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\newtheorem*{thm}{V\v{e}ta}
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\def\ce{\mathbb C} 
\def\en{\mathbb N}
\def\qe{\mathbb Q}
\def\zet{\mathbb Z}
\def\er{\mathbb R}
\def\Rr{\mathbb R^*}
\def\R{\er}
\def\N{\en}
\def\nec{Nech\v t\ }
\def\eqdef{\stackrel{\text{\tiny def}}{=}}
\def\Mod{\operatorname{mod}}
\def\lp{\displaystyle\lim_{n\to\infty}}
\newcommand\lx[1][0]{\displaystyle\lim_{x\to #1}}
\def \cotg {\operatorname{cotg}}
\def \tg {\operatorname{tg}}
\def \arctg {\operatorname{arctg}}
\def \suma {\sum_{n=1}^\infty}
\def \sumad {\sum_{k=2}^\infty}
\def \dx {\, \mathrm{d} x}
\def\lp{\displaystyle\lim_{n\to\infty}}
\def\C{\mathcal{C}}
\def\ov{\overline}
\def\Int{\operatorname{Int}}
\def\dist{\operatorname{dist}}
\def\setsep{:\;}

\newcounter{Priklad}
\addtocounter{Priklad}{1}
\newcommand{\pr}{\quad\alph{Priklad})\;\addtocounter{Priklad}{1}}
\newcommand{\resetpr}{\setcounter{Priklad}{1}}

\title{Cvi\v{c}en\'i - Matematick\'a anal\'yza 2, MFF}  % tyto dvÄ› poloĹľky jsou zde v podstatÄ› formĂˇlnÄ›, ve skuteÄŤnosti nejsou nikde 
\author{Marek C\'uth}

\begin{document} \pgfplotsset{width=7cm, compat=1.13}
\centerline{\MakeUppercase{i. l'Hospitalovo pravidlo a taylor\r{u}v polynom}}\ \\
\textbf{1. Spo\v{c}t\v{e}te limity (m\r{u}\v{z}ete pou\v{z}\'ivat l'Hospitalovo pravidlo)}\\[5pt]
\resetpr \pr $\lx \frac{\tg x -x }{x - \sin x}$ % 2
\pr $\lx \frac{3\tg 4x - 12 \tg x}{3\sin 4x-12\sin x}$ % -2
\pr $\lx \frac{x\cotg x-1}{x^2}$ % -1/3
\pr $\lx[\tfrac{\pi}{4}] \frac{\sqrt[3]{\tg x} - 1}{2\sin^2x - 1}$\\ % 1/3
\pr $\lx \frac{(1+x)^{1/x} - e}{x}$ % -e/2
\pr $\lx[1] \frac{x^x-x}{\ln x-x+1}$ % -2
\pr $\lx \left(\frac{1}{x} - \frac{1}{e^x - 1}\right)$ % 1/2
\pr $\lx \left(\frac{\arcsin x}{x}\right)^{1/x^2}$\\ % e^{1/6}
\pr $\lx[a] \frac{a^x - x^a}{x-a},\; a&gt;0$ % a^a (\ln a - 1)
\\[10pt]
 \textbf{P\v{r}ipome\v{n}me si:}\\[5pt]\resetpr
\pr Pro $x\in\er$ plat\'i:\quad $\sin(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$,\quad
$\cos(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}$,\quad
$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$.\\
\pr Pro $x\in (-1,1\rangle$:\quad $\log(1+x) = \sum_{n=1}^k (-1)^{n+1}\frac{x^n}{n}$\\
\pr Pro $x\in (-1,1)$ a $a\in\er$:\quad $(1+x)^a = \sum_{n=0}^k {a \choose n}x^n$,\;kde ${a \choose n} = \frac{a\cdot(a-1)\cdot(a-2)\cdots(a-n+1)}{n!}$.\\[10pt]
\textbf{2. Najd\v{e}te Taylor\r{u}v polynom $k$-t\'eho \v{r}\'adu v bod\v{e} 0 pro funkce:}\\[5pt]
a) $\tg(x),\;k=4$\quad
b) $\cos(\sin x),\;k=5$\quad
c) $\sin(\sin x),\;k=6$\quad
d) $\sin(1- \cos x),\;k=3$\quad
e) $\frac{1+x+x^2}{1-x+x^2},\;k=4$\\
f) $e^{2x-x^2},\;k=5$\quad
g) $\frac{x}{e^x-1},\;k=4$\quad
h) $\log(\cos x),\;k=6$\quad\\[10pt]
\textbf{3. Odhadn\v{e}te absolutn\'i chybu aproximace dan\'ych funkc\'i na dan\'ych intervalech:}\\[5pt]
a) $\sin x \approx x- \frac{x^3}{6},\;x\in[-1/2,1/2]$\quad
b) $\sqrt{1+x} \approx 1+ \frac{x}{2} - \frac{x^2}{8},\;x\in[0,1]$\\[10pt]
\textbf{4. Spo\v{c}t\v{e}te $\sqrt{e}$ s p\v{r}esnost\'i $10^{-2}$}\\[10pt]
\textbf{5. Spo\v{c}t\v{e}te limity}\\[5pt]
a) $\lx \frac{\cos x - e^{-\frac{x^2}{2}}}{x^4}$\quad
b) $\lx \frac{e^x\sin x - x(1+x)}{x^3}$\quad
c) $\lx[\infty] (\sqrt[6]{x^6+x^5} - \sqrt[6]{x^6-x^5})$\\
d) $\lx[\infty] x^{3/2}(\sqrt{x+1} + \sqrt{x-1} - 2\sqrt{x})$\quad
e) $\lx \frac{a^x+a^{-x}-2}{x^2}\;(a&gt;0)$\quad
f) $\lx\left(\tfrac{1}{x} - \tfrac{1}{\sin x}\right)$\\
g) $\lx[\infty] (x-x^2\log(1+\tfrac{1}{x}))$\quad
h) $\lx\tfrac{1}{x}\left(\tfrac{1}{x} - \tfrac{\cos x}{\sin x}\right)$\quad
i) $\lx[\infty] ((x^3-x^2+\tfrac{x}{2})e^{\frac{1}{x}} - \sqrt{x^6+1})$\\
j) $\lx \frac{1-(\cos x)^{\sin x}}{x^3}$
k) $\lx \frac{\sqrt{1-2x+x^3} - \sqrt[3]{1-3x+x^2}-\tfrac{x^2}{6}}{\sin x -x}$\\[10pt]
\textbf{6. Spo\v{c}t\v{e}te limity (p\v{r}\'iklady ze zkou\v{s}kov\'ych p\'isemek na FSV):}\\[5pt]\resetpr
\pr $\lx \frac{\sin(e^{(x^2)} - 1) + 2\sin(\cos(x) - 1)}{\log(1 + x^4)}$
\pr $\lx \frac{\log(1 + x\sin x) - \exp(x\sin x) + \cos(x\sin x)}{\sin^4 x}$
\newpage
\centerline{\MakeUppercase{ii. mocninn\'e \v{r}ady}}\ \\
\textbf{1. Nalezn\v{e}te polom\v{e}r konvergence n\'asleduj\'ic\'ich mocninn\'ych \v{r}ad. Ur\v{c}ete oblasti absolutn\'i
konvergence, neabsolutn\'i konvergence a divergence.}\\[5pt]\resetpr
\pr $\suma \frac{1}{n^p}z^n$ ($p\in\er$)
\pr $\suma \frac{3^n+(-2)^n}{n} (z+1)^n$
\pr $\suma (1+\tfrac{1}{n})^{n^2}z^n$
\pr $\suma \frac{1}{a^{\sqrt{n}}} z^n\;(a&gt;0)$\\
\pr $\suma \frac{(3+(-1)^n)^n}{n}z^n$
\pr $\suma \frac{z^n}{a^n + b^n}\;(a,b&gt;0)$
\pr $\suma \left(\frac{a^n}{n}+\frac{b^n}{n^2}\right)z^n\;(a,b&gt;0)$ 
\pr $\suma \frac{\big(1 + 2\cos \tfrac{\pi n}{4}\big)^n}{n}z^n$
\pr $\suma \frac{1}{2^n}z^{n^2}$\\[10pt]
\textbf{2. Se\v{c}t\v{e}te \v{r}ady uvnit\v{r} kruhu konvergence:}\\ \resetpr
	\pr $\sum_{n=0}^\infty \frac{z^{4n+5}}{n!}$
	\pr $\suma nz^n$
	\pr $\suma \frac{x^n}{n(n+1)}$
	% zacinaji priklady k procviceni
	\pr $\suma n^2 x^n$
	\pr $\suma (-1)^{n+1} n^2 x^n$
	\pr $\suma n(n+2)x^n$
	\pr $\suma (-1)^{n-1}\frac{x^{2n}}{n(2n-1)}$
	\pr $\suma \frac{n^2 - 1}{3^n\;n!}\;x^n$
	\pr $\sum_{n=0}^\infty \frac{(-1)^nnx^{2n+1}}{(2n+1)!}$
	\pr $\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$
	\pr $\sum_{n=0}^\infty \frac{x^{4n}}{(4n)!}$\\
	\pr $\suma (n-1)(n+3)x^{2n}$\\[5pt]
	m\r{u}\v{z}e se hodit: $\big((1-x)\log(1-x) - (1-x)\big)' = - \log(1-x)$; $\big(x\arctg x - \log(1+x^2)/2\big)' = \arctg x$\\[10pt]
\textbf{3. Vyj\'ad\v{r}ete n\'asleduj\'ic\'i funkce jako mocninnou \v{r}adu o st\v{r}edu 0:}\\ \resetpr
	\pr $e^{-z^2}$
	\pr $\arctg x$
	% zacinaji priklady k procviceni
	\pr $\sin^2 x$
	\pr $(1+x)\log(1+x)$
	\pr $\frac{z}{1 + z - 2z^2}$
	\pr $\arctg\left(\frac{2 - 2x}{1 + 4x}\right)$
	\pr $\frac{1}{(1-z)^2}$
	\pr $\log \sqrt{\frac{1+x}{1-x}}$\\[5pt]
%\textbf{4. Se\v{c}t\v{e}te \v{r}ady:}\\ \resetpr
	%\pr $\suma \frac{(-1)^n}{n}$
	%\pr $\suma \frac{n}{2^n}$
	%% zacinaji priklady k procviceni
	%\pr $\suma \frac{(-1)^n}{2n+1}$
	%\pr $\suma \frac{n(n+1)}{2^n}$
	%\pr $\suma \frac{(-1)^nn^3}{3^n}$
	%\pr $\suma \frac{(-1)^n}{4^n(2n+1)}$\\
	%\pr $\suma \frac{(-1)^{n-1}}{4n^2 - 1}$
	%%\pr $\sum_{n=0}^\infty (-1)^n\frac{2n+3}{(n+1)2^n}$
	%\pr $\suma \frac{(-1)^nn^2}{2^n\;n!}$\\[5pt]
%m\r{u}\v{z}e se hodit: $\big(x - \arctg x\big)' = \frac{x^2}{1 + x^2}$; $\big(\frac{x^2 + 1}{2}\arctg x - \frac{1}{2}x\big)' = x\arctg x$\\[10pt]
\textbf{4. Rozvi\v{n}te funkce $f(z) = \exp(z)$, $g(z) = z \exp(z)$, $h(z) = z^2 \exp(z)$ v mocninn\'e \v{r}ady se st\v{r}edem v bod\v{e} $z_0$, kde a) $z_0 = 0$, b) $z_0 = 1$, c) $z_0 = - 8$; a stanovte polom\v{e}r konvergence t\v{e}chto \v{r}ad.}\\[5pt]
\textbf{5. Rozvi\v{n}te funkce $f(z) = \sin(z)$, $g(z) = z \sin(z)$, $h(z) = z^2 \sin(z)$ v mocninn\'e \v{r}ady se st\v{r}edem v bod\v{e} $z_0$, kde a) $z_0 = 0$, b) $z_0 = \pi/6$, c) $z_0 = 2$; a stanovte polom\v{e}r konvergence t\v{e}chto \v{r}ad.}
\newpage
 \centerline{\MakeUppercase{iii. hled\'an\'i primitivn\'i funkce - \'uvod}}\ \\ 
\textbf{Vyj\'ad\v{r}ete primitivn\'i funkce na maxim\'aln\'ich intervalech existence} \\[10pt]
\textbf{1. P\v{r}\'iklady na integrov\'an\'i "p\v{r}\'imo":}\\
a) $\int x^9 + \frac{1}{x} - 5e^x + x^{-3} - \cos{x} \dx$ \quad
b) $\int 2e^{3x} - \sqrt[5]{5-x} \dx$ \quad
c) $\int \frac{x^2 + 3x + 6}{x^4} \dx$\quad
d) $\int x(1-x)^{10} \dx$\\[10pt]
\textbf{2. P\v{r}\'iklady, kde se mus\'i funkce "lepit":}\\
a) $\int |x| \dx$ \quad
b) $\int |\cos{x}| \dx$ \quad
c) $\int \sqrt{x^6} \dx$ \quad
d) $\int \sin{|2x-1|} \dx$
e) $\int |\sin{x} + \cos{x}| \dx$\\
f) $\int \max\{x,x^2\} \dx$\quad
g) $\int e^{-|x|} \dx$\quad
h) $\int |2x+1| \dx$\\[10pt]
\textbf{3. P\v{r}\'iklady na integrov\'an\'i pomoc\'i substituce:}\\
a) $\int \tg{x} \dx$ \quad
b) $\int \cotg{x} \dx$ \quad
c) $\int \frac{x^2}{\cos^2{x^3}} \dx$ \quad
d) $\int \frac{x}{1+x^4} \dx$ \quad
e) $\int \frac{1}{x\log{x}} \dx$ \quad
f) $\int \frac{x}{\sqrt{x^2+5}} \dx$ \quad
g) $\int \frac{1}{x\log{x}\log(\log{x})} \dx$ \\[10pt]
\textbf{4. Dal\v{s}\'i p\v{r}\'iklady k procvi\v{c}en\'i:}\\
a) $\int \left(1 - \frac {1}{x^2}\right)\sqrt{x\sqrt{x}} \dx$\quad
b) $\int \frac{\left(\sqrt{2x} - \sqrt[3]{3x}\right)^2}{x} \dx$\quad
c) $\int (2^x + 3^x)^2 \dx$\quad
d) $\int \frac{x^2}{1+x^2} \dx$\quad
e) $\int \frac{1}{\sqrt{2 - 5x}} \dx$\quad
f) $\int \frac{x^3}{x^8 + 1} \dx$\\
g) $\int \frac{1}{x^2}\sin(\frac 1x) \dx$\quad
h) $\int \cos^2x \dx$\quad
i) $\int \frac{2x}{2+2x^2+x^4} \dx$\quad
j) $\int \frac{\sin x}{1+\sin^4x}\cos x \dx$\quad
k) $\int \arcsin \sin \frac{4x}{x^2+1} \dx$ (t\v{e}\v{z}k\'e)\\[10pt]
\textbf{5. P\v{r}\'iklady na integrov\'an\'i "per partes":}\\
a) $\int x^3 \sin{x} \dx$ \quad
b) $\int e^x \cos{x} \dx$ \quad
c) $\int \log{x} \dx$ \quad
d) $\int x^n e^x \dx\;, n\in\en$ \quad
e) $\int x \log{x} \dx$ \quad
f) $\int x e^x \cos{x} \dx$  \\[10pt]
\textbf{6. Dal\v{s}\'i p\v{r}\'iklady k procvi\v{c}en\'i:}\\
a) $\int \frac{\sin{x}}{\sqrt{\cos^3x}} \dx$\quad
b) $\int \frac{x^2}{(8x^3 + 27)^{2/3}} \dx$\quad
c) $\int \frac{\arctg{x}}{1 + x^2} \dx$\quad
d) $\int \frac{2^{2x}}{9^x - 4^x} \dx$\quad
e) $\int \arctg{x} \dx$\quad
f) $\int x^2\sin(2x) \dx$\\
g) $\int \sqrt{x}\log^2x \dx$\quad
h) $\int x^2e^{-2x} \dx$\quad
i) $\int \left(\frac{\log{x}}{x}\right)^2 \dx$\quad
j) $\int x^5e^{x^3} \dx$\quad
k) $\int e^{\sqrt{x}} \dx$\quad
l) $\int x\sin{\sqrt{x}} \dx$\\[10pt]
\textbf{7. P\v{r}\'iklady na integrov\'an\'i pomoc\'i druh\'e v\v{e}ty o substituci:}\\
a) $\int \sqrt{4-x^2} \dx$ \quad
b) $\int \frac{1}{(1-x^2)^{3/2}} \dx$ \quad
c) $\int \frac{1}{(x^2 + a^2)^{3/2}} \dx\;(a&gt;0)$ \\
d) $\int \sqrt{\frac{x+a}{a-x}} \dx\;(a&gt;0)$ \quad
e) $\int x\sqrt{\frac{x}{2a-x}} \dx\;(a&gt;0)$ \\[10pt]
\newpage
 \centerline{\MakeUppercase{iv. hled\'an\'i primitivn\'i funkce - pokra\v{c}ov\'avn\'i}}\ \\ 
\textbf{Vyj\'ad\v{r}ete primitivn\'i funkce na maxim\'aln\'ich intervalech existence} \\[10pt]
\textbf{1. P\v{r}\'iklady na "integrov\'an\'i racion\'aln\'ich funkc\'i":}\\
a) $\int \frac{1}{(2x+3)(3x+2)(x+1)} \dx$\quad
b) $\int \frac{5x^3 + 3x^2 - x - 1}{x^2 + 2x + 1} \dx$\quad
c) $\int \frac{x^2 + 1}{x^2 - 1} \dx$\quad
d) $\int \frac{1}{x^4 + 1} \dx$\quad
e) $\int \frac{x^2 + 3x - 2}{(x - 1)(x^2 + x + 1)^2} \dx$\\
f) $\int \frac{x^5 + x^4 + x^3 + x^2 + x + 1}{x^3 - 3x^2 + 3x - 1} \dx$\\[10pt]
\textbf{2.}
a) $\int \frac{1}{1 + \sqrt{x}} \dx$\quad
b) $\int \frac{1 - \sqrt{x+1}}{1 + \sqrt[3]{x+1}} \dx$\quad
c) $\int \frac{x\sqrt[3]{2+x}}{x + \sqrt[3]{2+x}} \dx$\quad
d) $\int \frac{1}{(1+e^x)^2} \dx$\quad
e) $\int \frac{1}{e^{2x} + e^x - 2} \dx$\quad
f) $\int \frac{1}{1+e^{x/2} + e^{x/3} + e^{x/6}} \dx$\quad
g) $\int \frac{\sqrt{1+x}-\sqrt{x-1}}{\sqrt{1+x}+\sqrt{x-1}}$\\[10pt]
\textbf{3.}
a) $\int \frac{1}{\sin{x} + \tg{x}} \dx$\quad
b) $\int \frac{\tg^2x + 1}{\sin^2x + \sin{x}\cos{x}} \dx$\quad
c) $\int \frac{\sin^3x + \sin{x}}{\cos^3x + \cos{x}} \dx$\quad
d) $\int \frac{1}{\cos^2x(4\sin^2x - 1)} \dx$\quad
e) $\int \frac{1}{(2+\cos x)\sin x} \dx$\\
f) $\int \frac{\sin^2x}{1+\sin^2x} \dx$\quad
g) $\int \frac{\sin x}{\sin^3 x + \cos^3 x} \dx$\\[10pt]
\textbf{4.}
a) $\int \frac{1}{5+ \cos x} \dx$\quad
b) $\int \frac{\sin x \cos x}{1 + \sin^4 x} \dx$\quad
c) $\int \frac{1}{(\sin^2x + 2\cos^2x)^2} \dx$\quad
d) $\int \frac{1}{2\sin x - \cos x + 5} \dx$\\
%priklady na hyperbolicke substituce
e) $\int \sqrt{x^2 + a^2} \dx\;(a&gt;0)$\quad
f) $\int \sqrt{x^2 - a^2} \dx\;(a&gt;0)$\\[10pt]
\textbf{5.}
a) $\int \frac{x^2}{\sqrt{x^2 - 2}} \dx$\quad
b) $\int \frac{x^2}{\sqrt{x^2 + a^2}} \dx\;(a&gt;0)$\quad
c) $\int \frac{1}{1+\sqrt{x^2+2x+2}} \dx$\quad
d) $\int \frac{1}{(x-1)\sqrt{x^2-3x+2}} \dx$\\[10pt]
\textbf{6. P\v{r}eve\v{d}te ``$\int \sqrt{x^2 - 3x + 1} \dx$'' na integr\'al z racion\'aln\'i funkce pomoc\'i n\'asleduj\'ic\'ich substituc\'i:}
a) $t = \sqrt{x^2 - 3x + 1} - x$\quad
b) $t = \sqrt{x^2 - 3x + 1} + x$\quad
c) ``$t = \sqrt{\frac{x-x_1}{x-x_2}}$''\\[10pt]
\textbf{7. BONUS - p\v{r}\'iklady ze zkou\v{s}kov\'ych p\'isemek na FSV:}\\
a) $\int \frac{2x}{(2x+3)(\sqrt{2x+3} + \sqrt[3]{2x+3})} \dx$\quad
b) $\int \frac{\sin^3x}{\cos{x}(\sin^2x + 1)} \dx$\quad
c) $\int \frac{1}{(\sqrt{x^2+x+7} - x)^3 + \sqrt{x^2+x+7} - x} \dx$\quad
d) $\int \frac{\cos{x}-1}{\sin{x}-1} \dx$\\[10pt]
\textbf{8. BONUS - p\v{r}\'iklady ze zkou\v{s}kov\'ych p\'isemek na MFF:}\\
a) $\int \frac{1}{(e^{2x} + e^x - 2)(e^{2x}+1)} \dx$\quad
b) $\int \frac{\sin x}{\sin x + \cos x - 3} \dx$\quad
c) $\int \frac{1}{x+\sqrt{x^2 + x}} \dx$\quad
d) $\int \frac{2\log^2(x) + 3}{x\log^4(x) - x\log^2(x) - 6x} \dx$
\newpage
 \centerline{\MakeUppercase{v. newton\r{u}v integr\'al}}\ \\
\textbf{Vypo\v{c}t\v{e}te n\'asleduj\'ic\'i integr\'aly}\\ \resetpr
\textbf{1.}
\pr $\int_0^2 |1-x| \dx$
\pr $\int_0^{\log 2} xe^{-x} \dx$
\pr $\int_0^{2\pi} x^2\cos x \dx$
\pr $\int_{\frac1e}^e |\log x| \dx$
\pr $\int_0^{100\pi}\sqrt{1-\cos 2x} \dx$\\
\pr $\int_0^{\log 2} \sqrt{e^x-1} \dx$
\pr $\int_0^a x^2\sqrt{a^2-x^2} \dx$
\pr $\int_{0}^{2\pi}\frac{1}{1+\varepsilon\cos x} \dx$,
$\varepsilon\in[0,1)$
\pr $\int_{-1}^1 \frac{1}{a^2\sin^2 x+b^2\cos^2 x} \dx$, $ab\neq 0$\\
\pr $\int_{-1}^1\frac{x}{x^2+x+1} \dx$
\pr $\int_0^{2\pi} \frac{1}{\sin^4x+\cos^4x} \dx$\\[10pt]
\textbf{2.}\resetpr
%\pr $\int_{4}^\infty \frac{1}{x^2}\sqrt{\frac{x-2}{x-4}} \dx$
\pr $\int_0^\infty \frac{1}{x^2 + 3x + 2} \dx$
\pr $\int_{1}^3 \frac{1}{x\sqrt{-x^2+4x-3}} \dx$
\pr $\int_{-1}^\infty \frac{1}{\sqrt{x+1} + \sqrt{(x+1)^3}} \dx$
\pr $\int_{0}^{5\pi} \frac{1}{\sin^2 x + 2\cos^2 x} \dx$\\ 
%\pr $\int_{\pi/3}^{2\pi/3} \frac{1}{\sin x\sqrt{1+\cos x}} \dx$
\pr $\int_0^{6\pi} x\frac{\sin(x^2)}{2+\sin(x^2)} \dx$\\[10pt]
%\pr $\int_1^e \frac{1}{x}\tg^2 \sqrt{\log(x^2) + 1} \dx$ (t\v{e}\v{z}k\'e)\\[10pt]
\textbf{3.} Dal\v{s}\'i pou\v{c}n\'e p\v{r}\'iklady jsou nap\v{r}. na str\'ance doc. Kalendy zde (s\'erie IX):\\
\url{http://www.karlin.mff.cuni.cz/~kalenda/data/ma215-cv.pdf}
\newpage
\centerline{\MakeUppercase{vii. existence newtonova integr\'alu}}\ \\
\textbf{1. Vy\v{s}et\v{r}ete existenci n\'asleduj\'ic\'ich Newtonov\'ych integr\'al\r{u}}\\ \resetpr
\pr $\int_0^1 \frac{\sqrt{x}}{\sqrt{1-x^4}} \dx$
\pr $\int_0^1 \frac{x^2}{\sqrt[3]{(1-x^2)^5}} \dx$
\pr $\int_0^1 \frac{1}{e^{\sqrt x}-1} \dx$
\pr $\int_0^1 \frac{\sqrt{x}}{e^{\sin x}-1} \dx$
\pr $\int_0^1 \frac {1}{e^x-\cos x} \dx$
\pr $\int_0^{\frac\pi2} \frac{\log\sin x}{\sqrt x} \dx$\\
\pr $\int_0^{\frac\pi2} \frac{1}{\sin^p x\cos^q x} \dx$
\pr $\int_0^{\frac\pi2} \frac{1}{\sin^q x(1-\sin x)^p} \dx$
\pr $\int_0^\infty \frac{1}{\sqrt{x}\log(1+e^x)} \dx$
\pr $\int_0^1 x^{p-1}(1-x)^{q-1} \dx$
\pr $\int_0^\infty x^p e^{-\sqrt{x}} \dx$\\[10pt]
\textbf{2. Vy\v{s}et\v{r}ete existenci n\'asleduj\'ic\'ich Newtonov\'ych integr\'al\r{u} (v\v{s}echny parametry jsou re\'aln\'e)}\\ \resetpr
\pr $\int_0^{\frac\pi2}\frac{1-\cos x}{x^m} \dx$
%\pr $\int_1^\infty \frac{\sin x}{x^\alpha} \dx\;(\alpha &gt; 1)$
\pr $\int_1^\infty x^k\frac{x-\sin x}{x+\sin x} \dx$
\pr $\int_0^\infty x^\alpha\arctg^\beta x \dx$
\pr $\int_0^{\frac\pi2} (\tg x)^{\alpha} \dx$\\
\pr $\int_0^{\frac\pi2}x^\alpha\left(\frac\pi2-x\right)^\beta \tg^\gamma x\dx$
\pr $\int_2^\infty\frac{1}{x^p \ln^q x} \dx$\\[10pt]
\textbf{3. Pomoc\'i B-C podm\'inky uka\v{z}te, \v{z}e n\'asleduj\'ic\'i Newtonovy integr\'aly neexistuj\'i:}\\ \resetpr
\pr $\int_1^\infty \frac{\sin x}{x^\alpha} \dx\;(\alpha \leq 0)$
\pr $\int_1^\infty \frac{|\sin x|}{x} \dx$
\pr $\int_1^\infty \frac{|\cos x|}{x} \dx$
\pr $\int_\pi^\infty \frac{\sin^2 x}{x} \dx$\\[10pt]
%tohle bude na Patecnim cviceni
\textbf{4. Pomoc\'i metod ``per partes'' a ``substituce'' rozhodn\v{e}te o existenci n\'asleduj\'ic\'ich Newtonov\'ych integr\'al\r{u}:}\\ \resetpr
\pr $\int_1^\infty \frac{\sin x}{x^\alpha} \dx\;(\alpha\in(0,1])$
\pr $\int_0^\infty \sin(x^2) \dx$\\[10pt]
\textbf{5. Vy\v{s}et\v{r}ete existenci n\'asleduj\'ic\'ich Newtonov\'ych integr\'al\r{u}:}\\ \resetpr
\pr $\int_0^\infty 2x\cos\left(x^4\right)\dx$
\pr $\int_0^\infty \frac{\sqrt{x}\cos x}{x+8} \dx$
\pr $\int_0^\infty\frac{\sin^2x}{x^\alpha}\dx$
\pr $\int_1^\infty \frac{\sin x}{x} \frac{x^2}{x^2 + 1} \arctg x \dx$% (sta\v{c}\'i vy\v{s}et\v{r}it neabsolutn\'i konvergenci)
\pr $\int_0^\infty \frac{\log(1+x)}{x} \cos x \dx$\\
\pr $\int_0^\infty \frac{\sin\left(\tfrac{1}{x}\right)\arctg x}{x} \dx$
\pr $\int_0^\infty \frac{\sin\left(\tfrac{1}{x} + x\right)}{x^a} \dx$, ($a \in\er$)\\[10pt]
\textbf{6. BONUS - p\v{r}\'iklady ze zkou\v{s}kov\'ych p\'isemek na MFF:}\\[10pt] \resetpr
\pr $\int_0^\infty \left(1 + \frac 1x \right)^x \frac{\sin(\sqrt{x})}{x} \dx$
\begin{tabular}{l}
U tohoto p\v{r}\'ikladu m\r{u}\v{z}ete bez ov\v{e}\v{r}ov\'an\'i pou\v{z}\'it informaci,\\ \v{z}e existuje okol\'i nekone\v{c}na, na kter\'em je funkce $(1 + 1/x)^x$ rostouc\'i.
\end{tabular}\\[10pt]
\pr $\int_1^\infty \frac{(x^3 - 1)(\log x)^\alpha}{(2+x^2)^2} \dx\;(\alpha\in\er)$
\pr $\int_0^\infty \cos(x^\alpha + 1) \dx$
\newpage
\centerline{\MakeUppercase{viii. metrick\'e prostory}}\ \\
\textbf{1. Ov\v{e}\v{r}te, zda n\'asleduj\'ic\'i formule definuj\'i metriku na $\er$:}\\ \resetpr \pr $\rho(x,y) = |x^3 - y^3|$ \pr $\rho(x,y) = |x^2 - y^2|$  \pr $\rho(x,y) = (x - y)^2$\\[10pt]
\textbf{2. A\v{t} $\rho_1,\rho_2$ jsou metriky na mno\v{z}in\v{e} $P$. Mus\'i b\'yt i funkce $\rho$ definovan\'a n\'i\v{z}e metrikou na $P$?}\\ \resetpr \pr $\rho = \rho_1 + \rho_2$ \pr $\rho = \max\{\rho_1,\rho_2\}$ \pr $\rho = \min\{\rho_1,\rho_2\}$ \pr $\rho = \max\{\rho_1,1\}$ \pr $\rho = \min\{\rho_1,1\}$\\[10pt]
\textbf{3. Na mno\v{z}in\v{e} }$\mathcal{P} = \{A\subset\en:\;A\text{ je kone\v{c}n\'a}\}$\textbf{ definujme $\rho(A,B) = |(A\setminus B) \cup (B\setminus A)|$ pro $A,B\in \mathcal{P}$. Je $\rho$ metrika?}\\[10pt]
\textbf{4. Na prostoru $\C([0,1])$ uva\v{z}ujme supremovou metriku, tj. $\rho_s(f,g) = \max\{|f(t) - g(t)|:\;t\in[0,1]\}$}.\\ \resetpr \pr Spo\v{c}\'itejte $\rho_s(x,x^2)$.\\
\pr Pro $n\in\mathbb{N}$ a $x\in[0,1]$ definujme $f_n(x) = \tfrac{\sin(nx + 3)}{\sqrt{n + 1}}$ a $g_n(x) = \tfrac{nx}{1 + n^2x^2}$. Je posloupnost funkc\'i $(f_n)_{n=1}^{\infty}$ konvergentn\'i v metrice $\rho_s$ k funkci $f(x)=0$, $x\in[0,1]$? Plat\'i analogick\'e tvrzen\'i pro posloupnost $(g_n)_{n=1}^{\infty}$?\\[10pt]
\textbf{5. Na prostoru $\C([0,1])$ uva\v{z}ujme integr\'aln\'i metriku, tj. $\rho_i(f,g) = \int_0^1 |f(x) - g(x)| \dx$.}\\ \resetpr \pr Spo\v{c}\'itejte $\rho_i(x,x^2)$. \pr Pro jak\'e $a\in(0,1)$ je vzd\'alenost $\rho_i(ax,x^2)$ nejmen\v{s}\'i mo\v{z}n\'a?\\[10pt]
\textbf{6. Uva\v{z}ujme na $\er^2$ libovolnou metriku $\rho$. Kter\'a z n\'asleduj\'ic\'ich tvrzen\'i jsou pravdiv\'a? Kter\'a jsou pravdiv\'a, pokud $\rho(x,y) = \|x-y\|$ pro n\v{e}jakou normu na $\er^2$?}\\ \resetpr \pr $5U(0,1) = U(0,5)$ ($0$ zna\v{c}\'i nulov\'y vektor, $5A$ definujeme jako mnozinu $\{5a:\;a\in A\}$ pro $A\subset\er^2$)\\ \pr $5U(x,1) = x + U(0,5),\;x\in\er^2$\\[10pt]
\textbf{7. A\v{t} $P$ je metrick\'y prostor, $x\neq y$ jsou dva body z $P$. Doka\v{z}te:}\\ \resetpr \pr $\{x\}$ je uzav\v{r}en\'a mno\v{z}ina. \pr Existuj\'i disjunktn\'i otev\v{r}en\'e mno\v{z}iny $U$, $V$ \v{z}e $x\in U$ a $y\in V$.\\[10pt]
\textbf{8.} A\v{t} $(P,\rho)$ je metrick\'y prostor, $f:[0,\infty)\to[0,\infty)$ je funkce spl\v{n}uj\'ic\'i
\begin{itemize}
	\item[(i)] $f(x) = 0\leftrightarrow x=0$
	\item[(ii)] $r\leq s + t \rightarrow f(r) \leq f(s) + f(t)$
\end{itemize}
Polo\v{z}me $\sigma(x,y)=f(\rho(x,y))$. Doka\v{z}te, \v{z}e $\sigma$ je potom metrika na $P$.\\ \\
Pomoc\'i tohoto tvrzen\'i doka\v{z}te, \v{z}e $\sigma(x,y)=\frac{\rho(x,y)}{1+\rho(x,y)}$ je metrika na $P$.\\[10pt]
\textbf{9.} \v{R}ekneme, \v{z}e $\rho$ je \emph{pseudometrika} na $P$, pokud pro
ka\v{z}d\'e $x,y,z\in P$ plat\'i

\centerline{$\rho(x,x)=0$, $\rho(x,y)=\rho(y,x)$,
$\rho(x,z)\le\rho(x,y)+\rho(y,z)$.}\ \\
Nech\v{t} na mno\v{z}in\v{e} $P$ m\'ame
posloupnost pseudometrik $\rho_n,\,n\in\en$ spl\v{n}uj\'ic\'i
$$\forall x,y\in P,\;x\neq y\;\;\exists n\in\en:\quad\rho_n(x,y)\neq 0.$$
Doka\v{z}te, \v{z}e potom $\rho(x,y)=\sum_{n=1}^\infty\frac1{2^n} \min\{\rho_n(x,y),1\}$ definuje metriku na $P$.\\ \\
Toto tvrzen\'i aplikujte na d\r{u}kaz toho, \v{z}e pokud na $\C(\er)$ definuji $\rho_n(f,g) = \sup\{|f(x) - g(x)|:\;x\in[-n,n]\}$, pak $\rho(f,g)=\sum_{n=1}^\infty\frac1{2^n} \min\{\rho_n(f,g),1\}$ je metrika na $\C(\er)$.
\newpage
\textbf{10. Rozhodn\v{e}te, zda n\'asleduj\'{\i}c\'{\i} mno\v{z}iny jsou otev\v{r}en\'e (resp. uzav\v{r}en\'e). Zjist\v{e}te jejich vnit\v{r}ek, uz\'av\v{e}r a hranici.}\\ \resetpr
	\pr $\en$
	\pr $\qe$
	\pr $\{\tfrac{1}{n}\setsep n\in\en\}$	
	\pr  $\{[x,y]\in\er^2\setsep x &gt; 0,\,y\leq 0\}$
	\pr $\{[x,y]\in\er^2\setsep |x+y|&gt;x+y\}$\\
	\pr $\{[x,y]\in\er^2\setsep |x-y|=x-y\}$
	\pr $\{[x,y]\in\er^2\setsep x^2+y^2+2xy=5\}$\\
	\pr $\{[x,y,z]\in\er^3\setsep x\geq 0,y&gt;0,x+y=2,z\leq 0\}$
	\pr $\{f\in\C[0,1]:\; f(\frac12)=2\}$\\
	\pr $\{f\in\C[0,1]:\; f(\frac12)\in(0,2)\}$
	\pr $\{f\in\C[0,1]:\; \int_0^1f(x)\dx=0\}$\\[10pt]
\textbf{11. Plat\'{\i} v metrick\'ych prostorech n\'asleduj\'{\i}c\'{\i} rovnosti? Plat\'i alespo\v{n} jedna inkluze?}\\ \resetpr
\pr $\ov{A\cap B}=\ov{A}\cap\ov{B}$ \pr $\Int(A\setminus B) = \Int(A)\setminus\Int(B)$ \pr $\partial A = \ov{A}\setminus A$ (uva\v{z}ujte $A$ otev\v{r}enou)\\[10pt]
\textbf{12. Doka\v{z}te n\'asleduj\'ic\'i ekvivalence pro uzav\v{r}enou mno\v{z}inu $A$ v metrick\'em prostoru $P$:}
$$\Int(A) = \emptyset \leftrightarrow \ov{P\setminus A} = P,$$
$$\dist(x,A) = 0\leftrightarrow x\in A\qquad\text{(plat\'i toto tvrzen\'i i kdy\v{z} $A$ nen\'i uzav\v{r}en\'a?)}$$
\textbf{13. A\v{t} $A\subset P$ je podmno\v{z}ina metrick\'eho prostoru $(P,\rho)$. Doka\v{z}te:}\\ \resetpr
\pr $\Int(A) = \bigcup\{U\subset A:\;U\text{je otev\v{r}en\'a}\}$ \pr $\ov{A} = \bigcap\{F\supset A:\;F\text{ je uzav\v{r}en\'a}\}$
\newpage
\centerline{\MakeUppercase{ix. funkce v\'ice prom\v{e}nn\'ych - limity, derivace}}\ \\
\textbf{1. Lze n\'asleduj\'ic\'i funkce spojit\v{e} roz\v{s}\'i\v{r}it na $\er^2$?}\\ \resetpr \pr $\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$ \pr $\frac{\sqrt{x^2y^2+1}-1}{x^2+y^2}$  \pr $\frac{\sin(x^3+y^3)}{x^2+y^2}$ \pr $\frac{x^2y^2}{x^2y^2+(x-y)^2}$ \pr $\frac{x^3 - y^3}{x-y}$\\ \pr $(x+y)\sin\frac 1x\sin\frac 1y$ \pr $\frac{2xy}{x^2 + y^2}$ \pr $\frac{x+y}{\sqrt{x^2 + y^2}}$ \pr $\frac{xy}{\sqrt{x^2 + y^2}}$ \pr $(1 + x^2y^2)^{\frac{1}{x^2 + y^2}}$ \pr $(x^2 + y^2)^{x^2y^2}$\\[10pt]
\textbf{2. P\v{r}i\v{r}a\v{d}te grafy funkc\'i k p\v{r}edpis\r{u}m:}\\[10pt]
a)\hspace{170pt}b)\hspace{170pt}c)\\
\begin{tikzpicture}
\begin{axis}[colormap/blackwhite, hide axis]
\addplot3[surf, shader=interp] {x^2 - y^2};
\end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[colormap/blackwhite, hide axis]
\addplot3[surf, shader=interp] {x^2 + y^2};
\end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[colormap/blackwhite, hide axis]
\addplot3[surf, shader=interp] {x^4 + y};
\end{axis}
\end{tikzpicture}\\
d)\hspace{170pt}e)\hspace{170pt}f)\\
\begin{tikzpicture}
\begin{axis}[colormap/blackwhite, hide axis]
\addplot3[surf, shader=interp] {exp(x) + y};
\end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[colormap/blackwhite, hide axis]
\addplot3[surf, shader=interp] {-x^2-y^2};
\end{axis}
\end{tikzpicture}\hspace{30pt}
\begin{tikzpicture}
\begin{axis}[colormap/blackwhite, hide axis]
\addplot3[surf, shader=interp] {x + y};
\end{axis}
\end{tikzpicture}\\
i) $x^2 + y^2$\quad ii) $exp(x) + y$\quad iii) $-x^2 - y^2$\quad iv) $x +y$\quad v) $x^2 - y^2$\quad vi) $x^4 + y$\\[10pt]
\textbf{3. Ur\v{c}ete a nakreslete defini\v{c}n\'i obor a vrstevnice funkc\'i:}\\ \resetpr
%	\pr $f(x,y)=x+\sqrt y$
%	\pr $f(x,y)=\frac yx$
%	\pr $f(x,y)=x^2+y^2$
%	\pr $f(x,y)=x^2-y^2$
%	\pr $f(x,y)=\sqrt{xy}$
%	\pr $f(x,y)=\sqrt{1-x^2-y^2}$
	\pr $f(x,y)=\frac{1}{\sqrt{x^2+y^2-1}}$
	\pr $f(x,y)=\sqrt{(x^2+y^2-1)(4-x^2-y^2)}$
	\pr $f(x,y)=\sqrt{1-(x^2+y)^2}$
	\pr $f(x,y)=\sqrt{\sin(x^2+y^2)}$\\[5pt]
%	\pr $f(x,y)=\sgn(\sin x\cdot\sin y)$
%	\pr $f(x,y)=|x|+y$
\textbf{4. Spo\v{c}t\v{e}te parci\'aln\'i derivace funkc\'i v\v{s}ude, kde existuj\'i}\ \\[3pt] \resetpr
\pr $x^my^n$
\pr $e^{xy}$
\pr $xy+yz+zx$
\pr $|x|\cdot|y|$
\pr $|y+\cos x|$
\pr $|\sin y-\sin x|$
\pr $|\cos y-\sin x|$
\pr $\left(\frac xy\right)^z$\\
\pr $x^{\frac yz}$
\pr $\sin (x^y)$
\pr $f(x,y)=e^{-\pi\over x^2+3xy+3y^2}$ pro $(x,y)\neq 0$ , $f(0,0)=(0,0)$
\pr $\sqrt{x+y^2}$\\[5pt]
\textbf{5. Ur\v{c}ete a nakreslete defini\v{c}n\'i obor a vy\v{s}et\v{r}ete parci\'aln\'i derivace funkc\'i\\ (ze zkou\v{s}kov\'ych p\'isemek na FSV)}\ \\[3pt] \resetpr
\pr $\sqrt{e^{xy} - e}$
\pr $\log {\frac{1-|x|}{1-|y|}}$
\pr $\sqrt{e^{x^2 + y^2} - e^4}$
\pr $\sqrt{x^2 - y^2}$
\pr $\arcsin \frac{y^2 + 7}{x + 5}$
\pr $\log \frac{x^2 + y + 1}{1 - \sqrt{x}}$\\[10pt]
\textbf{6. Maj\'i n\'asleduj\'ic\'i funkce tot\'aln\'i diferenci\'al v bod\v{e} $[0,0]$? (pokud v tomto bod\v{e} funkce nen\'i definov\'ana, spojit\v{e} ji dodefinujte)}\\ \resetpr
  \pr $\sqrt{x^2 + y^2}$
  \pr $\sqrt[3]{x + y^2}$
  \pr $|xy|$
  \pr $\sqrt[3]{x^3 + y^3}$
  \pr $(x^2 + y^2)\sin\frac 1{x^2 + y^2}$
  \pr $e^{-\frac 1{x^2 + y^2}}$\\[10pt]
%  \pr $\sqrt[3]{x^2 + y}\ln(x^2 + y^2)$
\textbf{7. Vy\v{s}et\v{r}ete, ve kter\'ych bodech maj\'i n\'asleduj\'ic\'i funkce tot\'aln\'i diferenci\'al a spo\v{c}t\v{e}te jej}\\ \resetpr%\\ (ze zkou\v{s}kov\'ych p\'isemek na MFF)
  \pr $f(x,y) = \tfrac{x^5 - y^5}{x^4 + y^4}$ pro $[x,y]\neq [0,0]$, $f(0,0) = 0$
  \pr $\max\{x^3,y^3\}$
\end{document}
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