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

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 β„Ž(𝑒)=ο„Έβˆš3𝑑4𝑑+2𝑑οŠͺd.

  • Aβ„Žβ€²(𝑒)=βˆ’3(4π‘‘βˆ’2)2√3𝑑(4𝑑+2)
  • Bβ„Žβ€²(𝑒)=βˆ’3(4π‘‘βˆ’2)2√3𝑑(4𝑑+2)
  • Cβ„Žβ€²(𝑒)=√3𝑑4𝑑+2
  • Dβ„Žβ€²(𝑒)=βˆ’3(4π‘’βˆ’2)2√3𝑒(4𝑒+2)
  • Eβ„Žβ€²(𝑒)=√3𝑒4𝑒+2

Q2:

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

Q3:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑅(𝑦)=ο„Έ3𝑑2π‘‘π‘‘οŠ«ο˜οŠ¨sind.

  • A𝑅′(𝑦)=3𝑦2π‘¦οŠ¨sin
  • B𝑅′(𝑦)=βˆ’3𝑑2π‘‘οŠ¨sin
  • C𝑅′(𝑦)=6𝑑2π‘‘βˆ’6𝑑2π‘‘οŠ¨cossin
  • D𝑅′(𝑦)=6𝑑2𝑑+6𝑑2π‘‘οŠ¨cossin
  • E𝑅′(𝑦)=βˆ’3𝑦2π‘¦οŠ¨sin

Q4:

Find the derivative of the function 𝑔(π‘₯)=ο„Έ5π‘‘π‘‘π‘‘οŠ§οŠ°ο—οŠ§οŠ±οŠ¨ο—sind.

  • A𝑔′(π‘₯)=(5βˆ’2π‘₯)(1βˆ’2π‘₯)+(5+5π‘₯)(1+π‘₯)sinsin
  • B𝑔′(π‘₯)=(10βˆ’20π‘₯)(1βˆ’2π‘₯)+(5+5π‘₯)(1+π‘₯)sinsin
  • C𝑔′(π‘₯)=βˆ’(10βˆ’20π‘₯)(1βˆ’2π‘₯)βˆ’(5+5π‘₯)(1+π‘₯)sinsin
  • D𝑔′(π‘₯)=βˆ’(10βˆ’20π‘₯)(1βˆ’2π‘₯)+(5+5π‘₯)(1+π‘₯)sinsin
  • E𝑔′(π‘₯)=βˆ’(5βˆ’10π‘₯)(1βˆ’2π‘₯)+(5+5π‘₯)(1+π‘₯)sinsin

Q5:

Find the derivative of the function 𝑦(π‘₯)=ο„Έ(1βˆ’π‘£)𝑣οŠͺο—οŠ©ο—sincoslnd.

  • A𝑦′(π‘₯)=βˆ’(1βˆ’3π‘₯)+(1βˆ’4π‘₯)lncoslnsin
  • B𝑦′(π‘₯)=(1βˆ’3π‘₯)+(1βˆ’4π‘₯)lncoslnsin
  • C𝑦′(π‘₯)=3π‘₯(1βˆ’3π‘₯)+4π‘₯(1βˆ’4π‘₯)sinlncoscoslnsin
  • D𝑦′(π‘₯)=βˆ’3π‘₯(1βˆ’3π‘₯)+4π‘₯(1βˆ’4π‘₯)sinlncoscoslnsin
  • E𝑦′(π‘₯)=βˆ’3π‘₯(1βˆ’3π‘₯)βˆ’4π‘₯(1βˆ’4π‘₯)sinlncoscoslnsin

Q6:

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

  • A𝑓 is discontinuous somewhere in the interval (π‘Ž,𝑏).
  • B𝑓 is not differentiable everywhere on (π‘Ž,𝑏).
  • CThere is a mistake because whenever we integrate a function, it must be differentiable and 𝐹′(π‘₯)=𝑓(π‘₯).
  • D𝑓 is discontinuous everywhere on (π‘Ž,𝑏).
  • E𝑓 is continuous everywhere on (π‘Ž,𝑏).

Q7:

The figure shows the graph of the function 𝑓(𝑑)𝑑.ο—οŠ¦d

Which of the following is the graph of 𝑦=𝑓(π‘₯)?

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

Q8:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑦=ο„Έ2𝑑2+π‘‘π‘‘οŠ«ο—οŠ°οŠ©οŠͺd.

  • A𝑦′=2(5π‘₯+3)2+(5π‘₯+3)
  • B𝑦′=2(5𝑑+3)2+(5𝑑+3)
  • C𝑦′=2𝑑2+π‘‘οŠ«
  • D𝑦′=10(5𝑑+3)2+(5𝑑+3)
  • E𝑦′=10(5π‘₯+3)2+(5π‘₯+3)

Q9:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑦=ο„Έ5(5πœƒ)πœƒο—οŠ¨οŠ¨οŽ£cosd.

  • A𝑦′=5ο€Ή5π‘₯cosοŠͺ
  • B𝑦′=20π‘₯ο€Ή5π‘₯ο…οŠ©οŠ¨οŠͺcos
  • C𝑦′=βˆ’505πœƒ5πœƒsincos
  • D𝑦′=505πœƒ5πœƒsincos
  • E𝑦′=5(5πœƒ)cos

Q10:

Use the fundamental theorem of calculus to find the derivative of the function 𝑔(π‘₯)=ο„Έο€Ή1+π‘‘ο…π‘‘ο—οŠ©οŠ«lnd.

  • A𝑔′(π‘₯)=ο€Ή1+𝑑ln
  • B𝑔′(π‘₯)=11+π‘‘οŠ«
  • C𝑔′(π‘₯)=5π‘₯1+π‘₯οŠͺ
  • D𝑔′(π‘₯)=ο€Ή1+π‘₯ln
  • E𝑔′(π‘₯)=5𝑑1+𝑑οŠͺ

Q11:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑔(𝑠)=ο„Έο€Ή3π‘‘βˆ’4π‘‘ο…π‘‘οοŠ§οŠ©οŠ«οŠͺd.

  • A𝑔′(𝑠)=4ο€Ή9π‘ βˆ’20𝑠3π‘ βˆ’4π‘ ο…οŠ¨οŠͺ
  • B𝑔′(𝑠)=ο€Ή3π‘ βˆ’4π‘ ο…οŠ©οŠ«οŠͺ
  • C𝑔′(𝑠)=4ο€Ή9π‘‘βˆ’20𝑑3π‘‘βˆ’4π‘‘ο…οŠ¨οŠͺοŠͺ
  • D𝑔′(𝑠)=ο€Ή3π‘‘βˆ’4π‘‘ο…οŠ©οŠ«οŠͺ
  • E𝑔′(𝑠)=4ο€Ή9π‘‘βˆ’20𝑑3π‘‘βˆ’4π‘‘ο…οŠ¨οŠͺ

Q12:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝐹(π‘₯)=ο„Έβˆš2βˆ’3𝑑𝑑οŠͺsecd.

  • A𝐹′(π‘₯)=βˆ’βˆš2βˆ’3π‘₯sec
  • B𝐹′(π‘₯)=βˆ’3𝑑𝑑2√2βˆ’3𝑑sectansec
  • C𝐹′(π‘₯)=√2βˆ’3𝑑sec
  • D𝐹′(π‘₯)=√2βˆ’3π‘₯sec
  • E𝐹′(π‘₯)=3π‘₯π‘₯2√2βˆ’3π‘₯sectansec

Q13:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑔(π‘₯)=ο„Έβˆ’2π‘‘π‘‘ο—οŠ¨οŠͺd.

  • A𝑔′(π‘₯)=βˆ’8π‘₯οŠͺ
  • B𝑔′(π‘₯)=βˆ’2𝑑οŠͺ
  • C𝑔′(π‘₯)=βˆ’8π‘‘οŠ©
  • D𝑔′(π‘₯)=βˆ’8π‘₯
  • E𝑔′(π‘₯)=βˆ’2π‘₯οŠͺ

Q14:

Find the derivative of the function 𝑔(π‘₯)=ο„Έπ‘’βˆ’3𝑒+5𝑒οŠͺο—οŠ©ο—οŠ¨οŠ¨d.

  • A𝑔′(π‘₯)=4ο€Ή16π‘₯βˆ’316π‘₯+5βˆ’3ο€Ή9π‘₯βˆ’39π‘₯+5
  • B𝑔′(π‘₯)=4ο€Ή16π‘₯βˆ’316π‘₯+5+3ο€Ή9π‘₯βˆ’39π‘₯+5
  • C𝑔′(π‘₯)=βˆ’4ο€Ή16π‘₯βˆ’316π‘₯+5+3ο€Ή9π‘₯βˆ’39π‘₯+5
  • D𝑔′(π‘₯)=16π‘₯βˆ’316π‘₯+5+9π‘₯βˆ’39π‘₯+5
  • E𝑔′(π‘₯)=16π‘₯βˆ’316π‘₯+5βˆ’9π‘₯βˆ’39π‘₯+5

Q15:

Find the derivative of the function 𝐹(π‘₯)=ο„Έ2π‘’π‘‘οŠ¨ο—οŠ«ο—οοŽ‘οŽ‘d.

  • A𝐹′(π‘₯)=βˆ’8π‘₯𝑒+10𝑒οŠͺο—οŠ¨οŠ«ο—οŽ£οŽ‘
  • B𝐹′(π‘₯)=8π‘₯π‘’βˆ’10𝑒οŠͺο—οŠ¨οŠ«ο—οŽ£οŽ‘
  • C𝐹′(π‘₯)=2π‘₯π‘’βˆ’2𝑒οŠͺο—οŠ¨οŠ«ο—οŽ£οŽ‘
  • D𝐹′(π‘₯)=8π‘₯π‘’βˆ’10𝑒οŠͺο—οŠ¨οŠ«ο—οŽ‘οŽ‘
  • E𝐹′(π‘₯)=8π‘₯𝑒+10𝑒οŠͺο—οŠ¨οŠ«ο—οŽ£οŽ‘

Q16:

Given that 𝐹(π‘₯)=𝑑𝑑οŠͺο—βˆšο—οŠ±οŠ§tand, find 𝐹′(π‘₯).

  • A𝐹′(π‘₯)=4π‘₯βˆ’βˆšπ‘₯tantan
  • B𝐹′(π‘₯)=4π‘₯+√π‘₯tantan
  • C𝐹′(π‘₯)=44π‘₯+12√π‘₯√π‘₯tantan
  • D𝐹′(π‘₯)=44π‘₯βˆ’12√π‘₯√π‘₯tantan
  • E𝐹′(π‘₯)=βˆ’44π‘₯+12√π‘₯√π‘₯tantan

Q17:

Let 𝑦=ο„Έβˆš2+5π‘‘π‘‘οŠ¨οŠ¨ο—οŠ¨sind. Use the Fundamental Theorem of Calculus to find 𝑦′.

  • A𝑦′=βˆ’ο„2+52π‘₯sin
  • B𝑦′=2(2π‘₯)2+52π‘₯cossin
  • C𝑦′=√2+5π‘‘οŠ¨
  • D𝑦′=2+52π‘₯sin
  • E𝑦′=βˆ’2(2π‘₯)2+52π‘₯cossin

Q18:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑦=ο„Έ3πœƒ5πœƒπœƒο‘½οŽ’βˆšοŠ«ο—tand.

  • A𝑦′=3√5π‘₯ο€»5√5π‘₯tan
  • B𝑦′=βˆ’152ο€»5√5π‘₯tan
  • C𝑦′=3πœƒ5πœƒtan
  • D𝑦′=152ο€»5√5π‘₯tan
  • E𝑦′=βˆ’3√5π‘₯ο€»5√5π‘₯tan

Q19:

Use the Fundamental Theorem of Calculus to find the derivative of the function β„Ž(π‘₯)=ο„Έ3𝑧𝑧+2π‘§βˆšο—οŠͺοŠͺd.

  • Aβ„Žβ€²(π‘₯)=6𝑧+12π‘§βˆ’12𝑧(𝑧+2)οŠͺοŠͺ
  • Bβ„Žβ€²(π‘₯)=3βˆšπ‘§2(𝑧+2)
  • Cβ„Žβ€²(π‘₯)=3√π‘₯2(π‘₯+2)
  • Dβ„Žβ€²(π‘₯)=π‘₯π‘₯+2
  • Eβ„Žβ€²(π‘₯)=3𝑧𝑧+2οŠͺ

Q20:

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

  • A16π‘₯βˆ’5
  • B83π‘₯βˆ’52π‘₯+4π‘₯
  • C16
  • D8π‘₯βˆ’5π‘₯+4

Q21:

Use the Fundamental Theorem of Calculus to find the derivative of the function β„Ž(π‘₯)=ο„Έβˆ’π‘‘π‘‘οŒΎοŠ¨οŽ€ο‘lnd.

  • Aβ„Žβ€²(π‘₯)=βˆ’1𝑑
  • Bβ„Žβ€²(π‘₯)=βˆ’5𝑑
  • Cβ„Žβ€²(π‘₯)=βˆ’π‘‘ln
  • Dβ„Žβ€²(π‘₯)=βˆ’5π‘₯
  • Eβ„Žβ€²(π‘₯)=βˆ’25π‘₯π‘’οŠ«ο—

Q22:

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

Q23:

Use the Fundamental Theorem of Calculus to find the derivative of the function 𝑅(𝑦)=ο„Έβˆ’π‘‘3π‘‘π‘‘οŠ§ο˜οŠ¨sind.

  • A𝑅′(𝑦)=βˆ’π‘¦3π‘¦οŠ¨sin
  • B𝑅′(𝑦)=𝑑3π‘‘οŠ¨sin
  • C𝑅′(𝑦)=βˆ’3𝑑3𝑑+2𝑑3π‘‘οŠ¨cossin
  • D𝑅′(𝑦)=βˆ’3𝑑3π‘‘βˆ’2𝑑3π‘‘οŠ¨cossin
  • E𝑅′(𝑦)=𝑦3π‘¦οŠ¨sin

Q24:

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

  • Aπ‘Ž
  • B𝑏
  • C𝑐

Q25:

The figure shows the graph of the function 𝑓(𝑑)𝑑.ο—οŠ¦d

Which of the following is the graph of 𝑦=𝑓(π‘₯)?

  • A(c)
  • B(a)
  • C(b)
  • D(d)

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