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.

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