Calculate, estimate, and interpret average rate of change for quadratic functions.
Calculate and compare average rates of change of a quadratic over intervals
Problem
For \(f(x)=x^{2}\), calculate and compare the average rates of change on \([0,2]\) and \([2,4]\). Show all endpoint values and \(\text{both}~\text{difference}~\text{quotients}\), identify the larger signed rate, and find the rate difference.
Big Picture
What this problem is really about
Comparing average rates across two intervals requires two complete secant calculations, even when the intervals share an endpoint. We’ll evaluate each distinct endpoint once, form both difference quotients in the same direction, and then compare the signed rates and quantify their separation.
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