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Math Help - Is my book wrong?

  1. #1
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    Is my book wrong?

    This simplification came directly from my book. I think it is wrong, but I may just not understand it fully. Here it is:

     T'(t) = \\-4t(2t^2 + 1)^-2 <2t,2t^2,1> +\\ (2t^2 +1)^-1 <2,4t,0> \\ = (2t^2 + 1)^-2 <-8t^2 + 4t^2 + 2, \\-8t^3 + 8t^3 + 4t, -4t> \\ = 2(2t^2 + 1)^-2 <1-2t^2, 2t, -2t>

    I don't understand how they factored out the (2t^2 + 1)^-2. Is it wrong, or am I just missing something? Thanks.
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  2. #2
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    Quote Originally Posted by zellster87 View Post

    This simplification came directly from my book. I think it is wrong, but I may just not understand it fully. Here it is:



     T'(t) = \\-4t(2t^2 + 1)^-2 <2t, 2t^2, 1> +\\ (2t^2 +1)^-1 <2, 4t, 0> \\ = (2t^2 + 1)^-2 <-8t^2 + 4t^2 + 2, \\-8t^3 + 8t^3 + 4t, -4t> \\ = 2(2t^2 + 1)^-2 <1-2t^2, 2t, -2t>



    I don't understand how they factored out the (2t^2 + 1)^-2. Is it wrong, or am I just missing something? Thanks.
    Quote Originally Posted by zellster and edited by Mr F
    This simplification came directly from my book. I think it is wrong, but I may just not understand it fully. Here it is:


     T'(t) = \\-4t(2t^2 + 1)^{-2} <2t, \, 2t^2, \, 1> ~ + ~ (2t^2 +1)^{-1} <2, \, 4t, \, 0> \\

    = (2t^2 + 1)^{-2} <-8t^2 + 4t^2 + 2, \, -8t^3 + 8t^3 + 4t, \, -4t> \\

    = 2(2t^2 + 1)^{-2} <1-2t^2, \, 2t, \, -2t>

    I don't understand how they factored out the (2t^2 + 1)^{-2}. Is it wrong, or am I just missing something? Thanks.
    Now we can read the expression!
    Last edited by mr fantastic; June 2nd 2008 at 08:50 PM. Reason: Fixed some spacing
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  3. #3
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    Quote Originally Posted by zellster87 View Post
    This simplification came directly from my book. I think it is wrong, but I may just not understand it fully. Here it is:


     T'(t) = \\-4t(2t^2 + 1)^-2 <2t,2t^2,1> +\\ (2t^2 +1)^-1 <2,4t,0> \\ = (2t^2 + 1)^-2 <-8t^2 + 4t^2 + 2, \\-8t^3 + 8t^3 + 4t, -4t> \\ = 2(2t^2 + 1)^-2 <1-2t^2, 2t, -2t>

    I don't understand how they factored out the (2t^2 + 1)^-2. Is it wrong, or am I just missing something? Thanks.
    The expression is correct. The line in red should make this clear:

     T'(t) = -4t(2t^2 + 1)^{-2} ~ <2t, 2t^2, 1> ~ + ~ (2t^2 +1)^{-1} <2, 4t, 0>

    {\color{red}= (2t^2 + 1)^{-2} ~ <-8t^2, -8t^3, -4t> ~ + ~ (2t^2 +1)^{-2} <2(2t^2 + 1), 4t(2t^2 + 1), 0> }

    = (2t^2 + 1)^{-2} ~ <-8t^2 + 4t^2 + 2, -8t^3 + 8t^3 + 4t, -4t>

    = 2(2t^2 + 1)^{-2} ~ <1-2t^2, 2t, -2t>
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    Ahh I see. Thanks Mr. fantastic.
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