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
  • B49
  • Cβˆ’49
  • Dβˆ’1

Q5:

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

  • Aβˆ’950
  • B1710
  • C18289
  • Dβˆ’95
  • E5950

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.

  • A518
  • Bβˆ’536
  • C536
  • Dβˆ’518

Q8:

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

  • A57121
  • B57163
  • C9
  • D463
  • E5719

Q9:

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

  • A3
  • B110
  • C35
  • D65
  • E5

Q10:

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

  • A7π‘₯+14π‘₯β„Ž+7β„Ž+9
  • B14π‘₯
  • C7β„Ž+14π‘₯
  • D7β„Ž+14π‘₯β„ŽοŠ¨οŠ§

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.

  • A𝑓(1)=15, πœƒ=8611β€²βˆ˜
  • B𝑓(1)=7, πœƒ=8152β€²βˆ˜
  • C𝑓(1)=1, πœƒ=450β€²βˆ˜
  • D𝑓(1)=3, πœƒ=7134β€²βˆ˜

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?

  • Aβˆ’118πœ‹ cm2/cm
  • Bβˆ’59πœ‹ cm2/cm
  • C59πœ‹ cm2/cm
  • D118π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.

  • A𝑙=3+9𝑑
  • B𝑙=3𝑑
  • C𝑙=9+3𝑑
  • D𝑙=9𝑑

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.

  • A4763
  • B6163
  • C4721
  • D6121

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.

  • Aπ‘Ž=βˆ’6, 𝑏=10
  • Bπ‘Ž=6, 𝑏=βˆ’2
  • Cπ‘Ž=βˆ’1, 𝑏=3
  • Dπ‘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’2,10)

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.

  • A2√5
  • B3√520
  • C3√5

Q25:

Find the slope of the tangent to the curve 𝑦=95π‘₯1βˆ’5π‘₯tantan at π‘₯=βˆ’πœ‹6.

  • A180
  • B225
  • C45
  • D90

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