1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 4 Applications of Derivatives

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4.1 Extreme Values of Functions

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

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 5 Quick Review

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 8 What you’ll learn about Absolute (Global) Extreme Values Local (Relative) Extreme Values Finding Extreme Values …and why Finding maximum and minimum values of a function, called optimization, is an important issue in real-world problems.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 9 Absolute Extreme Values

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 10 The Extreme Value Theorem

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 11 Classifying Extreme Values

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 12 Local Extreme Values

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 13 Local Extreme Values

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 14 Critical Points

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 15 Example Finding Absolute Extrema

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 16 Example Finding Extreme Values

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4.2 Mean Value Theorem

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 18 Quick Review

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 19 Quick Review Solutions

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 20 What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences …and why The Mean Value Theorem is an important theoretical tool to connect the average and instantaneous rates of change.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 21 Mean Value Theorem for Derivatives

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 22 Example Explore the Mean Value Theorem

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 23 Increasing Function, Decreasing Function

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 24 Corollary: Increasing and Decreasing Functions

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 25 Example Determining Where Graphs Rise or Fall

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 26 Corollary: Functions with f’=0 are Constant

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 27 Corollary: Functions with the Same Derivative Differ by a Constant

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 28 Antiderivative

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 29 Example Finding Velocity and Position

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4.3 Connecting f ’ and f ” with the graph of f

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 31 Quick Review

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 32 Quick Review

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 33 Quick Review Solutions

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 34 Quick Review Solutions

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 35 What you’ll learn about First Derivative Test for Local Extrema Concavity Points of Inflection Second Derivative Test for Local Extrema Learning about Functions from Derivatives …and why Differential calculus is a powerful problem-solving tool precisely because of its usefulness for analyzing functions.

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 36 First Derivative Test for Local Extrema

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 37 First Derivative Test for Local Extrema

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 38 Example Using the First Derivative Test

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 39 Concavity

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 40 Concavity

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 41 Concavity Test

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 42 Example Determining Concavity

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 43 Point of Inflection

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 44 Example Finding Points of Inflection

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 45 Second Derivative Test for Local Extrema

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 46 Example Using the Second Derivative Test

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 47 Learning about Functions from Derivatives

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 48 Quick Quiz for Sections 4.1-4.3

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 49 Quick Quiz for Sections 4.1-4.3

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 50 Quick Quiz for Sections 4.1-4.3

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 51 Quick Quiz for Sections 4.1-4.3

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 52 Quick Quiz for Sections 4.1-4.3

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 53 Quick Quiz for Sections 4.1-4.3

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4.4 Modeling and Optimization

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 55 Quick Review

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 56 Quick Review Solutions

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 57 What you’ll learn about Examples from Mathematics Examples from Business and Industry Examples from Economics Modeling Discrete Phenomena with Differentiable Functions …and why Historically, optimization problems were among the earliest applications of what we now call differential calculus.

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 58 Strategy for Solving Max-Min Problems

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 59 Example Inscribing Rectangles

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 60 Maximum Profit

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 61 Example Maximizing Profit

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 62 Minimizing Average Cost

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4.5 Linearization and Newton’s Method

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 64 Quick Review

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 65 Quick Review Solutions

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 66 What you’ll learn about Linear Approximation Newton’s Method Differentials Estimating Change with Differentials Absolute, Relative, and Percent Change Sensitivity to Change …and why Engineering and science depend on approximation in most practical applications; it is important to understand how approximation techniques work.

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 67 Linearization

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 68 Example Finding a Linearization

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 69 Procedure for Newton’s Method

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 70 Procedure for Newton’s Method

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 71 Using Newton’s Method

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 72 Differentials

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 73 Example Finding the Differential dy

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 74 Differential Estimate of Change

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 75 Estimating Change with Differentials

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 76 Example Estimating Change with Differentials

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 4.6 Related Rates

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 78 Quick Review

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 79 Quick Review Solutions

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 80 What you’ll learn about Related Rate Equations Solution Strategy Simulating Related Motion …and why Related rate problems are at the heart of Newtonian mechanics; it was essentially to solve such problems that calculus was invented.

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 81 Strategy for Solving Related Rate Problems

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 82 Strategy for Solving Related Rate Problem

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 83 Example A Highway Chase

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 84 Example A Highway Chase (cont’d)

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 85 Quick Quiz for Sections 4.4-4.6

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 86 Quick Quiz for Sections 4.4-4.6

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 87 Quick Quiz for Sections 4.4-4.6

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 88 Quick Quiz for Sections 4.4-4.6

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 89 Quick Quiz for Sections 4.4-4.6

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 90 Quick Quiz for Sections 4.4-4.6

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 91 Chapter Test

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 92 Chapter Test

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 93 Chapter Test

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 94 Chapter Test

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 95 Chapter Test

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 96 Chapter Test Solutions

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 97 Chapter Test Solutions

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 98 Chapter Test Solutions

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 99 Chapter Test Solutions

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 100 Chapter Test Solutions