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

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Presentation transcript:

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

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

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

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

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

What would the graph look like? Time in minutes Distance in tenths of a mile If speed were constant Speed actually varied

What is my average speed from 2.5 minutes to 3 minutes? Time in minutes Distance in tenths of a mile

What is my average speed from 2 minutes to 2.5 minutes? Time in minutes Distance in tenths of a mile About 0.45 miles at 2 minutes About 0.62 miles at 2.5 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 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 Find the slope through the points (2, 0.45) and (2.5, 0.62) (2.5, 0.62) (2, 0.45)

What is my instantaneous speed at exactly 2.5 minutes? Time in minutes Distance in tenths of a mile 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:

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

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

About 0.62 miles at 2.5 minutes 0.62 — miles 2.5 — 2.49 minutes = 19 miles 100 minutes = 11.4 miles per hour About miles at 2.49 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. This limit is called the derivative of the distance with respect to time.

Instantaneous Speed Time in minutes Distance in tenths of a mile 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.

We will study Calculus through: Limits Derivatives Integrals Infinite series

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

Read Section P.1 pages 1-7 Start on your homework: p8 #1-75 odd, all, all