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Lesson Worksheet: Average Value of a Function Mathematics • Higher Education

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


The average value of ๐‘“(๐‘ฅ)=โˆ’6๐‘ฅ+6๐‘ฅโˆ’1๏Šจ on the interval [0,๐‘] is 0. Find all possible values of ๐‘.

  • A๐‘=โˆ’1 or ๐‘=โˆ’12
  • B๐‘=12 or ๐‘=1
  • C๐‘=โˆ’2 or ๐‘=โˆ’1
  • D๐‘=โˆ’โˆš36+12 or ๐‘=โˆš36+12


Find all ๐‘ such that ๐‘“(๐‘) equals the average value of ๐‘“(๐‘ฅ)=(๐‘ฅโˆ’2)๏Šจ over the interval [โˆ’1,5].

  • A5, โˆ’1
  • Bโˆš3+2, โˆ’โˆš3+2
  • C2+3โˆš2, โˆ’3โˆš2+2
  • D2+6โˆš3, โˆ’6โˆš3+2


Find the average value of ๐‘“(๐‘ฅ)=๐‘ฅโˆ’52๐‘ฅsinsin on the interval ๏“0,๐œ‹2๏Ÿ.

  • A2๐œ‹
  • Bโˆ’2๐œ‹
  • Cโˆ’4
  • Dโˆ’18๐œ‹
  • Eโˆ’8๐œ‹


Find the average value of ๐‘“(๐‘ฅ)=๐‘ฅ๐‘’๏Šฑ๏Šจ๏—๏Žก on the interval [โˆ’2,0].

  • A๐‘’๏Šฑ๏Šฎ
  • Bโˆ’14+14๐‘’๏Šฎ
  • Cโˆ’18+18๐‘’๏Šฎ
  • Dโˆ’12+12๐‘’๏Šฎ
  • Eโˆ’14๐‘’+14๏Šฎ


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

  • A15
  • B53
  • C135
  • Dโˆ’53
  • E199


Find the average value of ๐‘“(๐‘ฅ)=โˆš2๐‘ฅ on the interval [0,2].

  • A43
  • B2
  • C83
  • D1
  • E163


Determine the average value of ๐‘“(๐‘ฅ)=3๐‘ฅโˆ’2๐‘ฅ๏Šจ on the interval [โˆ’3,5].


Find the average value of ๐‘“(๐‘ฅ)=๐‘ฅ(๐‘ฅโˆ’5)๏Šจ๏Šฉ๏Šจ on the interval [โˆ’1,1].

  • A5288
  • B172
  • C136
  • D124
  • E118


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

  1. ๐‘“โ€ฒ(1)
  2. the average value of ๐‘“(๐‘ฅ) on 0โ‰ค๐‘ฅโ‰ค๐‘Ž
  3. the average value of the rate of change of ๐‘“(๐‘ฅ), for 0โ‰ค๐‘ฅโ‰ค๐‘Ž
  4. ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Œบ๏Šฆd
  • AA, C, B, D
  • BC, D, B, A
  • CA, D, C, B
  • DA, C, D, B
  • EB, D, C, A


Find the average value of ๐‘“(๐‘ข)=๐‘ข๐‘ขln on the interval [1,3].

  • Aln๏Šจ3
  • B94
  • Cln๏Šจ32
  • Dln๏Šจ34
  • Eln๏Šจ32

This lesson includes 9 additional questions and 140 additional question variations for subscribers.

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