12.3 Answers to 2013 test

2. a) The Laplace transform of $x^{2}e^{x}$ is $\frac{2}{(s-1)^{3}}$, and it converges for $s>1$.

b) i) $f(R)$ diverges as $R\rightarrow\infty$.

ii) $g(\delta)\rightarrow\log\frac{1}{2}=-\log 2$ as $\delta$ tends to $0$ from above.

3. i) False: the gradient (by implicit differentiation) is $\frac{6x-2}{3y^{2}}$, which equals $\frac{5}{6}$ at $(2,2)$.

ii) False, e.g. $f(x,y)=x+y$ has no stationary points, but $f_{x}=f_{y}$ everywhere.

iii) True: the Hessian discriminant $\Delta=f_{xx}f_{yy}-f_{xy}^{2}\leq f_{xx}f_{yy}<0$, so every stationary point is a saddle.

4. a) We obtain the estimate: $\int_{0}^{\frac{\pi}{2}}\sqrt{1-\frac{5}{9}\sin^{2}t}\,dt\approx\frac{\pi}{12}(\frac{5}{3}+4\sqrt{\frac{13}{18}})\approx 1.33$.

b) Using the parametrization and the arc-length formula we obtain that the perimeter of the ellipse is $12\int_{0}^{\frac{\pi}{2}}\sqrt{1-\frac{5}{9}\sin^{2}t}\,dt\approx 15.9$.