11.5 2011 test

1) (No longer relevant)

[6]

2) By using the method of partial fractions, evaluate the improper integral

[5]

3) Calculate the Laplace transform $F(s)=\cosh ax$ for $s>a>0$.

[4]

4) Let $x=t-\sin t$ and $y=1-\cos t$ for $0\leq t\leq 2\pi$ be a point on an arch $C$ of a cycloid.

i) Find a simple expression for $\frac{dy}{dx}$ in terms of $t$.

ii) Find the length $L$ of $C$.

[7]

5) Find the stationary points of the function

and determine their nature.

[8]

Total: 30