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AP Calculus Notes

AP Calculus Notes. Chapter P.1 – Graphs and Models Day 1. What is rate?. What is speed? Calculus allows us to analyze changing rates. My bike ride to school:. Distance = 0.8 miles Time = 4 minutes. What is my average speed?. 0.8. 1. mile. 0.8 miles. =. =. 5. minute. 4 minutes.

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AP Calculus Notes

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

  2. What is rate? What is speed? Calculus allows us to analyze changing rates

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

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

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

  6. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes What would the graph look like? If speed were constant Speed actually varied

  7. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes What is my average speed from 2.5 minutes to 3 minutes?

  8. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes 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

  9. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes 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 Find the slope through the points (2, 0.45) and (2.5, 0.62) y2 – y1 (2, 0.45) x2 – x1 (2.5, 0.62) 0.17 miles 17 miles 0.62 — 0.45 miles = = 0.5 minutes 2.5 — 2 minutes 50 minutes = 20.4 miles per hour

  10. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes What is my instantaneous speed at exactly 2.5 minutes? Computing average speed from 2.4 minutes to 2.5 minutes: About 0.6 miles at 2.4 minutes About 0.62 miles at 2.5 minutes

  11. Computing average speed from 2.4 minutes to 2.5 minutes: About 0.6 miles at 2.4 minutes About 0.62 miles at 2.5 minutes 0.02 miles 1 miles 0.62 — 0.6 miles = = 0.1 minutes 2.5 — 2.4 minutes 5 minutes = 12 miles per hour

  12. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes How can we get closer to the instantaneous speed? Computing average speed from 2.49 minutes to 2.5 minutes: About 0.6181 miles at 2.49 minutes About 0.62 miles at 2.5 minutes

  13. Computing average speed from 2.49 minutes to 2.5 minutes: About 0.6181 miles at 2.49 minutes About 0.62 miles at 2.5 minutes 19 miles 0.62 — 0.6181 miles = 100 minutes 2.5 — 2.49 minutes = 11.4 miles per hour

  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. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes 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. 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.

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

  17. 8 Distance in tenths of a mile 4 1 1 2 3 4 Time in minutes We will study Calculus using the following methods: • Graphical • Numerical • Algebraic • Verbal y2 – y1 x2 – x1 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

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