Compare function properties across representations, especially quadratic comparisons.
Compare the average rate of change of two functions on the same interval
Problem
Compare the average rates of change of \(A(x)=x^{2}\) and \(B(x)=3x\) on \([1,4]\). Evaluate \(\text{both}~\text{functions}\) at \(1\) and \(4\), divide each output change by \(4-1\), and compare the two signed rates rather than total changes.
Big Picture
What this problem is really about
Average rates are secant slopes, so both functions must use identical interval endpoints. We’ll find each pair of endpoint outputs, divide the signed output change by the common input change, and compare the resulting rates per input unit rather than raw changes.
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