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All courses Math III · F-IF.6 23 of 55
Calculate, estimate, and interpret average rate of change for advanced function types.

Find the average rate of change of a function on an interval

Problem
Find the average rate of change of f(x)=x³ on [1,3].
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Hint

Use the average rate of change formula on the interval [1,3]: \(\frac{f(3)-f(1)}{3-1}\). Start by finding \(f(3)\) and \(f(1)\) for \(f(x)=x^3\).

Average rate of change over \([a,b]\) is the slope of the secant line: \(\frac{f(b)-f(a)}{b-a}\).

Solution walkthrough

01

Write the average rate of change formula

\[\frac{f(3)-f(1)}{3-1}\]

Use change in output over change in input on the interval \([1,3]\).

02

Evaluate the function at the endpoints

\[f(3)~=~3^3~=~27~\quad~\text{and}~\quad~f(1)~=~1^3~=~1\]

Plug the interval endpoints into \(f(x)=x^3\).

03

Substitute and simplify

\[\frac{27-1}{3-1}~=~\frac{26}{2}~=~13\]

Subtract the function values, then divide by the difference in x-values.

04

State the result

\[\text{Average rate of change}~=~13\]

So the average rate of change over \([1,3]\) is 13.

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Another way

  1. You can view this as the slope between the points \((1,1)\) and \((3,27)\) on the graph of \(y=x^3\).

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Common mistake

A common mistake is to divide by 3 instead of \(3-1\). The denominator must be the change in x-values.