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\usepackage{verbatim} %pro vynechavani comment
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\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}
\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 - Teorie m\'iry a integr\'alu, MFF}  % tyto dvě položky jsou zde v podstatě formálně, ve skutečnosti nejsou nikde 
\author{Marek C\'uth}

\begin{document}
 \centerline{\textbf{V\'YSLEDKY}}\ \\	
% \centerline{\MakeUppercase{logick\'e h\'adanky}}\ \\
%\textbf{1.} \pr B padouch, C poctivec \pr A poctivec, B padouch \pr A poctivec, B poctivec \pr ano \resetpr
%\textbf{2.} \pr ano \pr ano \pr D vinen, u ostatn\'ich to nelze dok\'azat \resetpr
%\textbf{3.} \pr pondeli \pr to nem\r{u}\v{z}e \v{r}\'ict nikdy \pr pondeli nebo streda
%\textbf{4.} \pr plyne z toho, \v{z}e miluji Veroniku; neplyne z toho, \v{z}e miluji Janu \pr miluji ob\v{e} dv\v{e} \pr poctivec \resetpr
%\textbf{5.} \pr nelze ur\v{c}it jestli je A poctivec, ale na ostrov\v{e} je ur\v{c}it\v{e} poklad \pr ano, dozv\v{e}d\v{e}l bych se, \v{z}e na ostrov\v{e} poklad nen\'i \resetpr
%\textbf{6.} \pr rozumn\'y \v{c}lov\v{e}k \pr pomaten\'y up\'ir \pr ano, Dracula \v{z}ije \pr ano, Dracula je mrtev
\centerline{\MakeUppercase{i. opakov\'an\'i st\v{r}edo\v{s}kolsk\'e l\'atky}}\ \\
\textbf{1.} a) $(4;6]$\quad b) $(-6;-3)\cup(\tfrac{-1-\sqrt{13}}{2},\tfrac{-1+\sqrt{13}}{2})$\quad c) $[1;2]$\quad d) $\tfrac{4}{3}$\\ e) $\bigcup_{k\in\zet}{((-\frac \pi2 + 2k\pi; \frac \pi6 + 2k\pi)\cup(\frac \pi2 + 2k\pi; \frac {5\pi}{6} + 2k\pi))}$\\
 f) $(-\infty,3)\cup(3,2+\sqrt{6}]$\quad g) $[\tfrac{1}{2},\infty)$\\
 h) $\{0,\tfrac{2}{3}\}$\quad i) $(-10;-7)\cup(10;\infty)$\quad j) $\{7,\tfrac{\sqrt{7}}{7}\}$\\[5pt]
\textbf{3.} a) $c\leq 2\implies x\in(-\infty,\tfrac{-2-\sqrt{4-2c}}{2})\cup (\tfrac{-2+\sqrt{4-2c}}{2},\infty)$; $c > 2\implies x\in\er$\\
 b) $x\in (-e^{\pi/2-c},e^{-\pi/2 - c})\cup (e^{-\pi/2 - c},e^{\pi/2-c})$\\[10pt]
\centerline{\MakeUppercase{ii. v\'yroky, suprema, infima}}\ \\
\textbf{1. }\resetpr \pr Plat\'{\i}, negace $\exists x \in \en \ \forall y \in \en \
\exists z \in \en \, (z > x \ \&\ y \ge z)$;\\
 \pr Plat\'{\i}, negace $\forall y\in\en\ \exists x\in\en\ \exists z\in\en\ (z>x\ \&\ y\ge<z)$;\\
 \pr Neplat\'{\i}, negace $\forall x \in \en \ \exists y \in \en \ \exists z \in \en \, (z > x \ \&\ y \ge z)$;\\
 \pr Plat\'{\i}, negace $\forall x \in \en \ \exists y \in \en \ \exists z \in \en \, (z < x \ \&\ y \le z)$;\\
 \pr Plat\'{\i}, negace $\exists x\in\er \ \exists \varepsilon>0\ \forall\delta>0\ \exists y\in\er\, (|y-x|<\delta\ \&\ y\geq x+\frac \varepsilon3)$.\\[5pt]
\textbf{2. }\resetpr \pr $\sup A = 0, \min A = -1$;\quad
b) $\max B = \frac 32, \min B = 0$;\quad
c) $C_1$ nem\'a supremum ani infimum (nen\'i zdola, ani shora omezen\'a), $C_2$ nen\'i shora omezen\'a, $\min C_2 = 3$, $\max C_3 = 0$, $C_3$ nen\'i zdola omezen\'a;\quad
d) $\max D_1 = \frac 56$, $\inf D_1 = 0$, $D_2$ nen\'i shora omezen\'a, $\inf D_2 = 0$;\quad
e) $E$ nen\'i shora omezen\'a, $\inf E = 0$;\quad
f) $F$ nen\'i shora omezen\'a, $\inf F = 0$.\\[10pt]
\centerline{\MakeUppercase{iii. limity posloupnost\'i}}\ \\
\textbf{1.}\resetpr \pr 0 \pr 2 \pr 2 \pr 0 \pr 0 \pr Nem\'a limitu \pr 1/2 \pr 1 \pr $\tfrac{81}{2^{13} - 1}$ \pr 1/3 \pr 1/2 \pr 0 \pr 1/2 \pr 0 \pr 0 \pr 1 \pr 0\\
\textbf{2.}\resetpr \pr 0 \pr 2 \pr 0 \pr $-\tfrac{4}{5}$ \pr 5 \pr $\tfrac{1}{3}$ \pr $\sqrt{1+\sqrt{2}}$ \pr $\tfrac{2}{\sqrt{3}}$ \pr $\tfrac{3}{2}$\\
\textbf{3.}\resetpr \pr $-\frac{7^7}{3^6}$ \pr 30 \pr 2 \pr $-54\cdot 5^{10}\cdot\sqrt[6]{21}$ \pr $-5^{35}$  \pr $\frac{11}{9}$\\[10pt] %\pr $\frac 34$ \pr $-\frac{50}{99}$\\
\centerline{\MakeUppercase{v. limita funkc\'i}}\ \\
\textbf{1.}\resetpr \pr $\frac 12$ \pr 1 \pr $\tfrac{2}{3}$ \pr neexistuje \pr $+\infty$ \pr $\frac{3^{10}}{2^{10}}$ \pr $\infty$ \pr 0\\
\textbf{2.}\resetpr \pr 0 pro $n<1$, 1 pro $n=1$, pro $n>1$ sud\'e limita neexistuje a pro $n>1$ lich\'e je limita $\infty$ \pr 0 pro $n<1$, $-\frac{1}{12}$ pro $n=1$, pro $n>1$ sud\'e limita neexistuje a pro $n>1$ lich\'e je limita $-\infty$ \pr $\frac{12}{5}$ \pr $-\frac{1}{16}$\\
\textbf{3.}\resetpr \pr 2 \pr $-\frac 12$ \pr 1 \pr $\frac 32$ \pr neexistuje \pr $\frac 12$ \pr $-\frac {1}{12}$ \pr $\frac 14$ \pr $\frac 43$\\
\textbf{4.}\resetpr \pr 0 \pr $e^3$ \pr $\frac 15$ \pr neexistuje \pr $e$ \pr 1 \pr $\frac 32$ \pr $e^{-1}$ \pr $\infty$ \pr $\frac 23$ \pr $3\log 2$ \pr 1 \pr 0 pro $\alpha<\frac 12$, $\sqrt{2}$ pro $\alpha=\frac 12$, $\infty$ pro $\alpha>\frac 12$ \pr 2 \pr -1\\
\textbf{5.}\resetpr \pr $\sqrt{2}$ \pr $\frac {1}{3\log 3}$ \pr limita neexistuje (zprava $\sqrt {2}$, zleva $-\sqrt{2}$) \pr $\frac92$ \pr $\frac34$ \pr -6 \pr $\sqrt{e}$\\ \\
[10pt]
 \centerline{\MakeUppercase{vi. spojitosti a derivace}}\ \\ 
\textbf{1.}\resetpr \pr $f$ je definov\'ana a spojit\'a na $\er$, $f'(x)=87(x^2+51x+119)^{86}\cdot(2x+51)$.\\
  \pr $f$ je definov\'ana a spojit\'a na $\er$, $f'(x)= 3(x+15)^2(x-17)^{10}x^9 +10 (x+15)^3(x-17)^{9}x^9 +
    9(x+15)^3(x-17)^{10}x^8$.\\
  \pr $f$ je definov\'ana a spojit\'a na $(-\infty,-1)\cup(-1,0)\cup(0,+\infty)$; $\lx[0]f(x)=\lx[-1]f(x)=+\infty$,
nelze spojit\v{e} roz\v{s}\'{\i}\v{r}it. Pro $x\in D_f$ m\'ame $$f'(x)={\left((2x+1)e^{x^2+1}\cdot \cos x - e^{x^2+1}\cdot \sin x\right)\cdot (x+1)\cdot \log(x^2+1) - e^{x^2+1}\cdot \cos x\cdot\left(2\cdot \log(x^2+1)+{2x(x+1)\over x^2+1}\right) \over (x+1)^3\cdot \log^2(x^2+1)}.$$\\
  \pr $f$ je definov\'ana a spojit\'a na $\er$, $f'(x)={\frac{2x+1}{x^2+x+1}}$\\
  \pr $f$ je definov\'ana a spojit\'a na $\er$, $$f'(x)=-18\cos\left(\cos\left( (x^3+17x^2-56x+1)^{18} \right)
\right)\cdot\sin\left( (x^3+17x^2-56x+1)^{18} \right) \cdot (x^3+17x^2-56x+1)^{17}\cdot(3x^2+34x-56) $$.\\
  \pr $f$ je definov\'ana a spojit\'a na $(0,\infty)$, $\lx[0+]f(x)=+\infty$,
nelze spojit\v{e} roz\v{s}\'{\i}\v{r}it. $f'(x)=\left(\frac1x\right)^{\frac1x+2}\cdot(\log x-1)$ pro $x>0$.\\
  \pr $f$ je definov\'ana a spojit\'a na $\bigcup\limits_{k\in\zet}(2k\pi,(2k+1)\pi)$. $\lx[2k\pi+] f(x)=0$, $\lx[(2k+1)\pi-]
f(x)=+\infty$, lze tedy spojit\v{e} roz\v{s}\'{\i}\v{r}it na $\bigcup\limits_{k\in\zet}\langle 2k\pi,(2k+1)\pi)$. $f'(x)=(\sin x)^{\cos x}\cdot({\cos^2 x\over \sin x}-\sin x\log\sin x)$ pro $x\in \bigcup\limits_{k\in\zet}(2k\pi,(2k+1)\pi)$. Po dodefinov\'an\'{\i} je $f'_+(2k\pi)=1$.\\
  \pr $f$ je definov\'ana a spojit\'a na $\langle-1,1)$, $\lx[1-]f(x)=-\infty$. $f'(x)=-{1\over\arccos x\cdot\sqrt{1-x^2}}$ pro $x\in(-1,1)$, $f'_+(-1)=-\infty$.\\
  \pr $f$ je definov\'ana a spojit\'a na $\langle 0,+\infty)$. $f'(x)=\arcsin\sqrt{x\over x+1}$ pro $x>0$, $f'_+(0)=0$.\\
  \pr $f$ je definov\'ana a spojit\'a na $\er$, $f'(x)={e^x-1\over e^{2x}+1}$ pro $x\in\er$.\\
	\pr $f$ je definov\'ana a spojit\'a na $\er$, $f'(x)=5$ pro $x\in (1,4)$; $f'(x)=2x$ pro $x\in (-\infty,1)\cup (4,\infty)$; $f'_+(1)=f'_-(4)=5$, $f'_-(1)=2$, $f'_+(4)=8$.\\
	\pr $D_f = \er$, $f$ je spojit\'a v ka\v{z}d\'em bod\v{e} $\er\setminus(\zet\setminus\{−3, 3\})$, v bodech $\zet\setminus\{−3, 3\}$ je spojit\'a zprava a nespojit\'a zleva.
	$f'(x) = \tfrac{2}{3}x[x](x^2-9)^{-2/3}$ pro $x\in \er\setminus\zet$; $f'(-3)=f'(3)=\infty$; pro $k\in\zet\setminus\{-3,3\}$ je $f'_+(k)=\tfrac{2}{3}k^2(k^2-9)^{-2/3}$ a $f'_-(k)=\begin{cases}\infty,\quad& |k|>3,\\ -\infty,& |k|<3.\end{cases}$\\
	\pr $D_f = \bigcup_{k\in\zet}(-\tfrac{\pi}{2} + k\pi,\tfrac{\pi}{2} + k\pi)$, $f$ je spojit\'a v ka\v{z}d\'em bod\v{e} $D_f$. $f'(x) = -2\sin 2x\cdot(\tg x - 1)\cdot\sgn(\cos 2x) + |\cos 2x|\tfrac{1}{\cos^2x}$ pro $x\in\bigcup_{k\in\zet}(-\tfrac{\pi}{2} + k\pi,-\tfrac{\pi}{4} + k\pi)\cup(-\tfrac{\pi}{4} + k\pi, \tfrac{\pi}{4} + k\pi)\cup (\tfrac{\pi}{4} + k\pi,\tfrac{\pi}{2} + k\pi)$; $f'(\tfrac{\pi}{4} + k\pi) = 0$, $f'_+(-\tfrac{\pi}{4} + k\pi)=-4$, $f'_-(-\tfrac{\pi}{4} + k\pi)=4$ pro $k\in\zet$.\\[20pt]
	\centerline{\MakeUppercase{vii. pr\r{u}b\v{e}h funkce}}\ \\ 
	\textbf{1.}\resetpr \pr $D_f = \er\setminus\{-1,1\}$, $f$ je spojit\'a v ka\v{z}d\'em bod\v{e}	$D_f$ a je lich\'a, sta\v{c}\'i tedy vy\v{s}et\v{r}ovat na $\langle 0,+\infty)$, tj. na $\langle 0,1)\cup(1,+\infty)$. $f(0)=0$, limita v $1$ je $+\infty$, v $+\infty$ je $+\infty$. Pro $x\in \langle 0,1)\cup(1,+\infty)$ je $f'(x) = \tfrac{\sgn(x^4 - 1)x^2(x^4 - 3)}{\sqrt{|x^4 - 1|^3}}$, $f$ je rostouc\'i na $\langle 0, 1)$, klesaj\'ic\'i na $(1,\sqrt[4]{3}\rangle$, rostouc\'i na $\langle \sqrt[4]{3}, +\infty)$, v $\sqrt[4]{3}$ je lok\'aln\'i minimum, $H_f = \er$. Pro $x\in \langle 0,1)\cup(1,+\infty)$ je $f''(x) = \tfrac{6x(x^4 + 1)}{\sqrt{|x^4 - 1|^5}}$, $f$ je konvexn\'i na $(0,1)$ a na $(1,+\infty)$ (z lichosti konk\'avn\'i na $(-\infty,-1)$ a na $(-1,0)$). V bod\v{e} $0$ m\'a inflexn\'i bod. Asymptota v $+\infty$ i v $-\infty$ je $y=x$.\\[3pt]
	\pr $D_f = \er$, $f$ je spojit\'a v ka\v{z}d\'em bod\v{e}	$D_f$, limita v $-\infty$ je $+\infty$, v $+\infty$ je $0$. Pro $x\in \er\setminus\{-1,1\}$ je $f'(x) = \sgn(1-x^2)e^{-x}(x^2-2x-1)$, $f'_-(-1)=-2e$, $f'_+(-1)=2e$, $f'_-(1)=-2e^{-1}$, $f'_+(1)=2e^{-1}$, $f$ je klesaj\'ic\'i na $(-\infty,-1\rangle$, rostouc\'i na $\langle -1, 1-\sqrt{2}\rangle$, klesaj\'ic\'i na $\langle 1-\sqrt{2},1\rangle$, rostouc\'i na $\langle 1, 1+\sqrt{2})$, klesaj\'ic\'i na $\langle 1+\sqrt{2}, +\infty)$, v bodech $-1$ a $1$ m\'a minimum, v bodech $1-\sqrt{2}$ a $1+\sqrt{2}$ lok\'aln\'i maximum, $H_f = [0,\infty)$. Pro $x\in \er\setminus\{-1,1\}$ je $f''(x) = \sgn(1-x^2)e^{-x}(x^2-4x+1)$, $f$ je konvexn\'i na $(-\infty,-1 \rangle$, konk\'avn\'i na $\langle -1,2-\sqrt{3}\rangle$, konvexn\'i na $\langle 2-\sqrt{3}, 1 \rangle$, konk\'avn\'i na $\langle 1, 2+ \sqrt{3}\rangle$, konvexn\'i na $\langle 2 + \sqrt{3}, +\infty)$. V bodech $2-\sqrt{3}$ a $2+\sqrt{3}$ m\'a inflexn\'i body. Asymptotu v $-\infty$ nem\'a, v $+\infty$ m\'a asymptotu $y=0$.\\[3pt]
	\pr, \pr, \pr: viz. v\'ysledky zkou\v{s}kov\'e p\'isemky z roku 2008/2009, varianty E, D, B\\ \url{http://www.karlin.mff.cuni.cz/~kalenda/pis-fsv/0809/pis.htm}
\end{document}