1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 10 An Introductions to Calculus: Limits, Derivatives, and Integrals

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 10.1 Limits and Motion: The Tangent Problem

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 6 What you’ll learn about Average Velocity Instantaneous Velocity Limits Revisited The Connection to Tangent Lines The Derivative … and why The derivative allows us to analyze rates of change, which are fundamental to understanding physics, economics, engineering, and even history.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 7 Average Velocity Average velocity is the change in position divided by the change in time.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 8 Limits at a (Informal)

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 9 Example Finding the Slope of a Tangent Line

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 10 Example Finding the Slope of a Tangent Line

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 11 Average Rate of Change

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 12 Derivative at a Point

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 13 Derivative at a Point (easier for computing)

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 14 Example Finding a Derivative at a Point

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 15 Example Finding a Derivative at a Point

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 16 Derivative

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 17 Example Finding the Derivative of a Function

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 18 Example Finding the Derivative of a Function

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 10.2 Limits and Motion: The Area Problem

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 20 Quick Review

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 21 Quick Review Solutions

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 22 What you’ll learn about Distance from a Constant Velocity Distance from a Changing Velocity Limits at Infinity The Connection to Areas The Definite Integral … and why Like the tangent line problem, the area problem has many applications in every area of science, as well as historical and economic applications.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 23 Example Computing Distance Traveled A car travels at an average rate of 56 miles per hour for 3 hours and 30 minutes. How far does the car travel?

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 24 Example Computing Distance Traveled A car travels at an average rate of 56 miles per hour for 3 hours and 30 minutes. How far does the car travel?

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 25 Limits at Infinity (Informal)

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 26

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 27 Definite Integral

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 10.3 More on Limits

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 29 Quick Review

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 30 Quick Review Solutions

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 31 What you’ll learn about A Little History Defining a Limit Informally Properties of Limits Limits of Continuous Functions One-Sided and Two-Sided Limits Limits Involving Infinity … and why Limits are essential concepts in the development of calculus.

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 32 Limit at a (Informal)

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 33 Example Finding a Limit

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 34 Example Finding a Limit

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 35 Properties of Limits

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 36 Example Using the Limit Properties

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 37 Example Using the Limit Properties

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 38 One-sided and Two-sided Limits

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 39 Example Finding One-Sided Limits

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 40 Example Finding One-Sided Limits

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 41 Limits at Infinity

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 10.4 Numerical Derivatives and Integrals

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 43 Quick Review

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 44 Quick Review Solutions

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 45 What you’ll learn about Derivatives on a Calculator Definite Integrals on a Calculator Computing a Derivative from Data Computing a Definite Integral from Data … and why The numerical capabilities of a graphing calculator make it easy to perform many calculations that would have been exceedingly difficult in the past.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 46 Numerical Derivative

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 47 Example Computing a Numerical Derivative

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 48 Example Computing a Numerical Derivative

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 49 Numerical Integral

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 50 Chapter Test

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 51 Chapter Test

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 52 Chapter Test

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 53 Chapter Test Solutions

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 54 Chapter Test Solutions

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 10- 55 Chapter Test Solutions