Atividade: Derivadas de Ordem Superior

Nesta atividade, nós vamos praticar a calcular as derivadas de ordem superior de funções explícitas.

Q1:

Encontre as primeira e segunda derivadas da funรงรฃo ๐บ(๐‘Ÿ)=3โˆš๐‘Ÿโˆ’5โˆš๐‘Ÿ๏Žค.

  • A ๐บ โ€ฒ ( ๐‘Ÿ ) = 3 ๐‘Ÿ โˆ’ 5 ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Ž  ๏Žก ๏Žฃ ๏Žค , ๐บ โ€ฒ โ€ฒ ( ๐‘Ÿ ) = 3 ๐‘Ÿ โˆ’ 5 ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Žข ๏Žก ๏Žจ ๏Žค
  • B ๐บ โ€ฒ ( ๐‘Ÿ ) = 3 2 ๐‘Ÿ โˆ’ ๐‘Ÿ ๏Ž  ๏Žก ๏Ž  ๏Žค , ๐บ โ€ฒ โ€ฒ ( ๐‘Ÿ ) = โˆ’ 3 4 ๐‘Ÿ + 4 5 ๐‘Ÿ ๏Ž  ๏Žก ๏Ž  ๏Žค
  • C ๐บ โ€ฒ ( ๐‘Ÿ ) = 3 ๐‘Ÿ โˆ’ 5 ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Ž  ๏Žก ๏Žฃ ๏Žค , ๐บ โ€ฒ โ€ฒ ( ๐‘Ÿ ) = โˆ’ 3 2 ๐‘Ÿ + 4 ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Žข ๏Žก ๏Žจ ๏Žค
  • D ๐บ โ€ฒ ( ๐‘Ÿ ) = 3 2 ๐‘Ÿ โˆ’ ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Ž  ๏Žก ๏Žฃ ๏Žค , ๐บ โ€ฒ โ€ฒ ( ๐‘Ÿ ) = โˆ’ 3 4 ๐‘Ÿ + 4 5 ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Žข ๏Žก ๏Žจ ๏Žค
  • E ๐บ โ€ฒ ( ๐‘Ÿ ) = 3 2 ๐‘Ÿ โˆ’ ๐‘Ÿ ๏Ž  ๏Žก ๏Ž  ๏Žค , ๐บ โ€ฒ โ€ฒ ( ๐‘Ÿ ) = โˆ’ 3 4 ๐‘Ÿ + 4 5 ๐‘Ÿ ๏Šฑ ๏Šฑ ๏Ž  ๏Žก ๏Žฃ ๏Žค

Q2:

Sabendo que ๐‘ฆ=๐‘Ž๐‘ฅ+๐‘๐‘ฅ๏Šฉ๏Šจ, ๐‘ฆโ€ฒโ€ฒโ€ฒ=โˆ’18 e ๏—๐‘ฆ๐‘ฅ๏ฃ=โˆ’14dd๏Šจ๏Šจ๏—๏Šฒ๏Šจ, determine ๐‘Ž e ๐‘.

  • A ๐‘Ž = โˆ’ 3 , ๐‘ = โˆ’ 2 5
  • B ๐‘Ž = โˆ’ 6 , ๐‘ = โˆ’ 4 3
  • C ๐‘Ž = โˆ’ 6 , ๐‘ = 2 9
  • D ๐‘Ž = โˆ’ 3 , ๐‘ = 1 1

Q3:

Determine a terceira derivada da funรงรฃo ๐‘ฆ=44๐‘ฅ2๐‘ฅsen.

  • A 1 7 6 ๐‘ฅ 2 ๐‘ฅ โˆ’ 1 7 6 2 ๐‘ฅ s e n c o s
  • B โˆ’ 1 7 6 ๐‘ฅ 2 ๐‘ฅ + 1 7 6 2 ๐‘ฅ s e n c o s
  • C โˆ’ 8 ๐‘ฅ 2 ๐‘ฅ c o s
  • D 3 5 2 ๐‘ฅ 2 ๐‘ฅ + 5 2 8 2 ๐‘ฅ c o s s e n
  • E โˆ’ 3 5 2 ๐‘ฅ 2 ๐‘ฅ โˆ’ 5 2 8 2 ๐‘ฅ c o s s e n

Q4:

Dado ๐‘ฆ=โˆš๐‘ฅโˆ’9, determine dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A 1 2 ( ๐‘ฅ โˆ’ 9 ) ๏Žข ๏Žก
  • B โˆ’ 1 4 ( ๐‘ฅ โˆ’ 9 ) ๏Žข ๏Žก
  • C โˆ’ 4 ( ๐‘ฅ โˆ’ 9 ) ๏Žข ๏Žก
  • D 3 4 ( ๐‘ฅ โˆ’ 9 ) ๏Žข ๏Žก

Q5:

Dado ๐‘ฆ=(๐‘ฅโˆ’7)(4๐‘ฅ+7) e ๐‘ง=๐‘ฅ+5๐‘ฅ+9๏Šจ, determine dddd๏Šจ๏Šจ๏Šจ๏Šจ๐‘ฆ๐‘ฅ+๐‘ง๐‘ฅ.

Q6:

Se ๐‘“(๐‘ฅ)=๐‘Ž๐‘ฅ+7๐‘ฅโˆ’8๐‘ฅ+9๏Šฉ๏Šจ, e ๐‘“โ€ฒโ€ฒ(9)=โˆ’9, encontre ๐‘Ž.

  • A โˆ’ 1 6
  • B โˆ’ 8 9
  • C โˆ’ 2 3 5 4
  • D โˆ’ 1 6 9

Q7:

Determine o valor da segunda derivada de uma funรงรฃo ๐‘ฆ=12๐‘ฅโˆ’8๐‘ฅ para (1,4).

  • A โˆ’ 8
  • B16
  • C48
  • D โˆ’ 1 6

Q8:

Se ๐‘ฆ=5๐‘ฅsen, encontre 25๏€ฝ๐‘ฆ๐‘ฅ๏‰+๏€ฟ๐‘ฆ๐‘ฅ๏‹dddd๏Šจ๏Šจ๏Šจ๏Šจ.

Q9:

Sendo ๐‘ฆ=3๐‘ฅโˆ’52๐‘ฅ+7๏Šจ๏Šจ, determine dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A 6 2 ๏€น 7 โˆ’ 6 ๐‘ฅ ๏… ( 2 ๐‘ฅ + 7 ) ๏Šจ ๏Šจ ๏Šฉ
  • B 6 2 ๏€น 7 + 6 ๐‘ฅ ๏… ( 2 ๐‘ฅ + 7 ) ๏Šจ ๏Šจ ๏Šฉ
  • C 6 2 ๏€น 7 โˆ’ 6 ๐‘ฅ ๏… ( 2 ๐‘ฅ + 7 ) ๏Šจ ๏Šจ ๏Šช
  • D 6 2 ๐‘ฅ ( 2 ๐‘ฅ + 7 ) ๏Šจ ๏Šจ
  • E 7 โˆ’ 6 ๐‘ฅ ( 2 ๐‘ฅ + 7 ) ๏Šจ ๏Šจ ๏Šฉ

Q10:

Determine a terceira derivada da funรงรฃo ๐‘ฆ=โˆ’11๐‘ฅ+14๐‘ฅ.

  • A โˆ’ 1 4 ๐‘ฅ ๏Šช
  • B 2 8 ๐‘ฅ ๏Šฉ
  • C 8 4 ๐‘ฅ ๏Šช
  • D โˆ’ 8 4 ๐‘ฅ ๏Šช

Q11:

Encontre as primeira e segunda derivadas da funรงรฃo ๐‘“(๐‘ฅ)=0,003๐‘ฅโˆ’0,04๐‘ฅ๏Šฉ๏Šช.

  • A ๐‘“ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 9 ๐‘ฅ โˆ’ 0 , 1 6 ๐‘ฅ ๏Šจ ๏Šฉ , ๐‘“ โ€ฒ โ€ฒ ( ๐‘ฅ ) = 0 , 0 1 8 ๐‘ฅ โˆ’ 0 , 4 8 ๐‘ฅ ๏Šจ
  • B ๐‘“ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 9 ๐‘ฅ โˆ’ 0 , 1 6 ๐‘ฅ ๏Šช ๏Šซ , ๐‘“ โ€ฒ โ€ฒ ( ๐‘ฅ ) = 0 , 0 3 6 ๐‘ฅ โˆ’ 0 , 8 ๐‘ฅ ๏Šซ ๏Šฌ
  • C ๐‘“ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 9 ๐‘ฅ โˆ’ 0 , 1 6 ๐‘ฅ ๏Šฉ ๏Šช , ๐‘“ โ€ฒ โ€ฒ ( ๐‘ฅ ) = 0 , 0 2 7 ๐‘ฅ โˆ’ 0 , 6 4 ๐‘ฅ ๏Šฉ ๏Šช
  • D ๐‘“ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 3 ๐‘ฅ โˆ’ 0 , 0 4 ๐‘ฅ ๏Šจ ๏Šฉ , ๐‘“ โ€ฒ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 3 ๐‘ฅ โˆ’ 0 , 0 4 ๐‘ฅ ๏Šจ
  • E ๐‘“ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 3 ๐‘ฅ โˆ’ 0 , 0 4 ๐‘ฅ ๏Šช ๏Šซ , ๐‘“ โ€ฒ โ€ฒ ( ๐‘ฅ ) = 0 , 0 0 3 ๐‘ฅ โˆ’ 0 , 0 4 ๐‘ฅ ๏Šซ ๏Šฌ

Q12:

Dado ๐‘ฆ=โˆ’4๐‘ฅ2๐‘ฅ+42๐‘ฅcossen, determine dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ em ๐‘ฅ=5๐œ‹2.

  • A8
  • B0
  • C โˆ’ 4
  • D โˆ’ 4 0 ๐œ‹

Q13:

Dado que ๐‘ฆ=49๐‘ฅ5tg, determine ๐‘ฆโ€ฒโ€ฒ.

  • A 6 4 8 2 5 9 ๐‘ฅ 5 9 ๐‘ฅ 5 s e c t g ๏Šจ ๏Šจ
  • B 6 4 8 2 5 9 ๐‘ฅ 5 9 ๐‘ฅ 5 s e c t g
  • C 6 4 8 2 5 9 ๐‘ฅ 5 9 ๐‘ฅ 5 s e c t g ๏Šจ
  • D 3 6 5 9 ๐‘ฅ 5 s e c ๏Šจ

Q14:

Determine a terceira derivada da funรงรฃo ๐‘ฆ=๐‘ฅ+5๐‘ฅ+3๐‘ฅ+2๐‘ฅโˆ’๐‘ฅโˆ’9๏Šซ๏Šช๏Šฉ๏Šจ.

  • A ๐‘ฅ + 5 ๐‘ฅ + 3 ๏Šจ
  • B 2 0 ๐‘ฅ + 6 0 ๐‘ฅ + 1 8 ๐‘ฅ ๏Šฉ ๏Šจ
  • C 6 0 ๐‘ฅ + 1 2 0 ๐‘ฅ + 1 8 ๐‘ฅ ๏Šซ ๏Šช ๏Šฉ
  • D 6 0 ๐‘ฅ + 1 2 0 ๐‘ฅ + 1 8 ๏Šจ

Q15:

Sendo ๐‘ฆ=(โˆ’4๐‘ฅ+7)๏€นโˆ’7๐‘ฅโˆ’4๏…๏Šจ, determine dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A 7 ๐‘ฅ โˆ’ 4 9 ๐‘ฅ + 4 ๏Šจ
  • B 8 4 ๐‘ฅ โˆ’ 9 8 ๐‘ฅ + 1 6 ๏Šจ
  • C 1 4 ๐‘ฅ โˆ’ 4 9 ๐‘ฅ ๏Šฉ ๏Šจ
  • D 1 6 8 ๐‘ฅ โˆ’ 9 8

Q16:

Encontre a segunda derivada da funรงรฃo ๐‘ฆ=5๐‘ฅโˆ’42๐‘ฅโˆ’3 no ponto (2,6).

Q17:

Dados ๐‘ฆ=โˆ’๐‘ฅโˆ’8๐‘ฅโˆ’8๏Šจ e dd๏Šจ๏Šจ๐‘ฆ๐‘ฅโˆ’9๐‘˜+4=8. Encontre o valor de ๐‘˜.

  • A โˆ’ 1 4 9
  • B โˆ’ 2 3
  • C โˆ’ 2
  • D6

Q18:

Calcule ddsen๏Šซ๏Šง๏Šซ๏Šง๐‘ฅ(๐‘ฅ) determinando as primeiras derivadas e observando o padrรฃo que ocorre.

  • A s e n ๐‘ฅ
  • B 5 1 ๐‘ฅ ๐‘ฅ s e n c o s ๏Šซ ๏Šฆ
  • C โˆ’ ๐‘ฅ s e n
  • D c o s ๐‘ฅ
  • E โˆ’ ๐‘ฅ c o s

Q19:

Se ๐‘ฆ=๐‘ฅ๏Šฏ, encontre dd๏Šฎ๏Šฎ๐‘ฆ๐‘ฅ.

  • A ๐‘ฅ ( 8 ) !
  • B 9 !
  • C ๐‘ฅ ( 9 ) !
  • D 8 !

Q20:

Calcule ddddsec๐‘ฅ๏•โˆ’3๐‘ฅ+๐‘ฅ๏€น2๐‘ฅโˆ’9๐‘ฅ๏…๏ก๏Šฉ๏Šซ.

  • A 4 0 ๐‘ฅ โˆ’ 9 ๐‘ฅ โˆ’ 9 ๐‘ฅ ๐‘ฅ ๏Šฉ ๏Šจ t g s e c
  • B 4 0 ๐‘ฅ โˆ’ 9 ๐‘ฅ โˆ’ 9 ๐‘ฅ ๐‘ฅ โˆ’ 9 ๐‘ฅ ๏Šช ๏Šจ ๏Šจ ๏Šฉ t g s e c s e c
  • C 4 0 ๐‘ฅ โˆ’ 9 ๐‘ฅ โˆ’ 9 ๐‘ฅ ๐‘ฅ + 9 ๐‘ฅ ๏Šฉ ๏Šจ ๏Šจ ๏Šฉ t g s e c s e c
  • D 4 0 ๐‘ฅ โˆ’ 9 ๐‘ฅ โˆ’ 9 ๐‘ฅ ๐‘ฅ โˆ’ 9 ๐‘ฅ ๏Šฉ ๏Šจ ๏Šจ ๏Šฉ t g s e c s e c

Q21:

Sendo ๐‘ฆ=โˆš2๐‘ฅโˆ’5, determine ๐‘ฆโ€ฒโ€ฒโ€ฒ.

  • A 1 โˆš 2 ๐‘ฅ โˆ’ 5
  • B 3 ( 2 ๐‘ฅ โˆ’ 5 ) ๏Šจ
  • C 3 ๏„ ( 2 ๐‘ฅ โˆ’ 5 ) ๏Šซ
  • D โˆ’ 1 ๏„ ( 2 ๐‘ฅ โˆ’ 5 ) ๏Šฉ
  • E 3 8 ๏„ ( 2 ๐‘ฅ โˆ’ 5 ) ๏Šซ

Q22:

Se ๐‘ฆโˆถ๐‘ฆ=โˆ’๐‘ฅโˆ’1โˆ’๐‘ฅ+1๏Šซ๏Šซ, determine ๐‘ฆโ€ฒโ€ฒ.

  • A โˆ’ 6 0 ๐‘ฅ โˆ’ 4 0 ๐‘ฅ ( โˆ’ ๐‘ฅ + 1 ) ๏Šฎ ๏Šฉ ๏Šซ ๏Šจ
  • B โˆ’ 6 0 ๐‘ฅ โˆ’ 4 0 ๐‘ฅ ( โˆ’ ๐‘ฅ + 1 ) ๏Šฎ ๏Šฉ ๏Šซ ๏Šช
  • C โˆ’ 6 0 ๐‘ฅ โˆ’ 4 0 ๐‘ฅ ( โˆ’ ๐‘ฅ + 1 ) ๏Šฎ ๏Šฉ ๏Šซ ๏Šฉ
  • D โˆ’ 1 0 ๐‘ฅ ( โˆ’ ๐‘ฅ + 1 ) ๏Šช ๏Šซ ๏Šจ

Q23:

Dado que ๐‘ฆ=6๐‘ฅ+3๐‘ฅโˆ’7๐‘ฅ+6๏Šซ๏Šจ, determine dd๏Šจ๏Šจ๐‘ฆ๐‘ฅ.

  • A 3 0 ๐‘ฅ + 6 ๐‘ฅ โˆ’ 7 ๏Šช
  • B 3 0 ๐‘ฅ + 6 ๐‘ฅ โˆ’ 7 ๐‘ฅ ๏Šซ ๏Šจ
  • C 6 ๐‘ฅ + 3 ๐‘ฅ โˆ’ 7 ๏Šช
  • D 6 ๏€น 2 0 ๐‘ฅ + 1 ๏… ๏Šฉ

Q24:

Determine a segunda derivada da funรงรฃo ๐‘ฆ=โˆ’7๐‘ฅ+3๐‘ฅsencos em ๐‘ฅ=๐œ‹4.

  • A โˆ’ 2 โˆš 2
  • B โˆ’ 5 โˆš 2
  • C 2 โˆš 2
  • D 5 โˆš 2

Q25:

Determine a terceira derivada da funรงรฃo ๐‘ฆ=3๐‘ฅ+93๐‘ฅ๏Šจsen.

  • A โˆ’ 9 3 ๐‘ฅ c o s
  • B 8 1 3 ๐‘ฅ + 6 s e n
  • C โˆ’ 2 4 3 3 ๐‘ฅ c o s
  • D โˆ’ 8 1 3 ๐‘ฅ + 6 s e n
  • E 2 4 3 3 ๐‘ฅ c o s

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