(a) $$ f^{\prime}(x) =\frac{1}{1+(\sqrt{2}+x)^2} $$

$$ f^{\prime \prime}(x) =-\frac{2(\sqrt{2}+x)}{\left(1+(\sqrt{2}+x)^2\right)^2} $$

(b) $f(x)=0.955+0.333 x-0.157 x^2+\ldots$

(a) $f(x)=\tan ^{-1}(\sqrt{2}+x)$
$f^{\prime}(x)=\frac{1}{1+(\sqrt{2}+x)^2}$
$f^{\prime \prime}(x)=-\frac{2(\sqrt{2}+x)}{\left(1+(\sqrt{2}+x)^2\right)^2}$

(b)  $f(0)=0.955, f^{\prime}(0)=0.333, f^{\prime \prime}(0)=-0.314269$

Hence, $f(x)=0.955+0.333 x-0.157 x^2+\ldots$ (3 s.f.)