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PM5006 Week 6

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Aquí están las transparencias de la semana 6 de mi curso de Cálculo diferencial. Algunas transparencias las tomé de Dan Meyer y Dan Greene.

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PM5006 Week 6

1. 1. Day 26 1. Warm-up g(x) = 6x2 – x1) Find the domain and range of f(x).2) Evaluate: f(-2) = f(-3) = f(4) = f(-5)f(3) = 1 g(-2) = g( 2 ) = f(0) – g(-1) =
2. 2. Day 26 1. Warm-up g(x) = 6x2 – x Domain: x < 51) Find the domain and range of f(x). Range: y > -32) Evaluate: f(-2) = f(-3) = f(4) = f(-5)f(3) = 1 g(-2) = g( 2 ) = f(0) – g(-1) =
3. 3. Day 26 1. Warm-up g(x) = 6x2 – x Domain: x < 51) Find the domain and range of f(x). Range: y > -32) Evaluate: f(-2) = -3 f(-3) = -1 f(4) = -1.5 f(-5)f(3) = 1/9 1 g(-2) = 26 g( 2 ) = 1 f(0) – g(-1) = -10
4. 4. 2. Evaluating Composite Functions f( g(2) ) =
5. 5. 2. Evaluating Composite Functions f( g(2) ) = f( )=
6. 6. 2. Evaluating Composite Functions f( g(2) ) = f( )=
7. 7. 2. Evaluating Composite Functions f( g(2) ) = f( )=
8. 8. 2. Evaluating Composite Functions f( g(2) ) = f( )=
9. 9. 2. Evaluating Composite Functions -2 f( g(2) ) = f( )=
10. 10. 2. Evaluating Composite Functions f( g(2) ) = f( -2 ) =
11. 11. 2. Evaluating Composite Functions f( g(2) ) = f( -2 ) =
12. 12. 2. Evaluating Composite Functions f( g(2) ) = f( -2 ) =
13. 13. 2. Evaluating Composite Functions f( g(2) ) = f( -2 ) =
14. 14. 2. Evaluating Composite Functions f( g(2) ) = f( -2 ) =
15. 15. 2. Evaluating Composite Functions f( g(2) ) = f( -2 ) = -4
16. 16. 2. Evaluating Composite Functions g( f(-5) ) =
17. 17. 2. Evaluating Composite Functions g( f(-5) ) = g( )=
18. 18. 2. Evaluating Composite Functions g( f(-5) ) = g( )=
19. 19. 2. Evaluating Composite Functions g( f(-5) ) = g( )=
20. 20. 2. Evaluating Composite Functions g( f(-5) ) = g( )=
21. 21. 2. Evaluating Composite Functions5 g( f(-5) ) = g( )=
22. 22. 2. Evaluating Composite Functions g( f(-5) ) = g( 5 ) =
23. 23. 2. Evaluating Composite Functions g( f(-5) ) = g( 5 ) =
24. 24. 2. Evaluating Composite Functions g( f(-5) ) = g( 5 ) =
25. 25. 2. Evaluating Composite Functions g( f(-5) ) = g( 5 ) =
26. 26. 2. Evaluating Composite Functions g( f(-5) ) = g( 5 ) =
27. 27. 2. Evaluating Composite Functions g( f(-5) ) = g( 5 ) = 1
28. 28. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) =
29. 29. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )=
30. 30. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )=
31. 31. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )= Do theinside ﬁrst!
32. 32. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )= Do theinside ﬁrst! g(2) =
33. 33. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )= Do theinside ﬁrst! g(2) = 3( )2 –( )
34. 34. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )= Do theinside ﬁrst! g(2) = 3( 2 )2 – (2)
35. 35. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )= Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 =
36. 36. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( )= Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 = 12 – 2 = 10
37. 37. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( 10 ) = Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 = 12 – 2 = 10
38. 38. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( 10 ) = -2( ) + 1 = Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 = 12 – 2 = 10
39. 39. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( 10 ) = -2(10) + 1 = Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 = 12 – 2 = 10
40. 40. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( 10 ) = -2(10) + 1 = -19 Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 = 12 – 2 = 10
41. 41. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x f( g(2) ) = f( 10 ) = -2(10) + 1 = -19 Do theinside ﬁrst! g(2) =3( 2 )– (2) 2 = 3(4) – 2 = 12 – 2 = 10 Answer: f( g(2) ) = -19
42. 42. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =
43. 43. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( )
44. 44. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( )
45. 45. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( ) f(-3) =
46. 46. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( ) f(-3) = -2( ) + 1
47. 47. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( ) f(-3) = -2(-3) + 1
48. 48. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( ) f(-3) = -2(-3) + 1 = 6 + 1 =
49. 49. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( ) f(-3) = -2(-3) + 1 = 6 + 1 = 7
50. 50. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( 7 ) f(-3) = -2(-3) + 1 = 6 + 1 = 7
51. 51. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( 7 ) = 3( )2 –( ) f(-3) = -2(-3) + 1 = 6 + 1 = 7
52. 52. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( 7 ) = 3( 7 )2 – (7) f(-3) = -2(-3) + 1 = 6 + 1 = 7
53. 53. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( 7 ) = 3( 7 ) – (7) 2 = 3(49) – 7 f(-3) = -2(-3) + 1 = 6 + 1 = 7
54. 54. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( 7 ) = 3( 7 ) – (7) 2 = 3(49) – 7 = 140 f(-3) = -2(-3) + 1 = 6 + 1 = 7
55. 55. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –x g( f(-3) ) =g( 7 ) = 3( 7 ) – (7) 2 = 3(49) – 7 = 140 f(-3) = -2(-3) + 1 = 6 + 1 = 7 Answer: g( f(-3) ) = 140
56. 56. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –xYou try: g( f(5) ) = f( g(-1) ) = f( f(-4) ) =
57. 57. 2. Evaluating Composite Functions f(x) = -2x + 1 g(x) = 3x 2 –xYou try: g( f(5) ) = g( -9 ) = 252 f( g(-1) ) = f( 4 ) = -7 f( f(-4) ) = f( 9 ) = -17
58. 58. 3. ExercisesGiven f(x) = 4 - 2x, g(x) = 2x2 - 3x + 5 and h(x) = 3x − 2 x − 10 ,find the following:a) f(h(6))b) h(g(0))c) f(f(-5))d) g(f(1))e) f(h(g(3)))
59. 59. Day 271. Limits Common Sense Definition A limit is the intended height of a function.
60. 60. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right?
61. 61. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right?
62. 62. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right?
63. 63. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right?
64. 64. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right?
65. 65. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right?
66. 66. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right? a) 2
67. 67. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right? a) 2
68. 68. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right? a) 2
69. 69. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right? a) 2
70. 70. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right? a) 2
71. 71. 1. Limits Corregir la Answer the following: f(x) trayectoria de los a) To what value does puntos. f(x) approach when x approaches 1 from the left? b) To what value does f(x) approach when x approaches 1 from the right? a) 2 b) 2
72. 72. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right?
73. 73. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right?
74. 74. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right?
75. 75. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right?
76. 76. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1
77. 77. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1
78. 78. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1
79. 79. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1
80. 80. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1
81. 81. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1
82. 82. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches -1 from the left? b) To what value does f(x) approach when x approaches -1 from the right? a) 1 b) 2
83. 83. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
84. 84. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
85. 85. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
86. 86. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
87. 87. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
88. 88. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
89. 89. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
90. 90. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞
91. 91. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
92. 92. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
93. 93. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
94. 94. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
95. 95. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
96. 96. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
97. 97. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
98. 98. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0
99. 99. 1. Limits Answer the following: f(x) a) To what value does f(x) approach when x approaches ∞ b) To what value does f(x) approach when x approaches −∞ a) 0 b) 0
100. 100. 1. Limits Definition If f(x) approaches some finite number L as x approaches c, then we say that the limit of f(x) as x approaches c is L and symbolically write lim f (x) = L x→c
101. 101. 1. Limits One-sided limits lim f (x) − x approaches c from the left x→c La sesión terminó lim f (x) con estaapproaches c from the right x x→c + transparencia. Two-sided limitslim f (x) = L if and only if x→c− f (x) = L and x→c+ f (x) = L lim limx→c
102. 102. 1. Limits Example Find the following limits:
103. 103. 1. Limits Example Find the following limits: =1
104. 104. 1. Limits Example Find the following limits: =1 =DNE
105. 105. 1. Limits Example Find the following limits: =1 =DNE =2
106. 106. 1. Limits Example Find the following limits: =1 =DNE =2 = −∞
107. 107. 1. Limits Example True or False: ! DNE ! DNE ! DNE
108. 108. 1. Limits Example True or False: ! False DNE ! DNE ! DNE
109. 109. 1. Limits Example True or False: ! False DNE False ! DNE ! DNE
110. 110. 1. Limits Example True or False: ! False DNE False ! False DNE ! DNE
111. 111. 1. Limits Example True or False: ! False DNE False ! False DNE True ! DNE
112. 112. 1. Limits Example True or False: ! False DNE False ! False DNE True ! True DNE
113. 113. 1. Limits Example True or False: ! False DNE False ! False DNE True ! True DNE False