Question:
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.
My attempt:
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
$$A'(24)(24.2-24)$$
which equals $9.6 \pi$.
I'm just double-checking my solutions before showing it to my calculus students tomorrow.
Thanks,
Best Answer
Being a physicist, even if your solution is very correct, instead of using linear approximation, I suggest you use differentials (as written in the title of your post).
Let us start with a modified version of your example $$A=k\, r^n\implies \log(A)=\log(k)+n\log(r)$$ Differentiate $$\frac{dA}{A}=n\frac{dr}{r}\implies\frac{|\Delta A|}{A}=n\frac{|\Delta r|}{r}\implies|{\Delta A}|=n k r^{n-1} {|\Delta r|}$$