1. Accumulation AP Calculus AB Day 9
Instructional Focus: Explain the meaning of a definite integral when the rate is negative.

2. Exploration Negative Rate and Accumulation
Introduce definite integral notation and calculator use.
Introduce Fundamental Theorem of Calculus
Problems 2 and 3 can be assigned as homework then the next day have the students discuss with each other in a group then have group presentations.

3. Caren rides her bicycle along a straight road from home to school, starting at home at time t = 0 minutes and arriving at school at time t = 12 minutes. During the time interval 0≤𝑡≤12 minutes, her velocity 𝑣 𝑡 , in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.

4. Write a story about Caren’s 12-minute bicycle ride to school based on the graph of Caren’s velocity given above.
Answers will vary. Allow several students to read their story. Be sure they include how her trip during 0≤𝑡≤2 and during 2≤𝑡≤4 differ. How is the trop on these two intervals the same? What is happening during 4≤𝑡≤5? When is she accelerating? When is she riding at a constant velocity?

5. Shortly after leaving home, Caren realizes she left her calculus book at home, and she returns home to get it. At what time does she turn around to go back home? Give a reason for your answer.
Caren turns around after riding for 2 minutes, 𝑡=2. Her velocity is positive for the first 2 minutes, 0<𝑡<2. This means she is riding away from home because her distance from home (displacement) is increasing. From 2<𝑡<4, her velocity is negative so she is riding back home, her distance from home (displacement) is decreasing.

6. What time did she leave home the second time
What time did she leave home the second time? Give a reason for your answer.
Caren leaves home a second time at 𝑡=5 because her velocity becomes positive for 𝑡>5, which means her distance from home (displacement) is increasing.

7. Using correct units, explain the meaning of 0 2 𝑣 𝑡 𝑑𝑡 in terms of Caren’s bicycle ride to school.
0 2 𝑣 𝑡 𝑑𝑡 is the number of miles Caren rode her bike away from home during the first 2 minutes or how far she is from home after riding her bike the first 2 minutes.

8. Using correct units, explain the meaning of 2 4 𝑣 𝑡 𝑑𝑡 in terms of Caren’s bicycle ride to school.
2 4 𝑣 𝑡 𝑑𝑡 is the number of miles Caren rode her bike back toward home during the 2 minutes 2<𝑡<4.

9. If ℎ 𝑡 is Caren’s distance from home in miles t minutes after leaving home. Let 𝑡=0 be the time Caren left home the first time. What is the meaning of ℎ 4 in terms of Caren’s bicycle ride to school?
ℎ 4 is how far Caren is from home after 4 minutes of riding her bicycle, 0≤𝑡≤4.
According to the Fundamental Theorem of Calculus applied to this situation
ℎ 𝑡 =ℎ 𝑡 𝑣 𝑚 𝑑𝑚
Note: The variable inside the integrand should not be the same as the variable used as the upper limit.

10. Izzy Wright uses the above equation and calculates
ℎ 4 =ℎ 𝑣 𝑚 𝑑𝑚 = =0.4
Izzy’s twin sister Mae B. Wright uses the above equation and calculates
ℎ 4 =ℎ 𝑣 𝑚 𝑑𝑚 = 0 2 𝑣 𝑚 𝑑𝑚 − 2 4 𝑣 𝑚 𝑑𝑚 = − =0
Which Wright do you think is right? Explain your reasoning in terms of Caren’s bicycle ride to school.
Mae is correct because at the 4-minute mark she is back home looking for her book.

11. The definite integral of a rate
𝑎 𝑏 𝑓 ′ 𝑥 𝑑𝑥 =𝑓 𝑏 −𝑓 𝑎
measures the amount of change in 𝑓 𝑥 where 𝑎≤𝑏.

12. Suppose that 𝑓 ′ 𝑥 <0 on the interval 𝑎≤𝑥≤𝑏
Suppose that 𝑓 ′ 𝑥 <0 on the interval 𝑎≤𝑥≤𝑏. Is 𝑓 𝑥 increasing, decreasing, or both on 𝑎≤𝑥≤𝑏? How do you know?
𝑓 ′ 𝑥 <0 means that the function 𝑓 𝑥 is decreasing.

13. If 𝑓 𝑥 is decreasing for 𝑎≤𝑥≤𝑏, which is larger, 𝑓 𝑎 or 𝑓 𝑏
If 𝑓 𝑥 is decreasing for 𝑎≤𝑥≤𝑏, which is larger, 𝑓 𝑎 or 𝑓 𝑏 ? Is the value of 𝑓 𝑏 −𝑓 𝑎 greater than or less than zero? How do you know?
If 𝑓 𝑥 is decreasing for 𝑎≤𝑥≤𝑏, then 𝑓 𝑎 >𝑓 𝑏 and 𝑓 𝑏 −𝑓 𝑎 <0.

14. 𝑓 ′ 𝑥 <0 on the interval 𝑎≤𝑥≤𝑏 if and only if 𝑎 𝑏 𝑓 ′ 𝑥 𝑑𝑥 <0.

15. Returning to the context of Caren’s ride to school, why does 2 4 𝑣 𝑡 𝑑𝑡 have to be negative? What direction is Caren riding her bicycle during the time interval 2<𝑡<4 Explain the meaning of 2 4 𝑣 𝑡 𝑑𝑡 .
2 4 𝑣 𝑡 𝑑𝑡 has to be negative because Caren is riding her bicycle toward home during the time interval 2<𝑡<4, so the change in distance from home has to decrease, so 2 4 𝑣 𝑡 𝑑𝑡 is measuring the amount of decrease in distance from home.

16. How far does Caren live from school
How far does Caren live from school? Explain how you arrived at your answer. Write a definite integral that represents how far Caren lives from school.
Students arrive at this answer many ways.
5 12 𝑣 𝑡 𝑑𝑡 = triangle ± rectangle ± trapezoid ± rectangle ± triangle
1.4 miles

17. How far did Caren ride her bike during the 12 minutes, 0≤𝑡≤12
How far did Caren ride her bike during the 12 minutes, 0≤𝑡≤12? Is this the same answer as problem 11? Explain your reasoning.
During the first two minutes she rode 0 2 𝑣 𝑡 𝑑𝑡 = =0.2 miles, then she returned home so she rode 0.2 miles back home for a total of 0.4 miles in the first 4 minutes. She was at home for a minute getting her book then rode the 1.4 miles to school for a total of 1.8 miles.

18. Using correct units, explain the meaning of 0 12 𝑣 𝑡 𝑑𝑡 in terms of Caren’s bicycle ride to school?
The absolute value will turn the negative values of velocity to positive, so the integral 2 4 𝑣 𝑡 𝑑𝑡 will also be positive and not subtract from the total 𝑣 𝑡 𝑑𝑡 is the total distance in miles Caren rode during the 12 minutes, 0≤𝑡≤12. This is the answer arrived at in problem 12.