\begin{equation*} \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\N}{\mathbb{N}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\bar}{\overline} \renewcommand{\Lin}{\mathrm{Lin}} \renewcommand{\dd}{\mathrm{d}} \end{equation*}

1 Question 1

Evaluate

\begin{equation*} \iint_\Omega x^2y \dd\mathrm{Vol}(x,y,z) \end{equation*}

where \(\Omega\) is the volume obtained by interecting \(\Omega_1\) \(\Omega_2\) \(\Omega_3\) where

  • \(\Omega_1\) are the points \((x,y,z)\) between \(y=\sqrt{x}\) and \(y=0\),
  • \(\Omega_2\) are the points \((x,y,z)\) with \(x\) between \(1\) and \(3\).
  • \(\Omega_3\) are the points \((x,y,z)\) between the \(x,y\) plane and the plane \(x+y-z=0.\)

1.1 Asessment

Understanding the shape of the domain: 30% Setting up the integral: 50% Computing the integral: 20% Expected time: 6min work + 2 min explanation

2 Question 3

Indentify all local and global extremal points of

\begin{equation*} (x^{2}+1) (y^{2}+1)-\frac{1}{2} x^{2}-(\frac{1}{2}) y^{2}-y^{4} \end{equation*}

on \(\R^2\)

If there are no global maximums or minimums, explain why.

2.1 Asessment

  • Setting up : 50%
  • Computation: 30%
  • Explanation: 20%
  • Expected time: 6min work + 2 min explanation

3 Question 4

Find the global and local maxima and the minima of the function

\begin{equation*} f(x,y)= x^2+y^2 \end{equation*}

on the boundary of the region \(\Omega\) obtained by intersecting \(\Omega_1\) and \(\Omega_2\)

  • \(\Omega_1\) is the region between the lines \(x=-1\) and \(x=1\)
  • \(\Omega_2\) is the region between the lines \(y=0\) and \(y^2=\frac{100}{1+x^2/100}\).

If there are no global maximums or minimums, explain why.

3.1 Asessment

  • Setting up : 50%
  • Computation: 30%
  • Explanation: 20%
  • Expected time: 8min work + 2 min explanation