%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:04:32} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Integr\U{e1}lny po\U{10d}et funkci\U{ed} jednej re\U{e1}lnej premennej - Vlastnosti a existencia ur\U{10d}it\U{e9}ho integr\U{e1}lu\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Integr\'{a}lny po\v{c}et funkci\'{\i} jednej re\'{a}lnej premennej} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M7.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M71.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M71.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M73.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M73.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O7.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C7.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C7.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Vlastnosti a existencia ur\v{c}it\'{e}ho integr\'{a}lu} \begin{theorem} \label{7}\label{4}(Veta o vlastnostiach ur\v{c}it\'{e}ho integr\'{a}lu) Nech $f,\,g$ s\'{u} integrovate\v{l}n\'{e} funkcie na $\left\langle a,b\right\rangle $ a nech $k\in \mathbf{R}$. Potom plat\'{\i} 1) $kf+g$ je integrovate\v{l}n\'{a} a plat\'{\i} \[ \int_{a}^{b}\left( kf+g\right) \left( x\right) dx=k\int_{a}^{b}f\left( x\right) dx+\int_{a}^{b}g\left( x\right) dx\text{ \ (line\'{a}rnos\v{t})}. \] 2) Ak $f(x)\geq 0$, pre ka\v{z}d\'{e} $x\in \left\langle a,b\right\rangle $, potom $\int_{a}^{b}f\left( x\right) dx\geq 0$; ak $f(x)\geq g(x)$, pre ka% \v{z}d\'{e} $x\in \left\langle a,b\right\rangle $, potom $% \int_{a}^{b}f\left( x\right) dx\geq \int_{a}^{b}g\left( x\right) dx$. 3) Ak $m\leq f(x)\leq M$, pre ka\v{z}d\'{e} $x\in \left\langle a,b\right\rangle $, potom \[ m(b-a)\leq \int_{a}^{b}f\left( x\right) dx\leq M(b-a). \] 4) Ak $f$ je integrovate\v{l}n\'{a}, potom aj $\left| f\right| $ je integrovate\v{l}n\'{a} a plat\'{\i} \[ \left| \int_{a}^{b}f\left( x\right) dx\right| \leq \int_{a}^{b}\left| f\right| \left( x\right) dx. \] \end{theorem} $% \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO721.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO721.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2}$ \begin{example} Uk\'{a}\v{z}te, \v{z}e plat\'{\i} \[ 2\leq \int_{-1}^{1}\sqrt{1+x^{4}}dx\leq \frac{8}{3}. \] \end{example} \begin{solution} Plat\'{\i} nerovnica \[ 1\leq 1+x^{4}\leq 1+2x^{2}+x^{4}=(1+x^{2})^{2}, \]% teda \[ 1\leq \sqrt{1+x^{4}}\leq 1+x^{2}, \]% potom s vyu\v{z}it\'{\i}m predch\'{a}dzaj\'{u}cej vety a pr\'{\i}kladov %TCIMACRO{\hyperref{1}{}{}{M71.tex#3} }% %BeginExpansion \msihyperref{1}{}{}{M71.tex#3} %EndExpansion a %TCIMACRO{\hyperref{2}{}{}{M71.tex#4} }% %BeginExpansion \msihyperref{2}{}{}{M71.tex#4} %EndExpansion dost\'{a}vame \[ 2=\int_{-1}^{1}1dx\leq \int_{-1}^{1}\sqrt{1+x^{4}}dx\leq \int_{-1}^{1}\left( 1+x^{2}\right) dx=2+\frac{1^{3}-\left( -1\right) ^{3}}{3}=\frac{8}{3}% .\square \] \end{solution} Doteraz sme predpokladali, \v{z}e funkcia $f$ je ohrani\v{c}en\'{a} a integrovate\v{l}n\'{a}. Teraz sa budeme zauj\'{\i}ma\v{t} o podmienky za ak% \'{y}ch je funkcia integrovate\v{l}n\'{a}. Nie ka\v{z}d\'{a} ohrani\v{c}en% \'{a} funkcia mus\'{\i} by\v{t} integrovate\v{l}n\'{a} ako ukazuje nasleduj% \'{u}ci pr\'{\i}klad: \begin{example} Funkcia \[ f:\left\langle 0,1\right\rangle \longrightarrow \mathbf{R},\,f\left( x\right) =\left\{ \begin{tabular}{ccc} $1$ & pre & $x$ iracion\'{a}lne \\ $0$ & pre & $x$ racion\'{a}lne% \end{tabular}% .\right. \]% Uk\'{a}\v{z}te, \v{z}e $f$ nie je riemannovsky integrovate\v{l}n\'{a}. \end{example} \begin{solution} Ka\v{z}d\'{y} horn\'{y} s\'{u}\v{c}et $H\left( f,P\right) =1$ a ka\v{z}d\'{y} doln\'{y} s\'{u}\v{c}et $D\left( f,P\right) =0$, pre \v{l}ubovo\v{l}n\'{e} delenie $P$ intervalu $\left\langle 0,1\right\rangle $, preto\v{z}e ka\v{z}d% \'{y} jeho podinterval v\v{z}dy obsahuje racion\'{a}lne aj iracion\'{a}lne \v{c}\'{\i}slo. M\'{a}me: \[ \sup \left\{ D\left( f,P\right) :P\text{ je delenie }\left\langle 0,1\right\rangle \right\} =0,\,\inf \left\{ H\left( f,T\right) :T\text{ je delenie }\left\langle 0,1\right\rangle \right\} =1, \]% \textsl{neexistuje} \v{c}\'{\i}slo $I$, tak\'{e} aby \[ I=\sup \left\{ D\left( f,P\right) :P\text{ je delenie }\left\langle a,b\right\rangle \right\} =\inf \left\{ H\left( f,T\right) :T\text{ je delenie }\left\langle a,b\right\rangle \right\} , \]% \ funkcia $f$ nie je riemannovsky integrovate\v{l}n\'{a}. Okrem toho nie je spojit\'{a} v ka\v{z}dom bode z $\left\langle 0,1\right\rangle .\square $ \end{solution} V \v{c}asti spojitos\v{t} sme dok\'{a}zali %TCIMACRO{\hyperref{vetu}{}{}{M42.tex#11}}% %BeginExpansion \msihyperref{vetu}{}{}{M42.tex#11}% %EndExpansion , \v{z}e ka\v{z}d\'{a} spojit\'{a} funkcia definovan\'{a} na uzavretom intervale je %TCIMACRO{\hyperref{rovnomerne spojit\'{a}}{}{}{M42.tex#10}}% %BeginExpansion \msihyperref{rovnomerne spojit\'{a}}{}{}{M42.tex#10}% %EndExpansion . Teraz m\^{o}\v{z}eme sformulova\v{t} posta\v{c}uj\'{u}cu podmienku integrovate\v{l}nosti. \subsubsection{Posta\v{c}uj\'{u}ca podmienka integrovate\v{l}nosti funkcie} \begin{theorem} \label{9}(Posta\v{c}uj\'{u}ca podmienka integrovate\v{l}nosti funkcie) Ak $% f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je spojit\'{a} funkcia, potom je na $\left\langle a,b\right\rangle $\ riemannovsky integrovate\v{l}n\'{a}. \end{theorem} $% \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small D\^{o}kaz}}{}{}{DO723.tex} }% %BeginExpansion \msihyperref{{\small D\^{o}kaz}}{}{}{DO723.tex} %EndExpansion \\ \hline \end{tabular}% \label{5}$ \begin{theorem} \label{8}(Veta o adit\'{\i}vnosti ur\v{c}it\'{e}ho integr\'{a}lu) Nech $% f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je spojit\'{a} funkcia a nech $c\in \left( a,b\right) $. Potom \[ \int_{a}^{b}f\left( x\right) dx=\int_{a}^{c}f\left( x\right) dx+\int_{c}^{b}f\left( x\right) dx. \] \end{theorem} $% \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small D\^{o}kaz}}{}{}{DO722.tex} }% %BeginExpansion \msihyperref{{\small D\^{o}kaz}}{}{}{DO722.tex} %EndExpansion \\ \hline \end{tabular}% \label{3}$ \begin{example} Vypo\v{c}\'{\i}tajte $\int_{-1}^{2}\left| x\right| dx$. \end{example} \begin{solution} Funkcia $f\left( x\right) =\left| x\right| $ je spojit\'{a} na intervale $% \left\langle -1,2\right\rangle ,$\ ke\v{d} vyu\v{z}ijeme vetu o adit\'{\i}% vnosti ur\v{c}it\'{e}ho integr\'{a}lu a v\'{y}sledok %TCIMACRO{\hyperref{pr\'{\i}kladu}{}{}{M71.tex#4} }% %BeginExpansion \msihyperref{pr\'{\i}kladu}{}{}{M71.tex#4} %EndExpansion dostaneme: \[ \int_{-1}^{2}\left| x\right| dx=\int_{-1}^{0}\left| x\right| dx+\int_{0}^{2}\left| x\right| dx=\int_{-1}^{0}\left( -x\right) dx+\int_{0}^{2}xdx= \]% \[ =-\int_{-1}^{0}xdx+\int_{0}^{2}xdx=-\frac{0^{2}-\left( -1\right) ^{2}}{2}+% \frac{2^{2}-0^{2}}{2}=\frac{1}{2}+2=\frac{5}{2}.\square \] \end{solution} \subsubsection{Veta o strednej hodnote pre integr\'{a}ly} \begin{theorem} \label{10}\label{1}(Veta o strednej hodnote pre integr\'{a}ly). Nech $% f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je spojit\'{a} funkcia. Potom existuje bod $c\in \left\langle a,b\right\rangle $ tak\'{y}, \v{z}e \[ \int_{a}^{b}f\left( x\right) dx=f\left( c\right) \left( b-a\right) . \] \end{theorem} $% \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small D\^{o}kaz}}{}{}{DO724.tex} }% %BeginExpansion \msihyperref{{\small D\^{o}kaz}}{}{}{DO724.tex} %EndExpansion \\ \hline \end{tabular}% \label{6}$ \begin{quote} \bigskip Hodnota \[ f\left( c\right) =\frac{1}{b-a}\int_{a}^{b}f\left( x\right) dx \]% sa naz\'{y}va stredn\'{a} hodnota funkcie $f$ na intervale $\left\langle a,b\right\rangle $. \end{quote} \begin{example} N\'{a}jdime stredn\'{u} hodnotu funkcie $f:\left\langle 1,3\right\rangle \longrightarrow \mathbf{R,\,}f\left( x\right) =x.$\ \end{example} \begin{solution} M\'{a}me \[ \int_{1}^{3}xdx=\frac{3^{2}-1^{2}}{2}=4. \]% Pod\v{l}a vety o strednej hodnote existuje $c\in \left\langle 1,3\right\rangle $ tak\'{y}, \v{z}e $\int_{1}^{3}xdx=f\left( c\right) \left( 3-1\right) .$V na\v{s}om pr\'{\i}pade strednou hodnotou funkcie $f\left( x\right) =x$\ na intervale $\left\langle 1,3\right\rangle $ je hodnota \[ f\left( 2\right) =2, \]% preto\v{z}e \[ f\left( 2\right) =2=\frac{1}{3-1}\int_{1}^{3}xdx.\ \square \] \end{solution} \subsubsection{Pomocn\'{e} defin\'{\i}cie} V defin\'{\i}cii $\int_{a}^{b}f\left( x\right) dx$ sme explicitne predpokladali, \v{z}e $a