Problem

For \(f(x)=x^2\), compare the average rates of change over \([0,2]\) and \([2,4]\).

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What this problem is really about

The same nonlinear function can have different average rates on different intervals. We’ll evaluate every endpoint needed, build a separate output-change-over-input-change quotient for each interval, and compare the resulting rates only after both are simplified. Equal interval widths make the comparison especially clear, but they do not eliminate the need to divide by the input change.

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The average rates are equal because the function is x² on both intervals, so the rate of change stays the same.

On [0,2] the rate is 4 and on [2,4] the rate is 12, so [2,4] is greater because you only subtract the y-values.

On [0,2] the average rate of change is greater, because starting closer to 0 makes the graph rise faster there.

On [0,2] the average rate of change is 2 and on [2,4] it is 6, so [2,4] has the greater average rate.

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For \(f(x)=x^2\), compare the average rates of change over \([0,2]\) and \([2,4]\).

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