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 You Do Finding Absolute Extrema

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 17 Example Finding Extreme Values

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 18 You Do Finding Extreme Values

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

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 20 Quick Review

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 21 Quick Review Solutions

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 22 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.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 23 Rolles Theorem

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 24 Mean Value Theorem for Derivatives

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 25 Example Explore the Mean Value Theorem

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 26 You Do Explore the Mean Value Theorem

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 27 Increasing Function, Decreasing Function

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 28 Corollary: Increasing and Decreasing Functions

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 29 Example Determining Where Graphs Rise or Fall

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 30 You Do Determining Where Graphs Rise or Fall

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 31 Corollary: Functions with f’=0 are Constant

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 32 Corollary: Functions with the Same Derivative Differ by a Constant

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

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 34 Example Finding Velocity and Position

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall You Try Slide 4- 35

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

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 37 Quick Review

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 38 Quick Review

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

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

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

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 42 First Derivative Test for Local Extrema

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 43 First Derivative Test for Local Extrema

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 44 Example Using the First Derivative Test

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 45 You Do Using the First Derivative Test

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 46 Concavity

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 47 Concavity

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 48 Concavity Test

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 49 Example Determining Concavity

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 50 You Try Determining Concavity

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 51 Point of Inflection

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 52 Example Finding Points of Inflection

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 53 Example Finding Points of Inflection

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 54 Second Derivative Test for Local Extrema

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 55 Example Using the Second Derivative Test

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 56 You Do Using the Second Derivative Test

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 57 Learning about Functions from Derivatives

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 58 Quick Quiz for Sections 4.1-4.3

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 59 Quick Quiz for Sections 4.1-4.3

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 60 Quick Quiz for Sections 4.1-4.3

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 61 Quick Quiz for Sections 4.1-4.3

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 62 Quick Quiz for Sections 4.1-4.3

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 63 Quick Quiz for Sections 4.1-4.3

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

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

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 66 Quick Review Solutions

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

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 68 Strategy for Solving Max-Min Problems

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 69 Example Inscribing Rectangles

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 70 Example Boat Walk

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 71 Maximum Profit

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

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 73 You Do Maximizing Profit

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 74 Minimizing Average Cost

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

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 76 Quick Review

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 77 Quick Review Solutions

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

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

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 80 Example Finding a Linearization

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 81 Example Finding a Linearization

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 82 Procedure for Newton’s Method

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 83 Procedure for Newton’s Method

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 84 Using Newton’s Method

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 85 You Do Newton’s Method

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 86 Differentials

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 87 Example Finding the Differential dy

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 88 Differential Estimate of Change

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 89 Estimating Change with Differentials

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 90 Example Estimating Change with Differentials

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

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

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 93 Quick Review Solutions

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

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 95 Strategy for Solving Related Rate Problems

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 96 Strategy for Solving Related Rate Problem

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 97 Example A Highway Chase

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 98 Example A Highway Chase (cont’d)

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 99 Quick Quiz for Sections 4.4-4.6

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 100 Quick Quiz for Sections 4.4-4.6

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 101 Quick Quiz for Sections 4.4-4.6

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 102 Quick Quiz for Sections 4.4-4.6

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 103 Quick Quiz for Sections 4.4-4.6

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 104 Quick Quiz for Sections 4.4-4.6

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 105 Chapter Test

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 106 Chapter Test

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 107 Chapter Test

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 108 Chapter Test

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 109 Chapter Test

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 110 Chapter Test Solutions

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 111 Chapter Test Solutions

112. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 112 Chapter Test Solutions

113. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 113 Chapter Test Solutions

114. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 4- 114 Chapter Test Solutions