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Lesson Worksheet: The Fundamental Theorem of Calculus: Functions Defined by Integrals Mathematics • Higher Education

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

This lesson includes 20 additional questions and 175 additional question variations for subscribers.

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