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Math Help - parametric equation

  1. #1
    Junior Member
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    parametric equation

    I don't know how to prove it...

    (a) for parametric equations x = f(t) and y = g(t), where t is a parameter, show that
    <br />
\frac{d^2y}{dx^2}\:=<br />
\frac{\frac{d}{dt}(\frac{dy}{dx})} {\frac{dx}{dt}}


    (b) Given the parametric equations

    <br />
x = \frac{t^2}{2}\ and\ y = 1-t

    find <br />
\frac{d^2y}{dx^2}\
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  2. #2
    Member Glaysher's Avatar
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    (a)

    \frac{d}{dt}(\frac{dy}{dx}) = \frac{d^2y}{dx^2}\frac{dx}{dt}

    Divide by \frac{dx}{dt} and you have shown that the right hand side is the same as the left hand side

    (b)

    Find \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}

    Differentiate with respect to t and then divide by \frac{dx}{dt}
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  3. #3
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    Quote Originally Posted by ling_c_0202
    (b) Given the parametric equations

    <br />
x = \frac{t^2}{2}\ and\ y = 1-t

    find <br />
\frac{d^2y}{dx^2}\
    \frac{dx}{dt}=t and \frac{dy}{dt}=-1
    Thus,
    \frac{dy}{dx}=\frac{-1}{t}
    ---
    To find the second, take the derivative of this derivative you just found.
    \frac{d}{dt}(-1/t)=\frac{1}{t^2}
    And,
    \frac{dx}{dt}=t
    Thus,
    \frac{d^2y}{dx^2}=\frac{1}{t^3}
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