Section 1.1 Functions Given by Formulas

TRUE/FALSE

1. The functional notation means .

ANS: F PTS: 1 DIF: easy

2. The functional notation means the function whose name is evaluated at

ANS: T PTS: 1 DIF: 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 PTS: 1 DIF: easy

5. If represents the temperature hours after midnight, then indicates the temperature at 7:00 a.m.

ANS: T PTS: 1 DIF: easy

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

ANS: T PTS: 1 DIF: easy

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

ANS: F PTS: 1 DIF: easy

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

ANS: T PTS: 1 DIF: easy

9. If represents federal defense spending years after 2000, then representsfederal defense spending in 2012.

ANS: F PTS: 1 DIF: easy

Multiple Choice

1. If , calculate the value of b.

2. If , calculate the value of

3. If , calculate the value of .

4. If , calculate the value of

5. If , calculate the value of .

6. If , calculate the value of .

7. If , calculate the value of .

8. If , calculate the value of a. 2.71 b. 14.79 c. –1.65 d. 18.09

9. If , calculate the value of .

ANS: A PTS: 1 DIF: easy a. 0.64 b. 1 c. 1.84 d. 4.47

10. If , calculate the value of .

ANS: A PTS: 1 DIF: easy a. 5 miles b. 251.05 miles c. 63025 miles d. 178.06 miles

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.

ANS: B PTS: 1 DIF: medium a. Functional notation: Value: 118 outlets. b. Functional notation: Value: 136 outlets. c. Functional notation: . Value: 173 outlets. d. Functional notation: Value: 96 outlets.

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.

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 a. Functional notation: . Value is 17.33 pounds. b. Functional notation: Value is 156 pounds. c. Functional notation: d. None of the above. Value is 15 pounds.

Use functional notation to express the predicted weight of a puppy that weighs 5 pounds at age 15 weeks. Then calculate that value.

ANS: A PTS: 2 DIF: medium a. miles per hour b. miles per hour c. miles per hour d. miles per hour

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.

ANS: A PTS: 1 DIF: medium a. feet b. feet c. feet d. feet

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.

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 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.

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.

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 a. 632.51 grams b. 1.13 grams

How much of the chemical is removed from time to time ?

ANS: D PTS: 1 c. 24.34 grams d. DIF: 23.03 grams medium a. 4.38 kilograms b. 1.74 kilograms c. 7.18 kilograms d. 6.02 kilograms

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 ?

ANS: C PTS: 1 DIF: medium a. 0 b. 214 c. 384 d. 597

20. The number of armadillos in a certain area 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.

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 a. $110950.28 b. $9314.72

What is the balance after 10 years if the APR is 4 percent? (Be sure first to express the APR as a decimal.)

ANS: B PTS: 1

Short Answer

c. $6459.42 d. DIF: None of the above medium

1. If represents the library charges on books that are weeks overdue, explain in practical terms the meaning of

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 books that are weeks overdue, explain in practical terms the meaning of

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 miles per hour over the speed limit. Explain in practical terms the meaning of

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. Use functional notation to indicate the cost of 3 pizzas, 8 sodas, and 11 bags of chips.

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 hours later is given by .

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 seconds is given by .

A. Explain in practical terms the meaning of

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