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\def\ce{\mathbb C} 
\def\en{\mathbb N} 
\def\qe{\mathbb Q} 
\def\zet{\mathbb Z}
\def\er{\mathbb R}
\newcommand\lx[1][0]{\displaystyle\lim_{x\to #1}}
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\def \sgn {\operatorname{sgn}}
\def \dx {\, \mathrm{d} x}
\def \tg {\operatorname{tg}}
\def \arctg {\operatorname{arctg}}

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\title{Cvi\v{c}en\'i - Matematick\'a anal\'yza 1, 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{i. opakov\'an\'i st\v{r}edo\v{s}kolsk\'e l\'atky, v\'yroky}}\ \\
 \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]
	\begin{tabular}{ll}
	\textbf{4.}&\resetpr \pr $c > 1/4\implies x\in\er$; $c\in (0,1/4]\implies x\in (-\infty,\tfrac{-1-\sqrt{1-4c}}{2c})\cup (\tfrac{-1+\sqrt{1-4c}}{2c},\infty)$; $c=0\implies x > -1$;\\
		& \qquad $c < 0\implies x\in (\tfrac{-1+\sqrt{1-4c}}{2c}, \tfrac{-1-\sqrt{1-4c}}{2c})$\\
		& \pr $c > 0\implies x\in\emptyset$; $c = 0\implies x\in\er$; $c < 0\implies x < \log(-1/c)$\\
		& \pr $c<0\implies x\in\er$; $c = 0\implies x\in\er\setminus\{\pi/2 + k\pi:\;k\in\zet\}$;\\ &\qquad $c\in (0,1)\implies x\in\bigcup_{k\in\zet}(-\arccos c + k\pi,\arccos c + k\pi)$; $c\geq 1\implies x\in\emptyset$\\
		& \pr $x\in (-e^{\pi/2-c},e^{-\pi/2 - c})\cup (e^{-\pi/2 - c},e^{\pi/2-c})$\\
		& \\
 \textbf{5. }&\resetpr \pr $\forall m\in M\ \forall z\in Z: S(m,z)\Rightarrow L_1(m,z),$\\
      & \pr $\forall z\in Z\ \exists m\in M: L_1(m,z),$\\
      & \pr $\forall z\in Z\ \forall m_1\in M\ \forall m_2\in M : S(m_1,z)\ \&\ S(m_2,z)\Rightarrow m_1=m_2,$\\
      & \pr $\forall m\in M\ \forall z_1\in Z\ \forall z_2\in Z : S(m,z_1)\ \&\ S(m,z_2)\Rightarrow z_1=z_2$ (NE);\\
& \pr $\exists z\in Z\ \exists m\in M: S(m,z)$;\\
& \pr $ \exists m\in M\ \exists z\in Z : S(m,z)$ (ANO);\\
& \pr Existuje nev\v{e}rn\'a man\v{z}elka (aspo\v{n} jedna):\\
& $\qquad \exists  z\in Z\ \exists m_1\in M\ \exists m_2\in M (m_1\ne m_2 \ \&\ S(m_1,z)\ \&\ L_2(m_2,z))$,\\
& $\quad$ existuj\'{\i} nev\v{e}rn\'e man\v{z}elky (aspo\v{n} dv\v{e}):\\
& $\qquad \exists z_1\in Z\ \exists z_2\in Z \exists m_1\in M\ \exists m_2\in M\ \exists m_3\in M\ \exists m_4\in M$\\
& $\qquad \quad (z_1\ne z_2\ \& m_1\ne m_2 \ \&\ m_3\ne m_4\ \&\ S(m_1,z_1)\ \&\ L_2(m_2,z_1)\ \&\ S(m_3,z_2)\ \&\ L_2(m_4,z_2))$.\\
& \pr Existuje svobodn\'y mu\v{z}.\\
& \pr Existuje \v{z}ena, kter\'a \v{z}\'adn\'emu mu\v{z}i neop\v{e}tuje l\'asku.\\
& \pr Existuje \v{z}ena, j\'{\i}\v{z} \v{z}\'adn\'y mu\v{z} neop\v{e}tuje l\'asku.\\
& \pr Ka\v{z}d\'a \v{z}ena, kter\'a miluje n\v{e}jak\'eho mu\v{z}e, nemiluje n\v{e}kter\'eho mu\v{z}e, kter\'y miluje ji.\\
& \\
\textbf{6. }&\resetpr \pr (i), (ii), (iii) neplat\'i \pr (i), (ii) plat\'i; (iii) neplat\'i \pr (i) plat\'i; (ii), (iii) neplat\'i\\ &\pr (i), (ii), (iii) plat\'i\\
& \\
\textbf{7. }& \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 $\exists x\in\er \ \exists \varepsilon>0\ \forall\delta>0\ \exists y\in\er\, (|y-x|<\delta\ \&\ y\geq x+\frac \varepsilon3)$.\\
& \\
\textbf{8. } &\resetpr \pr $f(y)=15$ pro v\v{s}echna $y\in(2,12)$;\\
& \pr Funkce $f$ je na konstantn\'{\i} na $\er$.\\
\end{tabular}\ \\[10pt]
\centerline{\MakeUppercase{ii. d\r{u}kazov\'a technika, operace s mno\v{z}inami, bin\'arn\'i relace a zobrazen\'i}}\ \\
\textbf{3.}\resetpr \pr ano \pr ano \pr ne \pr ano \pr ne\\
\textbf{6.}\quad (i) plat\'i, (ii) neplat\'i, (iii) neplat\'i, (iv) neplat\'i\\
\textbf{7.}\quad (i) ano, (ii) ano, (iii) ne, (iv) ne, (v) ne, (vi) ne, (vii) ano, (viii) ne\\[10pt]
\centerline{\MakeUppercase{iii. suprema, infima}}\ \\
\textbf{1.}
a) $\sup A = 0, \min A = -1$;\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{iv. limity posloupnost\'i}}\ \\
\textbf{1.} a) $0$\quad b) $0$\quad c) $0$\quad d) $0$\\
\textbf{2.}\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 $\tfrac{1}{3}$ \pr $\tfrac{1}{2}$ \pr 0 \pr $\tfrac{1}{2}$ \pr 0 \pr 0 \pr 1 \pr 0\\
\textbf{3.}\resetpr \pr $\tfrac{1}{3}$ \pr $\tfrac{8}{50}$ \pr $\sqrt{1+\sqrt{2}}$ \pr Nem\'a limitu \pr 0 \pr 1 \pr 2 \pr 1 \pr 5 \pr $\tfrac{1}{3}$ \pr $\tfrac{2}{\sqrt{3}}$ \pr $\tfrac{3}{2}$ \pr Nem\'a limitu \pr 0 \pr 4 \pr $-\infty$ \pr $+\infty$ \pr 0\\
\textbf{4.}\resetpr \pr $-\frac{7^7}{3^6}$ \pr 30 \pr 2 \pr $-54\cdot 5^{10}\cdot\sqrt[6]{21}$ \pr $-5^{35}$  \pr $\frac{11}{9}$\\
\textbf{5.}\resetpr \pr 1 \pr 2 \pr $\sqrt{x}$ \pr 1 \pr $\sqrt{x}$ \pr $\sqrt[3]{x}$\\
\textbf{6.}\resetpr \pr $e^3$ \pr $\tfrac{1}{\sqrt{e}}$ \pr 1 \pr 0 \pr $\infty$\\
\textbf{7.}\resetpr \pr $\limsup_{n\to \infty} a_n = 1$, $\liminf_{n\to \infty} a_n = 0$
\pr $\limsup_{n\to \infty} a_n = \infty$, $\liminf_{n\to \infty} a_n = -\infty$
\pr $\limsup_{n\to \infty} a_n = 2$, $\liminf_{n\to \infty} a_n = 0$
\pr $\limsup_{n\to \infty} a_n = e$, $\liminf_{n\to \infty} a_n = -e$
\pr $\limsup_{n\to \infty} a_n = \infty$, $\liminf_{n\to \infty} a_n = 0$
\pr $\limsup_{n\to \infty} a_n = 1$, $\liminf_{n\to \infty} a_n = -\tfrac{1}{2}$
\pr $\limsup_{n\to \infty} a_n = 2$, $\liminf_{n\to \infty} a_n = 1$\\
\textbf{8.}\resetpr \pr $\{-\infty, 0, \infty\}$ \pr $\{-4,0,2,6\}$ \pr $\{-\infty, 1, \infty\}$ \pr $\{-e-\frac{\sqrt{2}}{2}, -e+\frac{\sqrt{2}}{2}, e, e - 1, e + 1\}$ \pr $\{-\frac{5}{3}, \frac{1}{3}, 1\}$\\
\textbf{9.}\resetpr \pr $\frac{1}{n}$ \pr $\frac{n^2}{3^n}$ \pr $\frac{1}{n^{2/3}}$ \pr $\frac{1}{n^{35/2}}$ \pr $\frac{1}{n^{5/36}}$ \pr $n$ \pr $\frac{1}{n}$\\
\textbf{10.} \textit{HINT: nejprve doka\v{z}te \v{z}e $|a_{n+2} - a_{n+1}|\leq \frac{1}{2}|a_{n+1} - a_n|$. Tuto nerovnost iterujte a doka\v{z}te $|a_{n+1} - a_n|\leq \frac{1}{2^{n-1}}|a_2 - a_1|$. To pou\v{z}ijte na odhad $|a_{n+k} - a_n|\leq |a_2-a_1| \cdot \sum_{i=n-1}^\infty \frac{1}{2^i}$ a s jeho pomoc\'i uka\v{z}te \v{z}e posloupnost je cauchyovsk\'a.}\\
\textbf{V\'ysledky uk\'azkov\'ych \'uloh:}\resetpr \pr 1 \pr $\frac{1}{e}$ \pr limita neexistuje \pr 0 \pr 3 \pr 0 \pr $+\infty$ \pr $\limsup a_n = e+1$, $\liminf a_n = -e-\frac{\sqrt{2}}{2}$ \\[10pt]
\centerline{\MakeUppercase{v. \v{r}ady}}\ \\
\textbf{1.}\resetpr \pr Diverguje (D) \pr D \pr D \pr Konverguje (K) \pr K \pr K \pr K \pr K \pr K \pr D \pr K \pr K \pr K \pr K \pr K\\
\textbf{2.}\resetpr \pr K \pr K \pr K pro $x < 0$, D pro $x \geq 0$ \pr K pro $x \leq 0$, D pro $x > 0$ \pr K \pr D \pr K \pr K \pr D \pr K \pr K \pr D \pr D \pr K\\
\textbf{3.}\resetpr \pr D \pr K \pr D \pr D\\
\textbf{4.}\resetpr \pr D (obecn\v{e}ji K $\iff a > 1$) \pr K (obecn\v{e}ji K $\iff a > 1$)\\
\textbf{5.}\resetpr \pr ne AK \pr AK \pr ne AK\\
\textbf{6.}\resetpr \pr K, ale ne absolutn\v{e} \pr D \pr K, ale ne absolutn\v{e} \pr K, ale ne absolutn\v{e} \pr D \pr K, ale ne absolutn\v{e} \pr AK\pr K, ale ne absolutn\v{e}\\
\textbf{7.}\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}\\
%\textbf{5.}\resetpr 
\pr D \pr K, ale ne absolutn\v{e} \pr K, ale ne absolutn\v{e} \pr K, ale ne absolutn\v{e} \pr K, ale ne absolutn\v{e} \pr AK\\[10pt]
\centerline{\MakeUppercase{vi. limita funkc\'i}}\ \\
\textbf{2.}\resetpr \pr 5 \pr $+\infty$ \pr 0 \pr 0 \pr neexistuje \pr $\lim_{x\to 2^+}f(x) = 0$, $\lim_{x\to 2^-}f(x) = 1$, $\lim_{x\to 2}f(x)$ neexistuje\\
\textbf{4.}\resetpr \pr $-\frac{1}{2}$ \pr $\frac{1}{2}$ \pr $1$ \pr $\frac{3^{10}}{2^{10}}$ \pr $\frac{49}{24}$
\textbf{5.}\resetpr \pr $+\infty$ \pr $\infty$ \pr 0\\
\textbf{6.}\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{7.}\resetpr \pr 2 \pr $\frac{1}{2}$ \pr $-\frac 12$ \pr 1 \pr $\frac 32$ \pr neexistuje \pr $\frac 12$ \pr 0 \pr $e^3$ \pr $e$ \pr 1 \pr $-\frac {1}{12}$ \pr $\frac 14$ \pr $\frac 43$ \pr -3 \pr $e^{-1}$ \pr $\frac{2}{3}$ \pr neexistuje\\
\textbf{8.}\resetpr \pr $\frac 15$ \pr $\frac{\sqrt{3}}{3}$ \pr 0 \pr 1 \pr $\frac 32$ \pr $\cos a$ \pr 0 \pr $\sqrt{2}$ \pr neexistuje \pr $3\log 2$ \pr 1 \pr -1 \pr $\frac 12$ \pr 1 \pr 2\\
\textbf{9.}\resetpr \pr $1$ \pr $2$ \pr AK \pr $\log 2$ \pr $\frac{1}{2}$ \pr AK\\
\textbf{10.}\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 $\sqrt{e}$\\
 \textbf{11.}\resetpr \pr $-\frac 15$ \pr 2\\
 \textbf{12.}\resetpr \pr AK \pr D \pr K, ale ne absolutn\v{e} \pr AK \pr K\\
 \textbf{13.}\resetpr \pr $\frac{1}{3\log 3}$ \pr $\sqrt[3]{6}$ \pr $\sqrt{2}$ \pr 1\\[10pt]
\centerline{\MakeUppercase{vii. spojitosti a derivace}}\ \\ 
\textbf{1.}\resetpr \pr spojit\'a na $\er$ \pr spojit\'a na $(0,\infty)\setminus\en$ a v bod\v{e} $x = 1$, v bodech $x\in \en\setminus\{1\}$ spojit\'a zprava a nespojit\'a zleva \pr spojit\'a v $(0,\pi)$, v bod\v{e} $x=0$ spojit\'a zprava (spojitost zleva nem\'a smysl vy\v{s}et\v{r}ovat), v bod\v{e} $x=\pi$ nen\'i spojit\'a zleva (spojitost zprava nem\'a smysl vy\v{s}et\v{r}ovat)\\
\textbf{2.} $A = - 1$\\
\textbf{3.}\resetpr \pr $f'(x) = \frac{2\sgn(x)}{1+x^2}$ pro $x\in\er\setminus\{0\}$, $f'_+(0) = 2$, $f'_-(0) = -2$.\\
\pr $f'(x) = 2x\exp(-|x-1|) - x^2\exp(-|x-1|)\sgn(x-1)$ pro $x\in\er\setminus\{1\}$, $f'_+(1) = 1$, $f'_-(1) = 3$.\\
\pr $f'(x) = \frac{xe^{-x^2}}{\sqrt{1-e^{-x^2}}}$ pro $x\in\er\setminus\{0\}$, $f'_+(0) = 1$, $f'_-(0) = -1$.\\
\pr $f'(x)=\arcsin\sqrt{x\over x+1}$ pro $x>0$, $f'_+(0)=0$.\\
\pr $f'(x)=-{1\over\arccos x\cdot\sqrt{1-x^2}}$ pro $x\in(-1,1)$, $f'_+(-1)=-\infty$.\\
\pr $f'(x)=(\sin x)^{\cos x}\cdot({\cos^2 x\over \sin x}-\sin x\log\sin x)$ pro $x\in (0,\pi)$, $f'_+(0)=1$.\\
\textbf{4.}\resetpr \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 $k\in\zet$.\\
	\pr $f'(x)=\begin{cases}2x(\sin\tfrac{1}{x} + \cos\tfrac{1}{x}) +  \sin\tfrac{1}{x} - \cos\tfrac{1}{x}& x\neq 0,\\
	0 & x = 0.\end{cases}$\\
	\pr $D_f = \er$, $f$ spojit\'a v ka\v{z}d\'em bod\v{e} $\er\setminus\{k\pi:\; k\in\zet\}$, v bodech $\{2k\pi:\; k\in\zet\}$ je spojit\'a zprava a nespojit\'a zleva, v bodech $\{(2k+1)\pi:\; k\in\zet\}$ je spojit\'a zleva a nespojit\'a zprava. Pro $k\in\zet$ m\'ame: $f'(x) = 0$ pro $x\in(2k\pi,(2k+1)\pi)$; $f'(x) = \sin x$ pro $x\in((2k-1)\pi,2k\pi)$; $f'_+(2k\pi) = f'_-((2k+1)\pi)=0$; $f'_-(2k\pi) = f'_+((2k+1)\pi)=+\infty$.\\
	\pr $f$ je definov\'ana a spojit\'a na $\er$, $f'(x) = (e + |x|)^x\cdot (\log(e+|x|) + \tfrac{|x|}{e+|x|})$ pro $x\in\er$\\(pozor, $f'(0) = 1$ je pot\v{r}eba spo\v{c}\'itat zvl\'a\v{s}\v{t})\\
	\centerline{\MakeUppercase{viii. aplikace derivac\'i}}\ \\ 
	\textbf{1. }\resetpr \pr 2 \pr -2 \pr $-\tfrac 13$ \pr $\tfrac 13$ \pr $-\tfrac{e}{2}$ \pr -2 \pr $\tfrac 12$ \pr $e^{1/6}$ \pr $a^a(\ln a - 1)$\\
	\textbf{2.} $\max f(x) = f(2) = 4$, $\min f(x) = f(-3) = -16$\\
	\textbf{3.} \resetpr \pr maximum neexistuje (ani lok\'aln\'i), glob\'aln\'i minimum v bod\v{e} $\tfrac{7}{5}$ \pr glob\'aln\'i maximum v bod\v{e} 1, glob\'aln\'i minimum v bod\v{e} 0, lok\'aln\'i maximum v bod\v{e} -1\\
	\textbf{6.} \resetpr \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}\\
	\pr, \pr: viz. sepsan\'e \'ulohy od Petra Po\v{s}ty (seps\'ano v roce 2010/11)\\ \url{http://www.karlin.mff.cuni.cz/~pposta/priklady/PrubFR.pdf}	
\end{document}
