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„Έ √ 3 𝑑 4 𝑑 + 2 𝑑 𝑒 4 d .

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

Q2:

Dado , determine .

Q3:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝑅 ( 𝑦 ) = ο„Έ 3 𝑑 2 𝑑 𝑑    s e n d .

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

Q4:

Encontre a derivada da função 𝑔 ( π‘₯ ) = ο„Έ 5 𝑑 𝑑 𝑑        s e n d .

  • A 𝑔 β€² ( π‘₯ ) = ( 5 βˆ’ 2 π‘₯ ) ( 1 βˆ’ 2 π‘₯ ) + ( 5 + 5 π‘₯ ) ( 1 + π‘₯ ) s e n s e n
  • B 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
  • 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‘₯ ) = ( 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 ( 1 βˆ’ 𝑣 ) 𝑣 4 π‘₯ 3 π‘₯ s e n c o s l n d .

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

Q6:

Use o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑦 = ο„Έ 5 ( 5 πœƒ ) πœƒ     c o s d .

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

Q7:

Utilize o Teorema Funnamental do CΓ‘lculo para determinar a derivada da função 𝑔 ( π‘₯ ) = ο„Έ ο€Ή 1 + 𝑑  𝑑    l n d .

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

Q8:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝑔 ( 𝑠 ) = ο„Έ ο€Ή 3 𝑑 βˆ’ 4 𝑑  𝑑 𝑠 1 3 5 4 d .

  • A 𝑔 β€² ( 𝑠 ) = 4 ο€Ή 9 𝑑 βˆ’ 2 0 𝑑  ο€Ή 3 𝑑 βˆ’ 4 𝑑  2 4 3 5 3
  • B 𝑔 β€² ( 𝑠 ) = ο€Ή 3 𝑑 βˆ’ 4 𝑑  3 5 4
  • C 𝑔 β€² ( 𝑠 ) = 4 ο€Ή 9 𝑠 βˆ’ 2 0 𝑠  ο€Ή 3 𝑠 βˆ’ 4 𝑠  2 4 3 5 3
  • D 𝑔 β€² ( 𝑠 ) = ο€Ή 3 𝑠 βˆ’ 4 𝑠  3 5 4
  • E 𝑔 β€² ( 𝑠 ) = 4 ο€Ή 9 𝑑 βˆ’ 2 0 𝑑  ο€Ή 3 𝑑 βˆ’ 4 𝑑  2 4 3 5 4

Q9:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝐹 ( π‘₯ ) = ο„Έ √ 2 βˆ’ 3 𝑑 𝑑 4 π‘₯ s e c d .

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

Q10:

Utilize o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑔 ( π‘₯ ) = ο„Έ βˆ’ 2 𝑑 𝑑 π‘₯ 2 4 d .

  • A 𝑔 β€² ( π‘₯ ) = βˆ’ 8 𝑑 3
  • B 𝑔 β€² ( π‘₯ ) = βˆ’ 2 𝑑 4
  • C 𝑔 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ 3
  • D 𝑔 β€² ( π‘₯ ) = βˆ’ 2 π‘₯ 4
  • E 𝑔 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ 4

Q11:

Encontre a derivada da função 𝑔 ( π‘₯ ) = ο„Έ 𝑒 βˆ’ 3 𝑒 + 5 𝑒 4 π‘₯ 3 π‘₯ 2 2 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 2 2 2 2
  • B 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 2 2 2 2
  • C 𝑔 β€² ( π‘₯ ) = 1 6 π‘₯ βˆ’ 3 1 6 π‘₯ + 5 βˆ’ 9 π‘₯ βˆ’ 3 9 π‘₯ + 5 2 2 2 2
  • D 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 2 2 2 2
  • E 𝑔 β€² ( π‘₯ ) = 1 6 π‘₯ βˆ’ 3 1 6 π‘₯ + 5 + 9 π‘₯ βˆ’ 3 9 π‘₯ + 5 2 2 2 2

Q12:

Encontre a derivada da função 𝐹 ( π‘₯ ) = ο„Έ 2 𝑒 𝑑 2 π‘₯ 5 π‘₯ 𝑑 2 2 d .

  • A 𝐹 β€² ( π‘₯ ) = 2 π‘₯ 𝑒 βˆ’ 2 𝑒 4 π‘₯ 2 5 π‘₯ 4 2
  • B 𝐹 β€² ( π‘₯ ) = 8 π‘₯ 𝑒 + 1 0 𝑒 4 π‘₯ 2 5 π‘₯ 4 2
  • C 𝐹 β€² ( π‘₯ ) = 8 π‘₯ 𝑒 βˆ’ 1 0 𝑒 4 π‘₯ 2 5 π‘₯ 2 2
  • D 𝐹 β€² ( π‘₯ ) = 8 π‘₯ 𝑒 βˆ’ 1 0 𝑒 4 π‘₯ 2 5 π‘₯ 4 2
  • E 𝐹 β€² ( π‘₯ ) = βˆ’ 8 π‘₯ 𝑒 + 1 0 𝑒 4 π‘₯ 2 5 π‘₯ 4 2

Q13:

Dado que 𝐹 ( π‘₯ ) = ο„Έ 𝑑 𝑑 οŠͺ  √    t g d , encontre 𝐹 β€² ( π‘₯ ) .

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

Q14:

Seja 𝑦 = ο„Έ √ 2 + 5 𝑑 𝑑 2 2 π‘₯ 2 s e n d . Utilize o Teorema Fundamental do CΓ‘lculo para encontrar 𝑦 β€² .

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

Q15:

Utilize o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função 𝑦 = ο„Έ 3 πœƒ 5 πœƒ πœƒ πœ‹ 3 √ 5 π‘₯ t g d .

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

Q16:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função β„Ž ( π‘₯ ) = ο„Έ 3 𝑧 𝑧 + 2 𝑧 √ π‘₯ 4 2 4 d .

  • A β„Ž β€² ( π‘₯ ) = 3 √ 𝑧 2 ( 𝑧 + 2 ) 2
  • B β„Ž β€² ( π‘₯ ) = 3 𝑧 𝑧 + 2 2 4
  • C β„Ž β€² ( π‘₯ ) = π‘₯ π‘₯ + 2 2
  • D β„Ž β€² ( π‘₯ ) = 3 √ π‘₯ 2 ( π‘₯ + 2 ) 2
  • E β„Ž β€² ( π‘₯ ) = 6 𝑧 + 1 2 𝑧 βˆ’ 1 2 𝑧 ( 𝑧 + 2 ) 4 5 4 2

Q17:

Dado que 𝑓 ( π‘₯ ) = ο„Έ ο€Ή 8 π‘₯ βˆ’ 5 π‘₯ + 4  π‘₯  d , encontre d d 𝑓 π‘₯ .

  • A16
  • B 8 3 π‘₯ βˆ’ 5 2 π‘₯ + 4 π‘₯  
  • C 1 6 π‘₯ βˆ’ 5
  • D 8 π‘₯ βˆ’ 5 π‘₯ + 4 

Q18:

Use o Teorema Fundamental do CΓ‘lculo para encontrar a derivada da função β„Ž ( π‘₯ ) = ο„Έ βˆ’ 𝑑 𝑑     l n d .

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

Q19:

Dado , determine .

Q20:

Utilize o Teorema Fundamental do CΓ‘lculo para determinar a derivada da função 𝑅 ( 𝑦 ) = ο„Έ βˆ’ 𝑑 3 𝑑 𝑑    s e n d .

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

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