Question

using derivative , find the approximate percentage increase in the area of the circle if its radius is increased by 2%

Answer 1

Have you considered the differential form of the first derivative?

\(A(r) = \pir^{2} \implies dA = 2\pi r dr\)

Answer 2

I would first find the change in terms of a fraction: amount of change divided by original area
\[ \frac{\Delta A}{\pi r^2} \]
As tk shows, by implicit differentiation
\[ dA = 2\pi r \ dr \]
replace the differentials with delta's (small but measurable quantities, to get the approximation:
\[ \Delta A \approx 2\pi r \ \Delta r \]

as shown above we want the fraction so divide both sides by pi r^2 (i.e. the original area)
\[ \frac{\Delta A}{\pi r^2} = \frac{2 \pi r \Delta r }{\pi r^2} \]
the right side simplifies to
\[ \frac{\Delta A}{\pi r^2} \approx 2\frac{ \Delta r }{ r} \]

the \(\Delta r\) /r is the fractional change i.e. 0.02
and thus we get the fractional change for the area:
\[ \frac{\Delta A}{\pi r^2} \approx 2\cdot 0.02 = 0.04 = 4% \]

Answer 3

* = 4%

Answer 4

... as tk showed poorly with bad LaTex. (he said sheepishly)