11.3 2013 test

1) (No longer relevant)

[4]

2) a) Find the Laplace transform $F(s)$ of $x^{2}e^{x}$, specifying for which values of $s$ it converges.

[6]

b) Determine whether each of the following expressions converge, and if it does, what the limit is:

i) $f(R)=3\log(R+13)-2\log(R^{2}+1)$ as $R\rightarrow\infty$,

ii) $g(\delta)=\log(\delta^{2}+\delta)-\log(\delta^{3}+2\delta)$ as $\delta\rightarrow 0^{+}$.

[2]

3) State whether each of the following statements is true or false. Briefly (i.e. in about one sentence) justify your answer.

i) The gradient $\frac{dy}{dx}$ of the curve $y^{3}-3x^{2}+2x=0$ at the point $(2,2)$ is $-10$.

ii) If $f(x,y)$ is a function of two variables and $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}$ at $P$, then $P$ is stationary point.

iii) If $f(x,y)$ is such that $f_{yy}=-f_{xx}\neq 0$ for all $x,y$ then every stationary point for $f$ is a saddle point.

[6]

4) a) Use Simpson’s rule to estimate the integral: $\int_{0}^{\frac{\pi}{2}}\sqrt{1-\frac{5}{9}\sin^{2}t}\,dt$ to 2 decimal places.

b) Hence, using the parametrization $x=3\sin t$, $y=2\cos t$ or otherwise, estimate the perimeter of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$.

[6]

5) (No longer relevant)

[6]

Total: 30