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Math Help - find T(t), N(t), at the given time t for the plane curve

  1. #1
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    find T(t), N(t), at the given time t for the plane curve

    find T(t), N(t), a_t ,  a_n at the given time t for the plane curve r(t)

    r(t)=  ti + \frac{1}{t} j,     t = 1

    The book gives this formula for T(t)

    T(t) =  \frac {r'(t)}{ || r'(t) ||}

    So, i find r' =  i - \frac {1}{t^2} j

    then the magnitude
     ||r'(t)|| = \sqrt{ (\frac{1^2 + 1^2}{t^2})^2} = \sqrt{ \frac{2}{t^4}}

    then, combining for T(t) the solutions manual has this which i cant seem to understand.

    T(t)  =  \frac {t^2} {\sqrt{  t^4 + 1}}  ( i - \frac{1} {\sqrt{ {t^2}}}j) = \frac{1} {\sqrt{ t^4 + 1}}(t^2i - j)

    I am completely lost!!!
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  2. #2
    Senior Member BAdhi's Avatar
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    Re: find T(t), N(t), at the given time t for the plane curve

    Quote Originally Posted by icelated View Post
    find T(t), N(t), a_t ,  a_n at the given time t for the plane curve r(t)

     ||r'(t)|| = \sqrt{ (\frac{1^2 + 1^2}{t^2})^2} = \sqrt{ \frac{2}{t^4}}
    this is wrong,

    ||r'(t)||=\sqrt{1^2+\left(\frac{1}{t^2}\right)^2}=  \frac{\sqrt{t^4+1}}{t^2} as in r=ai+bj and |r|=\sqrt{a^2+b^2}
    Last edited by BAdhi; July 10th 2012 at 09:10 AM.
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