Further Mathematics 7367: Numerical Methods Topical Past Paper Questions

A curve passes through the point $ (-2, 4.73) $ and satisfies the differential equation $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y^{2}-x^{2}}{2x+3y} $$ Use Euler's step by step method once, and then the midpoint formula $$ y_{r+1}=y_{r-1}+2h\mathrm{f}(x_{r},y_{r}),\ x_{r+1}=x_{r}…

Use I'Hôpital's rule to show that $$ \lim_{x\to\infty}(x\mathrm{e}^{-x})=0 $$ Fully justify your answer. [4 marks]

Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for $$ \begin{align*}&\frac{\sin x}{x}-\cos x\\ is\\&(-1)^{r+1}\frac{2r}{(2r+1)!}x^{2r}\\&\tag{4marks}\end{align*} $$ (b) Show that $$ \lim_{x\to0}\left[\frac{\frac{\sin x}{…