Worksheet: Average Value of a Function

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In this worksheet, we will practice using the mean value theorem for integrals to find the average value of a function.

Q1:

The average value of 𝑓(𝑥)=−6𝑥+6𝑥−1 on the interval [0,𝑏] is 0. Find all possible values of 𝑏.

  • A 𝑏 = − 1 or 𝑏=−12
  • B 𝑏 = 1 2 or 𝑏=1
  • C 𝑏 = − 2 or 𝑏=−1
  • D 𝑏 = − √ 3 6 + 1 2 or 𝑏=√36+12

Q2:

Find all 𝑐 such that 𝑓(𝑐) equals the average value of 𝑓(𝑥)=(𝑥−2) over the interval [−1,5].

  • A5, −1
  • B √ 3 + 2 , − √ 3 + 2
  • C 2 + 3 √ 2 , − 3 √ 2 + 2
  • D 2 + 6 √ 3 , − 6 √ 3 + 2

Q3:

Find the average value of 𝑓(𝑥)=𝑥−52𝑥sinsin on the interval 0,𝜋2.

  • A 2 𝜋
  • B − 2 𝜋
  • C − 4
  • D − 1 8 𝜋
  • E − 8 𝜋

Q4:

Find the average value of 𝑓(𝑥)=𝑥𝑒 on the interval [−2,0].

  • A 𝑒
  • B − 1 4 + 1 4 𝑒
  • C − 1 8 + 1 8 𝑒
  • D − 1 2 + 1 2 𝑒
  • E − 1 4 𝑒 + 1 4

Q5:

What is the average value of this function on the interval [−5,4]?

  • A15
  • B 5 3
  • C135
  • D − 5 3
  • E 1 9 9

Q6:

Find the average value of 𝑓(𝑥)=√2𝑥 on the interval [0,2].

  • A 4 3
  • B 2
  • C 8 3
  • D 1
  • E 1 6 3

Q7:

Determine the average value of 𝑓(𝑥)=3𝑥−2𝑥 on the interval [−3,5].

Q8:

Find the average value of 𝑓(𝑥)=𝑥(𝑥−5) on the interval [−1,1].

  • A 5 2 8 8
  • B 1 7 2
  • C 1 3 6
  • D 1 2 4
  • E 1 1 8

Q9:

Using the figure, arrange the following from least to greatest:

  1. 𝑓 ′ ( 1 )
  2. the average value of 𝑓(𝑥) on 0≤𝑥≤𝑎
  3. the average value of the rate of change of 𝑓(𝑥), for 0≤𝑥≤𝑎
  4. 𝑓 ( 𝑥 ) 𝑥 d
  • AA, C, B, D
  • BC, D, B, A
  • CA, D, C, B
  • DA, C, D, B
  • EB, D, C, A

Q10:

Find the average value of 𝑓(𝑢)=𝑢𝑢ln on the interval [1,3].

  • A l n 3
  • B 9 4
  • C l n 3 2
  • D l n 3 4
  • E l n 3 2

Q11:

Find the average value of 𝑓(𝑥)=43𝑥cos on the interval −5𝜋18,5𝜋18.

  • A 4 3
  • B0
  • C 1 2 5 𝜋
  • D − 1 2 5 𝜋
  • E 2 0 𝜋 2 7

Q12:

Find the average value of 𝑓(𝑥)=(2𝑥−5) on the interval [1,4].

Q13:

Find all 𝑐 such that 𝑓(𝑐) equals the average value of 𝑓(𝑥)=−52𝑥 over the interval [−5,−1].

  • A − 1 5 l n
  • B − 2 5 l n
  • C − 5 3
  • D − 4 5 l n
  • E − 1 4 5 l n

Q14:

Find the average value of 𝑓(𝑥)=𝑥𝑥sincos on the interval 0,𝜋2.

  • A 2 𝜋
  • B 1 5
  • C − 2 5 𝜋
  • D 2 5 𝜋
  • E 𝜋 1 0

Q15:

Find the average value of 𝑓(𝑡)=𝑒𝑡sincos on the interval 0,𝜋2.

  • A − 𝜋 2 + 𝑒 𝜋 2
  • B − 1 + 𝑒
  • C − 2 𝑒 𝜋 + 2 𝜋
  • D − 2 𝜋 + 2 𝑒 𝜋
  • E − 1

Q16:

Find the average value of 𝑔(𝑡)=2𝑡√2𝑡+2 on the interval [1,3].

  • A 4 √ 5 − 4
  • B √ 5 − 1
  • C √ 3 2 − 1 2
  • D √ 5 2 − 1 2

Q17:

Find the average value of 𝑓(𝑥)=−15𝑥 on the interval [−5,−1].

  • A l n 5 5
  • B 4 5 5 l n
  • C 1 2 5
  • D l n 5 2 0
  • E − 5 4 l n

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