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Math Help - How to apply differentiation here?

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    Post 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.
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    Re: How to apply differentiation here?

    Quote Originally Posted by Vinod View Post
    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.

    \frac{\Delta V}{V}

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

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

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

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

    \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

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