Q.The radius of a circular disc is given as 24 cm. with a maximum error in measurement of 0.02 cm. Estimate the maximum error in the calculated area of the disc and compute the relative error by using differentials.

When a quantity is approximated instead of computed exactly, the mismatch between the actual and approximate values is measured by three related notions of error, all built on linear approximation (equation $f(x_0+\Delta x)-f(x_0)\approx f'(x_0)\Delta x$).

Definition (Absolute Error).

If $h=x-x_0$ is the change in the input and $E(h)=f(x_0+h)-f(x_0)-f'(x_0)h$, then $E(h)$ is the absolute error made by using the linear approximation in place of the true value; $E(0)=0$ and $\displaystyle\lim_{h\to0}\frac{E(h)}{h}=0$ whenever $f$ is differentiable at $x_0$, i.e. the error vanishes faster than $h$ itself.

Absolute error alone doesn't say whether an approximation is good — an absolute error of $5$ is trivial if the actual value is $10{,}000$ but huge if the actual value is $6$. This is fixed by dividing out the actual value:

Definition (Relative and Percentage Error). If the actual value is nonzero,

Working method for an error problem. Given a quantity $Q=f(x)$ measured via $x$, and a measurement/change $dx$ in $x$:

1. Absolute error in $Q$ $\approx dQ = f'(x)\,dx$ (evaluated at the reference value of $x$).
2. Relative error $\approx \dfrac{dQ}{Q} = \dfrac{f'(x)}{f(x)}\,dx$.
3. Percentage error $= 100\times$ relative error. …