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Math Help - Finding the smallest root.

  1. #1
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    Finding the smallest root.

    How to find the smallest root of this equation?
    Finding the smallest root.-codecogseqn.gif
    Thanks.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Finding the smallest root.

    From inspection the left hand side of this equation must be less than or equal to 1, so therefore  e^{0.02x} \le 0.05, or  x \ge \frac {-ln(20)}{0.02}. So the smallest root must be greater than 149.8. I think you'll find that the root is around 160.9.
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  3. #3
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Finding the smallest root.

    It appears that x1=150. is the smallest root?

    Attached Thumbnails Attached Thumbnails Finding the smallest root.-smallest-root.png  
    Last edited by MaxJasper; October 11th 2012 at 05:31 AM.
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    Re: Finding the smallest root.

    Yes but how do I find it? I mean without plotting from graph
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  5. #5
    MHF Contributor ebaines's Avatar
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    Re: Finding the smallest root.

    Yes, the answer is near 150 (not exactly 150, but slightly more).

    You can find the solution using a numerical technique, such as Newton's method, to find the root for  f(x) = e^{-0.02x}+0.95+ \sin(2x) = 0. It involves first making a guess as to what the root is - call it  x_1 - and finding the error of this guess:

    Error =  e^{-0.02x_1}+0.95+ \sin(2x_1),

    Then calculate the slope of f(x_1):

    Slope =  -0.02 e^{-0.02x_1} + 2 \cos(2x_1)

    and set the next guess as x_2 = x_1 - \frac {error}{slope}. Iterate through this a few times and it will zero in on a value near 150.0007.
    Last edited by ebaines; October 11th 2012 at 06:02 AM.
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  6. #6
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    Re: Finding the smallest root.

    What do YOU mean by "finding" the root? You won't be able to write it in terms of elementary functions.
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    Re: Finding the smallest root.

    Thanks ebaines
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