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:
A:
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
25
31
36
39
42
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.
5
10
15
20
25
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
5
10
15
20
25
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.
0
5
10
15
20
17
24
27
31
34
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.
3
10
17
24
32
26
44
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.
5
10
15
20
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
0
5
10
15
20
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
20
30
40
50
60
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.