Worksheet: The Mean Value Theorem

In this worksheet, we will practice interpreting and using the mean value theorem and Rolle's theorem.

Q1:

Nader is not convinced that the mean value theorem is true because, he says, the function ๐‘“ ( ๐‘ฅ ) = | ๐‘ฅ | has the property that if we take ๐‘Ž = โˆ’ 2 and ๐‘ = 2 , we have ๐‘“ ( ๐‘ ) โˆ’ ๐‘“ ( ๐‘Ž ) ๐‘ โˆ’ ๐‘Ž = 0 , and yet there is no point ๐‘ฅ where ๐‘“ โ€ฒ ( ๐‘ฅ ) = 0 . What is his error?

  • AThe function should be strictly decreasing on the interval.
  • BThe theorem requires the domain to be an interval, which โ„ is not.
  • CThe function should be strictly increasing on the interval.
  • DThe function is not differentiable at ๐‘ฅ = 0 . The theorem requires diferentiability on an interval.
  • EThe function is not continuous. The theorem requires continuity on an interval.

Q2:

Mariam is not convinced that the mean value theorem is true because, she says, the function ๐‘“ ( ๐‘ฅ ) = | ๐‘ฅ | is certainly differentiable on โ„ โˆ’ { 0 } . But if we take ๐‘Ž = โˆ’ 1 and ๐‘ = 1 , we have ๐‘“ ( ๐‘ ) โˆ’ ๐‘“ ( ๐‘Ž ) ๐‘ โˆ’ ๐‘Ž = 0 , and yet there is no point ๐‘ฅ where ๐‘“ ( ๐‘ฅ ) = 0 ๏Ž˜ . What is her error?

  • AThe function should be strictly decreasing on the interval.
  • BThe theorem requires that the function be differentiable on its domain.
  • CThe function should be strictly increasing on the interval.
  • DThe theorem requires the domain to be an interval, which โ„ โˆ’ { 0 } is not.
  • EThe function should be continuous on the interval.

Q3:

Does the mean value theorem apply for the function ๐‘ฆ = 2 โˆš ๐‘ฅ โˆ’ 4 ๏Šจ over the interval [ โˆ’ 2 , 2 ] ?

  • AYes
  • BNo

Q4:

Does the mean value theorem apply to the function ๐‘ฆ = 2 | ๐‘ฅ + 2 | over the interval [ 0 , 6 ] ?

  • AYes
  • BNo

Q5:

Does the mean value theorem apply for the function ๐‘ฆ = 2 ๐‘ฅ โˆ’ 4 ๐‘ฅ + 7 ๏Šฉ over the interval [ 0 , 5 ] ?

  • AYes
  • BNo

Q6:

Does the mean value theorem apply for the function ๐‘ฆ = 3 + | 2 ๐‘ฅ | over the interval [ โˆ’ 2 , 2 ] ?

  • ANo
  • BYes

Q7:

For the function ๐‘“ ( ๐‘ฅ ) = โˆš ๐‘ฅ , find all the possible values of ๐‘ that satisfy the mean value theorem over the interval [ 4 , 9 ] .

  • A 5 2
  • B0
  • C 1 5
  • D 2 5 4
  • E ๏„ž 5 2

Q8:

For the function ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ + 2 ๐‘ฅ ๏Šจ , find all the values of ๐‘ that satisfy the mean value theorem over the interval [ โˆ’ 4 , 4 ] .

Q9:

Does the mean value theorem apply for the function ๐‘ฆ = 3 ๐‘ฅ 2 ๐œ‹ ๐‘ฅ s i n over the interval [ 0 , 2 ] ?

  • AYes
  • BNo

Q10:

For the function ๐‘“ ( ๐‘ฅ ) = 3 ๐‘ฅ ๏Šฉ , find all the possible values of ๐‘ that satisfy the mean value theorem over the interval [ 1 , 3 ] .

  • A2
  • B12
  • C 3 โˆš 3 9
  • D โˆš 3 9 3
  • E0

Q11:

For the function ๐‘“ ( ๐‘ฅ ) = ๐‘ฅ โˆ’ 4 ๐‘ฅ ๏Šฉ , find all the possible values of ๐‘ that satisfy the mean value theorem over the interval [ โˆ’ 2 , 2 ] .

  • A ๐‘ฅ = โˆ’ 2 โˆš 3
  • B ๐‘ฅ = 2 โˆš 3
  • C ๐‘ฅ = 2 , ๐‘ฅ = โˆ’ 2
  • D ๐‘ฅ = 2 โˆš 3 , ๐‘ฅ = โˆ’ 2 โˆš 3
  • E ๐‘ฅ = 0 , ๐‘ฅ = 2

Q12:

For the function ๐‘“ ( ๐‘ฅ ) = ( ๐‘ฅ โˆ’ 1 ) ๏Šฎ , find all the possible values of ๐‘ that satisfy the mean value theorem over the interval [ 0 , 2 ] .

Q13:

A rock is dropped from a height of 81 ft. Its position ๐‘ก seconds after it is dropped until it hits the ground is given by the function ๐‘  ( ๐‘ก ) = โˆ’ 1 6 ๐‘ก + 8 1 ๏Šจ .

Determine how long it will take for the rock to hit the ground.

  • A 8 1 1 6 s
  • B 0 s
  • C 9 2 โˆš 2 s
  • D 9 4 s
  • E 3 2 s

Find the average velocity, ๐‘ฃ a v g , of the rock from the point of release until it hits the ground.

Find the time ๐‘ก according to the mean value theorem when the instantaneous velocity of the rock is ๐‘ฃ a v g .

  • A 4 9 s
  • B 8 9 s
  • C 9 8 s
  • D 0 s
  • E 9 4 s

Q14:

Does the mean value theorem apply for the function ๐‘ฆ = ๐‘ฅ ๐‘ฅ + 1 c o s ๏Ž„ ๏Šจ over the interval [ 0 , 1 ] ?

  • AYes
  • BNo

Q15:

Does the mean value theorem apply for the function ๐‘ฆ = 2 3 ๐œ‹ ๐‘ฅ t a n over the interval [ 1 , 3 ] ?

  • ANo
  • BYes

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