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Q:
How to find the approximation of $ \sqrt[2]{x} – \sqrt{x} $ where $x > 1$?
I’m not quite sure how to approach this problem. I know that the approximation is the Taylor Series of the function. But the problem is I have to find the third-order approximation for a function of the following form: $f(x)= \sqrt[2]{x} – \sqrt{x} $
A:
If you use Taylor series (here with $f(x)=\sqrt[2]{x}-\sqrt{x}$) with $x=1+h$, you get
$$ f(x)=f(1)+\frac{f^{(2)}(1)}{2!}h+\frac{f^{(3)}(1)}{3!}h^2+\frac{f^{(4)}(1)}{4!}h^3+O(h^4) $$
so that
$$ f(1+h) = f(1)+h f'(1)+\frac{h^2}{2}f^{(2)}(1)+O(h^3) $$
To find $f'(1)$ you just have to square $f(x)=\sqrt[2]{x}-\sqrt{x}$.
Note that since $h\approx 0$ for small $h$ you will get the same result as if you had used the first order Taylor series with $f(x)=\sqrt[2]{x}-\sqrt{x}$.
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