Approximation with small parameter

[avikarto]

For some small parameter $\epsilon$, how would one go about making an approximation such as $\sqrt{k^2-\epsilon^2}\approx k-\frac{\epsilon^2}{2k}$? I was thinking that these types of approximations came from truncating Taylor series expansions, but I can't see how it would be obvious which parameter one would differentiate with respect to, a priori, or what value to expand about. Could someone please explain the general method for making such approximations? Thanks!

 

[Mark44]

For some small parameter $\epsilon$, how would one go about making an approximation such as $\sqrt{k^2-\epsilon^2}\approx k-\frac{\epsilon^2}{2k}$? I was thinking that these types of approximations came from truncating Taylor series expansions

More specifically, a binomial series expansion, writing $\sqrt{k^2-\epsilon^2}$ as $(k^2 - \epsilon^2)^{1/2}$. That's where I would start.

, but I can't see how it would be obvious which parameter one would differentiate with respect to, a priori, or what value to expand about. Could someone please explain the general method for making such approximations? Thanks!

 

[mfig]

You can use the binomial approximation by first pulling the k term out.

√(k^2 - e^2) = k⋅√(1 - (e/k)^2) = k⋅(1 - (e/k)^2)^(1/2)

Now look at the binomial approximation.