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Math Help - Differentiation ( Parametric)

  1. #1
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    Differentiation ( Parametric)

    Given x = t - ( 1/ t) , y = 2t + ( 1/t) with t is a non-zero parameter.
    a) Show that dy/dx = 2 - [ 3 / (t^2 + 1 ) ] ...... Done
    b) Hence, deduce that -1 < dy/dx < 2 .............. Undone

    I would be grateful if you can help me with the part B with explanations.

    Thanks in advance!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by ose90 View Post
    Given x = t - ( 1/ t) , y = 2t + ( 1/t) with t is a non-zero parameter.
    a) Show that dy/dx = 2 - [ 3 / (t^2 + 1 ) ] ...... Done
    b) Hence, deduce that -1 < dy/dx < 2 .............. Undone

    I would be grateful if you can help me with the part B with explanations.

    Thanks in advance!
    just observe..

    \frac{dy}{dx} = 2 - \frac{3}{t^2+1}

    as t approach \pm \infty, then \frac{3}{t^2+1} approach 0^+ but not zero.. thus, \frac{dy}{dx} = 2 - 0^+ < 2

    next, if t approach 0, then \frac{3}{t^2+1} approach 3^-.. hence, \frac{dy}{dx} = 2 - 3^- > -1..

    therefore, you get the desired conclusion..
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  3. #3
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    Thank you very much
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