= Solution
The required <derivatives> are $f'(x)=1/(2\sqrt x)$, $f''(x)=-1/(4x^{3/2})$, $f^{(3)}(x)=3/(8x^{5/2})$, and $f^{(4)}(x)=-15/(16x^{7/2})$. The four-term <Taylor polynomial> about one, evaluated at two, is
$$
\boxed{\sqrt2\approx1+\frac12-\frac18+\frac1{16}=\frac{23}{16}}.
$$
The <Taylor remainder> is $R_3(2)=-5/(128\xi^{7/2})$ for some $1<\xi<2$. Thus
$$
\boxed{\left|\sqrt2-\frac{23}{16}\right|\leq\frac5{128}}.
$$
Its negative sign also shows that this approximation is an overestimate. Bounding the fourth <derivative> throughout the interval establishes the requested error estimate without using a decimal value of $\sqrt2$.
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