Problem

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

Prompt graph of f(x)=x² with both specified interval secants and endpoint values.

Big Picture

What this problem is really about

Build one complete secant row for each interval before comparing anything: endpoint values, input width, output change, and quotient. Normalizing by each interval's own width makes the rates comparable even when raw changes differ. Then compare signed values and subtract in greater-minus-less order to report the exact rate gap.

Answer choices

Choose your path.

Preview only

I₁ row: endpoints (0,0),(2,4), output change 4, rate omitted. I₂ row: endpoints (2,4),(4,16), output change 12, rate omitted. Comparison: I₂ greater. Exact rate difference: not computed.

I₁ row: endpoints (0,0),(2,4), width 4, rate 1. I₂ row: endpoints (2,4),(4,16), width 4, rate 3. Comparison: I₂ greater. Exact difference: 3−1=2.

I₁ row: endpoints (0,0),(2,4), width 2, rate 2. I₂ row: endpoints (2,4),(4,16), width 2, rate 6. Comparison: I₁ greater. Exact difference: 2−6=−4.

I₁ row: endpoints (0,0),(2,4), width 2, rate 2. I₂ row: endpoints (2,4),(4,16), width 2, rate 6. Comparison: I₂ greater. Exact difference: 6−2=4.

The choices are visible. Checking your work and seeing which answer is correct happen inside Gozunta.

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

What this problem is really about

Turn the preview into practice.

Gozunta lets you study this problem—and more than 10,000 others.