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4.4 rates of change

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Robert Hammarstrand

This document contains examples of using graphs and tables to determine rates of change. Example 5 shows a table with time and cost for using an internet cafe, and calculates the rate of change in cost with respect to time as $3.50 per hour. Example 6 uses a graph to show attendance at a community theater over 10 weeks, finding different rates of change between weeks. Example 7 interprets a graph of distance over time to describe a student's commute of walking then waiting for the bus then riding the bus.

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EXAMPLE 5 Find a rate of change INTERNET CAFE The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time. Time (hours) 2 4 6 Cost (dollars) 7 14 21
EXAMPLE 5 Find a rate of change Rate of change = change in cost change in time 14– 7 4 – 2 = 7 2 = 3.5= ANSWER The rate of change in cost is $3.50 per hour. SOLUTION
Time(minute) 30 60 90 Distance (miles) 1.5 3 4.5 GUIDED PRACTICE for Example 5 The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. 7. EXERCISE ANSWER 0.05 mi/min
EXAMPLE 6 Use a graph to find and compare rates of change COMMUNITY THEATER A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time.
SOLUTION EXAMPLE 6 Use a graph to find and compare rates of change Find the rates of change using the slope formula. Weeks 1–4: 232 – 124 4 – 1 = 108 3 = 36 people per week Weeks 4–6: 204 – 232 6 – 4 = –28 2 = –14 people per week Weeks 6–10: 72 – 204 10 – 6 = –132 4 = –33 people per week ANSWER Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly.
EXAMPLE 7 Interpret a graph COMMUTING TO SCHOOL A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words.
EXAMPLE 7 Interpret a graph The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus. SOLUTION
EXAMPLE 7 Interpret a graphGUIDED PRACTICE for Examples 6 and 7 WHAT IF? How would the answer to Example 6 change if you knew that attendance was 70 people in week 12? 8. Sample answer: The attendance did not decrease as rapidly between weeks 10 and 12. ANSWER
EXAMPLE 7 Interpret a graphGUIDED PRACTICE for Examples 6 and 7 WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. 9. ANSWER
EXAMPLE 7 Interpret a graphGUIDED PRACTICE for Examples 6 and 7 WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. 9. ANSWER

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4.4 rates of change

  • 1. EXAMPLE 5 Find a rate of change INTERNET CAFE The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time. Time (hours) 2 4 6 Cost (dollars) 7 14 21
  • 2. EXAMPLE 5 Find a rate of change Rate of change = change in cost change in time 14– 7 4 – 2 = 7 2 = 3.5= ANSWER The rate of change in cost is $3.50 per hour. SOLUTION
  • 3. Time(minute) 30 60 90 Distance (miles) 1.5 3 4.5 GUIDED PRACTICE for Example 5 The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. 7. EXERCISE ANSWER 0.05 mi/min
  • 4. EXAMPLE 6 Use a graph to find and compare rates of change COMMUNITY THEATER A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time.
  • 5. SOLUTION EXAMPLE 6 Use a graph to find and compare rates of change Find the rates of change using the slope formula. Weeks 1–4: 232 – 124 4 – 1 = 108 3 = 36 people per week Weeks 4–6: 204 – 232 6 – 4 = –28 2 = –14 people per week Weeks 6–10: 72 – 204 10 – 6 = –132 4 = –33 people per week ANSWER Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly.
  • 6. EXAMPLE 7 Interpret a graph COMMUTING TO SCHOOL A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words.
  • 7. EXAMPLE 7 Interpret a graph The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus. SOLUTION
  • 8. EXAMPLE 7 Interpret a graphGUIDED PRACTICE for Examples 6 and 7 WHAT IF? How would the answer to Example 6 change if you knew that attendance was 70 people in week 12? 8. Sample answer: The attendance did not decrease as rapidly between weeks 10 and 12. ANSWER
  • 9. EXAMPLE 7 Interpret a graphGUIDED PRACTICE for Examples 6 and 7 WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. 9. ANSWER
  • 10. EXAMPLE 7 Interpret a graphGUIDED PRACTICE for Examples 6 and 7 WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. 9. ANSWER