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.

  • Aℎ′(𝑢)=−3(4𝑡−2)2√3𝑡(4𝑡+2)
  • Bℎ′(𝑢)=−3(4𝑡−2)2√3𝑡(4𝑡+2)
  • Cℎ′(𝑢)=√3𝑡4𝑡+2
  • Dℎ′(𝑢)=−3(4𝑢−2)2√3𝑢(4𝑢+2)
  • Eℎ′(𝑢)=√3𝑢4𝑢+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.

  • A𝑅′(𝑦)=3𝑦2𝑦sin
  • B𝑅′(𝑦)=−3𝑡2𝑡sin
  • C𝑅′(𝑦)=6𝑡2𝑡−6𝑡2𝑡cossin
  • D𝑅′(𝑦)=6𝑡2𝑡+6𝑡2𝑡cossin
  • E𝑅′(𝑦)=−3𝑦2𝑦sin

Q4:

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

  • A𝑔′(𝑥)=(5−2𝑥)(1−2𝑥)+(5+5𝑥)(1+𝑥)sinsin
  • B𝑔′(𝑥)=(10−20𝑥)(1−2𝑥)+(5+5𝑥)(1+𝑥)sinsin
  • C𝑔′(𝑥)=−(10−20𝑥)(1−2𝑥)−(5+5𝑥)(1+𝑥)sinsin
  • D𝑔′(𝑥)=−(10−20𝑥)(1−2𝑥)+(5+5𝑥)(1+𝑥)sinsin
  • E𝑔′(𝑥)=−(5−10𝑥)(1−2𝑥)+(5+5𝑥)(1+𝑥)sinsin

Q5:

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

  • A𝑦′(𝑥)=−(1−3𝑥)+(1−4𝑥)lncoslnsin
  • B𝑦′(𝑥)=(1−3𝑥)+(1−4𝑥)lncoslnsin
  • C𝑦′(𝑥)=3𝑥(1−3𝑥)+4𝑥(1−4𝑥)sinlncoscoslnsin
  • D𝑦′(𝑥)=−3𝑥(1−3𝑥)+4𝑥(1−4𝑥)sinlncoscoslnsin
  • E𝑦′(𝑥)=−3𝑥(1−3𝑥)−4𝑥(1−4𝑥)sinlncoscoslnsin

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?

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

  • A𝑦′=2(5𝑥+3)2+(5𝑥+3)
  • B𝑦′=2(5𝑡+3)2+(5𝑡+3)
  • C𝑦′=2𝑡2+𝑡
  • D𝑦′=10(5𝑡+3)2+(5𝑡+3)
  • E𝑦′=10(5𝑥+3)2+(5𝑥+3)

Q9:

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

  • A𝑦′=55𝑥cos
  • B𝑦′=20𝑥5𝑥cos
  • C𝑦′=−505𝜃5𝜃sincos
  • D𝑦′=505𝜃5𝜃sincos
  • E𝑦′=5(5𝜃)cos

Q10:

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

  • A𝑔′(𝑥)=1+𝑡ln
  • B𝑔′(𝑥)=11+𝑡
  • C𝑔′(𝑥)=5𝑥1+𝑥
  • D𝑔′(𝑥)=1+𝑥ln
  • E𝑔′(𝑥)=5𝑡1+𝑡

Q11:

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

  • A𝑔′(𝑠)=49𝑠−20𝑠3𝑠−4𝑠
  • B𝑔′(𝑠)=3𝑠−4𝑠
  • C𝑔′(𝑠)=49𝑡−20𝑡3𝑡−4𝑡
  • D𝑔′(𝑠)=3𝑡−4𝑡
  • E𝑔′(𝑠)=49𝑡−20𝑡3𝑡−4𝑡

Q12:

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

  • A𝐹′(𝑥)=−√2−3𝑥sec
  • B𝐹′(𝑥)=−3𝑡𝑡2√2−3𝑡sectansec
  • C𝐹′(𝑥)=√2−3𝑡sec
  • D𝐹′(𝑥)=√2−3𝑥sec
  • E𝐹′(𝑥)=3𝑥𝑥2√2−3𝑥sectansec

Q13:

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

  • A𝑔′(𝑥)=−8𝑥
  • B𝑔′(𝑥)=−2𝑡
  • C𝑔′(𝑥)=−8𝑡
  • D𝑔′(𝑥)=−8𝑥
  • E𝑔′(𝑥)=−2𝑥

Q14:

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

  • A𝑔′(𝑥)=416𝑥−316𝑥+5−39𝑥−39𝑥+5
  • B𝑔′(𝑥)=416𝑥−316𝑥+5+39𝑥−39𝑥+5
  • C𝑔′(𝑥)=−416𝑥−316𝑥+5+39𝑥−39𝑥+5
  • D𝑔′(𝑥)=16𝑥−316𝑥+5+9𝑥−39𝑥+5
  • E𝑔′(𝑥)=16𝑥−316𝑥+5−9𝑥−39𝑥+5

Q15:

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

  • A𝐹′(𝑥)=−8𝑥𝑒+10𝑒
  • B𝐹′(𝑥)=8𝑥𝑒−10𝑒
  • C𝐹′(𝑥)=2𝑥𝑒−2𝑒
  • D𝐹′(𝑥)=8𝑥𝑒−10𝑒
  • E𝐹′(𝑥)=8𝑥𝑒+10𝑒

Q16:

Given that 𝐹(𝑥)=𝑡𝑡√tand, find 𝐹′(𝑥).

  • A𝐹′(𝑥)=4𝑥−√𝑥tantan
  • B𝐹′(𝑥)=4𝑥+√𝑥tantan
  • C𝐹′(𝑥)=44𝑥+12√𝑥√𝑥tantan
  • D𝐹′(𝑥)=44𝑥−12√𝑥√𝑥tantan
  • E𝐹′(𝑥)=−44𝑥+12√𝑥√𝑥tantan

Q17:

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

  • A𝑦′=−2+52𝑥sin
  • B𝑦′=2(2𝑥)2+52𝑥cossin
  • C𝑦′=√2+5𝑡
  • D𝑦′=2+52𝑥sin
  • E𝑦′=−2(2𝑥)2+52𝑥cossin

Q18:

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

  • A𝑦′=3√5𝑥5√5𝑥tan
  • B𝑦′=−1525√5𝑥tan
  • C𝑦′=3𝜃5𝜃tan
  • D𝑦′=1525√5𝑥tan
  • E𝑦′=−3√5𝑥5√5𝑥tan

Q19:

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

  • Aℎ′(𝑥)=6𝑧+12𝑧−12𝑧(𝑧+2)
  • Bℎ′(𝑥)=3√𝑧2(𝑧+2)
  • Cℎ′(𝑥)=3√𝑥2(𝑥+2)
  • Dℎ′(𝑥)=𝑥𝑥+2
  • Eℎ′(𝑥)=3𝑧𝑧+2

Q20:

Given that 𝑓(𝑥)=8𝑥−5𝑥+4𝑥d, find dd𝑓𝑥.

  • A16𝑥−5
  • B83𝑥−52𝑥+4𝑥
  • C16
  • D8𝑥−5𝑥+4

Q21:

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

  • Aℎ′(𝑥)=−1𝑡
  • Bℎ′(𝑥)=−5𝑡
  • Cℎ′(𝑥)=−𝑡ln
  • Dℎ′(𝑥)=−5𝑥
  • Eℎ′(𝑥)=−25𝑥𝑒

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.

  • A𝑅′(𝑦)=−𝑦3𝑦sin
  • B𝑅′(𝑦)=𝑡3𝑡sin
  • C𝑅′(𝑦)=−3𝑡3𝑡+2𝑡3𝑡cossin
  • D𝑅′(𝑦)=−3𝑡3𝑡−2𝑡3𝑡cossin
  • E𝑅′(𝑦)=𝑦3𝑦sin

Q24:

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

  • A𝑎
  • B𝑏
  • C𝑐

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(c)
  • B(a)
  • C(b)
  • D(d)

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