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Math Help - URGENT Parametrics

  1. #1
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    URGENT Parametrics

    Let be the graph of the parametric equations





    What is the length of the smallest interval
    such that the graph of these equations for all produces the entire graph ?
    Last edited by orange; December 18th 2013 at 07:25 PM.
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  2. #2
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    Re: URGENT Parametrics

    Quote Originally Posted by orange View Post
    Let be the graph of the parametric equations





    What is the length of the smallest interval
    such that the graph of these equations for all produces the entire graph ?
    Cos[4t] needs (2pi)/4 = pi/2

    Sin[6t] needs (2pi)/6 = pi/3

    what's the least common multiple of 1/2 and 1/3 ? So what's the minimum interval needed for a full cycle?
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  3. #3
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    Re: URGENT Parametrics

    1 is their LCM. So I guess the minimum interval is just pi?
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  4. #4
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    Re: URGENT Parametrics

    give that man a kewpie doll!
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  5. #5
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    Re: URGENT Parametrics

    Hello, orange!

    I "eyeballed" the problem.
    I think I have the solution.


    Let G be the graph of the parametric equations: . \begin{Bmatrix}x &=& \cos(4t) \\ y &=& \sin(6t)\end{Bmatrix}

    What is the length of the smallest interval I such that the graph
    of the equations for all t \in I produces the entire graph of G ?

    When t = 0\!:\;\begin{Bmatrix}x &=& \cos(0) &=& 1 \\ y &=& \sin(0) &=& 0\end{Bmatrix}

    At the "start", the graph is at (1,0).


    Here is what I found:

    . . \begin{array}{c|c|c|} t & \cos(4t) & \sin(6t) \\ \hline 0 & 1 & 0 \\ \frac{\pi}{4} & \text{-}1 & \text{-}1 \\ \frac{\pi}{2} & 1 & 0 \\ \frac{3\pi}{4} & \text{-}1 & 1 \\ \hline \pi & 1 & 0 \\ \vdots & \vdots & \vdots\end{array}

    . . and the cycle repeats.


    Therefore: . I \,=\,[0,\,\pi]
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  6. #6
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    Re: URGENT Parametrics

    Hi,
    If a given parametric curve is periodic, finding its period is a mystery to me. Here's a problem that I tackled some time ago with its solution; you might try your hand at it (my proof was somewhat long and involved). I wish I knew some general procedure to find the period of such periodic functions.

    URGENT Parametrics-mhfcalc26.png
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