The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. Use differentials to estimate the maximum error in the calculated area of the disk.
Look at the linear approximation of $A(r) = \pi r^2$ near $r=24$.
We have that
$$A(r) \approx A(24) + A'(24)(x-24)$$
and maximizing the error in measurement of the radius gives
$$A(r) \approx A(24) + A'(24)(24.2-24)$$
and we read off the differential term
which equals $9.6 \pi$.
I'm just double-checking my solutions before showing it to my calculus students tomorrow.