d.

Exponential population growth Balance of an account subject to compound interest Radioactive decay All of the above

ANS: D

PTS: 1

DIF:

medium

28. Which of the following are described by an equation of change of the form a. Height of snow on the ground when snow is falling at a constant rate. b. Depth of water in a cylindrical glass when water is added at a constant rate.

c. Money in a penny bank when pennies are added on a regular basis. d. All of the above ANS: D

PTS: 1

DIF:

medium

SHORT ANSWER 1. Suppose that of

satisfies the equation of change

. The initial value of

is 2. Make a graph

. The initial value of

is 16. Make a

. Use a horizontal span of 0 to 5.

ANS: f 18 16 14 12 10 8 6 4 2 –1

–2

–4

PTS: 1

DIF: medium

2. Suppose that graph of

satisfies the equation of change

. Use a horizontal span of 0 to 5.

ANS: f 18 16 14 12 10 8 6 4 2 –1

–2

–4

PTS: 1

DIF: medium

3. Suppose that graph of

satisfies the equation of change

. The initial value of

is 16. Make a

. The initial value of

is 2.1. Make a

. Use a horizontal span of 0 to 5.

ANS: f 18 16 14 12 10 8 6 4 2 –1

–2

–4

PTS: 1

DIF: medium

4. Suppose that graph of

satisfies the equation of change

. Use a horizontal span of 0 to 5.

ANS: f 18 16 14 12 10 8 6 4 2 –1

–2

–4

PTS: 1

DIF: medium

ESSAY 1. If a rock is thrown downward near the surface of Mercury, then the downward velocity per second, of the rock after seconds satisfies the equation of change .

, in meters

A: What can you conclude about the function

B: If the rock is thrown downward with an initial velocity of 6 meters per second, find a formula that gives the velocity of the rock after seconds. C: What is the acceleration due to gravity on the surface of Mercury? ANS: A: is a linear function with slope 3.59. B: C: 3.59 meters per second per second PTS: 3

DIF: hard

2. Let denote the temperature, in degrees, of a cup of coffee seconds after it is placed on the counter to cool. Let denote, the difference, in degrees, between the temperature of the coffee and room temperature. Then satisfies the equation of change

A: What can you conclude about the function

B: If the temperature of the coffee is initially 197 degrees, find a formula that gives Write your answer using the alternative form for an exponential function. C: Find a formula that gives

in terms of .

ANS: A: is an exponential function with decay factor

B: C: PTS: 3

DIF: hard

3. Let

denote the temperature, in degrees, of a cake minutes after it is placed in the oven. Let denote the difference, in degrees, between the temperature of the oven and the temperature of the cake. Then satisfies the equation of change

A: What can you conclude about the function

B: If the temperature of the cake batter is initially 73 degrees, find a formula that gives . Write your answer using the alternative form for an exponential function. C: Find a formula that gives

in terms of

in terms of .

ANS: A: is an exponential function with decay factor

B: C: PTS: 3

DIF: hard

4. A certain animal grows to a maximum length of 24 inches. Its length, in inches, at age years is given by the function . The difference between the maximum length and the current length satisfies the equation of change

A: If the animal is 4 inches long when it is born, find a formula that gives the length difference D as a function of the age t. Write your answer using the alternative form for an exponential function. B: Find a formula that gives the length in terms of the age. C: What is the animal’s length at age 6 years? ANS: A: B: C: 9.18 inches PTS: 3

DIF: hard

5. There were 8 inches of snow on the ground before the latest snowstorm. Once the storm started, the depth , in inches, of snow on the ground satisfies the equation of change .

Here

is measured in hours.

A: Find a formula that gives the depth of snow

hours after the snow storm begins.

B: Make a graph of depth versus time over the first 5 hours of the storm. C: How long will it take for the depth of the snow to reach 20 inches? ANS: A: B: D 24 21 18 15 12 9 6 3 –1

–3

–6 –9

C: 4 hours PTS: 3

DIF: hard

6. The amount remaining , in grams, of a radioactive substance is a function of time years. The equation of change for is

measured in

A: What can you conclude about the function

B: If there are initially 44 grams of the substance present, find a formula that gives Write your answer using the alternative form for an exponential function. C: How long will it take before only 1/3 of the original amount is left? ANS: A: is an exponential function with decay factor B: C: 27.47 years PTS: 3

DIF: hard

in terms of .

7. If a rock is tossed upward from ground level, its upward velocity equation of change

, in feet per second, satisfies the

. Here the time t is measured in seconds. Suppose the rock is tossed upward with an initial velocity of 153 feet per second.

A: Write a formula that gives the velocity

seconds after the upward toss.

B: How many seconds after the toss does the rock reach the peak of its flight? Suggestion: What is the velocity of the rock when it reaches the peak of its flight? C: How many seconds after the rock is tossed upward does it strike the ground? ANS: A: B: 4.78 seconds C: 9.56 seconds PTS: 3

DIF: hard

8. You open an account by investing $646 with a financial institution that advertises an APR of 6% compounded continuously.

A: Write an equation of change satisfied by the balance B: Find a formula that gives the balance after an exponential function.

(t).

years. Write your answer using the alternative form for

C: How long will it be before the account value doubles? ANS: A: B: C: 11.55 years PTS: 3

DIF: hard

9. A balloon is leaking air, and the rate of change in volume is proportional to the volume. Let denote the volume, in liters, of air in the balloon after seconds, and take –0.04 to be the constant of proportionality. A: Write an equation of change satisfied by

B: If there are initially 6 liters of air in the balloon, find a formula that gives Write your answer using the alternative form for an exponential function.

as a function of .

C: How long will it take for 10% of the initial volume of air to leak from the balloon? ANS: A: B: C: 2.63 seconds PTS: 3

DIF: hard

10. Two magazines were introduced at the same time. Magazine 1 had an initial circulation of 200, and magazine 2 had an initial circulation of 700. The circulation for magazine 1 satisfies the equation of change . The circulation

for magazine 2 satisfies the equation of change .

Here is measured in months. In the following questions, if your answer involves an exponential function write it using the alternative form for an exponential function. A: Find a formula that gives the circulation of magazine 1 after months. B: Find a formula that gives the circulation of magazine 2 after months. C: When will the circulations for the two magazines be the same? ANS: A: B: C: 18.61 months PTS: 3

DIF: hard

Section 6.5 Equations of Change: Graphical Solutions TRUE/FALSE 1. Equilibrium or steady-state solutions for an equation of change are solutions that remain constant. ANS: T

PTS: 1

DIF:

easy

2. Equilibrium or steady-state solutions for an equation of change occur where the rate of change is 0. ANS: T

PTS: 1

DIF:

easy

3. The carrying capacity is an equilibrium solution for the equation of change for logistic growth. ANS: T

PTS: 1

DIF:

4. At the maximum value, a peak, for a function ANS: T

PTS: 1

, the rate of change

DIF:

5. At the minimum value, a valley, for a function ANS: T 6. When

PTS: 1 is negative, the function

ANS: F 7. When

PTS: 1 is positive, the function

ANS: T

PTS: 1

easy

easy , the rate of change

DIF:

easy

is also negative. DIF:

easy

is increasing. DIF:

easy

MULTIPLE CHOICE 1. Find the equilibrium solution of the equation of change a. b. ANS: A

c. d. PTS: 1

DIF:

medium

2. Find the equilibrium solution of the equation of change a.

is 0.

. c.

is 0.

ANS: A

PTS: 1

DIF:

medium

3. Find the equilibrium solution of the equation of change a. b.

c. d.

ANS: B

PTS: 1

DIF:

medium

4. Find the equilibrium solution of the equation of change a. b.

c. d.

ANS: D

PTS: 1

DIF:

medium

5. Find the equilibrium solutions of the equation of change a. b.

c. d.

and and

ANS: D

PTS: 1

DIF:

and and medium

6. Find the equilibrium solutions of the equation of change a. b.

and and

ANS: C

c. d. PTS: 1

DIF:

. and and

medium

7. Find the equilibrium solutions of the equation of change a. b.

and and

ANS: C

PTS: 1

c. d. DIF:

and and medium

8. Find the equilibrium solutions of the equation of change a. b. ANS: C

and and

c. d. PTS: 1

DIF:

. and and

medium

9. Find the equilibrium solutions of the equation of change a. b.

and and

ANS: C

c. d. PTS: 1

DIF:

and and medium

10. Find the equilibrium solution of the equation of change a. b.

c. d.

ANS: D

PTS: 1

DIF:

medium

11. Find the equilibrium solution of the equation of change

. Consider only positive

values of . a. b.

c. d.

ANS: C

PTS: 1

DIF:

medium

12. Find the equilibrium solution of the equation of change

. Consider only positive

values of . a. b.

c. d.

ANS: C 13. The function of

versus

PTS: 1

medium

satisfies the equation of change

. For what values of

is the graph

. For what values of

is the graph

increasing?

a. b.

c. d.

ANS: C 14. The function of

DIF:

versus

PTS: 1

medium

satisfies the equation of change increasing?

a. b. ANS: A

DIF:

c. d. PTS: 1

DIF:

medium

15. The function of

versus

satisfies the equation of change

. For what values of

is the graph

. For what value of

increasing?

a. b.

c. d.

ANS: A

PTS: 1

16. The function

DIF:

medium

satisfies the equation of change

increasing at the fastest rate? a. b.

c. d.

ANS: C

PTS: 1

17. The function

DIF:

medium

satisfies the equation of change

. For what value of

. For what values of

is the

increasing at the fastest rate? a. b.

c. d.

ANS: D

PTS: 1

18. The function graph of

versus

DIF:

medium

satisfies the equation of change decreasing?

a. b.

c. d.

ANS: B 19. If

PTS: 1

DIF:

medium

is an equilibrium solution of an equation of change involving

when

? c. –8 d. None of the above

a. 0 b. 8 ANS: A

, what is the value of

PTS: 1

20. Consider the equation of change

DIF:

medium

. What can you say about the graph of

? a. The graph is increasing. b. The graph is decreasing. ANS: B

PTS: 1

c. The graph reaches a maximum. d. The graph reaches a minimum. DIF:

medium

versus

when

21. Consider the equation of change

. What can you say about the graph of

versus

when

? a. The graph is increasing. b. The graph is decreasing. ANS: A

PTS: 1

22. The amount

c. The graph reaches a maximum. d. The graph reaches a minimum. DIF:

medium

, in liters, of liquid remaining in a tank after

hours satisfies the equation of change

. How much liquid is in the tank after a long time? a. 5 liters b. 30 liters

c. 1.2 liters d. 6 liters PTS: 1

ANS: B 23. The amount

DIF: medium

, in liters, of liquid remaining in a tank after

hours satisfies the equation of change

. What is happening to the liquid in the tank when there are 70 liters present? a. The amount of liquid is at a minimum. b. The amount of liquid in increasing. ANS: B

PTS: 1

24. The amount

c. The amount of liquid is deceasing. d. The amount of liquid is at a maximum. DIF:

medium

, in liters, of liquid remaining in a tank after

hours satisfies the equation of change

. What is happening to the liquid in the tank when there are 44 liters present? a. The amount of liquid is at a minimum. b. The amount of liquid in increasing. ANS: C

PTS: 1

c. The amount of liquid is deceasing. d. The amount of liquid is at a maximum. DIF:

medium

25. A certain population grows according to

. What is the carrying capacity of the

environment for this particular population? a. 572 b. 0.07 ANS: A

c. 40 d. 8171 PTS: 1

DIF: medium

26. A certain population grows according to

779 individuals. What can be said about the population at this time? a. b. c. d.

The population is increasing. The population is decreasing. The population has reached a maximum value. The population has reached a minimum value.

At a certain time the population is

ANS: B

PTS: 1

DIF:

medium

27. A certain population grows according to

At a certain time the population

is 1153 individuals. What can be said about the population at this time? a. b. c. d.

The population is increasing. The population is decreasing. The population has reached a maximum value. The population has reached a minimum value.

ANS: A 28. The weight

PTS: 1

pounds pounds

ANS: B

PTS: 1DIF:

c. 2.74 pounds d. None of the above medium

, in pounds of a certain animal at age .

years satisfies the equation of change

For what values of the weight w is the animal losing weight?

a. b.

years satisfies the equation of change

At what weight is the animal growing the fastest?

a. 0.41 pounds b. 2.44 pounds ANS: C

medium

, in pounds, of a certain animal at age .

29. The weight

DIF:

c. 1.22 pounds d. None of the above PTS: 1

DIF:

medium

30. A certain population grows according to an equation of change. What can be said of population growth when the population level is at a steady state solution of the equation of change? a. b. c. d.

The population is stable--neither increasing nor decreasing. The population is increasing. The population is decreasing. There is not sufficient information to determine how population grows.

ANS: A

PTS: 1

DIF:

medium

31. The weight of a certain animal satisfies an equation of change. At a specific weight, the equation of change indicates that

is positive. What can you conclude about how the animal’s weight can

be expected to change? a. The weight will stay the same. b. The animal will gain weight. ANS: B

PTS: 1

c. The animal will lose weight. d. The weight is at a minimum value. DIF:

medium

32. The weight

of a certain animal satisfies an equation of change. At a specific weight, the equation is negative. What can you conclude about how the animal’s weight can

of change indicates that be expected to change?

a. The weight will stay the same. b. The animal will gain weight. ANS: C

c. The animal will lose weight. d. The weight is at a maximum value.

PTS: 1

DIF:

medium

SHORT ANSWER 1. If

, make a graph of

versus

. Use a horizontal span of 0 to 5.

ANS: df/dx 30 25 20 15 10 5

–0.5 –5

0.5 1 1.5 2 2.5

3 3.5 4 4.5 5 5.5

–10

PTS: 1

DIF: medium

2. If

, make a graph of

versus

ANS: df/dx 30 25 20 15 10 5

–1 –5 –10

10 11

. Use a horizontal span of 0 to 10.

PTS: 1

DIF: medium

3. If

, make a graph of

versus

. Use a horizontal span of 0 to 5.

ANS: df/dx 60 50 40 30 20 10 –1

–10

–20 –30 –40

PTS: 1

4. If

DIF: medium

, make a graph of

versus

. Use a horizontal span of 0 to 5.

ANS: df/dx 50 45 40 35 30 25 20 15 10 5 –1

PTS: 1

DIF: medium

ESSAY 1. The equation of change for a certain population is .

A: Plot the graph of

versus

B: Explain in practical terms what the graph in part A shows us. C: What are the equilibrium solutions for this equation of change? D: What is happening to the population when it reaches the value of an equilibrium solution? ANS: A: dN/dt 11 10 9 8 7 6 5 4 3 2 1

–1

100 150 200 250 300 350 400 450 500

B: It shows the growth rate of the population at a given population level. C:

and

D: The population is not changing. It is neither growing nor declining. PTS: 4

DIF: hard

2. A certain animal population exhibits logistic population growth. The carrying capacity for this population is 499 individuals, and the value for this type of animal is 0.07.

A. Write the equation of change for logistic growth for this population. B: Make a graph of

versus

C: For what values of

would the population be expected to increase?

D: For what values of

would the population be expected to decrease?

ANS: A: B:

. Use population sizes up to 600.

dN/dt

10 8 6 4 2

–200 –100

100

200

300

400

500

600

–2 –4

C: For population sizes less than 499 D: For population sizes greater than 499 PTS: 4

DIF: hard

3. When an object falls subject to air resistance, its velocity

satisfies the equation of change .

Here time is measured in seconds and velocity in feet per second, and the variable drag coefficient.

is known as the

A: Terminal velocity for an average-sized man is 177 feet per second. Use this fact to calculate the value of the drag coefficient for an average-sized man. Suggestion: When terminal velocity is reached,

is 0.

B: An ordinary coffee filter has a terminal velocity of about 5 feet per second. Find the drag coefficient for a coffee filter. C: Based on your answer to parts A and B, would you expect the drag coefficient for a rock to be larger or smaller than that for a man? ANS: A:0.18 B: 6.4 C: Smaller PTS: 3

DIF: hard

4. The equation of change for a certain fish population is

Here in measured in tons of fish, and tons of fish per year taken by fishing.

is measured in years. The variable

is the number of

A: What number of tons of fish per year can be taken by fishing if the population is to be maintained at a level of 416 tons? That is, for what value of will 416 be an equilibrium solution? B: What will happen to the population if the population is 416 tons and the amount of fish taken by fishing is 46 tons per year?

C: Find the equilibrium solutions if 46 tons per year are taken by fishing. ANS: A: 37.63 tons per year B: The population will decline. C:

tons and

PTS: 3

tons

DIF: hard

5. Newspaper subscriptions

satisfy the following equation of change. .

Here is the time in months since the inaugural edition of the paper, when there were 59 subscriptions. A: According to this model, what is the maximum number of subscriptions that can be expected for this newspaper? B: At what subscription level are subscriptions growing at the fastest rate? Round your answer to the nearest whole number. ANS: A: 593 newspapers B: 297 newspapers PTS: 2

DIF: hard

6. The weight , in pounds, of a certain type of animal depends on its age in years. The weight satisfies the equation of change .

A: Explain in practical terms the meaning of B:

Make a graph of

versus

. Include weights up to 45 pounds.

C: At what weight is the animal gaining weight the fastest? D: To what weight does the animal grow? ANS: A: It is the growth rate of the animal, the number of pounds we expect the animal to gain in one year. B: dw/dt 6

–5

–2

–4

–6

C: 11.68 pounds D: 39.42 pounds PTS: 4

DIF: hard

7. The amount , in pounds, of food a certain grazing animal will consume in a day depends on the amount , in pounds per acre, of vegetation present. These variables satisfy the equation of change .

A: Explain in practical terms the meaning of

Make a graph of

versus

. Include consumption levels up to 3 pounds.

C: What is the most this animal will consume no matter how much vegetation is present? ANS:

A: It is the increase in the number of pounds consumed in a day for each extra pound per acre of vegetation present. dF/dV 0.5

0.4

0.3

0.2

0.1

–1

–0.5

0.5

1.5

2.5

–0.1

C: 2.88 pounds PTS: 3

DIF: hard