1. Two Questions of Calculus

2. Objective To determine the purpose for studying calculus. TS: Making decisions after reflection and review

3. Two Questions of Calculus I went on a bike ride.

4. Two Questions of Calculus Two-thirds of the way, I saw a sign.

5. Two Questions of Calculus Did I break the law?

6. Two Questions of Calculus Did I break the law, the instant I passed the sign?

7. First Question of Calculus How do you find instantaneous rates of change (velocities)?

8. Distance, Rate, & Time These are very useful for finding average rates, or for finding total distance from average rates, etc.

9. Distance, Rate, and Time D R T

10. Second Question of Calculus How do you find areas (or volumes) of exotic shapes?

11. Two Questions of Calculus How do you find instantaneous rates of change (velocities)? How do you find areas (or volumes) of exotic shapes? These two questions have related answers.

12. Rates of Change Velocity – the rate of change of position (Slope or Rate of change of distance v. time) Acceleration – the rate of change of velocity (Slope or Rate of change of velocity v. time) Population growth rates Unemployment rates Production rates

13. Two Types of Rate of Change Average – Rate of change between 2 points Ex: average speed traveled during a trip The average rate of change = the slope of the line containing the two points (so this is a secant slope)

14. Two Types of Rate of Change

15. Instantaneous – Rate of change at one point Ex: speed traveled at a single moment during a trip

16. Two Types of Rate of Change

17. Chapter II: The Derivative A common rate of change that we will be using is average velocity.

18. Chapter II: The Derivative Positive velocities are used for upward motion. Negative velocities used for downward motion. The absolute value of velocity is the speed of an object.