The binomial expansion of √(1 + x)

1. Evaluating at $0$: $f(0)=1$, $f'(0)=\frac12$, $f''(0)=-\frac14$, $f'''(0)=\frac38$. By Taylor–Young,
$$\sqrt{1+x}=1+\frac x2-\frac{x^2}8+\frac{x^3}{16}+o(x^3).$$

2. With $x=0.2$: $1+0.1-0.005+0.0005=1.0955$ (true value $1.095445\dots$).
With $x=-0.1$: $1-0.05-0.00125-0.0000625=0.9487\dots$ (true value $0.948683\dots$).

Remark. The general rule is $(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}2x^2+\frac{\alpha(\alpha-1)(\alpha-2)}6x^3+\cdots$, valid for all real $\alpha$.