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Math Help - Small problem

  1. #1
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    Small problem

    So I have this simple expression, which is part of long equation: sqrt(1-(r/R)^2)). By using r<<R1 and r<<R2 I can approximate and should end up with this expression: 1-1/2(r/R)^2.

    It's one of those things I could spend weeks on and still not figure out. I have really no clue how to understand how that approximation works. Any help appreciated. Maybe it's a Taylor series thing?
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  2. #2
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    Quote Originally Posted by hellfish View Post
    So I have this simple expression, which is part of long equation: sqrt(1-(r/R)^2)). By using r<<R1 and r<<R2 I can approximate and should end up with this expression: 1-1/2(r/R)^2.

    It's one of those things I could spend weeks on and still not figure out. I have really no clue how to understand how that approximation works. Any help appreciated. Maybe it's a Taylor series thing?
    let \displaystyle x = \frac{r}{R} ... note for small values of x , \displaystyle \sqrt{1-x^2} \approx 1 - \frac{x^2}{2} as depicted in the attached graph.

    yes, the 2nd degree Taylor polynomial approximation of y = \sqrt{1 - x^2} centered at x = 0 is \displaystyle T_2(x) = 1 - \frac{x^2}{2}
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