Worksheet: Average Value of a Function

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.

  • A𝑏=−1 or 𝑏=−12
  • B𝑏=12 or 𝑏=1
  • C𝑏=−2 or 𝑏=−1
  • D𝑏=−√36+12 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
  • C2+3√2, −3√2+2
  • D2+6√3, −6√3+2

Q3:

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

  • A2𝜋
  • B−2𝜋
  • C−4
  • D−18𝜋
  • E−8𝜋

Q4:

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

  • A𝑒
  • B−14+14𝑒
  • C−18+18𝑒
  • D−12+12𝑒
  • E−14𝑒+14

Q5:

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

  • A15
  • B53
  • C135
  • D−53
  • E199

Q6:

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

  • A43
  • B2
  • C83
  • D1
  • E163

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].

  • A5288
  • B172
  • C136
  • D124
  • E118

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)𝑥d
  • 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].

  • Aln3
  • B94
  • Cln32
  • Dln34
  • Eln32

Q11:

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

  • A43
  • B0
  • C125𝜋
  • D−125𝜋
  • E20𝜋27

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−15ln
  • B−25ln
  • C−53
  • D−45ln
  • E−145ln

Q14:

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

  • A2𝜋
  • B15
  • C−25𝜋
  • D25𝜋
  • E𝜋10

Q15:

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

  • A−𝜋2+𝑒𝜋2
  • B−1+𝑒
  • C−2𝑒𝜋+2𝜋
  • D−2𝜋+2𝑒𝜋
  • E−1

Q16:

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

  • A4√5−4
  • B√5−1
  • C√32−12
  • D√52−12

Q17:

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

  • Aln55
  • B455ln
  • C125
  • Dln520
  • E−54ln

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