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Worksheet: The Average Value of a Function

In this worksheet, we will practice calculating the average value of a function, also known as the mean, for continuous functions over intervals.

Q1:

Find the average value of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 2 π‘₯ s i n s i n on the interval  0 , πœ‹ 2  .

  • A βˆ’ 4
  • B βˆ’ 1 8 πœ‹
  • C βˆ’ 2 πœ‹
  • D βˆ’ 8 πœ‹
  • E 2 πœ‹

Q2:

Find all 𝑐 such that 𝑓 ( 𝑐 ) equals the average value of 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 2 ) 2 over the interval [ βˆ’ 1 , 5 ] .

  • A 2 + 3 √ 2 , βˆ’ 3 √ 2 + 2
  • B5, βˆ’ 1
  • C 2 + 6 √ 3 , βˆ’ 6 √ 3 + 2
  • D √ 3 + 2 , βˆ’ √ 3 + 2

Q3:

The average value of 𝑓 ( π‘₯ ) = βˆ’ 6 π‘₯ + 6 π‘₯ βˆ’ 1 2 on the interval [ 0 , 𝑏 ] is 0. Find all possible values of 𝑏 .

  • A 𝑏 = βˆ’ 2 or 𝑏 = βˆ’ 1
  • B 𝑏 = βˆ’ √ 3 6 + 1 2 or 𝑏 = √ 3 6 + 1 2
  • C 𝑏 = βˆ’ 1 or 𝑏 = βˆ’ 1 2
  • D 𝑏 = 1 2 or 𝑏 = 1

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