1. What is Calculus? What is Calculus? Using Distance, Speed and Acceleration to Understand the Very Very Basics

2. Example 1 – Constant Speed TimeDistanceSpeedAcceleration 150 0 2100500 3150500 4200500 5250500

3. Example 2 – Speeding up (constant acceleration) TimeDistanceSpeedAcceleration 110 2302010 3603010 41004010 51505010

4. Example 3 – Speeding up/slowing down (variable acceleration). TimeDistanceSpeedAcceleration 110 2302010 3603010 48020-10 59010-10

5. Example 4 – More Complicated Changes TimeDistanceSpeedAcceleration 115 232172 3397-10 449103 5742515

6. From Numbers to Functions! D(t) = distance traveled V(t) = speed = change in D(t)/unit of time A(t) = acceleration = change in V(t)/unit of time = change in the change in D(t)/unit of time!

7. BIG CONCEPT!! One of the BIG concepts of calculus is the RELATIONSHIP between these functions. Examples If you know distance, can you find speed? If you know speed, can you find acceleration? If you know speed, can you find distance? If you know acceleration, can you find distance?

8. These type of situations occur in many disciplines including physical and biological sciences, economics, business and engineering. Much of Calculus (Rules!) involve these relationships between: F(x) – the quantity of something at x F’(x) – the CHANGE in the quantity at time x F’’(x) – the CHANGE IN THE CHANGE in the quantity at time x. “Distance/Speed/Acceleration”

9. IPOD Collection Suppose the number of songs on your IPOD as a function of the time since you have owned the IPOD is: S = 200M + 25 Says you add (change) by 200 songs per month and you started with 25 (freebies?) So your rate of change is 200 songs/month. And your CHANGE in the CHANGE is 0.

10. IPOD Collection (more realistic?) But what if a better model for your collection would be S = 30M 2 - 10M + 25 What is your rate of change after month 1? Month 2? Does your rate of change change? Is the CHANGE in the CHANGE of your song collection 0? Can we find these answers exactly?

11. YES WE CAN!!!!! And we have many functions to work with!