Estimate an exponential and expand a shifted secant

(a) Let $f(x)=e^{\tan^{-1}(x/2)}$.

(i) Show $(4+x^2)f^{\prime}(x)=2f(x)$.1 mark

(ii) Use repeated differentiation to expand $f$ through $x^2$. Use it to estimate $e^{\pi/6}$ and show the error is about $3.3\%$.6 marks

(b) For $g(x)=1/[\sqrt2\cos(x/a+\pi/4)]$, where $a>1$, show that for sufficiently small $x$, $g(x)\approx1+px+qx^2$, finding $p,q$ in terms of $a$.4 marks