Further Mathematics 7367: Further Calculus Topical Past Paper Questions

The integral $ I_{n} $ is defined by $$ I_{n}=\int_{0}^{\frac{\pi}{4}}\cos^{n}x\mathrm{d}x\quad(n\geq0) $$ Show that $$ I_{n}=\left(\frac{n-1}{n}\right)I_{n-2}+\frac{1}{n\left(2^{\frac{n}{2}}\right)}\qquad(n\geq2) $$ [6 marks] Use the result from part (a) to show that $$ \int_{0}…

A bungee jumper of mass mkg is attached to an elastic rope. The other end of the rope is attached to a fixed point. The bungee jumper falls vertically from the fixed point. At time $t$ seconds after the rope first becomes taut, the extension of the rope is $x$ metres and the spee…

On an isolated island some rabbits have been accidentally introduced. In order to eliminate them, conservationists have introduced some birds of prey. At time $t$ years $(t \geq 0)$ there are $x$ rabbits and $y$ birds of prey. At time $t = 0$ there are 1755 rabbits and 30 birds o…