Worksheet: The Fundamental Theorem of Calculus: Functions Defined by Integrals

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In this worksheet, we will practice applying the fundamental theorem of calculus to find the derivative of a function defined by an integral.

Q1:

Use the fundamental theorem of calculus to find the derivative of the function ℎ(𝑢)=√3𝑡4𝑡+2𝑡d.

  • A ℎ ′ ( 𝑢 ) = − 3 ( 4 𝑢 − 2 ) 2 √ 3 𝑢 ( 4 𝑢 + 2 )
  • B ℎ ′ ( 𝑢 ) = √ 3 𝑡 4 𝑡 + 2
  • C ℎ ′ ( 𝑢 ) = √ 3 𝑢 4 𝑢 + 2
  • D ℎ ′ ( 𝑢 ) = − 3 ( 4 𝑡 − 2 ) 2 √ 3 𝑡 ( 4 𝑡 + 2 )
  • E ℎ ′ ( 𝑢 ) = − 3 ( 4 𝑡 − 2 ) 2 √ 3 𝑡 ( 4 𝑡 + 2 )

Q2:

Given that 𝑓(𝑥)𝑥=𝑥−7𝑥−𝑥+9+dC, find 𝑓′(1).

Q3:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑅(𝑦)=3𝑡2𝑡𝑡sind.

  • A 𝑅 ′ ( 𝑦 ) = 3 𝑦 2 𝑦 s i n
  • B 𝑅 ′ ( 𝑦 ) = − 3 𝑦 2 𝑦 s i n
  • C 𝑅 ′ ( 𝑦 ) = 6 𝑡 2 𝑡 + 6 𝑡 2 𝑡 c o s s i n
  • D 𝑅 ′ ( 𝑦 ) = 6 𝑡 2 𝑡 − 6 𝑡 2 𝑡 c o s s i n
  • E 𝑅 ′ ( 𝑦 ) = − 3 𝑡 2 𝑡 s i n

Q4:

Find the derivative of the function 𝑔(𝑥)=5𝑡𝑡𝑡sind.

  • A 𝑔 ′ ( 𝑥 ) = ( 1 0 − 2 0 𝑥 ) ( 1 − 2 𝑥 ) + ( 5 + 5 𝑥 ) ( 1 + 𝑥 ) s i n s i n
  • B 𝑔 ′ ( 𝑥 ) = ( 5 − 2 𝑥 ) ( 1 − 2 𝑥 ) + ( 5 + 5 𝑥 ) ( 1 + 𝑥 ) s i n s i n
  • C 𝑔 ′ ( 𝑥 ) = − ( 5 − 1 0 𝑥 ) ( 1 − 2 𝑥 ) + ( 5 + 5 𝑥 ) ( 1 + 𝑥 ) s i n s i n
  • D 𝑔 ′ ( 𝑥 ) = − ( 1 0 − 2 0 𝑥 ) ( 1 − 2 𝑥 ) + ( 5 + 5 𝑥 ) ( 1 + 𝑥 ) s i n s i n
  • E 𝑔 ′ ( 𝑥 ) = − ( 1 0 − 2 0 𝑥 ) ( 1 − 2 𝑥 ) − ( 5 + 5 𝑥 ) ( 1 + 𝑥 ) s i n s i n

Q5:

Find the derivative of the function 𝑦(𝑥)=(1−𝑣)𝑣sincoslnd.

  • A 𝑦 ′ ( 𝑥 ) = 3 𝑥 ( 1 − 3 𝑥 ) + 4 𝑥 ( 1 − 4 𝑥 ) s i n l n c o s c o s l n s i n
  • B 𝑦 ′ ( 𝑥 ) = − 3 𝑥 ( 1 − 3 𝑥 ) + 4 𝑥 ( 1 − 4 𝑥 ) s i n l n c o s c o s l n s i n
  • C 𝑦 ′ ( 𝑥 ) = − 3 𝑥 ( 1 − 3 𝑥 ) − 4 𝑥 ( 1 − 4 𝑥 ) s i n l n c o s c o s l n s i n
  • D 𝑦 ′ ( 𝑥 ) = − ( 1 − 3 𝑥 ) + ( 1 − 4 𝑥 ) l n c o s l n s i n
  • E 𝑦 ′ ( 𝑥 ) = ( 1 − 3 𝑥 ) + ( 1 − 4 𝑥 ) l n c o s l n s i n

Q6:

Suppose that 𝑓 is a function on the interval [𝑎,𝑏] and we are able to define 𝐹 by 𝐹(𝑥)=𝑓(𝑡)𝑡d. We find that 𝐹 is NOT differentiable on (𝑎,𝑏). What can we conclude?

  • AThere is a mistake because whenever we integrate a function, it must be differentiable and 𝐹′(𝑥)=𝑓(𝑥).
  • B 𝑓 is continuous everywhere on (𝑎,𝑏).
  • C 𝑓 is discontinuous somewhere in the interval (𝑎,𝑏).
  • D 𝑓 is discontinuous everywhere on (𝑎,𝑏).
  • E 𝑓 is not differentiable everywhere on (𝑎,𝑏).

Q7:

The figure shows the graph of the function 𝑓(𝑡)𝑡.d

Which of the following is the graph of 𝑦=𝑓(𝑥)?

  • A
  • Bnone of the above
  • C
  • D
  • E

Q8:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑦=2𝑡2+𝑡𝑡d.

  • A 𝑦 ′ = 2 ( 5 𝑥 + 3 ) 2 + ( 5 𝑥 + 3 )
  • B 𝑦 ′ = 1 0 ( 5 𝑡 + 3 ) 2 + ( 5 𝑡 + 3 )
  • C 𝑦 ′ = 2 𝑡 2 + 𝑡
  • D 𝑦 ′ = 2 ( 5 𝑡 + 3 ) 2 + ( 5 𝑡 + 3 )
  • E 𝑦 ′ = 1 0 ( 5 𝑥 + 3 ) 2 + ( 5 𝑥 + 3 )

Q9:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑦=5(5𝜃)𝜃cosd.

  • A 𝑦 ′ = 5 5 𝑥 c o s
  • B 𝑦 ′ = 5 0 5 𝜃 5 𝜃 s i n c o s
  • C 𝑦 ′ = 2 0 𝑥 5 𝑥 c o s
  • D 𝑦 ′ = 5 ( 5 𝜃 ) c o s
  • E 𝑦 ′ = − 5 0 5 𝜃 5 𝜃 s i n c o s

Q10:

Use the fundamental theorem of calculus to find the derivative of the function 𝑔(𝑥)=1+𝑡𝑡lnd.

  • A 𝑔 ′ ( 𝑥 ) = 1 + 𝑥 l n
  • B 𝑔 ′ ( 𝑥 ) = 5 𝑡 1 + 𝑡
  • C 𝑔 ′ ( 𝑥 ) = 5 𝑥 1 + 𝑥
  • D 𝑔 ′ ( 𝑥 ) = 1 + 𝑡 l n
  • E 𝑔 ′ ( 𝑥 ) = 1 1 + 𝑡

Q11:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑔(𝑠)=3𝑡−4𝑡𝑡d.

  • A 𝑔 ′ ( 𝑠 ) = 4 9 𝑡 − 2 0 𝑡 3 𝑡 − 4 𝑡
  • B 𝑔 ′ ( 𝑠 ) = 4 9 𝑠 − 2 0 𝑠 3 𝑠 − 4 𝑠
  • C 𝑔 ′ ( 𝑠 ) = 3 𝑡 − 4 𝑡
  • D 𝑔 ′ ( 𝑠 ) = 4 9 𝑡 − 2 0 𝑡 3 𝑡 − 4 𝑡
  • E 𝑔 ′ ( 𝑠 ) = 3 𝑠 − 4 𝑠

Q12:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝐹(𝑥)=√2−3𝑡𝑡secd.

  • A 𝐹 ′ ( 𝑥 ) = − 3 𝑡 𝑡 2 √ 2 − 3 𝑡 s e c t a n s e c
  • B 𝐹 ′ ( 𝑥 ) = √ 2 − 3 𝑡 s e c
  • C 𝐹 ′ ( 𝑥 ) = 3 𝑥 𝑥 2 √ 2 − 3 𝑥 s e c t a n s e c
  • D 𝐹 ′ ( 𝑥 ) = √ 2 − 3 𝑥 s e c
  • E 𝐹 ′ ( 𝑥 ) = − √ 2 − 3 𝑥 s e c

Q13:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑔(𝑥)=−2𝑡𝑡d.

  • A 𝑔 ′ ( 𝑥 ) = − 8 𝑡
  • B 𝑔 ′ ( 𝑥 ) = − 8 𝑥
  • C 𝑔 ′ ( 𝑥 ) = − 2 𝑡
  • D 𝑔 ′ ( 𝑥 ) = − 2 𝑥
  • E 𝑔 ′ ( 𝑥 ) = − 8 𝑥

Q14:

Find the derivative of the function 𝑔(𝑥)=𝑢−3𝑢+5𝑢d.

  • A 𝑔 ′ ( 𝑥 ) = − 4 1 6 𝑥 − 3 1 6 𝑥 + 5 + 3 9 𝑥 − 3 9 𝑥 + 5
  • B 𝑔 ′ ( 𝑥 ) = 1 6 𝑥 − 3 1 6 𝑥 + 5 + 9 𝑥 − 3 9 𝑥 + 5
  • C 𝑔 ′ ( 𝑥 ) = 1 6 𝑥 − 3 1 6 𝑥 + 5 − 9 𝑥 − 3 9 𝑥 + 5
  • D 𝑔 ′ ( 𝑥 ) = 4 1 6 𝑥 − 3 1 6 𝑥 + 5 + 3 9 𝑥 − 3 9 𝑥 + 5
  • E 𝑔 ′ ( 𝑥 ) = 4 1 6 𝑥 − 3 1 6 𝑥 + 5 − 3 9 𝑥 − 3 9 𝑥 + 5

Q15:

Find the derivative of the function 𝐹(𝑥)=2𝑒𝑡d.

  • A 𝐹 ′ ( 𝑥 ) = 2 𝑥 𝑒 − 2 𝑒
  • B 𝐹 ′ ( 𝑥 ) = 8 𝑥 𝑒 + 1 0 𝑒
  • C 𝐹 ′ ( 𝑥 ) = 8 𝑥 𝑒 − 1 0 𝑒
  • D 𝐹 ′ ( 𝑥 ) = 8 𝑥 𝑒 − 1 0 𝑒
  • E 𝐹 ′ ( 𝑥 ) = − 8 𝑥 𝑒 + 1 0 𝑒

Q16:

Given that 𝐹(𝑥)=𝑡𝑡√tand, find 𝐹′(𝑥).

  • A 𝐹 ′ ( 𝑥 ) = 4 4 𝑥 − 1 2 √ 𝑥 √ 𝑥 t a n t a n
  • B 𝐹 ′ ( 𝑥 ) = − 4 4 𝑥 + 1 2 √ 𝑥 √ 𝑥 t a n t a n
  • C 𝐹 ′ ( 𝑥 ) = 4 𝑥 + √ 𝑥 t a n t a n
  • D 𝐹 ′ ( 𝑥 ) = 4 𝑥 − √ 𝑥 t a n t a n
  • E 𝐹 ′ ( 𝑥 ) = 4 4 𝑥 + 1 2 √ 𝑥 √ 𝑥 t a n t a n

Q17:

Let 𝑦=√2+5𝑡𝑡sind. Use the Fundamental Theorem of Calculus to find 𝑦′.

  • A 𝑦 ′ = − 2 + 5 2 𝑥 s i n
  • B 𝑦 ′ = 2 + 5 2 𝑥 s i n
  • C 𝑦 ′ = − 2 ( 2 𝑥 ) 2 + 5 2 𝑥 c o s s i n
  • D 𝑦 ′ = √ 2 + 5 𝑡
  • E 𝑦 ′ = 2 ( 2 𝑥 ) 2 + 5 2 𝑥 c o s s i n

Q18:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑦=3𝜃5𝜃𝜃√tand.

  • A 𝑦 ′ = 3 𝜃 5 𝜃 t a n
  • B 𝑦 ′ = 1 5 2 5 √ 5 𝑥 t a n
  • C 𝑦 ′ = − 1 5 2 5 √ 5 𝑥 t a n
  • D 𝑦 ′ = 3 √ 5 𝑥 5 √ 5 𝑥 t a n
  • E 𝑦 ′ = − 3 √ 5 𝑥 5 √ 5 𝑥 t a n

Q19:

Use the Fundamental Theorem of Calculus to find the derivative of the function ℎ(𝑥)=3𝑧𝑧+2𝑧√d.

  • A ℎ ′ ( 𝑥 ) = 3 √ 𝑧 2 ( 𝑧 + 2 )
  • B ℎ ′ ( 𝑥 ) = 𝑥 𝑥 + 2
  • C ℎ ′ ( 𝑥 ) = 6 𝑧 + 1 2 𝑧 − 1 2 𝑧 ( 𝑧 + 2 )
  • D ℎ ′ ( 𝑥 ) = 3 𝑧 𝑧 + 2
  • E ℎ ′ ( 𝑥 ) = 3 √ 𝑥 2 ( 𝑥 + 2 )

Q20:

Given that 𝑓(𝑥)=8𝑥−5𝑥+4𝑥d, find dd𝑓𝑥.

  • A 1 6 𝑥 − 5
  • B16
  • C 8 𝑥 − 5 𝑥 + 4
  • D 8 3 𝑥 − 5 2 𝑥 + 4 𝑥

Q21:

Use the Fundamental Theorem of Calculus to find the derivative of the function ℎ(𝑥)=−𝑡𝑡lnd.

  • A ℎ ′ ( 𝑥 ) = − 5 𝑥
  • B ℎ ′ ( 𝑥 ) = − 5 𝑡
  • C ℎ ′ ( 𝑥 ) = − 2 5 𝑥 𝑒
  • D ℎ ′ ( 𝑥 ) = − 𝑡 l n
  • E ℎ ′ ( 𝑥 ) = − 1 𝑡

Q22:

Given that 𝑓(𝑥)𝑥=3𝑥+𝑥−8𝑥+5+dC, find 𝑓′(−1).

Q23:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑅(𝑦)=−𝑡3𝑡𝑡sind.

  • A 𝑅 ′ ( 𝑦 ) = 𝑦 3 𝑦 s i n
  • B 𝑅 ′ ( 𝑦 ) = − 3 𝑡 3 𝑡 − 2 𝑡 3 𝑡 c o s s i n
  • C 𝑅 ′ ( 𝑦 ) = − 𝑦 3 𝑦 s i n
  • D 𝑅 ′ ( 𝑦 ) = 𝑡 3 𝑡 s i n
  • E 𝑅 ′ ( 𝑦 ) = − 3 𝑡 3 𝑡 + 2 𝑡 3 𝑡 c o s s i n

Q24:

The graph of a function 𝑓 is shown. Which is the graph of an antiderivative of 𝑓?

  • A 𝑎
  • B 𝑐
  • C 𝑏

Q25:

The figure shows the graph of the function 𝑓(𝑡)𝑡.d

Which of the following is the graph of 𝑦=𝑓(𝑥)?

  • A(b)
  • B(a)
  • C(c)
  • D(d)

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