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Atividade: Derivadas de Ordem Superior de Equações Paramétricas

Nesta atividade, nós vamos praticar a determinação de derivadas de ordem superior (d²y/dx²) de equações paramétricas por aplicação da regra em cadeia.

Q1:

Dado que π‘₯ = 𝑑 + 5 3 e 𝑦 = 𝑑 βˆ’ 3 𝑑 2 , encontre d d 2 2 𝑦 π‘₯ .

  • A 𝑑 2 ( 3 βˆ’ 𝑑 )
  • B 2 ( 3 βˆ’ 𝑑 ) 𝑑
  • C 9 𝑑 2 ( 3 βˆ’ 𝑑 ) 5
  • D 2 ( 3 βˆ’ 𝑑 ) 9 𝑑 5
  • E 2 ( 3 βˆ’ 𝑑 ) 3 𝑑 ( 2 𝑑 βˆ’ 3 ) 3

Q2:

Dado que π‘₯ = 3 𝑑 + 1 3 e 𝑦 = 3 𝑑 βˆ’ 𝑑 2 , encontre d d 2 2 𝑦 π‘₯ .

  • A 𝑑 2 ( 1 βˆ’ 3 𝑑 )
  • B 2 ( 1 βˆ’ 3 𝑑 ) 𝑑
  • C 8 1 𝑑 2 ( 1 βˆ’ 3 𝑑 ) 5
  • D 2 ( 1 βˆ’ 3 𝑑 ) 8 1 𝑑 5
  • E 2 ( 1 βˆ’ 3 𝑑 ) 9 𝑑 ( 6 𝑑 βˆ’ 1 ) 3

Q3:

Dado que π‘₯ = 3 𝑑 + 1 3 e 𝑦 = 5 𝑑 βˆ’ 𝑑 2 , encontre d d 2 2 𝑦 π‘₯ .

  • A 𝑑 2 ( 1 βˆ’ 5 𝑑 )
  • B 2 ( 1 βˆ’ 5 𝑑 ) 𝑑
  • C 8 1 𝑑 2 ( 1 βˆ’ 5 𝑑 ) 5
  • D 2 ( 1 βˆ’ 5 𝑑 ) 8 1 𝑑 5
  • E 2 ( 1 βˆ’ 5 𝑑 ) 9 𝑑 ( 1 0 𝑑 βˆ’ 1 ) 3

Q4:

Dado que π‘₯ = 2 𝑒   e 𝑦 = 𝑑 𝑒    , encontre d d   𝑦 π‘₯ .

  • A 8 𝑒 4 𝑑 βˆ’ 3  
  • B 2 ( 4 𝑑 βˆ’ 3 )
  • C 3 βˆ’ 4 𝑑 8 𝑒  
  • D 4 𝑑 βˆ’ 3 8 𝑒  
  • E 2 ( 3 βˆ’ 4 𝑑 )

Q5:

Dado que π‘₯ = 𝑒  e 𝑦 = 4 𝑑 𝑒   , encontre d d   𝑦 π‘₯ .

  • A 𝑒 4 ( 2 𝑑 βˆ’ 3 )  
  • B 4 ( 2 𝑑 βˆ’ 3 )
  • C 4 ( 3 βˆ’ 2 𝑑 ) 𝑒  
  • D 4 ( 2 𝑑 βˆ’ 3 ) 𝑒  
  • E 4 ( 3 βˆ’ 2 𝑑 )

Q6:

Dado que π‘₯ = 𝑑 + 1 2 e 𝑦 = 𝑒 βˆ’ 1 𝑑 , encontre d d 2 2 𝑦 π‘₯ .

  • A 4 𝑑 𝑒 ( 𝑑 βˆ’ 1 ) 3 𝑑
  • B 𝑒 ( 𝑑 βˆ’ 1 ) 𝑑 𝑑
  • C 𝑒 ( 𝑑 βˆ’ 1 ) 2 𝑑 𝑑 3
  • D 𝑒 ( 𝑑 βˆ’ 1 ) 4 𝑑 𝑑 3
  • E 𝑑 βˆ’ 1 2 𝑑 2

Q7:

Dado que π‘₯ = 2 𝑑 + 4 2 e 𝑦 = 5 𝑒 βˆ’ 4 5 𝑑 , encontre d d 2 2 𝑦 π‘₯ .

  • A 1 6 𝑑 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 3 5 𝑑
  • B 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 𝑑 5 𝑑
  • C 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 8 𝑑 5 𝑑 3
  • D 2 5 𝑒 ( 5 𝑑 βˆ’ 1 ) 1 6 𝑑 5 𝑑 3
  • E 5 𝑑 βˆ’ 1 4 𝑑 2

Q8:

Determine d d 2 2 𝑦 π‘₯ , sendo π‘₯ = 6 𝑛 l n 5 e 𝑦 = βˆ’ 8 𝑛 3 .

  • A βˆ’ 8 𝑛 5 3
  • B βˆ’ 1 2 𝑛 5 2
  • C βˆ’ 4 𝑛 5 3
  • D βˆ’ 2 𝑛 2 5 3

Q9:

Dados que d d 𝑧 π‘₯ = 5 π‘₯ βˆ’ 6 e d d 𝑦 π‘₯ = 2 π‘₯ βˆ’ 1 2 , determine d d 2 2 𝑧 𝑦 em π‘₯ = 1 .

  • A βˆ’ 1
  • B14
  • C βˆ’ 1 4
  • D9

Q10:

Dados que d d 𝑧 π‘₯ = βˆ’ 7 π‘₯ + 7 e d d 𝑦 π‘₯ = 3 π‘₯ βˆ’ 1 2 , determine d d 2 2 𝑧 𝑦 em π‘₯ = 0 .

  • A βˆ’ 7
  • B42
  • C βˆ’ 4 2
  • D7

Q11:

Dados que d d 𝑧 π‘₯ = βˆ’ π‘₯ + 6 e d d 𝑦 π‘₯ = 2 π‘₯ + 5 2 , determine d d 2 2 𝑧 𝑦 em π‘₯ = βˆ’ 1 .

  • A1
  • B βˆ’ 2 6 3 4 3
  • C 2 6 3 4 3
  • D 3 7

Q12:

Se e , determine .

  • A
  • B
  • C
  • D
  • E

Q13:

Se e , determine .

  • A
  • B
  • C
  • D

Q14:

Se e , determine .

  • A
  • B
  • C
  • D

Q15:

Encontre d d 2 2 𝑦 π‘₯ se π‘₯ = βˆ’ 𝑒 4 𝑛 e 𝑦 = βˆ’ 2 𝑛 4 .

  • A βˆ’ 𝑒 𝑛 βˆ’ 4 𝑛 2
  • B 𝑒 𝑛 ( βˆ’ 8 𝑛 + 6 ) βˆ’ 4 𝑛 2
  • C 2 𝑒 𝑛 βˆ’ 4 𝑛 3
  • D 𝑛 2 𝑒 ( 4 𝑛 βˆ’ 3 ) 2 βˆ’ 8 𝑛

Q16:

Se 𝑦 = ( π‘₯ + 4 ) ο€Ή βˆ’ 4 π‘₯ βˆ’ 1  2 e 𝑧 = ( π‘₯ βˆ’ 5 ) ( π‘₯ + 4 ) , encontre ( 2 π‘₯ βˆ’ 1 ) 𝑦 𝑧 3 2 2 d d .

  • A βˆ’ 9 6 π‘₯ + 1 6 π‘₯ + 1 2 0 π‘₯ βˆ’ 8 4 π‘₯ + 1 6 4 3 2
  • B βˆ’ 4 8 π‘₯ + 7 2 π‘₯ + 4 4 π‘₯ βˆ’ 3 4 3 2
  • C βˆ’ 7 2 π‘₯ βˆ’ 1 0 4 π‘₯ + 3 0 2
  • D βˆ’ 2 4 π‘₯ + 2 4 π‘₯ + 3 4 2

Q17:

Se 𝑦 = ( βˆ’ π‘₯ + 2 ) ο€Ή 3 π‘₯ + 2  2 e 𝑧 = ( βˆ’ π‘₯ + 2 ) ( π‘₯ + 3 ) , encontre ( βˆ’ 2 π‘₯ βˆ’ 1 ) 𝑦 𝑧 3 2 2 d d .

  • A βˆ’ 7 2 π‘₯ βˆ’ 6 0 π‘₯ + 1 8 π‘₯ + 2 7 π‘₯ + 6 4 3 2
  • B βˆ’ 3 6 π‘₯ βˆ’ 5 4 π‘₯ + 1 4 π‘₯ + 1 6 3 2
  • C 5 4 π‘₯ βˆ’ 3 0 π‘₯ βˆ’ 8 2
  • D 1 8 π‘₯ + 1 8 π‘₯ βˆ’ 1 6 2

Aulas Pertinentes

Atividades Pertinentes