%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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 17:36:04} %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 II online - Funkcie, limita funkcie, spojit\U{e9} funkcie - Ot\U{e1}zky\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{Funkcie, limita funkcie, spojit\'{e} funkcie} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Ot\'{a}zky} Preverte si znalosti z\'{\i}skan\'{e} v tejto kapitole. Zodpovedajte v\v{s}% etky ot\'{a}zky. Ak neviete nejak\'{u} ot\'{a}zku zodpoveda\v{t} znovu pre% \v{s}tudujte pr\'{\i}slu\v{s}n\'{u} \v{c}as\v{t} a op\"{a}\v{t} odpovedajte. Po d\^{o}kladnom preveren\'{\i} Va\v{s}ich vedomost\'{\i} sa venujte po\v{c}% \'{\i}taniu pr\'{\i}kladov. \begin{enumerate} \item Definujte funkciu viacer\'{y}ch premenn\'{y}ch. \item Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{2}\right) \longrightarrow \mathbf{R}^{3},\,\mathbf{f}\left( \mathbf{x}\right) =\,\mathbf{f}\left( x_{1},x_{2}\right) =\left( x_{3},\sin \left( x_{1}x_{2}\right) ,\sqrt{% x_{1}^{2}+x_{2}^{2}}\right) .$ N\'{a}jdite defini\v{c}n\'{y} obor $A$ funkcie $\mathbf{f}$ a nap\'{\i}\v{s}te jej komponenty. \item Definujte limitu funkcie viacer\'{y}ch premenn\'{y}ch v bode. \item Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}\,$\ m\'{a} komponenty $\mathbf{f}=(f_{1},f_{2},...,f_{m}).$ Plat\'{\i} nasleduj\'{u}ce tvrdenie: $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b\Longleftrightarrow } $ $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f_{i}\left( \mathbf{x}\right) =b_{i},\,\forall i=1,2,...,m.$ \ Ak tvrdenie neplat\'{\i}, uve\v{d}te spr% \'{a}vne tvrdenie. \item Plat\'{\i} pre funkciu \ $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ nasleduj\'{u}ce tvrdenie: ak $% \lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}% \right) =\mathbf{b},\,$potom pre ka\v{z}d\'{u} podmno\v{z}inu $M\subset A$ defini\v{c}n\'{e}ho oboru funkcie $\mathbf{f},$ ktorej hromadn\'{y}m bodom je bod $\mathbf{a},$ plat\'{\i} \ $\lim_{\mathbf{x}\longrightarrow \mathbf{% a,x}\in M}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b.}$ Ak tvrdenie neplat% \'{\i}, uve\v{d}te pr\'{\i}klad. \item Za ak\'{y}ch predpokladov existuje s\'{u}\v{c}et dvoch funkci\'{\i} viacer\'{y}ch premenn\'{y}ch? Defini\v{c}n\'{y} obor s\'{u}\v{c}tu dvoch funkci\'{\i} sa rovn\'{a} zjednoteniu ich defini\v{c}n\'{y}ch oborov? Ak \'{a}no, zd\^{o}vodnite, ak nie uve\v{d}te pr\'{\i}klad. \item Za ak\'{y}ch predpokladov existuje s\'{u}\v{c}in dvoch funkci\'{\i} viacer\'{y}ch premenn\'{y}ch? Defini\v{c}n\'{y} obor s\'{u}\v{c}inu dvoch funkci\'{\i} sa rovn\'{a} kart\'{e}zskemu s\'{u}\v{c}inu ich defini\v{c}n% \'{y}ch oborov? Ak \'{a}no, zd\^{o}vodnite, ak nie uve\v{d}te pr\'{\i}klad. \item Definujte $\mathbf{f}\circ \mathbf{g.}$ Uve\v{d}te pr\'{\i}klad. \item Je skladanie funkci\'{\i} komutat\'{\i}vna oper\'{a}cia? Uve\v{d}te pr% \'{\i}klad. \item Plat\'{\i} pre funkciu \ $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ nasleduj\'{u}ce tvrdenie: ak pre dve podmno\v{z}iny $L,\,M\subset A$ defini\v{c}n\'{e}ho oboru funkcie $% \mathbf{f},$ ktor\'{y}ch hromadn\'{y}m bodom je bod $\mathbf{a},$ plat\'{\i} $\lim_{\mathbf{x}\longrightarrow \mathbf{a,x}\in L}\mathbf{f}\left( \mathbf{x% }\right) \neq \lim_{\mathbf{x}\longrightarrow \mathbf{a,x}\in M}\mathbf{f}% \left( \mathbf{x}\right) =\mathbf{b},$ potom $\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) $ neexistuje. Ak tvrdenie neplat\'{\i} uve\v{d}te pr\'{\i}klad. \item Limitu $\lim_{\left( x,y\right) \longrightarrow \left( 2,-1\right) }% \frac{x^{3}+y^{3}}{x^{2}+y^{2}},$ vypo\v{c}\'{\i}tame pou\v{z}it\'{\i}m vety o limite podielu? \item Limitu $\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }% \frac{x^{3}+y^{3}}{x^{2}+y^{2}},$ vypo\v{c}\'{\i}tame pou\v{z}it\'{\i}m\ vety o nerovnostiach pre limity? \item Definujte kedy je funkcia $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ spojit\'{a} v bode $\mathbf{a}% \in A.$ \item Plat\'{\i} tvrdenie: funkcia $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ je spojit\'{a} v bode $\mathbf{a}% \in A$ vtedy a len vtedy, ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}% \mathbf{f}\left( \mathbf{x}\right) =\mathbf{f}\left( \mathbf{a}\right) ?$ \item Plat\'{\i} pre funkciu $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ tvrdenie: $\mathbf{f}$ je spojit% \'{a} v bode $\mathbf{a}\in A$ vtedy a len vtedy ak pre ka\v{z}d\'{u} postupnos\v{t} $\left\{ \mathbf{x}^{\left( k\right) }\right\} _{k=1}^{\infty }\subset A,$ tak\'{u} \v{z}e $\lim_{k\longrightarrow \infty }\mathbf{x}% ^{\left( k\right) }=\mathbf{a\Longrightarrow }\lim_{k\longrightarrow \infty }% \mathbf{f}\left( \mathbf{x}^{\left( k\right) }\right) =\mathbf{f}\left( \mathbf{a}\right) ?$ \item Nech $K\subset \mathbf{R}^{n}$ je uzavret\'{a} mno\v{z}ina a nech $% \mathbf{f}:K\longrightarrow \mathbf{R}^{m}$ je funkcia spojit\'{a} na $K.$ Mno\v{z}ina $\mathbf{f}(K)$\ je uzavret\'{a} (aledbo otvoren\'{a})\ podmo% \v{z}ina priestoru $\mathbf{R}^{m}?$ \item Nech $K\subset \mathbf{R}^{n}$ je ohrani\v{c}en\'{a} mno\v{z}ina a nech $\mathbf{f}:K\longrightarrow \mathbf{R}^{m}$ je funkcia spojit\'{a} na $% K.$ Je $\mathbf{f}(K)$ ohrani\v{c}en\'{a} podmo\v{z}ina priestoru $\mathbf{R}% ^{m}?$ \item Nech $K\subset \mathbf{R}^{n}$ je kompaktn\'{a} mno\v{z}ina a nech $% \mathbf{f}:K\longrightarrow \mathbf{R}^{m}$ je funkcia spojit\'{a} na $K.$ Je $\mathbf{f}(K)$ kompaktn\'{a} podmo\v{z}ina priestoru $\mathbf{R}^{m}?$ \item Definujte minimum a maximum funkcie viecer\'{y}ch premenn\'{y}ch a uve% \v{d}te pr\'{\i}klad. \item Nech $K\subset \mathbf{R}^{n}$ je kompaktn\'{a} mno\v{z}ina a nech $% f:K\longrightarrow \mathbf{R}$ je funkcia spojit\'{a} na $K.$ Potom $f(K)$ m% \'{a} minimum aj maximum? \end{enumerate} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Funkcie, limita funkcie, spojit\'{e} funkcie} \end{document}