# Thread: How to apply differentiation here?

1. ## How to apply differentiation here?

Under ideal conditions, a perfect gas satisfies the equation PV=K ( where P is pressure, V is

volume and K is a constant ). If K=60 and pressure is found by measurement to be 1.5 units

with an error of 5% per unit,find approximately the error in calculating V.

2. ## Re: How to apply differentiation here?

Originally Posted by Vinod
Under ideal conditions, a perfect gas satisfies the equation PV=K ( where P is pressure, V is

volume and K is a constant ). If K=60 and pressure is found by measurement to be 1.5 units

with an error of 5% per unit,find approximately the error in calculating V.
The error is given by the exact change in volume divided by the volume.

$\displaystyle \frac{\Delta V}{V}$

This, in general, is hard to calculate so we approximate it with

$\displaystyle \frac{\Delta V}{V} \approx \frac{dV}{V}$

Now if we take the derivative using the product rule we get

$\displaystyle PdV+VdP=0 \iff dV=-\frac{VdP}{P}$ plugging this into the above gives

$\displaystyle \frac{\Delta V}{V} \approx \frac{dV}{V}=-\frac{\frac{Vdp}{P}}{V}=-\frac{dP}{P}=-\frac{.05}{1.5}=-\frac{1}{30}$

Since the 5% can be positive or negative we get that the approximate error is

$\displaystyle \pm \frac{1}{30} \approx \pm 3.\bar{3}\%$