Question

The relative change of \(f\) is \(\Delta f / f\)

For example, the relative change in the volume of a square cube with edges of length \(x\), when the change in the lengths of the cube is \(\Delta x\)

\(\frac{\Delta V}{V} = \frac{3 x^2 \Delta x}{x^3} = \frac{3 \Delta x}{x}\)

Consider a sphere will volume \(V = \frac{4}{3} \pi r^3\)

If \(r\) changes by \(\Delta r\), then \(\Delta V \simeq\) ? \(r^2 \Delta r\)

The relative change is approximately ? \(\frac{\Delta r}{r}\)

Answer 1

\[\frac{\Delta V}{V} = \frac{3 \times (4/3) \pi r^2 \Delta r}{(4/3) \pi r^3} = \frac{3 \Delta r}{r}\]