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Aula: Utilizando Propriedades de Integral Definida

Nesta aula, nós vamos aprender como utilizar propriedades de integrais definidas, como a ordem de limites de integração, limites de largura zero, somas e diferenças.

Atividade • 15 Questões

Q1:

Suponha 𝑓 tem valor de mΓ­nimo absoluto π‘š e valor de mΓ‘ximo absoluto 𝑀 . Dado que π‘Ž β©½ π‘₯ β©½ 𝑏 , que propriedade das integrais permite que vocΓͺ decida entre quais dois valores ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž d deve encontrar?

  • A π‘š ( 𝑏 βˆ’ π‘Ž ) β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑀 ( 𝑏 βˆ’ π‘Ž ) 𝑏 π‘Ž d
  • B π‘š ( 𝑏 βˆ’ π‘Ž ) β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ ( 𝑏 βˆ’ π‘Ž ) 𝑏 π‘Ž d
  • C π‘š β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑀 𝑏 π‘Ž d
  • D ( 𝑏 βˆ’ π‘Ž ) β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑀 ( 𝑏 βˆ’ π‘Ž ) 𝑏 π‘Ž d
  • E π‘Ž β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑏 𝑏 π‘Ž d

Q2:

Se ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 8 2 5 βˆ’ 4 d e ο„Έ 𝑔 ( π‘₯ ) π‘₯ = 7 4 5 βˆ’ 4 d , determine ο„Έ [ 2 𝑓 ( π‘₯ ) βˆ’ 4 𝑔 ( π‘₯ ) ] π‘₯ . 5 βˆ’ 4 d

Q3:

A função 𝑓 Γ© contΓ­nua em [ βˆ’ 4 , 4 ] e satisfaz ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 9 4 0 d . Determine ο„Έ [ 𝑓 ( π‘₯ ) βˆ’ 6 ] π‘₯ 4 0 d .

Q4:

A função 𝑓 Γ© contΓ­nua em ℝ . Dados ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 9 5 1 βˆ’ 2 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 7 1 8 d , quanto Γ© ο„Έ 𝑓 ( π‘₯ ) π‘₯ 8 βˆ’ 2 d ?

Q5:

A função 𝑓 Γ© contΓ­nua em ℝ . Dados ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 8 0 βˆ’ 4 βˆ’ 8 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 3 βˆ’ 4 7 d , quanto Γ© ο„Έ 𝑓 ( π‘₯ ) π‘₯ 7 βˆ’ 8 d ?

Q6:

A função 𝑓 Γ© contΓ­nua em ℝ . Dados ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 6 6 βˆ’ 5 βˆ’ 7 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 2 7 βˆ’ 5 2 d , quanto Γ© ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 βˆ’ 7 d ?

Q7:

Se ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 1 6 βˆ’ 2 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 1 1 9 βˆ’ 2 d , encontre ο„Έ 𝑓 ( π‘₯ ) π‘₯ 9 6 d .

Q8:

Se ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 2 , 4 2 βˆ’ 5 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 1 , 4 βˆ’ 1 βˆ’ 5 d , determine ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 βˆ’ 1 d .

Q9:

A função 𝑓 Γ© contΓ­nua em ℝ . Dados ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 1 8 4 βˆ’ 6 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 6 βˆ’ 6 βˆ’ 1 d , quanto Γ© ο„Έ 𝑓 ( π‘₯ ) π‘₯ 4 βˆ’ 1 d ?

Q10:

A função 𝑓 Γ© contΓ­nua em ℝ . Dados ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 8 6 7 1 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 3 7 1 5 d , quanto Γ© ο„Έ 𝑓 ( π‘₯ ) π‘₯ 7 5 d ?

Q11:

A função 𝑓 Γ© contΓ­nua em ℝ . Dados ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 9 1 βˆ’ 3 βˆ’ 9 d e ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 2 3 βˆ’ 9 βˆ’ 8 d , quanto Γ© ο„Έ 𝑓 ( π‘₯ ) π‘₯ βˆ’ 3 βˆ’ 8 d ?

Q12:

Se ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 7 1 βˆ’ 9 d e ο„Έ 𝑔 ( π‘₯ ) π‘₯ = βˆ’ 7 1 βˆ’ 9 d , determine o valor de ο„Έ [ 𝑓 ( π‘₯ ) + 𝑔 ( π‘₯ ) ] π‘₯ 1 βˆ’ 9 d .

Q13:

Se ο„Έ 𝑔 ( π‘₯ ) π‘₯ = 1 0 8 βˆ’ 7 d , determine o valor de ο„Έ 7 𝑔 ( π‘₯ ) π‘₯ βˆ’ 7 8 d .

Q14:

Suponha que em [ βˆ’ 2 , 5 ] , os valores de 𝑓 encontram-se no intervalo [ π‘š , 𝑀 ] . Entre quais limites ο„Έ 𝑓 ( π‘₯ ) π‘₯ 5 βˆ’ 2 d se encontra?

  • A 7 π‘š ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 7 𝑀 5 βˆ’ 2 d
  • B 7 π‘š ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 7 5 βˆ’ 2 d
  • C π‘š ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 𝑀 5 βˆ’ 2 d
  • D 7 ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 7 𝑀 5 βˆ’ 2 d
  • E βˆ’ 2 ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 5 5 βˆ’ 2 d

Q15:

Escreva ο„Έ 𝑓 ( π‘₯ ) π‘₯ + ο„Έ 𝑓 ( π‘₯ ) π‘₯ βˆ’ ο„Έ 𝑓 ( π‘₯ ) π‘₯ 3 βˆ’ 2 4 3 0 βˆ’ 2 d d d na forma ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž d .

  • A ο„Έ 𝑓 ( π‘₯ ) π‘₯ 4 0 d
  • B ο„Έ 𝑓 ( π‘₯ ) π‘₯ βˆ’ 2 3 d
  • C ο„Έ 𝑓 ( π‘₯ ) π‘₯ 0 4 d
  • D ο„Έ 𝑓 ( π‘₯ ) π‘₯ 0 3 d
  • E ο„Έ 𝑓 ( π‘₯ ) π‘₯ 3 0 d
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