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Lesson Worksheet: Average and Instantaneous Rates of Change Mathematics • Higher Education

In this worksheet, we will practice finding the average rate of change of a function between two x-values and using limits to find the instantaneous rate of change.

Q1:

Evaluate the average rate of change for the function 𝑓(π‘₯)=βˆ’7π‘₯βˆ’3π‘₯+3 when π‘₯ changes from 1 to 1.5.

Q2:

A farm’s production in kilograms 𝑦 as a function of the kilograms of insecticide π‘₯ is given by 𝑦=146βˆ’473π‘₯+8. Find the average rate of change in 𝑦 when π‘₯ varies from 13 to 17.

  • A347
  • B145
  • C359
  • D1259

Q3:

Determine the average rate of change function 𝐴(β„Ž) for 𝑓(π‘₯)=6π‘₯βˆ’3 at π‘₯=1.

  • A𝐴(β„Ž)=6β„Ž+12β„ŽοŠ¨
  • B𝐴(β„Ž)=βˆ’6β„Ž+12
  • C𝐴(β„Ž)=βˆ’6β„Žβˆ’12
  • D𝐴(β„Ž)=6β„Ž+12

Q4:

Find the average rate of change function 𝐴(β„Ž) of 𝑓(π‘₯)=2π‘₯+30 when π‘₯=π‘₯.

  • A2π‘₯+6β„Žπ‘₯+6β„Žπ‘₯+2β„Ž+30
  • B6π‘₯
  • C2β„Ž+6π‘₯β„Ž+6π‘₯β„ŽοŠ©οŠ§οŠ¨οŠ¨οŠ§
  • D2β„Ž+6π‘₯β„Ž+6π‘₯

Q5:

Determine the average rate of change function 𝐴(β„Ž) for 𝑓(π‘₯)=π‘₯+28π‘₯ when π‘₯ changes from π‘₯ to π‘₯+β„ŽοŠ§.

  • Aβ„Žπ‘₯+π‘₯βˆ’28β„Žπ‘₯+8π‘₯
  • Bβˆ’β„Žπ‘₯+π‘₯+28β„Žπ‘₯+8π‘₯
  • Cβ„Žπ‘₯+π‘₯+28β„Žπ‘₯+8π‘₯
  • Dβˆ’β„Žπ‘₯+π‘₯βˆ’28β„Žπ‘₯+8π‘₯

Q6:

Suppose a population is 𝑓(𝑑)=14𝑑+33,706 as a function of time 𝑑. What is the average rate of growth of this population when 𝑑 changes from π‘‘οŠ§ to 𝑑+β„ŽοŠ§?

  • A14β„Žβˆ’28π‘‘οŠ§
  • B14β„Ž+28π‘‘β„ŽοŠ¨οŠ§
  • C14β„Žβˆ’28π‘‘β„ŽοŠ¨οŠ§
  • D14β„Ž+28π‘‘οŠ§
  • E28π‘‘οŠ§

Q7:

For a function 𝑓(π‘₯), the average rate of change between a fixed point π‘₯ and another point π‘₯+β„Ž is 𝐴(β„Ž)=𝑓(π‘₯+β„Ž)βˆ’π‘“(π‘₯)β„Ž. Given that 𝑓(π‘₯)=π‘₯βˆ’6π‘₯+5, find 𝐴(0.5) when π‘₯=4.

Q8:

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

Q9:

Find the instantaneous rate of change of 𝑓(π‘₯)=√π‘₯ at π‘₯=π‘₯>0.

  • A12√π‘₯
  • Bβˆ’1√π‘₯
  • C1√π‘₯
  • Dβˆ’12√π‘₯
  • E12√π‘₯

Q10:

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?

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