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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

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