\(\def\Real{\mathbb{R}}

\def\Comp{\mathbb{C}}

\def\Rat{\mathbb{Q}}

\def\Field{\mathbb{F}}

\def\Fun{\mathbf{Fun}}

\def\e{\mathbf{e}}

\def\f{\mathbf{f}}

\def\bv{\mathbf{v}}

\)

- Consider the following subsets of \(\Real^2\):

\[

A=\{(x,y):xy>0\}; B=\{(x,y):x=0\}; C=\{(x,y):y=0\}; D=\{(0,0)\}.

\]

Which of the following sets is open? closed?

\[

A; B; A\cup C; A\cup B\cup C; A\cup D

\]**Answers**: open; closed; neither open nor closed; closed; neither open nor closed. - Consider the sequences of functions
- \(f_n=4^nx^n(1-x)^n\);
- \(f_n=x^n(1-x)^n\);
- \(f_n=\sin(nx)/\sqrt{n}\).

Which of them converge on \([0,1]\)? What is the limit? When the convergence is uniform?

**Answers**: a converges, but not uniformly; b,c converge uniformly. - Which of the following functions are continuous?
- \[

f(x,y)=\left\{

\begin{array}{ll} \frac{(x^3-y^3)^2}{(x^2+y^2)^3}, &\mathrm{if\quad} (x,y)\neq (0,0);\\

0&\mathrm{if\quad } (x,y)=(0,0);

\end{array}

\right.

\] - \[

f(x,y)=\left\{

\begin{array}{ll} \frac{xy}{\sqrt{x^2+y^2}}, &\mathrm{if\quad} (x,y)\neq (0,0);\\

0&\mathrm{if\quad } (x,y)=(0,0);

\end{array}

\right.

\] - \[

f(x,y)=h(x)-h(y),\]

where \(h(x)=\frac{\sin{x}}{x}\) if \(x\neq 0\) and \(0\) otherwise.

**Answers**: b and c are continuous; a – not. - \[
- If the function \(z=z(x,y)\) is given implicitly by the equation

\[

z^2+y^2+x^4-8x^2+12=0,

\]

find the differential \(dz\) at the point \(x=2,y=0,z=2\).**Answers**: \(\partial z/\partial x(2,0)=\partial z/\partial y(2,0)=0\). - Find the critical points of the function

\[

f(x,y)=x^3-3xy^2-8x-6y.

\]

Which of these points is (global) maximum? minimum?**Answer**: \(\pm(\sqrt{3},\sqrt{3}/3) \). Neither of the point is a global maximum or minimum as the function grows to \(\infty\) and decreases to \(-\infty\). - Maximize

\[

x_1x_2x_3\to\max

\]

subject to constraints

\[

\sum_{i=1}^3 x_i=0; \sum_{i=1}^3 x_i^2=1.

\]**Answer**: the maximum is attained at \((-\sqrt{6}/6,-\sqrt{6}/6,2\sqrt{6})/6\), or any permutaion of these coordinates.

Hi Professor. Can you post some exam logistics here, in case some students missed this information?

The exam is 11-12:15, during the regular class time on February 18. It is open book; no internet or texting.

For question 2, I think only part c converges uniformly to zero on [-1,1], and part b converges uniformly on [0,1] (not the whole interval [-1,1])

Anyone agrees? disagrees?

I feel both a and b converge uniformly to zero on [0,1] and c converges uniformly to zero on [-1,1]. Any remark/discussion is welcome

Answers posted – to corrected version.

Hi Professor, in problem 4, point (0,2,2) does not satisfy the equation. Is there something wrong with the problem?

Thank you! corrected…

Hi Professor, in problem 5, should the answer be +/-(\sqrt{3},-\sqrt{3}/3)?

In problem 6, should the answer be (-sqrt{6}/6, -sqrt{6}/6, 2*sqrt{6}/6)?

Corrected!

Hi professor. I don’t have a printed copy of the book; I use a PDF version on a tablet. Is that okay to use during the exam?

Sure – just keep it localy.

Hi professor, for question 3c, the function is not defined on x axis and y axis other than point (0,0)?

Well observed! Corrected.