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

1. Day 26 1. Warm-up
g(x) = 6x2 – x
1) 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. Day 26 1. Warm-up
g(x) = 6x2 – x
Domain: x < 5
1) Find the domain and range of f(x).
Range: y > -3
2) Evaluate:
f(-2) = f(-3) = f(4) = f(-5)f(3) =
1
g(-2) = g( 2 ) = f(0) – g(-1) =

3. Day 26 1. Warm-up
g(x) = 6x2 – x
Domain: x < 5
1) Find the domain and range of f(x).
Range: y > -3
2) 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. 2. Evaluating Composite Functions
f( g(2) ) =

5. 2. Evaluating Composite Functions
f( g(2) ) = f( )=

6. 2. Evaluating Composite Functions
f( g(2) ) = f( )=

7. 2. Evaluating Composite Functions
f( g(2) ) = f( )=

8. 2. Evaluating Composite Functions
f( g(2) ) = f( )=

9. 2. Evaluating Composite Functions
-2
f( g(2) ) = f( )=

10. 2. Evaluating Composite Functions
f( g(2) ) = f( -2 ) =

11. 2. Evaluating Composite Functions
f( g(2) ) = f( -2 ) =

12. 2. Evaluating Composite Functions
f( g(2) ) = f( -2 ) =

13. 2. Evaluating Composite Functions
f( g(2) ) = f( -2 ) =

14. 2. Evaluating Composite Functions
f( g(2) ) = f( -2 ) =

15. 2. Evaluating Composite Functions
f( g(2) ) = f( -2 ) = -4

16. 2. Evaluating Composite Functions
g( f(-5) ) =

17. 2. Evaluating Composite Functions
g( f(-5) ) = g( )=

18. 2. Evaluating Composite Functions
g( f(-5) ) = g( )=

19. 2. Evaluating Composite Functions
g( f(-5) ) = g( )=

20. 2. Evaluating Composite Functions
g( f(-5) ) = g( )=

21. 2. Evaluating Composite Functions
5
g( f(-5) ) = g( )=

22. 2. Evaluating Composite Functions
g( f(-5) ) = g( 5 ) =

23. 2. Evaluating Composite Functions
g( f(-5) ) = g( 5 ) =

24. 2. Evaluating Composite Functions
g( f(-5) ) = g( 5 ) =

25. 2. Evaluating Composite Functions
g( f(-5) ) = g( 5 ) =

26. 2. Evaluating Composite Functions
g( f(-5) ) = g( 5 ) =

27. 2. Evaluating Composite Functions
g( f(-5) ) = g( 5 ) = 1

28. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) =

29. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=

30. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=

31. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=
Do the
inside first!

32. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=
Do the
inside first!
g(2) =

33. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=
Do the
inside first!
g(2) = 3( )2 –( )

34. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=
Do the
inside first!
g(2) = 3( 2 )2 – (2)

35. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 =

36. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( )=
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 = 12 – 2 = 10

37. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( 10 ) =
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 = 12 – 2 = 10

38. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( 10 ) = -2( ) + 1 =
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 = 12 – 2 = 10

39. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( 10 ) = -2(10) + 1 =
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 = 12 – 2 = 10

40. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( 10 ) = -2(10) + 1 = -19
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 = 12 – 2 = 10

41. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
f( g(2) ) = f( 10 ) = -2(10) + 1 = -19
Do the
inside first!
g(2) =3( 2 )– (2)
2
= 3(4) – 2 = 12 – 2 = 10
Answer: f( g(2) ) = -19

42. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
g( f(-3) ) =

43. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
g( f(-3) ) =g( )

44. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
g( f(-3) ) =g( )

45. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
g( f(-3) ) =g( )
f(-3) =

46. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
g( f(-3) ) =g( )
f(-3) = -2( ) + 1

47. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
g( f(-3) ) =g( )
f(-3) = -2(-3) + 1

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. 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. 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. 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. 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. 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. 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. 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. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
You try:
g( f(5) ) =
f( g(-1) ) =
f( f(-4) ) =

57. 2. Evaluating Composite Functions
f(x) = -2x + 1 g(x) = 3x 2 –x
You try:
g( f(5) ) = g( -9 ) = 252
f( g(-1) ) = f( 4 ) = -7
f( f(-4) ) = f( 9 ) = -17

58. 3. Exercises
Given 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. Day 27
1. Limits
Common Sense Definition
A limit is the intended height of a function.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 limits
lim f (x) = L if and only if x→c− f (x) = L and x→c+ f (x) = L
lim lim
x→c

102. 1. Limits
Example
Find the following limits:

103. 1. Limits
Example
Find the following limits:
=1

104. 1. Limits
Example
Find the following limits:
=1 =DNE

105. 1. Limits
Example
Find the following limits:
=1 =DNE =2

106. 1. Limits
Example
Find the following limits:
=1 =DNE =2 = −∞

107. 1. Limits
Example
True or False:
!
DNE
!
DNE
!
DNE

108. 1. Limits
Example
True or False:
! False
DNE
!
DNE
!
DNE

109. 1. Limits
Example
True or False:
! False
DNE False
!
DNE
!
DNE

110. 1. Limits
Example
True or False:
! False
DNE False
! False
DNE
!
DNE

111. 1. Limits
Example
True or False:
! False
DNE False
! False
DNE True
!
DNE

112. 1. Limits
Example
True or False:
! False
DNE False
! False
DNE True
! True
DNE

113. 1. Limits
Example
True or False:
! False
DNE False
! False
DNE True
! True
DNE False