1. Rates of Change and Tangent Lines Chapter 2.4

2. Average Rates of Change 2

3. Example 1: Finding Average Rate of Change 3

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6. Example 2: Growing Drosophila in a Laboratory Experimental biologists often want to know the rates at which populations grow under controlled laboratory conditions. The figure on the next slide shows how the number of fruit flies (Drosophila) grew in a controlled 50-day experiment. The graph was made by counting flies at regular intervals, plotting a point for each count, and drawing a smooth curve through the plotted points (this can be done using a kind of regression known as logistic regression). 6

7. Example 2: Growing Drosophila in a Laboratory 7

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11. Instantaneous Rate of Change Suppose that, rather than average rate of change, we want to know the rate of change at on a particular day (say, day 23 from the previous example)? This is the instantaneous rate of change (which was first introduced in Chapter 2.1) In that chapter, we saw how we can find an instantaneous rate of change by starting with an approximation in the form of a secant line, and letting one point get closer to the desired point But we lacked, at that point, a way to formulate this idea mathematically; now we can do this (how?) 11

12. Instantaneous Rate of Change 12

13. Instantaneous Rate of Change 13

14. Example 3: Finding Slope and Tangent Line 14

15. Example 3: Finding Slope and Tangent Line 15

16. Example 3: Finding Slope and Tangent Line 16

17. Example 3: Finding Slope and Tangent Line 17

18. Slope of a Curve at a Point 18

19. Slope of a Curve at a Point 19

20. Example 4: Exploring Slope and Tangent 20

21. Example 4: Exploring Slope and Tangent 21

22. Example 4: Exploring Slope and Tangent 22

23. Example 4: Exploring Slope and Tangent 23

24. Example 4: Exploring Slope and Tangent 24

25. Normal to a Curve 25

26. Example 5: Finding a Normal Line 26

27. Example 5: Finding a Normal Line 27

28. Example 6: Finding Instantaneous Rate of Change 28

29. Example 6: Finding Instantaneous Rate of Change 29

30. Example 6: Finding Instantaneous Rate of Change 30

31. Example 7: Investigating Free Fall 31

32. Example 7: Investigating Free Fall 32

33. Exercise 2.4 33