Atividade: O Teorema Fundamental do Cálculo

Nesta atividade, nós vamos praticar a utilizar o teorema fundamental do cálculo para encontrar a derivada de uma função.

Q1:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função β„Ž(𝑒)=ο„Έβˆš3𝑑4𝑑+2𝑑οŠͺd.

  • A β„Ž β€² ( 𝑒 ) = βˆ’ 3 ( 4 𝑒 βˆ’ 2 ) 2 √ 3 𝑒 ( 4 𝑒 + 2 ) 
  • B β„Ž β€² ( 𝑒 ) = √ 3 𝑑 4 𝑑 + 2
  • C β„Ž β€² ( 𝑒 ) = √ 3 𝑒 4 𝑒 + 2
  • D β„Ž β€² ( 𝑒 ) = βˆ’ 3 ( 4 𝑑 βˆ’ 2 ) 2 √ 3 𝑑 ( 4 𝑑 + 2 )
  • E β„Ž β€² ( 𝑒 ) = βˆ’ 3 ( 4 𝑑 βˆ’ 2 ) 2 √ 3 𝑑 ( 4 𝑑 + 2 ) 

Q2:

Dado 𝑓(π‘₯)π‘₯=π‘₯βˆ’7π‘₯βˆ’π‘₯+9+dC, determine 𝑓′(1).

Q3:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝑅(𝑦)=ο„Έ3𝑑2π‘‘π‘‘οŠ«ο˜οŠ¨send.

  • A 𝑅 β€² ( 𝑦 ) = 3 𝑦 2 𝑦  s e n
  • B 𝑅 β€² ( 𝑦 ) = βˆ’ 3 𝑑 2 𝑑  s e n
  • C 𝑅 β€² ( 𝑦 ) = 6 𝑑 2 𝑑 βˆ’ 6 𝑑 2 𝑑  c o s s e n
  • D 𝑅 β€² ( 𝑦 ) = 6 𝑑 2 𝑑 + 6 𝑑 2 𝑑  c o s s e n
  • E 𝑅 β€² ( 𝑦 ) = βˆ’ 3 𝑦 2 𝑦  s e n

Q4:

Encontre a derivada da função 𝑔(π‘₯)=ο„Έ5π‘‘π‘‘π‘‘οŠ§οŠ°ο—οŠ§οŠ±οŠ¨ο—send.

  • A 𝑔 β€² ( π‘₯ ) = ( 1 0 βˆ’ 2 0 π‘₯ ) ( 1 βˆ’ 2 π‘₯ ) + ( 5 + 5 π‘₯ ) ( 1 + π‘₯ ) s e n s e n
  • B 𝑔 β€² ( π‘₯ ) = ( 5 βˆ’ 2 π‘₯ ) ( 1 βˆ’ 2 π‘₯ ) + ( 5 + 5 π‘₯ ) ( 1 + π‘₯ ) s e n s e n
  • C 𝑔 β€² ( π‘₯ ) = βˆ’ ( 5 βˆ’ 1 0 π‘₯ ) ( 1 βˆ’ 2 π‘₯ ) + ( 5 + 5 π‘₯ ) ( 1 + π‘₯ ) s e n s e n
  • D 𝑔 β€² ( π‘₯ ) = βˆ’ ( 1 0 βˆ’ 2 0 π‘₯ ) ( 1 βˆ’ 2 π‘₯ ) + ( 5 + 5 π‘₯ ) ( 1 + π‘₯ ) s e n s e n
  • E 𝑔 β€² ( π‘₯ ) = βˆ’ ( 1 0 βˆ’ 2 0 π‘₯ ) ( 1 βˆ’ 2 π‘₯ ) βˆ’ ( 5 + 5 π‘₯ ) ( 1 + π‘₯ ) s e n s e n

Q5:

Encontre a derivada da função 𝑦(π‘₯)=ο„Έ(1βˆ’π‘£)𝑣οŠͺο—οŠ©ο—sencoslnd.

  • A 𝑦 β€² ( π‘₯ ) = 3 π‘₯ ( 1 βˆ’ 3 π‘₯ ) + 4 π‘₯ ( 1 βˆ’ 4 π‘₯ ) s e n l n c o s c o s l n s e n
  • B 𝑦 β€² ( π‘₯ ) = βˆ’ 3 π‘₯ ( 1 βˆ’ 3 π‘₯ ) + 4 π‘₯ ( 1 βˆ’ 4 π‘₯ ) s e n l n c o s c o s l n s e n
  • C 𝑦 β€² ( π‘₯ ) = βˆ’ 3 π‘₯ ( 1 βˆ’ 3 π‘₯ ) βˆ’ 4 π‘₯ ( 1 βˆ’ 4 π‘₯ ) s e n l n c o s c o s l n s e n
  • D 𝑦 β€² ( π‘₯ ) = βˆ’ ( 1 βˆ’ 3 π‘₯ ) + ( 1 βˆ’ 4 π‘₯ ) l n c o s l n s e n
  • E 𝑦 β€² ( π‘₯ ) = ( 1 βˆ’ 3 π‘₯ ) + ( 1 βˆ’ 4 π‘₯ ) l n c o s l n s e n

Q6:

Suponha que 𝑓 Γ© uma função no intervalo [π‘Ž,𝑏] e definimos 𝐹 como 𝐹(π‘₯)=𝑓(𝑑)π‘‘ο—οŒΊd. Descobrimos que 𝐹 NΓƒO Γ© derivΓ‘vel em (π‘Ž,𝑏). O que podemos concluir?

  • AHΓ‘ um erro, porque sempre integramos uma função, ela deve ser derivΓ‘vel e 𝐹′(π‘₯)=𝑓(π‘₯).
  • B 𝑓 Γ© contΓ­nua em todo o (π‘Ž,𝑏).
  • C 𝑓 nΓ£o Γ© contΓ­nua algures no intervalo (π‘Ž,𝑏).
  • D 𝑓 nΓ£o Γ© contΓ­nua em todo o (π‘Ž,𝑏).
  • E 𝑓 nΓ£o Γ© derivΓ‘vel em todo o (π‘Ž,𝑏).

Q7:

A figura mostra a representação grΓ‘fica da função 𝑓(𝑑)𝑑.ο—οŠ¦d

Qual das seguintes Γ© a representação grΓ‘fica de 𝑦=𝑓(π‘₯)?

  • A
  • Bnenhuma das anteriores
  • C
  • D
  • E

Q8:

Use o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑦=ο„Έ2𝑑2+π‘‘π‘‘οŠ«ο—οŠ°οŠ©οŠͺd.

  • A 𝑦 β€² = 2 ( 5 π‘₯ + 3 ) 2 + ( 5 π‘₯ + 3 ) 
  • B 𝑦 β€² = 1 0 ( 5 𝑑 + 3 ) 2 + ( 5 𝑑 + 3 ) 
  • C 𝑦 β€² = 2 𝑑 2 + 𝑑 
  • D 𝑦 β€² = 2 ( 5 𝑑 + 3 ) 2 + ( 5 𝑑 + 3 ) 
  • E 𝑦 β€² = 1 0 ( 5 π‘₯ + 3 ) 2 + ( 5 π‘₯ + 3 ) 

Q9:

Use o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑦=ο„Έ5(5πœƒ)πœƒο—οŠ¨οŠ¨οŽ£cosd.

  • A 𝑦 β€² = 5 ο€Ή 5 π‘₯  c o s  οŠͺ
  • B 𝑦 β€² = 5 0 5 πœƒ 5 πœƒ s e n c o s
  • C 𝑦 β€² = 2 0 π‘₯ ο€Ή 5 π‘₯    οŠͺ c o s
  • D 𝑦 β€² = 5 ( 5 πœƒ ) c o s 
  • E 𝑦 β€² = βˆ’ 5 0 5 πœƒ 5 πœƒ s e n c o s

Q10:

Utilize o Teorema Funnamental do CΓ‘lculo para determinar a derivada da função 𝑔(π‘₯)=ο„Έο€Ή1+π‘‘ο…π‘‘ο—οŠ©οŠ«lnd.

  • A 𝑔 β€² ( π‘₯ ) = ο€Ή 1 + π‘₯  l n 
  • B 𝑔 β€² ( π‘₯ ) = 5 𝑑 1 + 𝑑 οŠͺ 
  • C 𝑔 β€² ( π‘₯ ) = 5 π‘₯ 1 + π‘₯ οŠͺ 
  • D 𝑔 β€² ( π‘₯ ) = ο€Ή 1 + 𝑑  l n 
  • E 𝑔 β€² ( π‘₯ ) = 1 1 + 𝑑 

Q11:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝑔(𝑠)=ο„Έο€Ή3π‘‘βˆ’4π‘‘ο…π‘‘οοŠ§οŠ©οŠ«οŠͺd.

  • A 𝑔 β€² ( 𝑠 ) = 4 ο€Ή 9 𝑑 βˆ’ 2 0 𝑑  ο€Ή 3 𝑑 βˆ’ 4 𝑑   οŠͺ   οŠͺ
  • B 𝑔 β€² ( 𝑠 ) = 4 ο€Ή 9 𝑠 βˆ’ 2 0 𝑠  ο€Ή 3 𝑠 βˆ’ 4 𝑠   οŠͺ   
  • C 𝑔 β€² ( 𝑠 ) = ο€Ή 3 𝑑 βˆ’ 4 𝑑    οŠͺ
  • D 𝑔 β€² ( 𝑠 ) = 4 ο€Ή 9 𝑑 βˆ’ 2 0 𝑑  ο€Ή 3 𝑑 βˆ’ 4 𝑑   οŠͺ   
  • E 𝑔 β€² ( 𝑠 ) = ο€Ή 3 𝑠 βˆ’ 4 𝑠    οŠͺ

Q12:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝐹(π‘₯)=ο„Έβˆš2βˆ’3𝑑𝑑οŠͺsecd.

  • A 𝐹 β€² ( π‘₯ ) = βˆ’ 3 𝑑 𝑑 2 √ 2 βˆ’ 3 𝑑 s e c t g s e c
  • B 𝐹 β€² ( π‘₯ ) = √ 2 βˆ’ 3 𝑑 s e c
  • C 𝐹 β€² ( π‘₯ ) = 3 π‘₯ π‘₯ 2 √ 2 βˆ’ 3 π‘₯ s e c t g s e c
  • D 𝐹 β€² ( π‘₯ ) = √ 2 βˆ’ 3 π‘₯ s e c
  • E 𝐹 β€² ( π‘₯ ) = βˆ’ √ 2 βˆ’ 3 π‘₯ s e c

Q13:

Utilize o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑔(π‘₯)=ο„Έβˆ’2π‘‘π‘‘ο—οŠ¨οŠͺd.

  • A 𝑔 β€² ( π‘₯ ) = βˆ’ 8 𝑑 
  • B 𝑔 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ 
  • C 𝑔 β€² ( π‘₯ ) = βˆ’ 2 𝑑 οŠͺ
  • D 𝑔 β€² ( π‘₯ ) = βˆ’ 2 π‘₯ οŠͺ
  • E 𝑔 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ οŠͺ

Q14:

Encontre a derivada da função 𝑔(π‘₯)=ο„Έπ‘’βˆ’3𝑒+5𝑒οŠͺο—οŠ©ο—οŠ¨οŠ¨d.

  • A 𝑔 β€² ( π‘₯ ) = βˆ’ 4 ο€Ή 1 6 π‘₯ βˆ’ 3  1 6 π‘₯ + 5 + 3 ο€Ή 9 π‘₯ βˆ’ 3  9 π‘₯ + 5    
  • B 𝑔 β€² ( π‘₯ ) = 1 6 π‘₯ βˆ’ 3 1 6 π‘₯ + 5 + 9 π‘₯ βˆ’ 3 9 π‘₯ + 5    
  • C 𝑔 β€² ( π‘₯ ) = 1 6 π‘₯ βˆ’ 3 1 6 π‘₯ + 5 βˆ’ 9 π‘₯ βˆ’ 3 9 π‘₯ + 5    
  • D 𝑔 β€² ( π‘₯ ) = 4 ο€Ή 1 6 π‘₯ βˆ’ 3  1 6 π‘₯ + 5 + 3 ο€Ή 9 π‘₯ βˆ’ 3  9 π‘₯ + 5    
  • E 𝑔 β€² ( π‘₯ ) = 4 ο€Ή 1 6 π‘₯ βˆ’ 3  1 6 π‘₯ + 5 βˆ’ 3 ο€Ή 9 π‘₯ βˆ’ 3  9 π‘₯ + 5    

Q15:

Encontre a derivada da função 𝐹(π‘₯)=ο„Έ2π‘’π‘‘οŠ¨ο—οŠ«ο—οοŽ‘οŽ‘d.

  • A 𝐹 β€² ( π‘₯ ) = 2 π‘₯ 𝑒 βˆ’ 2 𝑒 οŠͺ      
  • B 𝐹 β€² ( π‘₯ ) = 8 π‘₯ 𝑒 + 1 0 𝑒 οŠͺ      
  • C 𝐹 β€² ( π‘₯ ) = 8 π‘₯ 𝑒 βˆ’ 1 0 𝑒 οŠͺ      
  • D 𝐹 β€² ( π‘₯ ) = 8 π‘₯ 𝑒 βˆ’ 1 0 𝑒 οŠͺ      
  • E 𝐹 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ 𝑒 + 1 0 𝑒 οŠͺ      

Q16:

Dado que 𝐹(π‘₯)=𝑑𝑑οŠͺο—βˆšο—οŠ±οŠ§tgd, encontre 𝐹′(π‘₯).

  • A 𝐹 β€² ( π‘₯ ) = 4 π‘₯ βˆ’ √ π‘₯ t g t g    
  • B 𝐹 β€² ( π‘₯ ) = 4 π‘₯ + √ π‘₯ t g t g    
  • C 𝐹 β€² ( π‘₯ ) = 4 4 π‘₯ + 1 2 √ π‘₯ √ π‘₯ t g t g    
  • D 𝐹 β€² ( π‘₯ ) = 4 4 π‘₯ βˆ’ 1 2 √ π‘₯ √ π‘₯ t g t g    
  • E 𝐹 β€² ( π‘₯ ) = βˆ’ 4 4 π‘₯ + 1 2 √ π‘₯ √ π‘₯ t g t g    

Q17:

Seja 𝑦=ο„Έβˆš2+5π‘‘π‘‘οŠ¨οŠ¨ο—οŠ¨send. Utilize o Teorema Fundamental do CΓ‘lculo para encontrar 𝑦′.

  • A 𝑦 β€² = βˆ’ √ 2 + 5 2 π‘₯ s e n 
  • B 𝑦 β€² = √ 2 + 5 2 π‘₯ s e n 
  • C 𝑦 β€² = βˆ’ 2 ( 2 π‘₯ ) √ 2 + 5 2 π‘₯ c o s s e n 
  • D 𝑦 β€² = √ 2 + 5 𝑑 
  • E 𝑦 β€² = 2 ( 2 π‘₯ ) √ 2 + 5 2 π‘₯ c o s s e n 

Q18:

Utilize o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑦=ο„Έ3πœƒ5πœƒπœƒο‘½οŽ’βˆšοŠ«ο—tgd.

  • A 𝑦 β€² = 3 πœƒ 5 πœƒ t g
  • B 𝑦 β€² = 1 5 2 ο€» 5 √ 5 π‘₯  t g
  • C 𝑦 β€² = βˆ’ 1 5 2 ο€» 5 √ 5 π‘₯  t g
  • D 𝑦 β€² = 3 √ 5 π‘₯ ο€» 5 √ 5 π‘₯  t g
  • E 𝑦 β€² = βˆ’ 3 √ 5 π‘₯ ο€» 5 √ 5 π‘₯  t g

Q19:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função β„Ž(π‘₯)=ο„Έ3𝑧𝑧+2π‘§βˆšο—οŠͺοŠͺd.

  • A β„Ž β€² ( π‘₯ ) = 3 √ 𝑧 2 ( 𝑧 + 2 ) 
  • B β„Ž β€² ( π‘₯ ) = π‘₯ π‘₯ + 2 
  • C β„Ž β€² ( π‘₯ ) = 6 𝑧 + 1 2 𝑧 βˆ’ 1 2 𝑧 ( 𝑧 + 2 ) οŠͺ  οŠͺ 
  • D β„Ž β€² ( π‘₯ ) = 3 𝑧 𝑧 + 2  οŠͺ
  • E β„Ž β€² ( π‘₯ ) = 3 √ π‘₯ 2 ( π‘₯ + 2 ) 

Q20:

Dado que 𝑓(π‘₯)=ο„Έο€Ή8π‘₯βˆ’5π‘₯+4π‘₯d, encontre dd𝑓π‘₯.

  • A 1 6 π‘₯ βˆ’ 5
  • B16
  • C 8 π‘₯ βˆ’ 5 π‘₯ + 4 
  • D 8 3 π‘₯ βˆ’ 5 2 π‘₯ + 4 π‘₯  

Q21:

Use o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função β„Ž(π‘₯)=ο„Έβˆ’π‘‘π‘‘οŒΎοŠ¨οŽ€ο‘lnd.

  • A β„Ž β€² ( π‘₯ ) = βˆ’ 5 π‘₯
  • B β„Ž β€² ( π‘₯ ) = βˆ’ 5 𝑑
  • C β„Ž β€² ( π‘₯ ) = βˆ’ 2 5 π‘₯ 𝑒  
  • D β„Ž β€² ( π‘₯ ) = βˆ’ 𝑑 l n
  • E β„Ž β€² ( π‘₯ ) = βˆ’ 1 𝑑

Q22:

Dado 𝑓(π‘₯)π‘₯=3π‘₯+π‘₯βˆ’8π‘₯+5+dC, determine 𝑓′(βˆ’1).

Q23:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝑅(𝑦)=ο„Έβˆ’π‘‘3π‘‘π‘‘οŠ§ο˜οŠ¨send.

  • A 𝑅 β€² ( 𝑦 ) = βˆ’ 𝑦 3 𝑦  s e n
  • B 𝑅 β€² ( 𝑦 ) = 𝑑 3 𝑑  s e n
  • C 𝑅 β€² ( 𝑦 ) = βˆ’ 3 𝑑 3 𝑑 + 2 𝑑 3 𝑑  c o s s e n
  • D 𝑅 β€² ( 𝑦 ) = βˆ’ 3 𝑑 3 𝑑 βˆ’ 2 𝑑 3 𝑑  c o s s e n
  • E 𝑅 β€² ( 𝑦 ) = 𝑦 3 𝑦  s e n

Q24:

O grΓ‘fico de uma função 𝑓 Γ© dado. Qual Γ© o grΓ‘fico de uma antiderivada de 𝑓?

  • A π‘Ž
  • B 𝑏
  • C 𝑐

Q25:

A figura mostra o grΓ‘fico da função 𝑓(𝑑)𝑑.ο—οŠ¦d

Qual dos seguintes Γ© o grΓ‘fico de 𝑦=𝑓(π‘₯)?

  • A(b)
  • B(a)
  • C(c)
  • D(d)

A Nagwa usa cookies para garantir que vocΓͺ tenha a melhor experiΓͺncia em nosso site. Saiba mais sobre nossa PolΓ­tica de privacidade.