1. Velocity and Other Rates of Change Chapter 3.4

2. Instantaneous Rates of Change 2 *https://en.wikipedia.org/wiki/Archetype

3. Instantaneous Rates of Change By “understanding instantaneous rate of change conceptually” is meant that you are able to understand and interpret The derivative of a function analytically (i.e., when you take a derivative) The derivative of a function from its graph (or the function from a graph of its derivative) The derivative of a function from a table of data The derivative of a function from a verbal description of the data 3

4. Instantaneous Rate of Change 4

5. Instantaneous Velocity 5

6. 6

7. 7

8. Example 2: Vertical Motion 8

9. 9

10. Speed Velocity is a vector quantity, meaning that it has both magnitude and direction When movement is either vertical or horizontal (with respect to some axes), then velocity is either positive or negative With our usual axes, velocity is positive when movement is upward and negative when movement is downward Horizontally, velocity is positive when movement is to the right and negative when movement is to the left The speed of an object is the value of the velocity without regard to direction 10

11. Speed 11

12. Example 3: Reading a Velocity Graph A student walks around in front of a motion detector that records her velocity at 1-second intervals for 36 seconds. She stores the data in her graphing calculator and uses it to generate the time-velocity graph shown below. Describe her motion as a function of time by reading the velocity graph. When is her speed a maximum? 12

13. Example 3: Reading a Velocity Graph She walks forward (away from the detector) for the first 14 seconds, moves backward for the next 12 seconds, stands still for 6 seconds, and then moves forward again. Her maximum speed occurs at about 20 seconds, while walking backward. 13

14. Acceleration 14

15. Acceleration Due to Gravity 15

16. Example 4: Modeling Vertical Motion 16

17. Example 4: Modeling Vertical Motion 17

18. Example 4: Modeling Vertical Motion 18

19. Example 4: Modeling Vertical Motion 19

20. Example 4: Modeling Vertical Motion 20

21. Example 5: Studying Particle Motion 21

22. Example 5: Studying Particle Motion 22

23. Example 5: Studying Particle Motion 23

24. Example 5: Studying Particle Motion 24

25. Example 5: Studying Particle Motion 25

26. Example 5: Studying Particle Motion 26

27. Example 1: Enlarging Circles 27

28. Example 1: Enlarging Circles 28

29. Example 1: Enlarging Circles 29

30. Example 1: Enlarging Circles 30

31. Example 1: Enlarging Circles 31

32. Example 1: Enlarging Circles 32

33. Derivatives in Economics 33

34. Derivatives in Economics 34

35. Derivatives in Economics 35

36. Example 7: Derivatives in Economics 36

37. Example 7: Derivatives in Economics 37

38. Linear Density 38

39. Linear Density 39

40. Example 8: Linear Density 40

41. Example 8: Linear Density 41

42. Exercise 3.4 42