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 versus can help you answer this question.

a. Distance increases as period increases.

b. Distance decreases as period increases.

c. Distance is at a maximum.

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

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.

ANS: B PTS: 1 DIF: medium a. Larger stars live 2.5 times as long as smaller stars. b. Larger stars have longer life expectancy. c. Larger stars have shorter life expectancy. d. All of the above.

3. The life expectancy , in solar lifetimes, of certain stars depends on their mass , in solar masses. The relationship is .

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

ANS: C PTS: 1 DIF: medium

4. The life expectancy , in solar lifetimes, of certain stars depends on their mass , in solar masses. The relationship is a. Life expectancy decreases at a decreasing rate. b. Life expectancy decreases at an increasing rate. c. Life expectancy reaches a point of inflection. d. None of the above.

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.

ANS: A PTS: 1 DIF: medium 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.

5. When a car skids to a stop, the length , in feet, of the skid marks is related to the speed , in miles per hour, of the car by the equation .

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.

ANS: A PTS: 1 DIF: medium a. About 360 centimeters tall c. About 378 centimeters tall b. About 18 centimeters tall d. About 364.3 centimeters tall

6. The height h, in centimeters, of a sunflower days after the seed emerges is given by Use a graph to estimate the tallest the sunflower will ever be.

ANS: A PTS: 1 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 c. About 0.91 milligrams per deciliter b. About 1.75 milligrams per deciliter d. About 2.58 milligrams per deciliter

ANS: B PTS: 1 DIF: medium

8. The concentration of a drug, in milligrams per deciliter, in the blood hours after an injection is given by a. About 4.03 milligrams per deciliter c. About 4.51 milligrams per deciliter b. About 8.54 milligrams per deciliter d. 0 milligrams per deciliter

What is the eventual concentration of the drug in the blood?

ANS: D PTS: 1 DIF: medium a. About 4.01 nanograms per liter c. About 5.82 nanograms per liter b. About 1.81 nanograms per liter d. 0 nanograms per liter

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

What is the eventual amount of adrenaline in the bloodstream?

ANS: B PTS: 1 DIF: medium

10. The amount , in grams, of a radioactive substance remaining after years is given by a. About 0.54 grams b. About 19 grams c. About 19.54 grams d. 0 grams

What is the limiting value for the amount of the radioactive substance?

ANS: D PTS: 1 DIF: medium

11. Dye is being added to a liquid mixture. The amount , in grams, of dye in the mixture after minutes is given by a. About 171.32 grams b. About 184 grams c. About 0.88 grams d. 178.12 grams

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

ANS: B PTS: 1 DIF: medium

12. Which of the following is the graph of on a horizontal span of 0 to 4.5?

ANS: C PTS: 1 DIF: medium

13. Which of the following is the graph of on a horizontal span of 0 to 4.5?

ANS: D PTS: 1 DIF: medium a. From month 3 to the end of the year b. From month 6 to the end of the year c. Over the first 3 months d. Over the first 6 months

14. Sales , in thousands of dollars, 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.

ANS: C PTS: 1 DIF: medium

15. The population , in thousands, of a certain city years after 2000 is given by a. From 2005 to 2010 c. From 2000 to 2005 b. From 2007 to 2010 d. From 2000 to 2007

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.

ANS: A PTS: 1 DIF: medium

Short Answer

1. Make the graph of Use a horizontal span of 0 to 10.

ANS:

PTS: 1 DIF: easy

2. Make the graph of . Use a horizontal span of 0 to 5.

ANS:

PTS: 1 DIF: easy

3. Make the graph of Use a horizontal span of 0 to 5.

ANS:

PTS: 1 DIF: easy

4. Make the graph of Use a horizontal span of 0 to 5.

ANS:

PTS: 1 DIF: easy

5. Make the graph of Use a horizontal span of 0 to 100.

ANS:

PTS: 1 DIF: medium

6. Make the graph of . Use a horizontal span of to 5.

ANS:

PTS: 1 DIF: medium

7. Make the graph of . Use a horizontal span of 0 to 0.9.

ANS:

PTS: 1 DIF: medium

8. Make the graph of Use a horizontal span of 0 to 5.

ANS:

PTS: 1 DIF: easy

9. Make the graph of Use a horizontal span of 0 to 3.

ANS:

PTS: 1 DIF: easy

10. Make the graph of . Use a horizontal span of 0 to 5.

ANS:

PTS: 1 DIF: easy

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

ANS:

PTS: 1 DIF: easy

12. Sketch a graph that is decreasing and concave down.

ANS:

PTS: 1 DIF: easy

13. When a car skids to a stop, the length , in feet, of the skid marks is related to the speed , in miles per hour, of the car by the equation .

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 DIF: medium

15. The life expectancy , in solar lifetimes, of certain stars depends on their mass , in solar masses. The relationship is .

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 months after a diet and exercise program was discontinued is given by

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

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.

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 , in square meters, can be estimated from the weight measured in kilograms. The relation is .

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:

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 miles per hour, for the trip depends on the time , in hours, spent driving. The relationship is

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.

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 , in degrees, after minutes of baking is

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.

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 , in pounds per acre, of vegetation present. The relationship is

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:

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.

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.

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.

B. The required number of rotations decreases as the radius increases.

C. 5.05 revolutions per minute.

PTS: 3 DIF: hard

10. The fraction of the surface of Earth that is visible from an altitude of kilometers above the surface is given by

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.

B. The fraction is 0.04. This is equivalent to 4% of Earth’s surface.

C. 0.5 or 50%.

PTS: 3 DIF: hard