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Lesson: The Fundamental Theorem of Calculus

Worksheet • 23 Questions

Q1:

Suppose that 𝑓 is a function on the interval [ π‘Ž , 𝑏 ] and we are able to define 𝐹 by 𝐹 ( π‘₯ ) = ο„Έ 𝑓 ( 𝑑 ) 𝑑 π‘₯ π‘Ž d and find that 𝐹 is NOT differentiable on ( π‘Ž , 𝑏 ) . What can we conclude?

  • A 𝑓 is discontinuous somewhere in the interval ( π‘Ž , 𝑏 ) .
  • BThere is a mistake because whenever we integrate a function, it must be differentiable and 𝐹 β€² ( π‘₯ ) = 𝑓 ( π‘₯ ) .
  • C 𝑓 is discontinuous everywhere on ( π‘Ž , 𝑏 ) .
  • D 𝑓 is not differentiable everywhere on ( π‘Ž , 𝑏 ) .
  • E 𝑓 is not differentiable somewhere in the interval ( π‘Ž , 𝑏 ) .

Q2:

Use the Fundamental Theorem of Calculus to find the derivative of the function β„Ž ( 𝑒 ) = ο„Έ √ 3 𝑑 4 𝑑 + 2 𝑑 𝑒 4 d .

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

Q3:

Let 𝑦 = ο„Έ √ 2 + 5 𝑑 𝑑 2 2 π‘₯ 2 s i n d . Use the Fundamental Theorem of Calculus to find 𝑦 β€² .

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