Problem

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Given that when $y=\arcsin 2x$

$$\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x}=2(1-4x^2)^{-\frac12} \end{align*}$$

(a) show that

$$\begin{align*} \frac{\mathrm{d}^3y}{\mathrm{d}x^3} =\frac{Ax^2+8}{(1-4x^2)^{\frac52}} \end{align*}$$

where $A$ is a constant to be determined.

(b) Hence determine the Maclaurin series expansion for $\arcsin 2x$ in ascending powers of $x$ up to and including the term in $x^3$

The Maclaurin series expansion for $e^x$ is given by

$$\begin{align*} e^x=1+x+\frac{x^2}{2}+\ldots+\frac{x^r}{r!}+\ldots \end{align*}$$

(c) Use the Maclaurin series expansion for $e^{3x}$ and the answer to part (b) to show that, for small values of $x$

$$\begin{align*} e^{3x}\arcsin 2x\approx Cx+Dx^2+Ex^3 \end{align*}$$

where $C$, $D$ and $E$ are constants to be determined.