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

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