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Worksheet: Moving from Average Rates of Change to Instantaneous Rates of Change

Q1:

What is the average rate of change function 𝐴 ( β„Ž ) for the function 𝑓 ( π‘₯ ) = 7 5 π‘₯ ?

  • A βˆ’ 7 βˆ’ 5 β„Ž π‘₯ + 5 π‘₯ 2
  • B 7 5 β„Ž π‘₯ + 5 π‘₯ 2
  • C 7 βˆ’ 5 β„Ž π‘₯ + 5 π‘₯ 2
  • D βˆ’ 7 5 β„Ž π‘₯ + 5 π‘₯ 2

Q2:

Determine the average rate of change function 𝐴 ( β„Ž ) for 𝑓 ( π‘₯ ) = π‘₯ + 2 8 π‘₯ 2 when π‘₯ changes from π‘₯ 1 to π‘₯ + β„Ž 1 .

  • A βˆ’ β„Ž π‘₯ + π‘₯ + 2 8 β„Ž π‘₯ + 8 π‘₯ 1 2 1 1 2 1
  • B β„Ž π‘₯ + π‘₯ + 2 8 β„Ž π‘₯ + 8 π‘₯ 1 2 1 1 2 1
  • C βˆ’ β„Ž π‘₯ + π‘₯ βˆ’ 2 8 β„Ž π‘₯ + 8 π‘₯ 1 2 1 1 2 1
  • D β„Ž π‘₯ + π‘₯ βˆ’ 2 8 β„Ž π‘₯ + 8 π‘₯ 1 2 1 1 2 1

Q3:

Find the average rate of change of 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 6 π‘₯ + 2 2 when π‘₯ varies from 1.3 to 2.

Q4:

For a function 𝑓 ( π‘₯ ) , the rate of change at a fixed π‘₯ is 𝐴 ( β„Ž ) = 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž . Given that 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 6 π‘₯ + 5 2 , find 𝐴 ( 0 . 5 ) when π‘₯ = 4 .

Q5:

The average rate of change of a function 𝑓 between π‘₯ and π‘₯ + β„Ž is 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž . Compute this quantity for 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 8 2 at π‘₯ = βˆ’ 4 , and for β„Ž = 0 . 3 .

Q6:

If the function 𝑓 ( π‘₯ ) = βˆ’ 3 π‘₯ βˆ’ 5 9 , find l i m β„Ž β†’ 0 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž .

  • A βˆ’ 2 7
  • B βˆ’ 2 7 π‘₯ 9
  • C βˆ’ 3 π‘₯ 9
  • D βˆ’ 2 7 π‘₯ 8
  • E βˆ’ 2 7 π‘₯ βˆ’ 5 β„Ž 8

Q7:

If the function 𝑓 ( π‘₯ ) = 3 π‘₯ + 7 3 , find l i m β„Ž β†’ 0 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž .

  • A9
  • B 9 π‘₯ 3
  • C 3 π‘₯ 3
  • D 9 π‘₯ 2
  • E 9 π‘₯ + 7 β„Ž 2

Q8:

If the function 𝑓 ( π‘₯ ) = βˆ’ π‘₯ + 3 9 , find l i m β„Ž β†’ 0 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž .

  • A βˆ’ 9
  • B βˆ’ 9 π‘₯ 9
  • C βˆ’ π‘₯ 9
  • D βˆ’ 9 π‘₯ 8
  • E βˆ’ 9 π‘₯ + 3 β„Ž 8

Q9:

If the function 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 2 6 , find l i m β„Ž β†’ 0 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž .

  • A12
  • B 1 2 π‘₯ 6
  • C 2 π‘₯ 6
  • D 1 2 π‘₯ 5
  • E 1 2 π‘₯ βˆ’ 2 β„Ž 5

Q10:

If the function 𝑓 ( π‘₯ ) = 4 π‘₯ + 1 0 7 , find l i m β„Ž β†’ 0 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž .

  • A28
  • B 2 8 π‘₯ 7
  • C 4 π‘₯ 7
  • D 2 8 π‘₯ 6
  • E 2 8 π‘₯ + 1 0 β„Ž 6

Q11:

If the function 𝑓 ( π‘₯ ) = π‘₯ + 1 6 4 , find l i m β„Ž β†’ 0 𝑓 ( π‘₯ + β„Ž ) βˆ’ 𝑓 ( π‘₯ ) β„Ž .

  • A4
  • B 4 π‘₯ 4
  • C π‘₯ 4
  • D 4 π‘₯ 3
  • E 4 π‘₯ + 1 6 β„Ž 3

Q12:

Consider the average rate of change of the function 𝑓 ( π‘₯ ) = 1 π‘₯ over the interval [ 3 , 3 + β„Ž ] with small values of β„Ž .

Simplify the expression 𝑓 ( 3 + β„Ž ) βˆ’ 𝑓 ( 3 ) ( 3 + β„Ž ) βˆ’ 3 .

  • A βˆ’ 1 β„Ž + 9
  • B 1 3 β„Ž + 9
  • C 1 β„Ž + 9
  • D βˆ’ 1 3 β„Ž + 9
  • E βˆ’ 1 3 β„Ž + 3

The average rate of change gets closer and closer to βˆ’ 1 9 as β„Ž becomes smaller and smaller. Simplify the expression for the difference 𝛿 ( β„Ž ) between 𝑓 ( 3 + β„Ž ) βˆ’ 𝑓 ( 3 ) ( 3 + β„Ž ) βˆ’ 3 and βˆ’ 1 9 .

  • A β„Ž 9 β„Ž + 2 7
  • B βˆ’ β„Ž 9 β„Ž + 2 7
  • C β„Ž β„Ž + 2 7
  • D βˆ’ 6 + β„Ž 9 β„Ž + 2 7
  • E β„Ž 9 β„Ž βˆ’ 2 7

For what values of β„Ž is the difference 𝛿 ( β„Ž ) exactly 1 1 0 , 1 1 0 0 , 1 1 0 4 . Give your answer as a fraction.

  • A βˆ’ 2 7 , βˆ’ 2 7 9 1 , 2 7 9 9 9 1
  • B 2 7 1 9 , 1 0 9 2 7 , 1 0 0 0 9 2 7
  • C27, 2 7 9 1 , 2 7 9 9 9 1
  • D 1 2 7 , 9 1 2 7 , 9 9 9 1 2 7
  • E27, 2 7 1 0 9 , 2 7 1 0 0 0 9