Atividade: Séries de Maclaurin

Nesta atividade, nós vamos praticar a representação de funções exponenciais e trigonométricas como séries de potências, a determinação da expansão em torno de zero e a determinação do intervalo de convergência da série.

Q1:

Considere a função 𝑓 ( π‘₯ ) = 𝑒 π‘₯ .

Encontre 𝑓 ( π‘₯ ) β€² .

  • A 𝑒 π‘₯ π‘₯ l n
  • B 𝑒 π‘₯ βˆ’ 1
  • C 𝑒 π‘₯ π‘₯ βˆ’ 1 l n
  • D 𝑒 π‘₯
  • E l n π‘₯

Encontre 𝑓 ( π‘₯ ) ( 𝑛 ) , onde 𝑓 ( 𝑛 ) representa a 𝑛 (enΓ©sima) derivada de 𝑓 em relação a π‘₯ .

  • A 𝑒 π‘₯
  • B 𝑒 π‘₯ + 𝑒 ( βˆ’ 1 ) ( 𝑛 βˆ’ 2 ) ! π‘₯ π‘₯ βˆ’ 𝑛 π‘₯ βˆ’ 1 𝑛 ( 𝑛 βˆ’ 1 ) l n para 𝑛 > 1
  • C ( βˆ’ 1 ) ( 𝑛 βˆ’ 2 ) ! π‘₯ 𝑛 ( 𝑛 βˆ’ 1 ) para 𝑛 > 1
  • D 𝑒 π‘₯ βˆ’ 𝑛
  • E 𝑒 π‘₯ + 𝑒 ( βˆ’ 1 ) ( 𝑛 βˆ’ 2 ) ! π‘₯ π‘₯ π‘₯ 𝑛 ( 𝑛 βˆ’ 1 ) l n para 𝑛 > 1

E entΓ£o, derive a sΓ©rie Maclaurin para 𝑒 π‘₯ .

  • A 𝑒 = ο„š 𝑓 ( π‘Ž ) ( π‘₯ βˆ’ π‘Ž ) 𝑛 ! π‘₯ ∞ 𝑛 = 1 ( 𝑛 ) 𝑛
  • B 𝑒 = ο„š π‘₯ 𝑛 ! π‘₯ ∞ 𝑛 = 1 𝑛
  • C 𝑒 = ο„š π‘₯ 𝑛 ! π‘₯ ∞ 𝑛 = 0 𝑛
  • D 𝑒 = ο„š 𝑒 𝑛 ! π‘₯ ∞ 𝑛 = 0 𝑛
  • E 𝑒 = ο„š 𝑓 ( π‘Ž ) ( π‘₯ βˆ’ π‘Ž ) 𝑛 ! π‘₯ ∞ 𝑛 = 0 ( 𝑛 ) 𝑛

Qual Γ© o raio de convergΓͺncia 𝑅 da sΓ©rie Maclaurin para 𝑒 π‘₯ ?

  • A 𝑅 = + ∞
  • B 𝑅 = 1 0 0
  • C 𝑅 = 1
  • D 𝑅 = 𝑒
  • ENΓ£o converge.

Q2:

Considere a função 𝑓 ( π‘₯ ) = π‘₯ s e n .

Quais sΓ£o as quatro primeiras derivadas de 𝑓 em relação a π‘₯ ?

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ s e n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) s e n
  • B 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s e n , 𝑓 β€² β€² β€² ( π‘₯ ) = π‘₯ c o s , e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) s e n
  • C 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s e n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , e 𝑓 ( π‘₯ ) = βˆ’ π‘₯ ( 4 ) s e n
  • D 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s e n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) s e n
  • E 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ s e n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) s e n

Escreva a fΓ³rmula geral para a 𝑛 (enΓ©sima) derivada de 𝑓 em relação a π‘₯ .

  • A 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) s e n
  • B 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) s e n
  • C 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) c o s
  • D 𝑓 ( π‘₯ ) = ( π‘₯ + 𝑛 πœ‹ ) ( 𝑛 ) s e n
  • E 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) c o s

E entΓ£o, derive a sΓ©rie Maclaurin para s e n π‘₯ .

  • A ∞ 𝑛 = 0 2 𝑛 𝑛 ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !
  • B ∞ 𝑛 = 0 𝑛 2 𝑛 ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 ) !
  • C ∞ 𝑛 = 0 𝑛 2 𝑛 + 1 ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 + 1 ) !
  • D ∞ 𝑛 = 0 2 𝑛 + 1 2 𝑛 + 1 ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 + 1 ) !
  • E ∞ 𝑛 = 0 𝑛 𝑛 ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !

Qual Γ© o raio 𝑅 de convergΓͺncia da sΓ©rie Maclaurin para s e n π‘₯ ?

  • A 𝑅 = + ∞
  • B 𝑅 = 2 πœ‹
  • C 𝑅 = πœ‹
  • D 𝑅 = 1
  • E 𝑅 = πœ‹ 2

Q3:

Considere a função 𝑓 ( π‘₯ ) = π‘₯ c o s .

Quais sΓ£o as primeiras quatro derivadas de 𝑓 em ordem a π‘₯ ?

  • A 𝑓 β€² ( π‘₯ ) = π‘₯ s e n , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = π‘₯ s e n e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) c o s
  • B 𝑓 β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s e n , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) s e n
  • C 𝑓 β€² ( π‘₯ ) = π‘₯ s e n , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s e n e 𝑓 ( π‘₯ ) = βˆ’ π‘₯ ( 4 ) c o s
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ s e n , 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = π‘₯ s e n e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) c o s
  • E 𝑓 β€² ( π‘₯ ) = βˆ’ π‘₯ s e n , 𝑓 β€² β€² ( π‘₯ ) = π‘₯ c o s , 𝑓 β€² β€² β€² ( π‘₯ ) = βˆ’ π‘₯ s e n e 𝑓 ( π‘₯ ) = π‘₯ ( 4 ) c o s

Escreva a forma geral da 𝑛 -Γ©sima derivada de 𝑓 em ordem a π‘₯ .

  • A 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) c o s
  • B 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) s e n
  • C 𝑓 ( π‘₯ ) = ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) s e n
  • D 𝑓 ( π‘₯ ) = βˆ’ ο€» π‘₯ + 𝑛 πœ‹ 2  ( 𝑛 ) c o s
  • E 𝑓 ( π‘₯ ) = ( π‘₯ + 𝑛 πœ‹ ) ( 𝑛 ) c o s

Por fim, derive a sΓ©rie de Maclaurin de c o s π‘₯ .

  • A ∞ 𝑛 = 0 2 𝑛 𝑛 ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !
  • B ∞ 𝑛 = 0 𝑛 2 𝑛 + 1 ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 + 1 ) !
  • C ∞ 𝑛 = 0 𝑛 2 𝑛 ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 ) !
  • D ∞ 𝑛 = 0 2 𝑛 2 𝑛 ο„š ( βˆ’ 1 ) π‘₯ ( 2 𝑛 ) !
  • E ∞ 𝑛 = 0 𝑛 𝑛 ο„š ( βˆ’ 1 ) π‘₯ 𝑛 !

Qual Γ© o raio 𝑅 de convergΓͺncia da sΓ©rie de Maclaurin de c o s π‘₯ ?

  • A 𝑅 = + ∞
  • B 𝑅 = 2 πœ‹
  • C 𝑅 = πœ‹
  • D 𝑅 = 1
  • E 𝑅 = πœ‹ 2

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