Worksheet: Rate of Change and Derivatives

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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‘₯)=2π‘₯+9 at π‘₯=βˆ’3.

Q2:

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

Q3:

Find the rate of change of 5π‘₯βˆ’18 with respect to π‘₯ when π‘₯=2.

Q4:

Evaluate the rate of change of 𝑓(π‘₯)=βˆ’94π‘₯βˆ’7 at π‘₯=4.

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

Q5:

If the function 𝑓(π‘₯)=5π‘₯+74π‘₯+2, determine its rate of change when π‘₯=2.

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

Q6:

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

Q7:

Determine the rate of change of the function 𝑓(π‘₯)=59π‘₯ at π‘₯=√2.

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

Q8:

Evaluate the rate of change of 𝑓(π‘₯)=9π‘₯βˆ’47π‘₯ at π‘₯=3.

  • A 5 7 1 2 1
  • B 5 7 1 6 3
  • C9
  • D 4 6 3
  • E 5 7 1 9

Q9:

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

  • A3
  • B 1 1 0
  • C 3 5
  • D 6 5
  • E5

Q10:

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

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

Q11:

Find the rate of change of 𝑓(π‘₯)=4π‘₯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 5  , πœƒ = 8 6 1 1 β€² ∘
  • B 𝑓 ( 1 ) = 7  , πœƒ = 8 1 5 2 β€² ∘
  • C 𝑓 ( 1 ) = 1  , πœƒ = 4 5 0 β€² ∘
  • D 𝑓 ( 1 ) = 3  , πœƒ = 7 1 3 4 β€² ∘

Q12:

A circular disk 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 βˆ’ 1 1 8 πœ‹ cm2/cm
  • B βˆ’ 5 9 πœ‹ cm2/cm
  • C 5 9 πœ‹ cm2/cm
  • D 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 + 9 𝑑
  • B 𝑙 = 3 𝑑
  • C 𝑙 = 9 + 3 𝑑
  • D 𝑙 = 9 𝑑

Q14:

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

Q15:

Evaluate the rate of change of 𝑓(π‘₯)=6π‘₯+77π‘₯ at π‘₯=3.

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

Q16:

The biomass of a bacterial culture in milligrams as a function of time in minutes is given by 𝑓(𝑑)=71𝑑+63. What is the rate of growth of the culture when 𝑑=2minutes?

Q17:

Let 𝑓(π‘₯)=5+π‘Žπ‘₯+𝑏π‘₯. 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’ 6 , 𝑏 = 1 0
  • B π‘Ž = 6 , 𝑏 = βˆ’ 2
  • C π‘Ž = βˆ’ 1 , 𝑏 = 3
  • D π‘Ž = βˆ’ 3 , 𝑏 = 5

Q18:

A particle moves along the curve 𝑦=3π‘₯βˆ’2π‘₯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 ( 1 , βˆ’ 5 )
  • D ( βˆ’ 2 , 1 0 )

Q19:

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

Q20:

Find the rate of change of 𝑓(π‘₯)=5π‘₯+17 when π‘₯=3.

Q21:

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

Q22:

The distance in meters traveled by a body in 𝑑 seconds is 𝑆=9𝑑+5𝑑+7. What is the rate of change of 𝑆 with respect to 𝑑 when 𝑑=11?

Q23:

A population’s size after 𝑑 days is given by 𝑓(𝑑)=11𝑑+35,923. Find the rate of change in the population when 𝑑=12.

Q24:

Evaluate the rate of change of 𝑓(π‘₯)=√3π‘₯βˆ’1 at π‘₯=7.

  • A 2 √ 5
  • B 3 √ 5 2 0
  • C 3 √ 5

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