apostila___limites_e_derivadas
DESCRIPTION
LimitesTRANSCRIPT
LIMITES
LIMITES
1. Calcule os limites:
a)
b)
c)
d)
e)
_____________________________________
2. Determine:
a)
b)
c)
d)
e)
f)
_____________________________________
3. Calcule:
a)
b)
c)
d)
_____________________________________
4. Ache o valor de:
a)
b)
c)
d)
_____________________________________
5. Calcule os limites:
a)
b)
_____________________________________
6. Calcule , em cada caso:
a)
b)
c)
d)
e)
_____________________________________
7. Dada a funo , calcule:
a)
b)
c)
d)
e)
_____________________________________
8. Dada a funo diga se f(x) contnua nos pontos:
a) x = 0b) x = 1c) x = 2
_____________________________________
9. Seja m R e f: R R a funo definida por:
Calcular o valor de m para que f(x) seja contnua em x = 3.
_____________________________________
10. Dada a funo , diga se f(x) contnua nos pontos:
a) x = 5
b) x = 2
_____________________________________
11. Seja R e seja f: R R a funo definida por
Calcule para que f(x) seja contnua em x=3.
12. Determine se a funo f , definida por:
contnua ou descontnua nos pontos:
a) x = 1
b) x = 3
_____________________________________
13. Mostre se a funo contnua ou descontnua em x = 3.
_____________________________________
14. Considere a funo, definida em R por:
Calcular o valor de k para que a funo seja contnua em x = 1.
_____________________________________
15. Dada a funo:
Determinar m para que f(x) seja contnua em x = 2.
Sugesto: multiplicar o numerador e denominador pelo conjugado .
_____________________________________
16. A funo contnua y = f(x) est definida no intervalo [ 4, 8] por:
Sendo a e b nmeros reais.
Calcule os valores de a e b e esboce o grfico cartesiano da funo dada.
_____________________________________
17. Determine:
a)
b)
18. Determine:
a)
b)
_____________________________________
19. Ache o valor de:
a)
b)
_____________________________________
20. Calcule:
a)
b)
c)
_____________________________________
21. Calcule:
a)
e)
b)
f)
c)
g)
d)
h)
_____________________________________
22. Calcule
_____________________________________
23. Determine:
a)
c)
b)
d)
24. Calcule
_____________________________________
25. Determine:
a)
c)
b)
d)
_____________________________________
26. Calcule:
a)
c)
b)
d)
_____________________________________
27. Calcule:
a)
b)
c)
_____________________________________
28. Calcular
_____________________________________
29. Ache o valor de
_____________________________________
30. Calcular
_____________________________________
31. Determine:
a)
b)
_____________________________________
32. Determine
33. Calcule
_____________________________________
34. Calcule
_____________________________________
35. Calcule:
a)
c)
b)
d)
_____________________________________
36. Calcule:
a)
b)
c)
d)
e)
f)
_____________________________________
37. Fatore as expresses e simplifique as fraes para obter o valor de:
a)
c)
b)
d)
_____________________________________
38. Calcular o valor de
_____________________________________
39. Determine:
a)
b)
c)
d)
e)
_____________________________________
40. Calcule
_____________________________________
41. Multiplique o numerador e o denominador pelo conjugado de um deles para determinar:
a)
b)
_____________________________________
42. Calcular o valor de
_____________________________________
43. Calcular o valor da expresso
_____________________________________
44. Determine o valor de
Sugesto: multiplicar o denominador e numerador pelos conjugados de ambos.
_____________________________________
45. Calcular
_____________________________________
46. Calcule
_____________________________________
47. Calcule:
a)
b)
_____________________________________
48. Determine:
_____________________________________
49. Calcular o valor de
_____________________________________
50. Dada a funo
, calcule:
a)
b)
51. Calcule:
a)
c)
b)
d)
_____________________________________
52. Calcule
_____________________________________
53. Ache o valor de
_____________________________________
54. Calcule:
a)
c)
b)
d)
_____________________________________
55. Calcular para:
a) k = 0
b) k 0
_____________________________________
56. Calcular:
a)
b)
c)
_____________________________________
57. Determine
_____________________________________
58. Sejam R e a R, a 0. Determine:
a)
b)
59. A funo f: R R, com
contnua para x = 4. Calcular o valor de m.
_____________________________________
60. A funo no est definida para x = 1. Seja f(1) = k. Calcular o valor de k para que a funo f(x) seja contnua no ponto x = 1.
_____________________________________
61. Esboce o grfico da funo e determine o limite:
a)
d)
b)
e)
c)
f)
_____________________________________
62. Calcule:
a)
b)
_____________________________________
63. Calcular os limites:
a)
b)
_____________________________________
64. Calcule:
a)
b)
_____________________________________
65. Esboce o grfico da funo e determine o limite:
a)
b)
_____________________________________
66. Calcular:
a)
b)
67. Esboce o grfico da funo e d o valor de:
a)
b)
_____________________________________
68. Calcule
_____________________________________
69. Calcule:
a)
c)
b)
d)
_____________________________________
70. Determinar
_____________________________________
71. Calcular o valor de
_____________________________________
72. Ache o valor de
_____________________________________
73. Determine
_____________________________________
74. Determine:
a)
b)
_____________________________________
75. Calcule o valor de
_____________________________________
76. Calcule:
a)
b)
_____________________________________
77. Sabendo que , calcule
78. Aplicando o limite exponencial fundamental, calcule:
a)
c)
b)
d)
_____________________________________
79. Ache o valor de
_____________________________________
80. Calcule
_____________________________________
81. Calcule
_____________________________________
82. Determine
_____________________________________
83. Calcule
_____________________________________
84. Se , calcule ln a.
_____________________________________
85. Calcule .
Sugesto:
_____________________________________
RECORDANDO
1. Calcule:
a)
b)
c)
2. Ache o valor de .
_____________________________________
3. Seja um nmero real e seja f: R R a funo tal que:
Calcule para que exista
_____________________________________
4. Sabendo-se que , x m, ento podemos afirmar que:
a) m maior do que 4
b) m menor do que 4
c) m [1, 4]
d) m [ 4, 1]
e) no existe m, tal que
_____________________________________
5. Seja f definida por
o valor de :
a) 1b) 2c) 3d) 4e) 5
_____________________________________
6. Determine:
a)
b)
c)
d)
_____________________________________
7. Calcule
_____________________________________
8. Determine m para que
9. Determine:
a)
b)
_____________________________________
10. O valor de :
a) zero b) + c) d) 2 e) 1
_____________________________________
11. Determine:
a)
b)
c)
d)
_____________________________________
12.Calcule:
a)
b)
c)
d)
_____________________________________
13. Dada a funo f: R R, definida por , calcule
_____________________________________
14. Calcule
_____________________________________
15. Determine
_____________________________________
16. Calcule:
a)
b)
_____________________________________
17. Determine:
a)
b)
18. Dada a funo f: R R tal que
Determinar o valor de m de modo que f(x) seja contnua em x = 1.
_____________________________________
19. Calcule
_____________________________________
20. Sabe-se que
. Conclui-se que :
a)
b) 0
c) infinito
d) indeterminado
e) no existe
_____________________________________
21. Calcule
_____________________________________
22. Calcule:
a)
b)
_____________________________________
23. Determinar
Sugesto:
_____________________________________
24. Calcular
Sugesto:
_____________________________________
25. Determine
DERIVADAS1. Aplicando a definio, calcule a derivada da funo f(x) = x2 + x no ponto de abscissa:
a) x = 3
b) x = 2
_____________________________________
2. Dada a funo f(x) = x2 5x + 6. Calcule:
a) f (1)
b) f ( 4)
_____________________________________
3. Dada a funo f(x) = 2 x3, calcule f( 2)
_____________________________________
4. Dada a funo , determine, se existir, a derivada da funo no ponto de abscissa:
a) x = 1
b) x = 0
_____________________________________
5. Dada a funo , determine a derivada de f(x) no ponto x = 1.
_____________________________________
6. Usando a definio, calcule a derivada da funo f(x) = 3x + 1_____________________________________
7. Usando a definio, calcule f(x) em cada caso:
a) f(x) = 5x2
b)
_____________________________________
8. Dada a funo , determine a derivada de f(x) para x = 4.
_____________________________________
9. Calcule a derivada f(x) das seguintes funes:
a) f(x) = 8
f)
b)
g)
c) f(x) = x6
h)
d) f(x) = x-5
i) f(x) = 7x2e)
j) f(x) = 4x10. Ache a derivada das seguintes funes:
a)
c)
b)
d)
_____________________________________
11. Dada a funo , calcule a derivada de f(x) no ponto x = 8.
_____________________________________
12. Ache a derivada f(x) das seguintes funes:
a)
c)
b)
d)
_____________________________________
13. Dada a funo . Calcular a derivada da funo para:
a) x = 1
c) x = 3
b) x = 4
d) x = 6
_____________________________________
14. Ache a derivada f(x) das seguintes funes:
a)
b)
c)
d)
e)
f)
_____________________________________
15. Considere as funes definidas em R por g(x)= 4x + 1 e h(x) = 2x 3.
a) Calcule f(x), sabendo que f(x) = g[h(x)]
b) Calcule f(2)
_____________________________________
16. Se , calcule f().
_____________________________________
17. Determinar a derivada f(x) das funes para x = 2 nos seguintes casos:
a) f(x) = 6x3 5x2 + 2x 1
b) f(x) = 5x4 2x2 + 18
c) f(x) = 2x5 3x2 + 4x 2
18. Determine a derivada das funes:
a)
b)
c)
d)
_____________________________________
19. Dada a funo de R em R definida por f(x) = x3 12x + 7, determine o valor de sua derivada para x = 3.
_____________________________________
20. Calcule f(x) das seguintes funes:
a) f(x) = 3x . sen x
b) f(x) = sen x . cos x
c) f(x) = x2 . cos x
d) f(x) = x3 . (2x2 3x)
_____________________________________
21. Calcule a derivada f(x) das seguintes funes:
a) f(x) = (x + 4) (x 2)
b) f(x) = (x 1) (2x 3)
c) f(x) = (x3 7) (2x2 + 3)
d) f(t) = (t2 1) (t2 + 1)
_____________________________________
22. Em cada caso, calcule a derivada f(t):
a) f(t) = (t2 + 1) . (t3 2)
b) f(t) = (t5 2t3) . (t2 + t 2)
_____________________________________
23. Dada a funo
f(x) = (x2 1) . (x2 + x 2) . (1 x)
Calcule a derivada f(x) para:
a) x = 0c)
b) x = 1d) x = 2
_____________________________________
24. Determine a derivada f(x) das seguintes funes:
a)
b)
_____________________________________
25. Calcule a derivada das funes para x = 2 nos seguintes casos:
a)
b)
_____________________________________
26. Considere a funo definida em R por
a) Determine as razes de f(x)
b) Calcule f(1) e f( 1)
c) Resolva a inequao f(x) < 0
27. Dada a funo , determine f(x).
_____________________________________
28. Aplicando a derivada do quociente, demonstre que:
a) Se f(x) = cotg x, ento f(x) = cosec2 x
b) Se f(x) = sec x, ento f(x) = tg x . sec x
c) Se f(x) = cosec x, ento f(x) = cotg x . cosec x
_____________________________________
29. Dado , calcular
_____________________________________
30. Quais os valores de x que anulam a derivada f(x) da funo
_____________________________________
31. Calcule a derivada das funes:
a) f(x) = cos 6x
b) f(x) = sen (3x + 1)
c) f(x) = sen 3x cos 2x
d) f(x) = sen 2x + sen 4x
_____________________________________
32. Dada a funo , calcule f(x)
_____________________________________
33. Calcule a derivada das funes:
a) f(x) = sen2 x
b) f(x) = sen2 (1 x2)
_____________________________________
34. Determinar a derivada das funes:
a) f(x) = (x2 1)3b) f(x) = (x3 2x)2c) f(x) = (x4 3x2 + 1)2_____________________________________
35. Considere a funo definida em R {2} por . Calcule:
a) f(x)
b) f(3)
_____________________________________
36. Ache a derivada das funes:
a)
b)
_____________________________________
37. Dada a funo ,determinar:
a) f(x)
b) f(3)
38. Calcular a derivada da funo
para x = 2.
_____________________________________
39. Sabendo que , determinar f(1).
_____________________________________
40. Determinar a derivada f(x) das funes:
a)
b)
_____________________________________
41. Calcule a derivada da funo
para x = 2.
_____________________________________
42. Determine a derivada das funes:
a)
d)
b)
e)
c)
f)
_____________________________________
43. Dada a funo , calcule f(2).
_____________________________________
44. Dada a funo , determinar f(1).
_____________________________________
45. Dado , calcule f(1).
_____________________________________
46. Sabendo que , determine f(x)
_____________________________________
47. Calcule a derivada f(x) das seguintes funes:
a)
c)
b)
d)
_____________________________________
48. Se f(x) = ln (x2 4x + ). Calcule f(x).
_____________________________________
49. Se , determine f(x).
_____________________________________
50. Determine f(x), sabendo que .
51. Determine f(x) sabendo que .
_____________________________________
52. Calcule o valor da derivada de:
a) para x = 2
b) para x = 1
c) para x = 0
d) para x = 1
_____________________________________
53. Dada a funo . Calcule:
a) f(4)
b) f(6)
c) f(10)
_____________________________________
54. Ache as quatro primeiras derivadas da funo f(x) = x5 x4 + x3 x2 + x 1.
_____________________________________
55. Se f(x) = sen x + cos x, determine f(4)(x).
_____________________________________
56. Determine a derivada segunda de
f(x) = 4x3 5x2 + 2x 1 no ponto x = 0.
_____________________________________
57. Calcule a derivada terceira da funo para x = 2.
_____________________________________
58. Seja a funo f(x) = 4x3 + 2x2 5x + 2, calcule f(0) + f(0) + f(0).
_____________________________________
59. Obtenha as leis das duas primeiras funes derivadas de .
_____________________________________
60. Dada a funo f(x) = sen x cos x. Calcule:
a) f
b) f
c) f
_____________________________________
61. Calcule o coeficiente angular da tangente ao grfico das funes a seguir nos pontos de abscissa tambm indicados:
a) para x = 1
b) para x = 4
c) para x = 8
62. Determine a equao da reta tangente ao grfico da funo f(x) = x2 6x + 5 no ponto de abscissa x = 0.
_____________________________________
63. Seja a curva de equao y = x3 12x. Determine a equao da reta tangente curva no ponto (4, 16).
_____________________________________
64. Qual a equao da reta tangente ao grfico da funo no ponto ?
_____________________________________
65. Considere a funo f: R R definida por f(x) = x3 3x2 + x + 2. Calcule as coordenadas dos pontos do grfico dessa funo nos quais a reta tangente tem coeficiente angular igual a 1.
_____________________________________
66. Determine a equao da reta tangente ao grfico da funo f(x) = x2 4 e que seja paralela reta de equao y = 2x 1.
_____________________________________
67. Determinar um ponto sobre a curva f(x) = x3 1 de tal modo que a reta tangente curva nesse ponto seria paralela reta y = 12x + 1.
_____________________________________
68. Determine a equao da reta tangente ao grfico de f(x) = 3 cos x no ponto em que .
_____________________________________
69. Determinar a equao da reta tangente curva y = 2x2 1, no ponto de abscissa x = 1.
_____________________________________
70. Em que ponto da curva f(x) = x2 3x 4 a reta tangente paralela ao eixo Ox?
_____________________________________
71. Determine a equao da reta tangente ao grfico de f(x) = x2 4x + 1, que perpendicular reta 2y + x 5 = 0.
_____________________________________
72. Determinar a equao da reta tangente curva no ponto de abscissa x = 10.
_____________________________________
Aplicando a regra de LHospital, resolva:
73.
74.
_____________________________________
75.
_____________________________________
76.
_____________________________________
77.
_____________________________________
78. Determine os intervalos de crescimento e decrescimento das funes:
a)
b)
c)
d)
e)
_____________________________________
79. Dada a funo , determine k para que f(x) seja crescente em R.
_____________________________________
80. Dada a funo , determine:
a) o ponto em que o grfico corta o eixo y
b) os pontos em que a reta tangente ao grfico de f(x) paralela ao eixo x
c) um esboo do grfico de f(x)
d) o conjunto em que f(x) crescente
e) o conjunto em que f(x) decrescente
f) um esboo do grfico de f(x)
_____________________________________
81. Considerando a concavidade da parbola, classifique os pontos cujas abscissas so os pontos crticos das funes quadrticas:
a) f(x) = x2 x + 1 b) f(x) = x x2_____________________________________
82. Determine os pontos cujas abscissas so pontos crticos da funo
f(x) = x4 4x3 + 4x2 + 2
_____________________________________83. Calcule os pontos , sendo que o ponto crtico das funes:
a) f(x) = 2x3 + 3x2 + 1 b) f(x) = x3 3x
c) f(x) = (x2 1)2 + 3
_1184403144.unknown
_1184489404.unknown
_1184504884.unknown
_1184589461.unknown
_1184590899.unknown
_1184592388.unknown
_1185017729.unknown
_1185018029.unknown
_1185018327.unknown
_1185018522.unknown
_1185020046.unknown
_1185020124.unknown
_1185018535.unknown
_1185018409.unknown
_1185018257.unknown
_1185018281.unknown
_1185018234.unknown
_1185017955.unknown
_1185017991.unknown
_1185017797.unknown
_1184677202.unknown
_1184677687.unknown
_1184677863.unknown
_1184677203.unknown
_1184676923.unknown
_1184676951.unknown
_1184676871.unknown
_1184592120.unknown
_1184592308.unknown
_1184592373.unknown
_1184592257.unknown
_1184590995.unknown
_1184591050.unknown
_1184590932.unknown
_1184590411.unknown
_1184590683.unknown
_1184590816.unknown
_1184590858.unknown
_1184590718.unknown
_1184590545.unknown
_1184590546.unknown
_1184590443.unknown
_1184589797.unknown
_1184589890.unknown
_1184589904.unknown
_1184589844.unknown
_1184589648.unknown
_1184589697.unknown
_1184589549.unknown
_1184574993.unknown
_1184576944.unknown
_1184577344.unknown
_1184589392.unknown
_1184589427.unknown
_1184577366.unknown
_1184577258.unknown
_1184577287.unknown
_1184577045.unknown
_1184576036.unknown
_1184576929.unknown
_1184576930.unknown
_1184576494.unknown
_1184575748.unknown
_1184575787.unknown
_1184575747.unknown
_1184505758.unknown
_1184574741.unknown
_1184574842.unknown
_1184574866.unknown
_1184574841.unknown
_1184574123.unknown
_1184574740.unknown
_1184505782.unknown
_1184505090.unknown
_1184505557.unknown
_1184505662.unknown
_1184505305.unknown
_1184504968.unknown
_1184505007.unknown
_1184504928.unknown
_1184492579.unknown
_1184503521.unknown
_1184503942.unknown
_1184504144.unknown
_1184504178.unknown
_1184504856.unknown
_1184504145.unknown
_1184504023.unknown
_1184504053.unknown
_1184503989.unknown
_1184503753.unknown
_1184503813.unknown
_1184503832.unknown
_1184503774.unknown
_1184503593.unknown
_1184503634.unknown
_1184503563.unknown
_1184493077.unknown
_1184502425.unknown
_1184502779.unknown
_1184502805.unknown
_1184502618.unknown
_1184493179.unknown
_1184493305.unknown
_1184493129.unknown
_1184492900.unknown
_1184493005.unknown
_1184493018.unknown
_1184493004.unknown
_1184492711.unknown
_1184492769.unknown
_1184492602.unknown
_1184490859.unknown
_1184491407.unknown
_1184492186.unknown
_1184492332.unknown
_1184492382.unknown
_1184492221.unknown
_1184492046.unknown
_1184492061.unknown
_1184491486.unknown
_1184491086.unknown
_1184491234.unknown
_1184491322.unknown
_1184491137.unknown
_1184491017.unknown
_1184491039.unknown
_1184490899.unknown
_1184490048.unknown
_1184490412.unknown
_1184490524.unknown
_1184490746.unknown
_1184490511.unknown
_1184490248.unknown
_1184490372.unknown
_1184490095.unknown
_1184489692.unknown
_1184489930.unknown
_1184489991.unknown
_1184489762.unknown
_1184489583.unknown
_1184489674.unknown
_1184489561.unknown
_1184485710.unknown
_1184487379.unknown
_1184488357.unknown
_1184488856.unknown
_1184489288.unknown
_1184489349.unknown
_1184489166.unknown
_1184488512.unknown
_1184488745.unknown
_1184488424.unknown
_1184488028.unknown
_1184488272.unknown
_1184488333.unknown
_1184488126.unknown
_1184487941.unknown
_1184487953.unknown
_1184487401.unknown
_1184486526.unknown
_1184486994.unknown
_1184487218.unknown
_1184487294.unknown
_1184487072.unknown
_1184486927.unknown
_1184486959.unknown
_1184486800.unknown
_1184486244.unknown
_1184486449.unknown
_1184486469.unknown
_1184486393.unknown
_1184485777.unknown
_1184486109.unknown
_1184485755.unknown
_1184404669.unknown
_1184405316.unknown
_1184405524.unknown
_1184405538.unknown
_1184485675.unknown
_1184405525.unknown
_1184405334.unknown
_1184405426.unknown
_1184405317.unknown
_1184404929.unknown
_1184404948.unknown
_1184404995.unknown
_1184404930.unknown
_1184404847.unknown
_1184404848.unknown
_1184404724.unknown
_1184404103.unknown
_1184404558.unknown
_1184404623.unknown
_1184404646.unknown
_1184404593.unknown
_1184404270.unknown
_1184404397.unknown
_1184404161.unknown
_1184403721.unknown
_1184403915.unknown
_1184404046.unknown
_1184403734.unknown
_1184403535.unknown
_1184403575.unknown
_1184403222.unknown
_1184251136.unknown
_1184400176.unknown
_1184401902.unknown
_1184402483.unknown
_1184402814.unknown
_1184403092.unknown
_1184403107.unknown
_1184402988.unknown
_1184403078.unknown
_1184402945.unknown
_1184402597.unknown
_1184402799.unknown
_1184402510.unknown
_1184402201.unknown
_1184402392.unknown
_1184402432.unknown
_1184402228.unknown
_1184402087.unknown
_1184402154.unknown
_1184401957.unknown
_1184400990.unknown
_1184401361.unknown
_1184401622.unknown
_1184401801.unknown
_1184401598.unknown
_1184401218.unknown
_1184401346.unknown
_1184401047.unknown
_1184400629.unknown
_1184400744.unknown
_1184400938.unknown
_1184400672.unknown
_1184400478.unknown
_1184400561.unknown
_1184400381.unknown
_1184398634.unknown
_1184399440.unknown
_1184399807.unknown
_1184400080.unknown
_1184400133.unknown
_1184400024.unknown
_1184399579.unknown
_1184399696.unknown
_1184399741.unknown
_1184399664.unknown
_1184399543.unknown
_1184398999.unknown
_1184399308.unknown
_1184399439.unknown
_1184399264.unknown
_1184398768.unknown
_1184398821.unknown
_1184398713.unknown
_1184251846.unknown
_1184397569.unknown
_1184397767.unknown
_1184397882.unknown
_1184397729.unknown
_1184397515.unknown
_1184397532.unknown
_1184397484.unknown
_1184251619.unknown
_1184251790.unknown
_1184251824.unknown
_1184251735.unknown
_1184251272.unknown
_1184251400.unknown
_1184251210.unknown
_1184245671.unknown
_1184249205.unknown
_1184250412.unknown
_1184250756.unknown
_1184250777.unknown
_1184251020.unknown
_1184250766.unknown
_1184250517.unknown
_1184250553.unknown
_1184250714.unknown
_1184250467.unknown
_1184249812.unknown
_1184250042.unknown
_1184250182.unknown
_1184249828.unknown
_1184249524.unknown
_1184249546.unknown
_1184249346.unknown
_1184247537.unknown
_1184248491.unknown
_1184248798.unknown
_1184249049.unknown
_1184248743.unknown
_1184248091.unknown
_1184248200.unknown
_1184247680.unknown
_1184246670.unknown
_1184247419.unknown
_1184247497.unknown
_1184246765.unknown
_1184245932.unknown
_1184246372.unknown
_1184245687.unknown
_1184244222.unknown
_1184244900.unknown
_1184245515.unknown
_1184245626.unknown
_1184245655.unknown
_1184245602.unknown
_1184245027.unknown
_1184245060.unknown
_1184244972.unknown
_1184244635.unknown
_1184244805.unknown
_1184244843.unknown
_1184244676.unknown
_1184244361.unknown
_1184244394.unknown
_1184244317.unknown
_1183456630.unknown
_1183456792.unknown
_1183456843.unknown
_1183456938.unknown
_1183456814.unknown
_1183456703.unknown
_1183456736.unknown
_1183456658.unknown
_1183454980.unknown
_1183456577.unknown
_1183456599.unknown
_1183455022.unknown
_1183454933.unknown
_1183454956.unknown
_1183454889.unknown