\documentclass[12pt,notitlepage]{report}

\frenchspacing % aktivuje použití některých českých typografických pravidel

\usepackage[T1]{fontenc}
\usepackage{amsthm,amsmath,amssymb,amscd,mathrsfs,enumerate,multicol,verbatim,tikz,pgfplots,url}
\usepackage[top=1cm, left=1cm]{geometry} % nastavení dané velikosti okrajů
\pagestyle{empty}
\newtheorem*{thm}{V\v{e}ta}
\theoremstyle{definition}
\newtheorem*{defin}{Definice}

\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\C{\mathcal{C}}
\def\R{\er}
\def\N{\en}
\def\nec{Nech\v t\ }
\def\ov{\overline}
\def\Int{\operatorname{Int}}
\def\dist{\operatorname{dist}}
\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 \sumaO {\sum_{n=0}^\infty}
\def \sumad {\sum_{n=2}^\infty}
\def \dx {\, \mathrm{d} x}
\def\lp{\displaystyle\lim_{n\to\infty}}
\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 dvě položky jsou zde v podstatě formálně, ve skutečnosti nejsou nikde 
\author{Marek C\'uth}

\begin{document} \pgfplotsset{width=7cm, compat=1.13}
 \centerline{\MakeUppercase{i. taylor\r{u}v polynom}}\ \\
 \textbf{P\v{r}ipome\v{n}me si definice element\'arn\'ich funkc\'i:}\\[5pt]
a) $\sin(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$\qquad
b) $\cos(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}$\qquad
c) $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$\\[10pt]
%\textbf{1. Doka\v{z}te, \v{z}e Taylor\r{u}v polynom $k$-t\'eho \v{r}\'adu v bod\v{e} 0 pro funkce $f$ je roven polynomu $P$:}\\
%(tyto v\'ysledky lze d\'ale pova\v{z}ovat za zn\'am\'e rozvoje a pou\v{z}\'ivat je v dal\v{s}\'ich v\'ypo\v{c}tech)\\[5pt]
%a) $f(x) = (1+x)^a\;(a\in\er),\;P(x) = \sum_{n=0}^k {a \choose n}x^n$,\;kde ${a \choose n} = \frac{a\cdot(a-1)\cdot(a-2)\cdots(a-n+1)}{n!}$\\
%b) $f(x) = \log(1+x),\;P(x) = \sum_{n=1}^k (-1)^{n+1}\frac{x^n}{n}$\\[10pt]
\textbf{1. 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 $\sqrt{5}$ s p\v{r}esnost\'i $10^{-4}$}\\[10pt]
\textbf{2. 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) $\lx \frac{a^x+a^{-x}-2}{x^2}\;(a>0)$\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{3. 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 $\lx \frac{\log(1 + x\sin x) - \exp(x\sin x) + \cos(x\sin x)}{\sin^4 x}$\\[10pt]
\textbf{4. Vy\v{s}et\v{r}ete konvergenci \v{r}ad}\\[5pt]
a) $\suma \left[\tfrac{1}{\sqrt{n}} + \log\left(\sqrt{1+\tfrac{1}{n}} - \tfrac{1}{\sqrt{n}}\right)\right]$\quad
b) $\suma \left[2\tg\left(\tfrac{1}{\sqrt[3]{n}}\right)-\sin\left(\tfrac{1}{\sqrt[3]{n}}\right)-\tfrac{1}{n}\right]$\\
c) $\suma \left[\sin\left(\tfrac{1}{\sqrt{n}}\right) - \log\left(1+\tfrac{1}{\sqrt[3]{n}}\right)\right]$\quad
d) $\suma (e^{1/n}-1-\tfrac{1}{n})(\arcsin \tfrac{1}{n} - \tfrac{1}{\sqrt{n}})$\\
e) $\suma \arctg\left(\tfrac{1}{n} - \tfrac{1}{\sqrt{n}}\right)$\quad
f) $\suma \left(\sin \tfrac{1}{n} - \arcsin \tfrac{1}{n}\right)$\quad
g) $\suma \sin\left(\tfrac{1}{n} - \arcsin \tfrac{1}{n}\right)$\\[10pt]
\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 $\suma \frac{1}{n^p}x^n$ ($p\in\er$)
\pr $\suma \frac{3^n+(-2)^n}{n} (x+1)^n$
\pr $\suma (1+\tfrac{1}{n})^{n^2}x^n$
\pr $\suma \frac{1}{a^{\sqrt{n}}} x^n\;(a>0)$\\
\pr $\suma \frac{(3+(-1)^n)^n}{n}x^n$
\pr $\suma \frac{x^n}{a^n + b^n}\;(a,b>0)$
\pr $\suma \left(\frac{a^n}{n}+\frac{b^n}{n^2}\right)x^n\;(a,b>0)$ 
\pr $\suma \frac{\big(1 + 2\cos \tfrac{\pi n}{4}\big)^n}{n}x^n$
\pr $\suma \frac{1}{2^n}x^{n^2}$\\[10pt]
\textbf{2. Rozvi\v{n}te funkce $f(x) = \frac{1}{1-x}$, $g(x) = \frac{1}{(1-x)^2}$, $h(z) = \frac{x}{(1-x)^2}$ v mocninn\'e \v{r}ady se st\v{r}edem v bod\v{e} $x_0$, kde a) $x_0 = 0$, b) $x_0 = - 1$, c) $x_0 = 7$; a stanovte polom\v{e}r konvergence t\v{e}chto \v{r}ad.}\\[5pt]
\textbf{3. Rozvi\v{n}te funkce $f(x) = \exp(x)$, $g(x) = x \exp(x)$, $h(x) = x^2 \exp(x)$ v mocninn\'e \v{r}ady se st\v{r}edem v bod\v{e} $x_0$, kde a) $x_0 = 0$, b) $x_0 = 1$, c) $x_0 = - 8$; a stanovte polom\v{e}r konvergence t\v{e}chto \v{r}ad.}\\[5pt]
\textbf{4. Rozvi\v{n}te funkce $f(x) = \sin(x)$, $g(x) = x \sin(x)$, $h(x) = x^2 \sin(x)$ v mocninn\'e \v{r}ady se st\v{r}edem v bod\v{e} $x_0$, kde a) $x_0 = 0$, b) $x_0 = \pi/6$, c) $x_0 = 2$; a stanovte polom\v{e}r konvergence t\v{e}chto \v{r}ad.}\\[5pt]
\textbf{5. Vyj\'ad\v{r}ete n\'asleduj\'ic\'i funkce jako mocninnou \v{r}adu o st\v{r}edu 0:}\\ \resetpr
	\pr $e^{-x^2}$
	\pr $\arctg x$
	% zacinaji priklady k procviceni
	\pr $\sin^2 x$
	\pr $(1+x)\log(1+x)$
	%\pr $\frac{x}{1 + x - 2x^2}$
	\pr $\arctg\left(\frac{2 - 2x}{1 + 4x}\right)$
	%\pr $\frac{1}{(1-x)^2}$
	\pr $\log \sqrt{\frac{1+x}{1-x}}$
	\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) $\int x\sin{\sqrt{x}} \dx$\\[20pt]
\centerline{\MakeUppercase{iv. hled\'an\'i primitivn\'i funkce - pokra\v{c}ov\'an\'i}}\ \\
\textbf{Vyj\'ad\v{r}ete primitivn\'i funkce na maxim\'aln\'ich intervalech existence} \\[10pt]
\textbf{1.}\resetpr
\pr $\int \sqrt{4-x^2} \dx$
\pr $\int \frac{1}{(x^2 + a^2)^{3/2}} \dx\;(a>0)$
\pr $\int \sqrt{x^2 + a^2} \dx\;(a>0)$
\pr $\int \frac{5x^3 + 3x^2 - x - 1}{x^2 + 2x + 1} \dx$\\
\pr $\int \frac{1}{(2x+3)(3x+2)(x+1)} \dx$
\pr $\int \frac{x^5 + x^4 + x^3 + x^2 + x + 1}{x^3 - 3x^2 + 3x - 1} \dx$
\pr $\int \sqrt{x^2 - a^2} \dx\;(a>0)$
\pr $\int \frac{x^2}{\sqrt{x^2 - 2}} \dx$\\[5pt]
\textbf{2.}\resetpr
\pr $\int \frac{x^2 + 3x - 2}{(x - 1)(x^2 + x + 1)^2} \dx$
\pr $\int \frac{1}{x^4 + 1} \dx$
\pr $\int \frac{1}{e^{2x} + e^x - 2} \dx$
\pr $\int \frac{1}{1+e^{x/2} + e^{x/3} + e^{x/6}} \dx$
\pr $\int \frac{2\log^2(x) + 3}{x\log^4(x) - x\log^2(x) - 6x} \dx$\\[5pt]
\textbf{3.}\resetpr
\pr $\int \frac{1}{\sin{x} + \tg{x}} \dx$
\pr $\int \frac{\tg^2x + 1}{\sin^2x + \sin{x}\cos{x}} \dx$
\pr $\int \frac{\sin^3x + \sin{x}}{\cos^3x + \cos{x}} \dx$
\pr $\int \frac{\sin^2x}{1+\sin^2x} \dx$
\pr $\int \frac{\sin x}{\sin^3 x + \cos^3 x} \dx$\\
\pr $\int \frac{1}{5+ \cos x} \dx$
\pr $\int \frac{1}{2\sin x - \cos x + 5} \dx$\\[5pt]
\textbf{4.}\resetpr
\pr $\int \frac{1}{1 + \sqrt{x}} \dx$
\pr $\int \frac{1}{1+\sqrt{x^2+2x+2}} \dx$
\pr $\int \frac{1}{(x-1)\sqrt{x^2-3x+2}} \dx$\\[5pt]
\textbf{5. 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:}\resetpr \pr $t = \sqrt{x^2 - 3x + 1} - x$ \pr ``$t = \sqrt{\frac{x-x_1}{x-x_2}}$''\\[5pt]
\textbf{6.}\resetpr
\pr $\int \frac{\sin x}{\sin x + \cos x - 3} \dx$
\pr $\int \frac{1}{x+\sqrt{x^2 + x}} \dx$
\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 $\int_{1}^3 \frac{1}{x\sqrt{-x^2+4x-3}} \dx$
\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 $\int_0^{6\pi} x\frac{\sin(x^2)}{2+\sin(x^2)} \dx$\\
\pr $\int_0^\pi \frac{x\sin x}{1 + \cos^2x} \dx$
\pr $\int_{e^{-2\pi n}}^1 |\cos \log \frac{1}{x}|$
\pr $\int_0^{\pi/4} \left(\frac{\sin x - \cos x}{\sin x + \cos x}\right)^{2n+1} \dx$
\pr $\int_{\pi/6}^{5\pi/6} \frac{\cos 2x}{|\sin x - \cos x|} \dx$\\[20pt]
\centerline{\MakeUppercase{vi. mocninn\'e \v{r}ady - s\v{c}\'it\'an\'i \v{c}\'iseln\'ych \v{r}ad}}\ \\
\textbf{1. Se\v{c}t\v{e}te \v{r}ady v\v{s}ude na intervalu konvergence:}\\ \resetpr
\pr $\sumaO \frac{x^{4n+5}}{n!}$
\pr $\suma nx^n$
\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 $\sumaO \frac{x^{2n}}{(2n)!}$
%\pr $\sumaO \frac{x^{4n}}{(4n)!}$
%\pr $\suma (n-1)(n+3)x^{2n}$
\pr $\suma (-1)^{n-1}\frac{x^{2n}}{n(2n-1)}$
%\pr $\suma \frac{n^2 - 1}{3^n\;n!}\;x^n$
\\[20pt]
\textbf{2. 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 $\suma (-1)^n\frac{2n+3}{(n+1)2^n}$
\pr $\suma \frac{(-1)^nn^2}{2^n\;n!}$\\
\newpage
\centerline{\MakeUppercase{vii. konvergence newtonova integr\'alu}}\ \\
\textbf{1. Vy\v{s}et\v{r}ete konvergenci n\'asleduj\'ic\'ich 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 konvergenci n\'asleduj\'ic\'ich integr\'al\r{u}}\\ \resetpr
\pr $\int_0^{\frac\pi2}\frac{1-\cos x}{x^m} \dx$
\pr $\int_1^\infty \frac{\sin x}{x^\alpha} \dx\;(\alpha > 1)$
\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 integr\'aly nejsou konvergentn\'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]
\textbf{4. Pomoc\'i metod ``per partes'' a ``substituce'' rozhodn\v{e}te o konvergenci n\'asleduj\'ic\'ich 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 konvergenci (absolutn\'i i neabsolutn\'i) n\'asleduj\'ic\'ich 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 $\int_0^\infty \frac{\log(1+x)}{x} \cos x \dx$\\[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 $\int_0^\infty \cos(x^\alpha + 1) \dx$\\[10pt]
Dal\v{s}\'i p\v{r}\'iklady ze zkou\v{s}kov\'ych p\'isemek jsou nap\v{r}. na str\'ance doc. Zelen\'eho zde:\\
\url{http://www.karlin.mff.cuni.cz/~zeleny/mff/MA_2/index_MA_2.php}\\[20pt]
\centerline{\MakeUppercase{viii. aplikace ur\v{c}it\'eho integr\'alu}}\ \\
\textbf{1. Spo\v{c}t\v{e}te obsah plochy vymezen\'e k\v{r}ivkami:}\\ \resetpr
\pr $y=\frac{1}{1+x^2}\quad$ a $\quad y = \frac{x^2}{2}$
\pr $y = x^2\quad$ a $\quad y = 2 - x$
\pr $y = x^2\quad$ a $\quad y = \sqrt{x}$\\
\pr $y^2 = 2x+1\quad$ a $\quad x - y - 1 = 0$
\pr $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$\\[10pt]
\textbf{2. Spo\v{c}t\v{e}te d\'elku grafu funkce:}\resetpr \pr $f(x) = x^{3/2},\;x\in[0,4]$ \pr $f(x) = \frac{x^2}{2},\;x\in[0,1]$\\[10pt]
\textbf{3. Spo\v{c}t\v{e}te d\'elky n\'asleduj\'ic\'ich k\v{r}ivek:}\resetpr
\pr $y = \log x,\;x\in[\sqrt{3},\sqrt{8}]$
\pr $x = \frac{y^2}{4} - \frac{\log y}{2},\;y\in[1,e]$
\pr $x^2 + y^2 = 1$
\pr $x^{2/3} + y^{2/3} = a^{2/3}$\\[10pt]
\textbf{4. Ur\v{c}ete objem a povrch pl\'a\v{s}t\v{e} t\v{e}lesa vznikl\'eho rotac\'i mno\v{z}iny}
\[\{(x,y)\in\er^2:\;x^2 + (y-b)^2\leq a^2\}\qquad(0\leq a \leq b)\]
\textbf{5. Vy\v{s}et\v{r}ete konvergenci \v{r}ad}\resetpr \pr $\sumad \frac{1}{n\log n}$ \pr $\sumad \frac{1}{n\log^2n}$
\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 $\rho(x,y) = |x^2 - y^2|$\\[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 $\rho = \max\{\rho_1,\rho_2\}$ \pr $\rho = \min\{\rho_1,\rho_2\}$ \pr $\rho = \min\{\rho_1,1\}$\\[10pt]
\textbf{3. 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{4. 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{5. 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 $5B(0,1) = B(0,5)$ ($0$ zna\v{c}\'i nulov\'y vektor, $5A$ definujeme jako mnozinu $\{5a:\;a\in A\}$ pro $A\subset\er^2$)\\ \pr $5B(x,1) = x + B(0,5),\;x\in\er^2$\\[10pt]
\textbf{6. 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  $\{[x,y]\in\er^2\setsep x > 0,\,y\leq 0\}$
	\pr $\{[x,y]\in\er^2\setsep |x+y|>x+y\}$\\
	\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 $\{[x,y,z]\in\er^3\setsep x\geq 0,y>0,x+y=2,z\leq 0\}$
	\pr $\{f\in\C[0,1]:\; f(\frac12)=2\}$\\
	\pr $\{f\in\C[0,1]:\; f(\frac12)\in(0,2)\}$
	\pr $\{f\in\C[0,1]:\; \int_0^1f(x)\dx=0\}$
\newpage
\centerline{\MakeUppercase{ix. funkce v\'ice prom\v{e}nn\'ych - limity, spojitost, derivace}}\ \\
\textbf{1.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{2. Zjist\v{e}te, zda existuj\'i limity a pokud ano, spo\v{c}t\v{e}te je:}\\ \resetpr
	%Ja udelam tyhle:
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{\sqrt{x^2y^2+1}-1}{x^2+y^2}$
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2y^2+(x-y)^2}$
	\pr $\lim\limits_{\substack{(x,y)\to(0,0)\\ x+y\neq 0}} \frac{x-y}{x+y}$
	\pr $\lim\limits_{\substack{(x,y)\to(0,0)\\ xy\neq 0}} (x+y)\sin\frac 1x\sin\frac 1y$\\
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{\sin(x^3+y^3)}{x^2+y^2}$
	%Oni samostatne mohou zkouset tyhle (mozna jeden na mocninu bych mohl udelat take .. uvidi se podle casu):
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{yx^2}{x^4 + y^2}$
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{x+y}{\sqrt{x^2 + y^2}}$\\
	\pr $\lim\limits_{(x,y)\to(0,0)} (1 + x^2y^2)^{\frac{1}{x^2 + y^2}}$
	\pr $\lim\limits_{(x,y)\to(0,0)} (x^2 + y^2)^{x^2y^2}$
	\pr $\lim\limits_{(x,y)\to(0,3)} \frac{\sin(xy)}{x}$
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{x^6}{x^3 + y^3}$\\
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{x(x^2 + y^2)^2}{1 - \cos(x^2+y^2)}$
	\pr $\lim\limits_{(x,y)\to(0,0)} \frac{1 - \cos(x^2 + y^2)}{(x^2+y^2)x^2y^2}$ 
	\pr	$\lim\limits_{(x,y)\to(0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}$\\[10pt]
\textbf{3. Lze n\'asleduj\'ic\'i funkce spojit\v{e} roz\v{s}\'i\v{r}it na $\er^2$?}\\ \resetpr
	\pr $\frac{x^3 - y^3}{x-y}$
	\pr $(x+y)\sin\frac 1x\sin\frac 1y$
	\pr $\frac{\sin(xy)}{x}$
	\pr $\frac{1 - \cos(x^2 + y^2)}{(x^2+y^2)x^2y^2}$\\[10pt]
\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 $\log \frac{x^2 + y + 1}{1 - \sqrt{x}}$\newpage
\noindent \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 $\max\{x^3,y^3\}$\\[10pt]
\textbf{8.} A\v{t} $F = (F_1, F_2): \er^3\to \er^2$ je zobrazen\'i definovan\'e p\v{r}edpisem
  $$F(x,y,z) = \begin{cases} (x\sin y \sin z, x^2\sin\frac{1}{x^2+y^2+z^2}) & \text{pro }(x,y,z)\in\er^3\setminus\{(0,0,0)\}\\
		  (0,0)& \text{pro }(x,y,z)=(0,0,0).
               \end{cases}
  $$
  a) Uka\v{z}te, \v{z}e v bod\v{e} $(\pi,1,0)$ existuje derivace zobrazen\'i $F$ a spo\v{c}t\v{e}te jej\'i reprezentuj\'ic\'i matici.\\
  b) Spo\v{c}t\v{e}te $\frac{\partial F_2}{\partial x}(0,0,0)$, pokud existuje.\\[10pt]
\textbf{9.} A\v{t} $F = (F_1, F_2): \er^3\to \er^2$ je zobrazen\'i definovan\'e p\v{r}edpisem
  $$F(x,y,z) = ((x+1)(y+1)^2(z+1)^3, \sin x \cos(y+2z)).$$
  Zobrazen\'i $G:\er^2\to \er^3$ m\'a v bod\v{e} $(1,0)$ derivaci reprezentovanou matic\'i
    $$\left( \begin{array}{cc}
       2 & 1\\
       -1 & 1\\
       0 & 1\\
      \end{array}\right).
$$
a) Uka\v{z}te, \v{z}e zobrazen\'i $G\circ F$ m\'a v bod\v{e} $(0,0,0)$ derivaci a spo\v{c}t\v{e}te p\v{r}\'islu\v{s}nou reprezentuj\'ic\'i matici.\\
b) Spo\v{c}t\v{e}te derivaci funkce $F_1$ v bod\v{e} $(0,0,0)$ podle vektoru $(2,0,1)$.\\[10pt]
\textbf{10.} A\v{t} $F = (F_1, F_2, F_3, F_4): \er^2\to \er^4$ je zobrazen\'i definovan\'e p\v{r}edpisem
  $$F(x,y) = \left(\arctg(x+2y), x+y, (x^2+y^2)\frac{|x|}{1+|y|},ye^x\right).$$
  a) Uka\v{z}te, \v{z}e v bod\v{e} $(1,-1)$ existuje derivace funkce $F$ a spo\v{c}t\v{e}te jej\'i reprezentuj\'ic\'i matici.\\
  b) Spo\v{c}t\v{e}te $\frac{\partial F_3}{\partial x}(0,0)$, pokud existuje.\\[10pt]
\textbf{11. Spo\v{c}t\v{e}te parci\'aln\'i derivace funkce $F$}\ \\[3pt]\resetpr
\pr $f:\er^2\to \er$ d\'ana p\v{r}edpisem $f(x,y) = x^2 + y^2$; $\varphi_1,\varphi_2:\er^3\to\er$ d\'any p\v{r}edpisy $\varphi_1(r,s,t) = 2r + s$, $\varphi_2(r,s,t) = t$; $F:\er^3\to \er$ d\'ana p\v{r}edpisem $F(r,s,t) = f(\varphi_1(r,s,t),\varphi_2(r,s,t))$\\
\pr $f:\er^3\to \er$ d\'ana p\v{r}edpisem $f(x,y,z) = x + y^2 + 2z$; $\varphi_1,\varphi_2,\varphi_3:\er^2\to\er$ d\'any p\v{r}edpisy $\varphi_1(s,t) = 3st$, $\varphi_2(s,t) = s^2 + 2t$, $\varphi_3(s,t) = t$; $F:\er^2\to \er$ d\'ana p\v{r}edpisem $F(s,t) = f(\varphi_1(s,t),\varphi_2(s,t), \varphi_3(s,t))$
\end{document}
