1. AP Calculus Notes Chapter P.1 – Graphs and Models Day 1

2. WHAT IS SPEED? CALCULUS ALLOWS US TO ANALYZE CHANGING RATES What is rate?

3. My bike ride to school: Distance = 0.8 miles Time = 4 minutes What is my average speed? 0.8 miles 4 minutes = 0.8 4 = 1 mile minute 5 What is this in miles per hour?

4. Some Unit Analysis! 1 miles minute 5  60 1 minutes hour =12

5. My bike ride to school: Distance = 0.8 miles Time = 4 minutes What is my average speed? 0.8 miles 4 minutes = 0.8 4 = 1 miles minute 5 = 12 miles per hour

6. What would the graph look like? Time in minutes Distance in tenths of a mile 1234 1 4 8 If speed were constant Speed actually varied

7. What is my average speed from 2.5 minutes to 3 minutes? Time in minutes Distance in tenths of a mile 1234 1 4 8

8. What is my average speed from 2 minutes to 2.5 minutes? Time in minutes Distance in tenths of a mile 1234 1 4 8 About 0.45 miles at 2 minutes About 0.62 miles at 2.5 minutes

9. What is my average speed from 2 minutes to 2.5 minutes? About 0.45 miles at 2 minutes About 0.62 miles at 2.5 minutes 0.62 — 0.45 miles 2.5 — 2 minutes = 0.17 miles 0.5 minutes = = 20.4 miles per hour 17 miles 50 minutes y 2 – y 1 x 2 – x 1 Time in minutes Distance in tenths of a mile 1234 1 4 8 Find the slope through the points (2, 0.45) and (2.5, 0.62) (2.5, 0.62) (2, 0.45)

10. What is my instantaneous speed at exactly 2.5 minutes? Time in minutes Distance in tenths of a mile 1234 1 4 8 About 0.6 miles at 2.4 minutes About 0.62 miles at 2.5 minutes Computing average speed from 2.4 minutes to 2.5 minutes:

11. About 0.6 miles at 2.4 minutes About 0.62 miles at 2.5 minutes 0.62 — 0.6 miles 2.5 — 2.4 minutes = 0.02 miles 0.1 minutes = = 12 miles per hour 1 miles 5 minutes

12. How can we get closer to the instantaneous speed? Time in minutes Distance in tenths of a mile 1234 1 4 8 About 0.6181 miles at 2.49 minutes About 0.62 miles at 2.5 minutes Computing average speed from 2.49 minutes to 2.5 minutes:

13. About 0.62 miles at 2.5 minutes 0.62 — 0.6181 miles 2.5 — 2.49 minutes = 19 miles 100 minutes = 11.4 miles per hour About 0.6181 miles at 2.49 minutes

14. Instantaneous speed The instantaneous speed at 2.5 minutes is the limit of the average rate (slope) between two points as the time interval approaches 0. This limit is called the derivative of the distance with respect to time.

15. Instantaneous Speed Time in minutes Distance in tenths of a mile 1234 1 4 8 The instantaneous speed at 2.5 minutes is the limit of the average rate (slope) between two points as the time interval approaches 0. This limit is called the derivative of the distance with respect to time. The instantaneous speed at 2.5 minutes is the limit of the average rate (slope) between two points as the time interval approaches 0.

16. We will study Calculus through: Limits Derivatives Integrals Infinite series

17. We will study Calculus using the following methods: Graphical Numerical Algebraic Verbal TimeDistance 20.45 2.40.6 2.490.6181 2.50.62 Time in minutes Distance in tenths of a mile 1234 1 4 8 y 2 – y 1 x 2 – x 1 This limit is called the derivative of the distance with respect to time. Blah blah blah

18. Read Section P.1 pages 1-7 Start on your homework: p8 #1-75 odd, 77-79 all, 81- 85 all