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 Slide 2- 10 Definition of Limit

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 11 Definition of Limit continued

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 12 Definition of Limit continued

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 13 Properties of Limits

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 14 Properties of Limits continued Product Rule: Constant Multiple Rule:

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

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 16 Example Properties of Limits

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 17 Polynomial and Rational Functions

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 18 Example Limits

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

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

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 21 Example Limits [-6,6] by [-10,10]

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 22 One-Sided and Two-Sided Limits

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 23 One-Sided and Two-Sided Limits continued

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 24 Example One-Sided and Two-Sided Limits o 12 3 4 Find the following limits from the given graph.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 25 Sandwich Theorem

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 26 Sandwich Theorem

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.2 Limits Involving Infinity

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 28 Quick Review

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

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 30 Quick Review

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 31 Quick Review

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 32 Quick Review Solutions [-12,12] by [-8,8][-6,6] by [-4,4]

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 33 Quick Review Solutions

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 34 Quick Review Solutions

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 35 Quick Review Solutions

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 36 What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 37 Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 38 Horizontal Asymptote

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 39 [-6,6] by [-5,5] Example Horizontal Asymptote

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 40 Example Sandwich Theorem Revisited

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 41 Properties of Limits as x→±∞

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 42 Properties of Limits as x→±∞ Product Rule: Constant Multiple Rule:

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 43 Properties of Limits as x→±∞

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 44 Infinite Limits as x→a

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 45 Vertical Asymptote

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 46 Example Vertical Asymptote [-6,6] by [-6,6]

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 47 End Behavior Models

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 48 Example End Behavior Models

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

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 50 End Behavior Models

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 51 Example “Seeing” Limits as x→±∞

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 52 Quick Quiz Sections 2.1 and 2.2

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 53 Quick Quiz Sections 2.1 and 2.2

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 54 Quick Quiz Sections 2.1 and 2.2

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 55 Quick Quiz Sections 2.1 and 2.2

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 56 Quick Quiz Sections 2.1 and 2.2

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 57 Quick Quiz Sections 2.1 and 2.2

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Continuity

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 59 Quick Review

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 60 Quick Review

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 61 Quick Review

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 62 Quick Review

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 63 Quick Review Solutions

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 64 Quick Review Solutions

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 65 Quick Review Solutions

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 66 Quick Review Solutions

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 67 What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions …and why Continuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 68 Continuity at a Point

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 69 Example Continuity at a Point o

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 70 Continuity at a Point

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 71 Continuity at a Point If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 72 Continuity at a Point The typical discontinuity types are: a)Removable(2.21b and 2.21c) b)Jump(2.21d) c)Infinite(2.21e) d)Oscillating (2.21f)

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 73 Continuity at a Point

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 74 Example Continuity at a Point [-5,5] by [-5,10]

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 75 Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 76 Continuous Functions [-5,5] by [-5,10]

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 77 Properties of Continuous Functions

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 78 Composite of Continuous Functions

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 79 Intermediate Value Theorem for Continuous Functions

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 80 Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.4 Rates of Change and Tangent Lines

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 82 Quick Review

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 83 Quick Review

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 84 Quick Review

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 85 Quick Review Solutions

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 86 Quick Review Solutions

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 87 Quick Review Solutions

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 88 What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited …and why The tangent line determines the direction of a body’s motion at every point along its path.

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 89 Average Rates of Change The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. Also, the average rate of change can be thought of as the slope of a secant line to a curve.

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 90 Example Average Rates of Change

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 91 Tangent to a Curve In calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. The problem with this is that we only have one point and our usual definition of slope requires two points.

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 92 Tangent to a Curve The process becomes: 1.Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. 2.Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3.Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 93 Example Tangent to a Curve

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 94 Example Tangent to a Curve

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 95 Slope of a Curve To find the tangent to a curve y = f(x) at a point P(a,f(a)) calculate the slope of the secant line through P and a point Q(a+h, f(a+h)). Next, investigate the limit of the slope as h→0. If the limit exists, it is the slope of the curve at P and we define the tangent at P to be the line through P with this slope.

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 96 Slope of a Curve

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 97 Slope of a Curve at a Point

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 98 Slope of a Curve

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 99 Normal to a Curve The normal line to a curve at a point is the line perpendicular to the tangent at the point. The slope of the normal line is the negative reciprocal of the slope of the tangent line.

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 100 Example Normal to a Curve

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 101 Speed Revisited

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 102 Quick Quiz Sections 2.3 and 2.4

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 103 Quick Quiz Sections 2.3 and 2.4

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 104 Quick Quiz Sections 2.3 and 2.4

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 105 Quick Quiz Sections 2.3 and 2.4

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 106 Quick Quiz Sections 2.3 and 2.4

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 107 Quick Quiz Sections 2.3 and 2.4

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 108 Chapter Test

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 109 Chapter Test

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 110 Chapter Test

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 111 Chapter Test

112. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 112 Chapter Test

113. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 113 Chapter Test

114. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 114 Chapter Test Solutions

115. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 115 Chapter Test Solutions

116. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 116 Chapter Test Solutions

117. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 117 Chapter Test Solutions

118. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 118 Chapter Test Solutions

119. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 119 Chapter Test Solutions

120. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 120 Chapter Test Solutions

121. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 121 Chapter Test Solutions