# Thread: Small problem

1. ## 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?

2. Originally Posted by hellfish
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 $\displaystyle \displaystyle x = \frac{r}{R}$ ... note for small values of $\displaystyle x$ , $\displaystyle \displaystyle \sqrt{1-x^2} \approx 1 - \frac{x^2}{2}$ as depicted in the attached graph.

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