IAL 2022 Jan Q5 | 國際數學站

Given that

\begin{align*} y =\,& \sqrt{4 + \ln x} \qquad x > \frac{1}{2}\\[2mm] \end{align*}

(a) Show that

\begin{align*} \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} =\,& -\frac{9 + 2\ln x}{4x^2(4 + \ln x)^{\frac{3}{2}}}\\[2mm] \end{align*}

(5)

(b) Hence, or otherwise, determine the Taylor series expansion about $x = 1$ for $y$ , in ascending powers of $(x - 1)$ , up to and including the term in $(x - 1)^2$ , giving each coefficient in simplest form.

(3)