# mock midterm 1

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

1. 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.

2. Consider the sequences of functions
1. $$f_n=4^nx^n(1-x)^n$$;
2. $$f_n=x^n(1-x)^n$$;
3. $$f_n=\sin(nx)/\sqrt{n}$$.

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.

3. Which of the following functions are continuous?
1. $f(x,y)=\left\{ \begin{array}{ll} \frac{(x^3-y^3)^2}{(x^2+y^2)^3}, &\mathrm{if\quad} (x,y)\neq (0,0);\\ 0&\mathrm{if\quad } (x,y)=(0,0); \end{array} \right.$
2. $f(x,y)=\left\{ \begin{array}{ll} \frac{xy}{\sqrt{x^2+y^2}}, &\mathrm{if\quad} (x,y)\neq (0,0);\\ 0&\mathrm{if\quad } (x,y)=(0,0); \end{array} \right.$
3. $f(x,y)=h(x)-h(y),$
where $$h(x)=\frac{\sin{x}}{x}$$ if $$x\neq 0$$ and $$0$$ otherwise.

Answers: b and c are continuous; a – not.

4. 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: $$\partial z/\partial x(2,0)=\partial z/\partial y(2,0)=0$$.

5. 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 $$-\infty$$.

6. 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.