Problem

For \(x=0,1,2,3\) and \(y=3,6,12,25\), use successive ratios to fit \(y=a\cdot~b^x\) informally. State \(a\) and \(b\), give the model and predictions, calculate residuals, and report the maximum absolute residual.

Source modeling plot: For x=0,1,2,3, data y=3,6,12,25. Fit y=a·b^x informally. No fitted model or assessment is annotated.

Big Picture

What this problem is really about

For equal input steps, exponential behavior shows up through nearly constant output ratios rather than constant differences. Estimate the common factor, use the output at zero as the initial value, and test the resulting model by listing predictions and observed-minus-predicted residuals.

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ratios=2,2,25/12≈2.083; b≈2; a=y(0)=3; model y=3·2ˣ; predictions=3,6,12,24; residuals=0,0,0,1; max absolute residual=1

ratios=2,2,25/12≈2.083; b≈3; a=y(0)=3; model y=3·3ˣ; predictions=3,9,27,81; residuals=0,−3,−15,−56; max absolute residual=56

ratios=2,2,25/12≈2.083; b≈1; a=y(0)=3; model y=3·1ˣ; predictions=3,3,3,3; residuals=0,3,9,22; max absolute residual=22

ratios=2,2,25/12≈2.083; b≈0.5; a=y(0)=3; model y=3·0.5ˣ; predictions=3,1.5,0.75,0.375; residuals=0,4.5,11.25,24.625; max absolute residual=24.625

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For \(x=0,1,2,3\) and \(y=3,6,12,25\), use successive ratios to fit \(y=a\cdot~b^x\) informally. State \(a\) and \(b\), give the model and predictions, calculate residuals, and report the maximum absolute residual.

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