Problem

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

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

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I₁ row: endpoints (1,0),(e,1), width e, rate 1/e. I₂ row: endpoints (e,1),(e²,2), width e, rate 1/e. Comparison: equal. Difference: 0.

I₁ row: endpoints (1,0),(e,1), width e−1, rate 1/(e−1). I₂ row: endpoints (e,1),(e²,2), width e²−e, rate 1/(e²−e). Comparison: I₁ greater. Difference: 1/(e−1)−1/(e²−e).

I₁ row: endpoints (1,0),(e,1), output change 1, rate omitted. I₂ row: endpoints (e,1),(e²,2), output change 1, rate omitted. Comparison: equal. Exact rate difference: not computed.

I₁ row: endpoints (1,0),(e,1), width e−1, rate 1/(e−1). I₂ row: endpoints (e,1),(e²,2), width e²−e, rate 1/(e²−e). Comparison: I₂ greater. Difference: reversed.

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

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