Problem

For \(f(x)~=~\sqrt{x}\), compute complete average-rate rows on \(I_{1}~=~[0,~1]\) and \(I_{2}~=~[4,~9]\), including endpoints, interval widths, and rates; compare the rates and give their exact difference.

Prompt graph of f(x)=√x with both specified interval secants and endpoint values.

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I₁ row: endpoints (0,0),(1,1), width 1, rate 1. I₂ row: endpoints (4,2),(9,3), width 13, rate 1/13. Comparison: I₁ greater. Exact difference: 12/13.

I₁ row: endpoints (0,0),(1,1), output change 1, rate omitted. I₂ row: endpoints (4,2),(9,3), output change 1, rate omitted. Comparison: equal. Exact rate difference: not computed.

I₁ row: endpoints (0,0),(1,1), width 1, rate 1. I₂ row: endpoints (4,2),(9,3), width 5, rate 1/5. Comparison: I₂ greater. Exact difference: 1/5−1=−4/5.

I₁ row: endpoints (0,0),(1,1), width 1, rate 1. I₂ row: endpoints (4,2),(9,3), width 5, rate 1/5. Comparison: I₁ greater. Exact difference: 1−1/5=4/5.

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For \(f(x)~=~\sqrt{x}\), compute complete average-rate rows on \(I_{1}~=~[0,~1]\) and \(I_{2}~=~[4,~9]\), including endpoints, interval widths, and rates; compare the rates and give their exact difference.

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