1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 2 Limits and Continuity

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.1 Rates of Change and Limits

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 Quick Review

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Average and Instantaneous Speed

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the average and instantaneous speeds of an object in free fall at t = 4 seconds. Verify algebraically and with a table. Slide 2- 10

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

12. With the table we can see as we get closer and closer to exactly 4 seconds the speed gets closer and closer to 128. This gets us thinking about the idea of a limit. Slide 2- 12

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Delta – Epsilon Definition of Limit

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall We will not rely too heavily upon the Delta-Epsilon definition for limits. It will be sufficient (in most cases) to understand the limit of a function as the input approaches a certain number, c, to be the value the function approaches when the input is very close to c. Slide 2- 14

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 15 Definition of Limit continued

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 Definition of Limit continued

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 Properties of Limits

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Properties of Limits continued Product Rule: Constant Multiple Rule:

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 19 Properties of Limits continued

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 20 Example Properties of Limits

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 21 Example Properties of Limits Use the properties of limits to determine

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 22 Example Properties of Limits Use the properties of limits to determine

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 23 Theorem 2: Polynomial and Rational Functions

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 24 Example Limits

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use substitution to determine the limit of the function below and then support your answer with a graph or a table. Slide 2- 25

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use substitution to determine the limit of the function below and then support your answer with a graph or a table. Slide 2- 26

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use substitution to determine the limit of the function below and then support your answer with a graph or a table. Slide 2- 27

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 28 Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 29 Example Limits

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Slide 2- 30

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function. Slide 2- 31

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall When the limit of a function cannot be found algebraically, we can use a table or a graph to determine the limit. Slide 2- 32

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit. Slide 2- 33 This is a really, really important limit so remember it!

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 34 Example Limits [-6,6] by [-10,10]

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Confirm using another method. Slide 2- 35

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Confirm using another method. Slide 2- 36

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Confirm using another method. Slide 2- 37

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Confirm using another method. Slide 2- 38

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Confirm using another method. Slide 2- 39

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below. Confirm using another method. Slide 2- 40

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Assignment 2.1.1 page 66, # 3 – 27 multiples of 3 Slide 2- 41

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the limit of the function below using your graphing calculator. Slide 2- 42

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 43 One-Sided and Two-Sided Limits

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 44 One-Sided and Two-Sided Limits continued

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 45 Example One-Sided and Two-Sided Limits o 12 3 4 Find the following limits from the given graph.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Match the function with the table. Slide 2- 46 a.a. b.b. d.d. c.c.

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Match the function with the table. Slide 2- 47 a.b. d.c.

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Operations with Limits Slide 2- 48

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 49

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall a. Draw the graph of f(x). b. At what points c in the domain of f does lim f(x) exist? c. At what points c does only the left handed limit exist? d. At what points c does only the right handed limit exist? Slide 2- 50 xcxc

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 51 Sandwich Theorem

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 52 Sandwich Theorem

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use the Sandwich Theorem to determine the limit below. Slide 2- 53

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall A water balloon dropped from a window high above the ground falls y = 4.9t 2 m in t sec. Find the balloon’s average speed during the first 3 seconds of fall. Slide 2- 54

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall A water balloon dropped from a window high above the ground falls y = 4.9t 2 m in t sec. Find the balloon’s speed at the instant t = 3. Slide 2- 55

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Assignment 2.1.2 page 66 – 68, #30, 33, 36, 38, 47, 50, 51, 54, 55, 61, 64 Slide 2- 56