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All courses Algebra II · F-IF.6 30 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 \(f(x)~=~x^3\) 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 secant-slope formula

\[\text{average}~\text{rate}=(f(3)-f(1))/(3-1)\]

Average rate of change on [a,b] is the change in output divided by the change in input. Here a=1 and b=3 come from the stated interval.

02

Evaluate both endpoints

\[f(3)=3^3=27;~f(1)=1^3=1\]

Substitute the interval endpoints into f(x)=x³ before forming the differences.

03

Compute and interpret the slope

\[(27-1)/(3-1)=26/2=13\]

The output rises 26 while the input rises 2, giving 13 units of output per input unit.

04

State the answer

\[13\]

The average rate of change of x cubed on [1,3] is 13, choice A.

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Another way

  1. Use the secant line through the points (1,1) and (3,27).

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Common mistake

Keep the endpoint order consistent in numerator and denominator; reversing only one difference changes the sign.

Solution walkthrough video