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Thread: Practical application of a rational function with a parabolic Does asymptote?

  1. #1
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    Practical application of a rational function with a parabolic Does asymptote?

    Does anyone know of any real-world situations (in science, engineering, etc) where a parabolic (or otherwise curvilinear) asymptote comes into relevance?

    Thanks...
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    MHF Contributor ebaines's Avatar
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    Re: Practical application of a rational function with a parabolic Does asymptote?

    Here's one: when a string hangs under its own weight it forms a catenery, which is a constant times the hyperbolic cosine:  y = \frac A 2 (e^x + e^{-x}). For large values of x this assymptotically approaches  y = \frac A 2 e^x .
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    Re: Practical application of a rational function with a parabolic Does asymptote?

    Thanks so much, ebaines.
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    Re: Practical application of a rational function with a parabolic Does asymptote?

    Would you please clarify what variables x and y represent in this situation?
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    MHF Contributor ebaines's Avatar
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    Re: Practical application of a rational function with a parabolic Does asymptote?

    y= deflection of the string, and x = position along the horizontal axis, where x=0 is at the midpoint of the string. See the attached figure - the blue line is the hangng string and the red line is the graph of (1/2)e^x.

    Practical application of a rational function with a parabolic Does asymptote?-catenery.jpg
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    Re: Practical application of a rational function with a parabolic Does asymptote?

    Does the constant A represent the length of the string?
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    Re: Practical application of a rational function with a parabolic Does asymptote?

    Also, I've found several websites that present it this way:

     y = \frac{A}{2} \left( e^{x/a} + e^{-x/a} \right)

    Is this what you meant, or is this something different?
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