Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.
Start Practicing

Worksheet: The Rate of Change as a Derivative

Q1:

If the function 𝑓 ( π‘₯ ) = 1 1 π‘₯ + 1 6 3 , find the rate-of-change function when π‘₯ = π‘₯ 1 .

  • A 1 1 π‘₯ + 3 3 β„Ž π‘₯ + 3 3 β„Ž π‘₯ + 1 1 β„Ž + 1 6 3 1 2 1 2 1 3
  • B 1 1 β„Ž + 3 3 π‘₯ β„Ž + 3 3 π‘₯ 2 1 2 1
  • C 1 1 β„Ž + 3 3 π‘₯ β„Ž + 3 3 π‘₯ β„Ž 3 1 2 2 1
  • D 3 3 π‘₯ 2 1
  • E 3 3 π‘₯ 3

Q2:

If the function 𝑓 ( π‘₯ ) = 5 π‘₯ βˆ’ 1 8 3 , find the rate-of-change function when π‘₯ = π‘₯ 1 .

  • A 5 π‘₯ + 1 5 β„Ž π‘₯ + 1 5 β„Ž π‘₯ + 5 β„Ž βˆ’ 1 8 3 1 2 1 2 1 3
  • B 5 β„Ž + 1 5 π‘₯ β„Ž + 1 5 π‘₯ 2 1 2 1
  • C 5 β„Ž + 1 5 π‘₯ β„Ž + 1 5 π‘₯ β„Ž 3 1 2 2 1
  • D 1 5 π‘₯ 2 1
  • E 1 5 π‘₯ 3

Q3:

The biomass of a bacterial culture in milligrams as a function of time in minutes is given by . What is the rate of growth of the culture when minutes?

Q4:

The biomass of a bacterial culture in milligrams as a function of time in minutes is given by . What is the rate of growth of the culture when minutes?

Q5:

The distance in meters traveled by a body in 𝑑 seconds is 𝑆 = 9 𝑑 + 5 𝑑 + 7 2 . What is the rate of change of 𝑆 with respect to 𝑑 when 𝑑 = 1 1 ?

  • A248
  • B238
  • C 1 015
  • D203

Q6:

The distance in meters traveled by a body in 𝑑 seconds is 𝑆 = 7 𝑑 + 6 𝑑 + 5 2 . What is the rate of change of 𝑆 with respect to 𝑑 when 𝑑 = 1 5 ?

  • A237
  • B225
  • C648
  • D216

Q7:

A particle moves along the curve 𝑦 = 3 π‘₯ βˆ’ 2 π‘₯ βˆ’ 6 2 . At what point is the rate of change in its 𝑦 -coordinate four times the rate of change of its π‘₯ -coordinate?

  • A ( 2 , 2 )
  • B ( 0 , βˆ’ 6 )
  • C ( βˆ’ 2 , 1 0 )
  • D ( 1 , βˆ’ 5 )

Q8:

Find the rate of change of the gradient of the tangent of function at .

  • A
  • B
  • C
  • D

Q9:

The output in mg of a chemical reaction after 𝑑 seconds is given by 𝑦 = 4 𝑑 3 . What is the rate of production of this reaction at 𝑑 = 2 seconds?

Q10:

Let 𝑓 ( π‘₯ ) = 5 + π‘Ž π‘₯ + 𝑏 π‘₯ 2 . Suppose that the change in 𝑓 ( π‘₯ ) as π‘₯ goes from βˆ’ 1 to 2 is 6 and that the rate of change of 𝑓 ( π‘₯ ) at π‘₯ = 2 is 17. Determine π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 , 𝑏 = 3
  • B π‘Ž = βˆ’ 6 , 𝑏 = 1 0
  • C π‘Ž = 6 , 𝑏 = βˆ’ 2
  • D π‘Ž = βˆ’ 3 , 𝑏 = 5

Q11:

Find the rate of change of 𝑓 ( π‘₯ ) = 5 π‘₯ + 1 7 3 when π‘₯ = 3 .

Q12:

Determine the rate of change of the function 𝑓 ( π‘₯ ) = π‘₯ + 4 8 π‘₯ + 3 when π‘₯ = π‘₯ 1 .

  • A βˆ’ 2 9 ( 8 π‘₯ + 3 ) ( 8 β„Ž + 8 π‘₯ + 3 ) 1 1
  • B β„Ž + π‘₯ + 4 8 β„Ž + 8 π‘₯ + 3 1 1
  • C βˆ’ 2 9 β„Ž ( 8 π‘₯ + 3 ) ( 8 β„Ž + 8 π‘₯ + 3 ) 1 1
  • D βˆ’ 2 9 ( 8 π‘₯ + 3 ) 1 2

Q13:

Evaluate the rate of change of 𝑓 ( π‘₯ ) = 6 π‘₯ + 7 7 π‘₯ 2 at π‘₯ = 3 .

  • A 4 7 2 1
  • B 6 1 6 3
  • C 6 1 2 1
  • D 4 7 6 3

Q14:

What is the rate of change for the function 𝑦 = βˆ’ 5 π‘₯ βˆ’ 9 ?