Worksheet: Rate of Change and Derivatives

In this worksheet, we will practice finding the instantaneous rate of change for a function using derivatives and applying this in real-world problems.

Q1:

Evaluate the rate of change of 𝑓 ( π‘₯ ) = 2 π‘₯ + 9 2 at π‘₯ = βˆ’ 3 .

Q2:

Evaluate the rate of change of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 3 π‘₯ + 2 2 at π‘₯ = 5 .

Q3:

Find the rate of change of 5 π‘₯ βˆ’ 1 8 3 with respect to π‘₯ when π‘₯ = 2 .

Q4:

Evaluate the rate of change of 𝑓 ( π‘₯ ) = βˆ’ 9 4 π‘₯ βˆ’ 7 at π‘₯ = 4 .

  • A4
  • B βˆ’ 1
  • C βˆ’ 4 9
  • D 4 9

Q5:

If the function 𝑓 ( π‘₯ ) = 5 π‘₯ + 7 4 π‘₯ + 2 , determine its rate of change when π‘₯ = 2 .

  • A 5 9 5 0
  • B 1 7 1 0
  • C βˆ’ 9 5
  • D βˆ’ 9 5 0
  • E 1 8 2 8 9

Q6:

What is the rate of change for the function 𝑦 = 4 π‘₯ + 7 ?

Q7:

Determine the rate of change of the function 𝑓 ( π‘₯ ) = 5 9 π‘₯ at π‘₯ = √ 2 .

  • A βˆ’ 5 3 6
  • B 5 1 8
  • C 5 3 6
  • D βˆ’ 5 1 8

Q8:

Evaluate the rate of change of 𝑓 ( π‘₯ ) = 9 π‘₯ βˆ’ 4 7 π‘₯ at π‘₯ = 3 .

  • A 5 7 1 9
  • B9
  • C 5 7 1 2 1
  • D 5 7 1 6 3
  • E 4 6 3

Q9:

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

  • A3
  • B 6 5
  • C5
  • D 3 5
  • E 1 1 0

Q10:

Evaluate the rate of change of 𝑓 ( π‘₯ ) = 7 π‘₯ + 9 2 at π‘₯ = π‘₯ 1 .

  • A 7 β„Ž + 1 4 π‘₯ β„Ž 2 1
  • B 7 π‘₯ + 1 4 π‘₯ β„Ž + 7 β„Ž + 9 2 1 1 2
  • C 7 β„Ž + 1 4 π‘₯ 1
  • D 1 4 π‘₯ 1

Q11:

Find the rate of change of 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ π‘₯ βˆ’ 3  when π‘₯ = 1 and determine, to the nearest minute, the positive angle between the tangent at ( 1 , 0 ) and the positive π‘₯ -axis.

  • A 𝑓 ( 1 ) = 1  , πœƒ = 4 5 0 β€² ∘
  • B 𝑓 ( 1 ) = 3  , πœƒ = 7 1 3 4 β€² ∘
  • C 𝑓 ( 1 ) = 1 5  , πœƒ = 8 6 1 1 β€² ∘
  • D 𝑓 ( 1 ) = 7  , πœƒ = 8 1 5 2 β€² ∘

Q12:

A circular disc preserves its shape as it shrinks. What is the rate of change of its area with respect to radius when the radius is 59 cm?

  • A 5 9 πœ‹ cm2/cm
  • B 1 1 8 πœ‹ cm2/cm
  • C βˆ’ 5 9 πœ‹ cm2/cm
  • D βˆ’ 1 1 8 πœ‹ cm2/cm

Q13:

A length 𝑙 is initially 9 cm and increases at a rate of 3 cm/s. Write the length as a function of time, 𝑑 .

  • A 𝑙 = 3 𝑑
  • B 𝑙 = 9 𝑑
  • C 𝑙 = 3 + 9 𝑑
  • D 𝑙 = 9 + 3 𝑑

Q14:

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

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

Q15:

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

Q16:

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

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

Q17:

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

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

Q18:

The biomass of a bacterial culture in milligrams as a function of time in minutes is given by 𝑓 ( 𝑑 ) = 7 1 𝑑 + 6 3  . What is the rate of growth of the culture when 𝑑 = 2 m i n u t e s ?

Q19:

Let 𝑓 ( π‘₯ ) = 5 + π‘Ž π‘₯ + 𝑏 π‘₯  . 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

Q20:

A particle moves along the curve 𝑦 = 3 π‘₯ βˆ’ 2 π‘₯ βˆ’ 6  . 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 )

Q21:

Find the rate of change of the slope of the tangent of function 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 3 at π‘₯ = 8 .

Q22:

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

Q23:

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

Q24:

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

Q25:

A population’s size after 𝑑 days is given by 𝑓 ( 𝑑 ) = 1 1 𝑑 + 3 5 9 2 3 2 . Find the rate of change in the population when 𝑑 = 1 2 .

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