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Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values x 1 and x 2 .

  1. Calculate the difference y 2 y 1 = Δ y .
  2. Calculate the difference x 2 x 1 = Δ x .
  3. Find the ratio Δ y Δ x .

Computing an average rate of change

Using the data in [link] , find the average rate of change of the price of gasoline between 2007 and 2009.

In 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is

Δ y Δ x = y 2 y 1 x 2 x 1 = $ 2.41 $ 2.84 2009 2007 = $ 0.43 2  years = $ 0.22  per year
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Using the data in [link] , find the average rate of change between 2005 and 2010.

$ 2.84 $ 2.31 5  years = $ 0.53 5  years = $ 0.106 per year.

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Computing average rate of change from a graph

Given the function g ( t ) shown in [link] , find the average rate of change on the interval [ 1 , 2 ] .

Graph of a parabola.

At t = 1 , [link] shows g ( −1 ) = 4. At t = 2 , the graph shows g ( 2 ) = 1.

Graph of a parabola with a line from points (-1, 4) and (2, 1) to show the changes for g(t) and t.

The horizontal change Δ t = 3 is shown by the red arrow, and the vertical change Δ g ( t ) = 3 is shown by the turquoise arrow. The output changes by –3 while the input changes by 3, giving an average rate of change of

1 4 2 ( 1 ) = 3 3 = −1
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Computing average rate of change from a table

After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in [link] . Find her average speed over the first 6 hours.

t (hours) 0 1 2 3 4 5 6 7
D ( t ) (miles) 10 55 90 153 214 240 282 300

Here, the average speed is the average rate of change. She traveled 282 miles in 6 hours, for an average speed of

292 10 6 0 = 282 6 = 47

The average speed is 47 miles per hour.

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Computing average rate of change for a function expressed as a formula

Compute the average rate of change of f ( x ) = x 2 1 x on the interval [2, 4].

We can start by computing the function values at each endpoint of the interval.

f ( 2 ) = 2 2 1 2 f ( 4 ) = 4 2 1 4 = 4 1 2 = 16 1 4 = 7 2 = 63 4

Now we compute the average rate of change.

Average rate of change = f ( 4 ) f ( 2 ) 4 2                                       = 63 4 7 2 4 2                                      = 49 4 2                                      = 49 8
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Find the average rate of change of f ( x ) = x 2 x on the interval [ 1 , 9 ] .

1 2

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Finding the average rate of change of a force

The electrostatic force F , measured in newtons, between two charged particles can be related to the distance between the particles d , in centimeters, by the formula F ( d ) = 2 d 2 . Find the average rate of change of force if the distance between the particles is increased from 2 cm to 6 cm.

We are computing the average rate of change of F ( d ) = 2 d 2 on the interval [ 2 , 6 ] .

Average rate of change  = F ( 6 ) F ( 2 ) 6 2 = 2 6 2 2 2 2 6 2 Simplify . = 2 36 2 4 4 = 16 36 4 Combine numerator terms . = 1 9 Simplify

The average rate of change is 1 9 newton per centimeter.

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Finding an average rate of change as an expression

Find the average rate of change of g ( t ) = t 2 + 3 t + 1 on the interval [ 0 , a ] . The answer will be an expression involving a .

We use the average rate of change formula.

Average rate of change = g ( a ) g ( 0 ) a 0 Evaluate .                                      = ( a 2 + 3 a + 1 ) ( 0 2 + 3 ( 0 ) + 1 ) a 0 Simplify .                                      = a 2 + 3 a + 1 1 a Simplify and factor .                                      = a ( a + 3 ) a   Divide by the common factor  a .                                      = a + 3

This result tells us the average rate of change in terms of a between t = 0 and any other point t = a . For example, on the interval [ 0 , 5 ] , the average rate of change would be 5 + 3 = 8.

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Questions & Answers

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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