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Worksheet: Average Rates of Change and Secant Lines

In this worksheet, we will practice finding the average rates of change and secant lines, which are the equivalent geometric definition of the average rates of change.

Q1:

Evaluate the average rate of change for the function 𝑓 ( π‘₯ ) = βˆ’ 7 π‘₯ βˆ’ 3 π‘₯ + 3 2 when π‘₯ changes from 1 to 1.5.

Q2:

The height, in feet, of a projectile as a function of time, in seconds, is given by 𝑠 ( 𝑑 ) = βˆ’ 1 6 𝑑 + 9 2 𝑑 2 . Find the average rate of change of height with respect to time between 1 and 1.5 seconds.

Q3:

A triangular lamina with a base twice its height expands while maintaining its shape. Find the average rate of change in its area when its height changes from 14 cm to 23 cm.

Q4:

The graph below is of the sales revenue 𝑓 ( 𝑑 ) in millions of pounds after 𝑑 months. Find the average rate of change in revenue per month between the 9 t h and 1 2 t h months.

Q5:

Determine the average rate of change function 𝐴 ( β„Ž ) for 𝑓 ( π‘₯ ) = 5 π‘₯ + 2 5 π‘₯ when π‘₯ changes from π‘₯ 1 to π‘₯ + β„Ž 1 .

  • A 2 5 π‘₯ βˆ’ 2 5 π‘₯ 2 1 2 1
  • B 2 5 π‘₯ β„Ž + 2 5 π‘₯ β„Ž βˆ’ 2 β„Ž 5 π‘₯ + 5 π‘₯ β„Ž 2 1 1 2 2 1 1
  • C 2 5 π‘₯ + 5 0 π‘₯ β„Ž + 2 5 β„Ž + 2 5 π‘₯ + 5 π‘₯ β„Ž 2 1 1 2 2 1 1
  • D 2 5 π‘₯ + 2 5 π‘₯ β„Ž βˆ’ 2 5 π‘₯ + 5 π‘₯ β„Ž 2 1 1 2 1 1

Q6:

From the graph of shown, on what interval is the average rate of change of the greatest?

  • A
  • B
  • C
  • D

Q7:

Determine the average rate of change for 𝑓 ( π‘₯ ) = 6 π‘₯ βˆ’ 8 2 when π‘₯ changes from 8 to 8.4.

Q8:

Determine the average rate of change function 𝐴 ( β„Ž ) for 𝑓 ( π‘₯ ) = 4 π‘₯ + 3 π‘₯ + 2 2 at π‘₯ = 1 .

  • A 𝐴 ( β„Ž ) = βˆ’ 4 β„Ž βˆ’ 1 1
  • B 𝐴 ( β„Ž ) = 4 β„Ž + 1 1 β„Ž 2
  • C 𝐴 ( β„Ž ) = βˆ’ 4 β„Ž + 1 1
  • D 𝐴 ( β„Ž ) = 4 β„Ž + 1 1

Q9:

A spherical balloon preserves that shape as it expands. Determine the average rate of change of its surface area when its radius changes from 49 cm to 119 cm.

  • A 3 9 2 πœ‹
  • B 3 6 4 πœ‹
  • C 3 3 6 πœ‹
  • D 6 7 2 πœ‹
  • E βˆ’ 3 6 4 πœ‹

Q10:

Determine the average rate of change function for at .

  • A
  • B
  • C
  • D

Q11:

Determine the average rate of change function 𝐴 ( β„Ž ) for 𝑓 ( π‘₯ ) = √ 3 π‘₯ + 1 when π‘₯ changes from π‘₯ 1 to π‘₯ + β„Ž 1 .

  • A 3 2 √ 3 π‘₯ + 1 1
  • B 3 β„Ž √ 3 π‘₯ + 3 β„Ž + 1 + √ 3 π‘₯ + 1 1 1
  • C √ 3 π‘₯ + 3 β„Ž + 1 1
  • D 3 √ 3 π‘₯ + 3 β„Ž + 1 + √ 3 π‘₯ + 1 1 1

Q12:

A farm’s production in kilograms 𝑦 as a function of the kilograms of insecticide π‘₯ is given by 𝑦 = 1 4 6 βˆ’ 4 7 3 π‘₯ + 8 . Find the average rate of change in 𝑦 when π‘₯ varies from 13 to 17.

  • A 1 2 5 9
  • B 3 4 7
  • C145
  • D 3 5 9

Q13:

Find the average rate of change function 𝐴 ( β„Ž ) of 𝑓 ( π‘₯ ) = 2 π‘₯ + 3 0 3 when π‘₯ = π‘₯ 1 .

  • A 6 π‘₯ 2 1
  • B 2 π‘₯ + 6 β„Ž π‘₯ + 6 β„Ž π‘₯ + 2 β„Ž + 3 0 3 1 2 1 2 1 3
  • C 2 β„Ž + 6 π‘₯ β„Ž + 6 π‘₯ β„Ž 3 1 2 2 1
  • D 2 β„Ž + 6 π‘₯ β„Ž + 6 π‘₯ 2 1 2 1

Q14:

A soap bubble maintains its spherical shape as it expands. Determine the average rate of change in its surface area when its radius changes from 10 cm to 12 cm.

  • A 1 7 6 πœ‹ cm2/cm
  • B 8 0 πœ‹ cm2/cm
  • C 4 0 0 πœ‹ cm2/cm
  • D 8 8 πœ‹ cm2/cm

Q15:

The distance travelled by a body in 𝑑 seconds is 𝑆 = 5 𝑑 + 3 𝑑 + 7 2 . What is the average rate of change of 𝑆 when 𝑑 changes from 9 to 13 seconds?

Q16:

Consider the function 𝑓 ( π‘₯ ) = ο€Ή π‘₯ βˆ’ 7 5 π‘₯  5 0 3 . What is the average rate of change in 𝑓 ( π‘₯ ) over the interval [ 2 , 4 ] ?

  • A47
  • B 4 7 5 0
  • C βˆ’ 4 7
  • D βˆ’ 4 7 5 0
  • E βˆ’ 7 1 2 5

Q17:

Find the average rate of change in the volume of a cube when its side length changes from 8 cm to 9.5 cm.

  • A512
  • B345.38
  • C857.38
  • D230.25

Q18:

Evaluate the average rate of change of 𝑓 ( π‘₯ ) = √ 2 π‘₯ + 9 when π‘₯ varies from π‘₯ 1 to π‘₯ + β„Ž 1 .

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

Q19:

For the function 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 1 4 π‘₯ + 7 2 , list the average rates of change of 𝑓 over the interval  3 , 3 + 1 1 0  π‘˜ , where π‘˜ = 1 , 2 , 3 , 4 , evaluated to 4 decimal places at most.

  • A3.3, 3.03, 3.003, 3.0003
  • B4.6, 4.06, 4.006, 4.0006
  • C3.6, 3.06, 3.006, 3.0006
  • D 4.3, 4.03, 4.003, 4.0003
  • E3.4, 3.04, 3.004, 3.0004

Q20:

Evaluate the average rate of change of 𝑓 ( π‘₯ ) = √ 2 π‘₯ βˆ’ 1 when π‘₯ varies from 5 to 5.62.

  • A 2 3
  • B 1 5
  • C 1 6 5
  • D 1 0 3 1

Q21:

From the graph of , determine the intervals where the average rate of change of is constant.

  • A ,
  • B ,
  • C ,
  • D ,

Q22:

The average rate of change of 𝑓 as π‘₯ varies from 2 to 2.6 is βˆ’ 1 . 6 7 . If 𝑓 ( 2 ) = βˆ’ 1 3 , what is 𝑓 ( 2 . 6 ) ?

Q23:

Let 𝑓 ( π‘₯ ) = βˆ’ 3 π‘₯ + 7 π‘₯ βˆ’ 2 2 . Compute the average rate of change function of 𝑓 as π‘₯ varies from 5 to 5.1.

Q24:

If the average rate of change of the function 𝑓 is 6.67 when π‘₯ varies from 2 to 2.3, find the change in 𝑓 .

Q25:

A metallic cube expands but preserves its shape as it is heated. What is the average rate of change of its surface area when its sides change from 69 cm to 69.7 cm?

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