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Math Help - need help generating formula to fit desired results

  1. #1
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    need help generating formula to fit desired results

    I am unsure if this is the correct forum for my question. However, Calculus 1 is the highest level of math I have completed, and I am able to define the requirements of the formula with limits. Feel free to move the thread if it is in the wrong forum.

    I need to define i.
    There are three variables: x, y, and z.
    However, x and y are only considered in the form (x+2y), so we can replace that with a single variable for simplicity, let's say w.
    All of these variables are integers.
    I'm fairly certain that i can be defined as i = w/(some expression).
    w >= 0
    w < z
    z >= 1
    0 < i < 1

    as w -> 0, i -> w/z
    as w -> infinity, i -> w/w
    Notice that the denominator is the only part that changes, and that it defines i to be between 0 and 1. But I need one function that will work for both limits.

    For most circumstances, using i=w/z would work just fine. But when w gets larger, the formula becomes less reliable. As w increases, the z variable increases at a growing ratio. It needs to be a curved relationship, such that values of w < 100 or so still are much closer to using i=w/z. But for values of w > 1000, it should already be very close to i=w/w.
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  2. #2
    Super Member
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    How about something like

     i = \frac{w}{w + ze^{-w}} ?
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