\documentclass[12pt,notitlepage]{report}

\frenchspacing % aktivuje použití některých českých typografických pravidel

\usepackage{amsthm,amsmath,amssymb,amscd,mathrsfs,tikz,pgfplots,hyperref,url}%,showkeys,layout}
\usepackage{verbatim} %pro vynechavani comment
\usepackage[top=1cm, left=1cm]{geometry} % nastavení dané velikosti okrajů
\pagestyle{empty}

\def\ce{\mathbb C}
\def\C{\mathcal{C}}
\def\en{\mathbb N} 
\def\zet{\mathbb Z}
\def\er{\mathbb R}
\def\qe{\mathbb Q}
\def\ov{\overline}
\def\vn{\operatorname{Int}}
\def\setsep{:\;}
\newcommand\lx[1][0]{\displaystyle\lim_{x\to #1}}
\def \sgn {\operatorname{sgn}}
\def \dx {\, \mathrm{d} x}
\def \tg {\operatorname{tg}}
\def \arctg {\operatorname{arctg}}
\newcounter{Priklad}
\addtocounter{Priklad}{1}
\newcommand{\pr}{\quad\alph{Priklad})\;\addtocounter{Priklad}{1}}
\newcommand{\resetpr}{\setcounter{Priklad}{1}}
\def \suma {\sum_{n=1}^\infty}
\def \sumaO {\sum_{n=0}^\infty}
\newcommand\pdx[1][f]{\frac{\partial #1}{\partial x}}
\newcommand\pdy[1][f]{\frac{\partial #1}{\partial y}}
\newcommand\pdz[1][f]{\frac{\partial #1}{\partial z}}

\title{Cvi\v{c}en\'i - Matematika 2, FSV}  % tyto dvě položky jsou zde v podstatě formálně, ve skutečnosti nejsou nikde 
\author{Marek C\'uth}

\begin{document}
\pgfplotsset{width=7cm, compat=1.13}
 \centerline{\textbf{\MakeUppercase{v\'ysledky}}}\ \\	
\centerline{\MakeUppercase{i. vrstevnice funkce, otev\v{r}en\'e a uzav\v{r}en\'e mno\v{z}iny}}\ \\[20pt]
\textbf{1.}\resetpr \pr $D_f=\{[x,y]\in\er^2\setsep y\ge0\}$\hspace{30pt}
\pr $D_f=\{[x,y]\in\er^2\setsep x\ne0\}$\\[10pt]
\begin{tikzpicture}%[domain=0:4]
\begin{axis}[
        %title={$x + \sqrt{y}$},
       domain=-4:4,
	y domain=0:4,
	xlabel=$x$, ylabel=$y$,
%       xmin=-5,xmax=5,ymin=0,ymax=5,
%       enlarge x limits,
        view={0}{90}%,
    ]
   \addplot3[contour gnuplot={number=16, labels over line},thick]
          {x + sqrt(y)};
    \end{axis}
%    \begin{axis}
			%\addplot+[sharp plot] coordinates {(0,0) (1,2) (2,3)};
%			\addplot gnuplot[id=sin]{sin(x)};
%		\end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[
        %title={$\tfrac{y}{x}$},
       domain=-2:2,
	restrict y to domain=-2:2,
	xlabel=$x$, ylabel=$y$,
%	 unbounded coords=jump,
        view={0}{90}%,
    ]
    \addplot [color=red]    {2*x};
    \addplot [color=blue]    {x};
     \addplot [color=green]    {-x};
     \addplot [color=yellow]    {-2*x};
      \addplot [color=orange]    {0.5*x};
       \addplot [color=black]    {-0.5*x};                
      \addplot [color=black]    {0};
    \end{axis}
\end{tikzpicture}\\
\pr $D_f=\er^2$\hspace{170pt}
\pr $D_f=\er^2$\\[10pt]
\begin{tikzpicture}
\begin{axis}[
    %    title={$x^2 + y^2$},
       domain=-4:4,
	y domain=-4:4,
	xlabel=$x$, ylabel=$y$,
        view={0}{90}
    ]
   \addplot3[contour gnuplot={number=8, labels over line},thick]
          {x^2 + y^2};
    \end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[
      %  title={$x^2 - y^2$},
       domain=-4:4,
	y domain=-4:4,
	xlabel=$x$, ylabel=$y$,
        view={0}{90}
    ]
   \addplot3[contour gnuplot={number=8, labels over line},thick]
          {x^2 - y^2};
    \end{axis}
\end{tikzpicture}\\[10pt]
\pr $D_f=\{[x,y]\setsep (x\ge0\ \&\ y\ge0) \vee (x\le0\ \&\ y\le
0)\}$\hspace{30pt}
\pr $D_f=\{[x,y]\setsep x^2+y^2\le 1\}$\\[10pt]
\begin{tikzpicture}
\begin{axis}[
       % title={$\sqrt{xy}$},
       domain=-4:4,
	y domain=-4:4,
	xlabel=$x$, ylabel=$y$,
        view={0}{90},
        unbounded coords=jump
    ]
   \addplot3[contour gnuplot={number=8, labels over line},thick]
          {sqrt(x*y)};
    \addplot3[contour gnuplot={levels={0}, labels over line},thick]
          {x*y};
    \end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[
      %  title={$\sqrt{1 - x^2 - y^2}$},
       domain=-1:1,
	y domain=-1:1,
	xlabel=$x$, ylabel=$y$,
        view={0}{90},
        unbounded coords=jump
    ]
   \addplot3[contour gnuplot={number=8, labels over line},thick]
          {sqrt(1 - x^2 - y^2)};
    \end{axis}
\end{tikzpicture}\newpage\ \\
\pr $D_f=\{[x,y]\setsep x^2+y^2> 1\}$, vrstevnice jsou kru\v{z}nice.
\pr $D_f=\{[x,y]\setsep 1\le x^2+y^2\le 4\}$, vrstevnice jsou dvojice kru\v{z}nic, v jednom p\v{r}\'{\i}pad\v{e} kru\v{z}nice.
\pr $D_f=\{[x,y]\setsep -x^2-1\le y\le -x^2+1\}$, vrstevnice jsou dvojice parabol, v jednom p\v{r}\'{\i}pad\v{e} parabola.
\pr $D_f=\{[x,y]\setsep 2k\pi\le x^2+y^2\le (2k+1)\pi\text{ pro
n\v{e}jak\'e }k=0,1,2,\dots\}$, vrstevnice jsou posloupnosti kru\v{z}nic.
\pr $D_f=\er^2$, jsou t\v{r}i vrstevnice - \v{c}tverce $\bigcup_{k,l\in\en} (2k\pi,(2k+1)\pi)\times (2l\pi,(2l+1)\pi)$, sjednocen\'i jejich hranic a zbytek.
\pr $D_f=\er^2$, vrstevnice jsou grafy funkc\'{\i} $y=k-|x|$.\\[10pt]
\textbf{2.} a)-v), b)-i), c)-vi), d)-ii), e)-iii), f)-iv)\\[10pt]
\textbf{3.}\resetpr
\pr $\en$ je uzav\v{r}en\'a, $\vn\en=\emptyset$, $H(\en)=\overline{\en}=\en$
\pr $\vn \qe=\emptyset,\,H(\qe)=\overline{\qe}=\er$, $\qe$ nen\'{\i} otev\v{r}en\'a ani uzav\v{r}en\'a.
\pr Mno\v{z}ina nen\'{\i} uzav\v{r}en\'a ani otev\v{r}en\'a, vnit\v{r}ek je pr\'azdn\'y,
hranice i uz\'av\v{e}r jsou $\{\frac1n\setsep n\in\en\}\cup\{0\}$.
\pr Mno\v{z}ina je uzav\v{r}en\'a, nen\'{\i} otev\v{r}en\'a, vnit\v{r}ek je pr\'azdn\'y, hranice a uz\'av\v{e}r jsou $\{-1,1\}$.
\pr Mno\v{z}ina nen\'{\i} uzav\v{r}en\'a ani otev\v{r}en\'a. Vnit\v{r}ek je
$\{[x,y]\in\er^2\setsep x>0,\,y< 0\}$, uz\'av\v{e}r $\{[x,y]\in\er^2\setsep
x\ge0,\,y\le 0\}$,  hranice$\{[x,y]\in\er^2\setsep
x\ge0\ \&\ y\le 0\ \&\ (x=0\vee y=0)\}$.
\pr Otev\v{r}en\'a, uz\'av\v{e}r $\{[x,y]\in\er^2\setsep x^2+y^2\le1\}$,
hranice $\{[x,y]\in\er^2\setsep x^2+y^2=1\}$.
\pr Uzav\v{r}en\'a, vnit\v{r}ek $\{[x,y]\in\er^2\setsep x^2+y^2>1\}$,
hranice $\{[x,y]\in\er^2\setsep x^2+y^2=1\}$.
\pr Otev\v{r}en\'a, hranice $\{[x,y]\in\er^2\setsep x^2+e^y=17\}$,
uz\'av\v{e}r $\{[x,y]\in\er^2\setsep x^2+e^y\geq 17\}$
\pr Otev\v{r}en\'a, uz\'av\v{e}r $\{[x,y]\setsep x+y\le0\}$, hranice
$\{[x,y]\setsep x+y=0\}$.
\pr Uzav\v{r}en\'a, vnit\v{r}ek $\{[x,y]\setsep x > y\}$, hranice $\{[x,y]\setsep
x = y\}$.
\pr Uzav\v{r}en\'a, pr\'azdn\'y vnit\v{r}ek.
\pr Ani uzav\v{r}en\'a ani otev\v{r}en\'a, vnit\v{r}ek pr\'azdn\'y, hranice i uz\'av\v{e}r
$\{[x,y,z]\in\er^3\setsep x\ge0,y\ge0,x+y=2,z\le0\}$.\\[10pt]
\centerline{\MakeUppercase{ii. parci\'aln\'i derivace}}\ \\
\textbf{1.}\resetpr
	\pr $\pdx=m x^{m-1} y^n$, $\pdy=n x^m y^{n-1}$ pro $(x,y)\in\er^2$.
	\pr $\pdx=y e^{xy}$, $\pdy=x e^{xy}$ pro $(x,y)\in\er^2$.
	\pr $\pdx=y+z$, $\pdy=x+y$, $\pdz=x+y$ pro $(x,y,z)\in\er^3$.
	\pr $\pdx(x,y)= |y|\sgn x$ pro $x\ne 0$. $\pdy(x,y)=|x|\sgn y$ pro $y\ne 0$. $\pdx(0,0)=\pdy(0,0)=0$. $\pdx(0,y)$ pro $y\ne 0$ a $\pdy(x,0)$ pro $x\ne0$ neexistuj\'{\i}.
	\pr $\pdx(x,y)=-\sgn(y+\cos x)\cdot\sin x$, $\pdy(x,y)=\sgn(y+\cos x)$, pokud $y\neq -\cos x$. $\pdy(x,-\cos x)$ neexistuje pro $x\in\er$. $\pdx(k\pi,(-1)^{k+1})=0$ pro $k\in\zet$. $\pdx(x,-\cos x)$ neexistuje pro $x\neq k\pi$.
	\pr $\pdx(x,y)=\cos x\sgn(\sin x-\sin y)$, $\pdy(x,y)=-\cos y\sgn(\sin x-\sin y)$, pokud $\sin x\ne\sin y$. $\pdx(\frac\pi2+k\pi,\frac\pi2+l\pi)=\pdy(\frac\pi2+k\pi,\frac\pi2+l\pi)=0$. V ostatn\'{\i}ch bodech parci\'aln\'{\i} derivace neexistuj\'{\i}.
	\pr $\pdx(x,y)=-\cos x\sgn(\cos y-\sin x)$, $\pdy(x,y)=-\sin y\sgn(\cos y-\sin x)$, pokud $\sin x\neq\cos x$. 
$\pdx(\frac\pi2+k\pi,(k+2l)\pi)=\pdy(\frac\pi2+k\pi+2l\pi,k\pi)=0$. V ostatn\'{\i}ch bodech parci\'aln\'{\i} derivace neexistuj\'{\i}.
\pr Pokud $x,y>0$ nebo $x,y<0$, pak $\pdx= \frac zy\cdot \left(\frac xy\right)^{z-1}$; $\pdy=-\frac{zx}{y^2}\cdot \left(\frac xy\right)^{z-1}$; $\pdz=\left(\frac xy\right)^z\cdot\log\frac xy$.
\pr Pokud $x>0$ a $z\ne 0$, pak $\pdx=\frac yz \cdot x^{\frac yz-1}$; $\pdy=x^{\frac yz}\cdot\log x\cdot \frac1z$;
$\pdz=-x^{\frac yz}\cdot\log x\cdot \frac y{z^2}$.
\pr Pokud $x>0$, pak $\pdx=\cos(x^y)\cdot x^{y-1}\cdot y$; $\pdy=\cos(x^y)\cdot x^y\cdot \log x$.
\pr $\pdx=e^{-\frac{\pi}{x^2+3xy+3y^2}} \cdot \frac{\pi(2x+3y)}{(x^2+3xy+3y^2)^2}$, $\pdy=e^{-\frac{\pi}{x^2+3xy+3y^2}} \cdot \frac{\pi(3x+6y)}{(x^2+3xy+3y^2)^2}$ pro $(x,y)\ne(0,0)$; v bod\v{e} $(0,0)$ jsou ob\v{e} parci\'aln\'{\i} derivace nulov\'e.
\pr Pokud $x>-y^2$, pak $\pdx=\frac{1}{2\sqrt{x+y^2}}$; $\pdy=\frac{y}{\sqrt{x+y^2}}$. Jinak parci\'aln\'i derivace nemaj\'i smysl.\\[10pt]
\textbf{2.} viz. v\'ysledky zkou\v{s}kov\'ych p\'isemek zde: \url{http://www.karlin.mff.cuni.cz/~kalenda/edu.php?edutype=archpis}\\[10pt]
\textbf{3.}\resetpr
\pr $\frac{\partial F}{\partial r} = 8r + 4s$, $\frac{\partial F}{\partial s} = 4r + 2s$, $\frac{\partial F}{\partial t} = 2t$
\pr $\frac{\partial F}{\partial s} = 4s^3 + 8st + 3t$, $\frac{\partial F}{\partial t} = 4s^2 + 3s + 8t + 2$\\[10pt]
\centerline{\MakeUppercase{iii. implicitn\'i funkce}}\ \\
\textbf{1.} $f'(0) = 2$, $f''(0) = -14$, rovnice te\v{c}ny je $y = -4x$, funkce je na okol\'i bodu $0$ konk\'avn\'i\\[10pt]
\textbf{2.} viz. v\'ysledky zkou\v{s}kov\'ych p\'isemek zde: \url{http://www.karlin.mff.cuni.cz/~kalenda/edu.php?edutype=archpis}\\[10pt]
\textbf{3.} te\v{c}n\'a rovina je $z = \frac{7}{5}(y+2) + 1$\\[10pt]
\textbf{4.}\resetpr \pr $1 - \frac{1}{2}(y+1) + \frac{1}{2}z$ \pr $1 - \frac{4}{\pi - 2e}(y - \tfrac \pi4) + \frac{\tfrac{\pi^2}{8e} - 4}{\pi - 2e}z$\\[10pt]
\textbf{5.}\resetpr \pr $f'(-1) = 1/2$, $g'(-1) = -1/4$ \pr $f'(1) = -1/3$, $g'(1) = 1/3$\pr $f'(1) = \frac{5}{3}$, $g'(1) = \frac{7}{3}$\\[10pt]
\centerline{\MakeUppercase{iv. extr\'emy funkc\'i v\'ice prom\v{e}nn\'ych}}\ \\
\textbf{1.}\resetpr \pr max $-2$ v bod\v{e} $[1,0]$, min $-5$ v bod\v{e} $[0,1]$
	\pr max $5$ v bodech $[\pm1,\pm1,1]$, min $-1$ v bod\v{e} $[0,0,-1]$
	\pr $\sup$ neexistuje, $\inf$ neexistuje
	\pr max $5$ v bod\v{e} $[\frac 34,0]$
	\pr max $1$ v bodech $[\pm 1,0]$, $[0,\pm 1]$, min $0$ v bod\v{e} $[0,0]$
	\pr max ${\frac{\sqrt{a^2+b^2}}{ab}}$ v bod\v{e} $[\frac{b}{\sqrt{a^2+b^2}},\frac{a}{\sqrt{a^2+b^2}}]$, min $-\frac{\sqrt{a^2+b^2}}{ab}$ v bod\v{e} $[-\frac{b}{\sqrt{a^2+b^2}},-\frac{a}{\sqrt{a^2+b^2}}]$
	\pr max $\frac 1e$ v bodech kru\v{z}nice $x^2+y^2 = 1$, min $0$ v bod\v{e} $[0,0]$
	\pr max $\frac 5e$ v bodech $[0,\pm 1]$, min $0$ v bod\v{e} $[0,0]$
	\pr max $1$ v bodech $[\pm 1,0]$, min $\frac 14$ v bodech $[0,\pm \frac 12]$
	\pr sup $\frac1{2e}$, nenab\'yv\'a se, inf $0$, nenab\'yv\'a se
	\pr max $\frac {25}4$ v bod\v{e} $[\frac 52,\frac 52]$, $\inf$ neexistuje
	\pr max $300$ v bodech $[0,\pm 10, 0]$, min $-100$ v bodech $[0,0,\pm 10]$\\[10pt]
\textbf{2.}\resetpr \pr max $3$ v $[\frac13,-\frac23,\frac23]$, min $-3$ v $[-\frac13,\frac23,-\frac23]$
	\pr max $\sqrt{\frac {26}{3}}$ v $[\frac2{\sqrt{78}},-\frac7{\sqrt{78}}, \frac5{\sqrt{78}}]$, min  $-\sqrt{\frac {26}{3}}$ v $[-\frac2{\sqrt{78}},\frac7{\sqrt{78}}, -\frac5{\sqrt{78}}]$
	\pr max $\frac18$ v $[\frac\pi6,\frac\pi6,\frac\pi6]$, inf $0$, nenab\'yv\'a se
	\pr max ${\frac{a^6}{6^6}}$ v $[\frac a6,\frac a6,\frac a6]$, inf $-\infty$
	\pr max 2 v $[1, 1]$, min 0 v $[0,0]$
	\pr max $\sqrt{102}$ v $[\frac1{\sqrt{102}},-\frac1{\sqrt{102}},\frac{10}{\sqrt{102}}]$;
 min $-\sqrt{102}$ v $[-\frac1{\sqrt{102}},\frac1{\sqrt{102}},-\frac{10}{\sqrt{102}}]$\\[10pt]
\textbf{3.}\resetpr \pr max $7 + 2\sqrt{6}$ v $[-\frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}]$; min $7 - \frac{10}{\sqrt{45}}$ v $[-\frac{5}{\sqrt{45}},\frac{2}{\sqrt{45}},\frac{4}{\sqrt{45}}]$
	\pr max v bodech $[\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2},\sqrt{\frac{5}{2}}]$ a $[ - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2},\sqrt{\frac{5}{2}}]$; min v bodech $[\frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}, - \sqrt{\frac{5}{2}}]$ a $[ - \frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}, - \sqrt{\frac{5}{2}}]$
	\pr max v bodech $[-\frac{1}{\sqrt{2}}, - \frac{2}{\sqrt{2}}, \pm \sqrt{\frac{3}{2}}]$; min v bodech $[\frac{2}{3}, \frac{4}{3}, \pm \frac{\sqrt{11}}{3}]$
	\pr max $\sqrt{6}$ v bod\v{e} $[\sqrt{2}, \frac{2}{\sqrt{3}}, \frac{1}{\sqrt{3}}]$; min $-\frac{3}{25}(16 + \sqrt{6})$ v bod\v{e} $[\frac{2  -3\sqrt{6}}{5}, \frac{2  + 2\sqrt{6}}{5}, \frac{1 + \sqrt{6}}{5}]$
	\pr max $9$ v bod\v{e} $[3,0,0]$; min $-\frac{25}{3}$ v bodech $[\frac{1}{3}, \pm \frac{2}{3}\sqrt{19}, \frac{2}{3}]$\\[10pt]
	\centerline{\MakeUppercase{v. matice}}\ \\
	\textbf{1.}\resetpr \pr 3 \pr 3 pro $a\neq 1$, 1 pro $a = 1$ \pr 3 pro $a\neq0,-1,2$, jinak 2 \pr 3 pokud $x=6$ nebo $y=\tfrac{497}{23}$, jinak 4 \pr 2 pokud $(x,y)=(11,13)$, jinak 3\\[5pt]
\textbf{2.}\resetpr \pr $AB = \begin{pmatrix}7&1&1&7\\9&3&-3&-3\\2&0&1&4\end{pmatrix}$, $BA$ nen\'i definov\'ana \pr $AB = \begin{pmatrix}7&6&1\\4&-2&2\\1&8&3\end{pmatrix}$, $BA = \begin{pmatrix}9&0&11\\3&0&7\\-1&6&-1\end{pmatrix}$\\[5pt]
\textbf{3.}\resetpr \pr $\begin{pmatrix} 3 & -1 & 3 \\ -2 & 1 & -2\\ -2 & 1 & -1 \end{pmatrix}$ \pr $\begin{pmatrix} -8 & 29 & -11 \\ -5 & 18 & -7\\ 1 & -3 & 1 \end{pmatrix}$ \pr $\begin{pmatrix}2& 2& 2& -3\\ -1& -1& -1& 2\\ 1& 1& 2& -2\\ 2& 1& 2& -3\end{pmatrix}$ \pr $\begin{pmatrix}-2& 1& x& -1\\ -3& 1& x+2& -1\\ \tfrac{13}{2}& -\tfrac{11}{4}& -3x-2& 3\\ 0& 0& \tfrac{1}{7}& 0\end{pmatrix}$\\[5pt]
\textbf{4.}\resetpr \pr -36 \pr 1 \pr 1 \pr $2x - 14$, $\operatorname{det}A = \operatorname{det}(A^{1})$ pro $x\in\{\tfrac{15}{2},\tfrac{13}{2}\}$\\[5pt]
\textbf{5.}\resetpr \pr $x=5$, $y=-3$, $z=-2$ \pr nem\'a \v{r}e\v{s}en\'i \pr \v{R}e\v{s}en\'im jsou vektory $(\tfrac{-25-18t}{11},\tfrac{37 + 2t}{11},t)$, $t\in\er$ \pr \v{R}e\v{s}en\'im jsou vektory $(1+t/4,-t/4,1,t)$, $t\in\er$\\[5pt]
\textbf{6.} Pro \textbf{b}$_1: (6,-28,-16,19)$; pro \textbf{b}$_2: (-14,-160,-75,111)$; pro \textbf{b}$_3: (3,10,4,-7)$\\[5pt]
\textbf{7.} Determinant je nula pro $p\in\{0,-\tfrac{7}{4}\}$. Pro $p = -\tfrac{7}{4}$ soustava nem\'a \v{r}e\v{s}en\'i, pro $p=0$ jsou \v{r}e\v{s}en\'im soustavy vektory $(-t,-2-3t,t,1)$, $t\in\er$.\\[10pt]
\centerline{\MakeUppercase{vi. \v{r}ady}}\ \\
\noindent \textbf{1.}\resetpr \pr K \pr K \pr D \pr D \pr K \pr K \pr K \pr D \pr K \pr D \pr K \pr D \pr K pro $A\in(0,1)$, jinak D \pr K \pr K pro $x < -1$, jinak D\\[5pt]
\textbf{2.}\resetpr \pr K, ale ne absolutn\v{e} \pr AK \pr D \pr K, ale ne absolutn\v{e} \pr D \pr AK \pr K, ale ne absolutn\v{e}\\[5pt]
\textbf{3.}\resetpr \pr AK pro $|x| < 3$, K neabsolutn\v{e} pro $x = -3$ a jinak D \pr AK pro $x > 1/2$, jinak K neabsolutn\v{e} \pr D (nen\'i spln\v{e}na nutn\'a podm\'inka konvergence) \pr AK pro $x\in (0,1)$, K neabsolutn\v{e} pro $x = 1$, jinak D \pr D \pr AK pro $x < -2$, jinak D
\end{document}