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Thread: convergent equation

  1. #1
    Dec 2017
    Tel Aviv

    convergent equation

    Can be a convergent equation (Term that I invent) like this example:

    Find x in the equation:
    X(cos(X)) = C
    when C [Constant] = 3.

    I use the calculator and choose x (x is in degree not Radians) the value 4.
    4(cos(4)) = 3.99...
    I need to decrease the value to find X.
    Let x be 3:
    3(cos(3)) = 2.99...
    So the x is between 3 to 4.
    You can continue to find x in that way by increasing and decreasing the value of X by calculator.
    And Also:
    You can choose any number of the equation:
    X(cos(x)) = C
    with different value of C.

    Can you give me more examples of this phenomenon?
    Is there a reason to this phenomenon?
    Why it happen?
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  2. #2
    Senior Member
    Jan 2009

    Re: convergent equation

    Here's the plot for the equation x * cos(x * pi/180) from -5 to 5 and from -1000 to 1000.

    Wolfram|Alpha: Computational Intelligence
    Wolfram|Alpha: Computational Intelligence

    Why is the curve of the equation close to the equation y = x between -5 and 5? It's because cos(0) = 1 so any number x close to 0 in radians will make cos(x) a value close to 1.
    Hence x * cos(x) will be approximately x * cos(0) = x

    Conclude two things:
    1) The function is bounded above and below by curves y = x and y = -x. This is because cos(x) is bounded between -1 and 1
    2) In the neighborhood when cos(x * pi/180) is close to 1, the function x * cos(x * pi/180) will be approximately x.
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