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Lesson Worksheet: Rate of Change and Derivatives Mathematics • Higher Education

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 instantaneous 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§

Q2:

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

  • A4
  • B49
  • Cβˆ’49
  • Dβˆ’1

Q3:

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

  • A3
  • B110
  • C35
  • D65
  • E5

Q4:

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

  • A4763
  • B6163
  • C4721
  • D6121

Q5:

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 instantaneous rate of growth of the culture when 𝑑=2minutes?

Q6:

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

Q7:

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?

  • A118πœ‹ cm2/cm
  • Bβˆ’59πœ‹ cm2/cm
  • C59πœ‹ cm2/cm
  • Dβˆ’118πœ‹ cm2/cm

Q8:

Let 𝑓(π‘₯)=π‘Žπ‘₯+π‘οŠ¨ with constants π‘Ž, 𝑏. Determine the constants for which the slope of the tangent to the curve 𝑦=𝑓(π‘₯) at (βˆ’2,29) is βˆ’24.

  • Aπ‘Ž=12, 𝑏=βˆ’19
  • Bπ‘Ž=6, 𝑏=5
  • Cπ‘Ž=βˆ’3, 𝑏=41

Q9:

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

Q10:

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

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

This lesson includes 46 additional questions and 441 additional question variations for subscribers.

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