\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 13:23:10} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - D\U{f4}kazy\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{Taylorova veta} \subsection{Taylorova veta pre re\'{a}lnu funkciu viacer\'{y}ch premenn\'{y}% ch.} \textbf{D\^{o}kaz: }Nech $\Phi (t)=f(\mathbf{a}+t\mathbf{h}).$ Potom $\Phi $ je definovan\'{e} pre ka\v{z}d\'{e} $t$ tak\'{e}, \v{z}e $\mathbf{a}+t% \mathbf{h}\in G.$ Pod\v{l}a predpokladov vety $\mathbf{a}+t\mathbf{h}\in G,$ pre v\v{s}etky $t\in \left\langle 0,1\right\rangle $ potom m\^{o}\v{z}me aplikova\v{t} Taylorovu vetu pre re\'{a}lnu funkciu re\'{a}lnej premennej na funkciu $\Phi :\left\langle 0,1\right\rangle \longrightarrow \mathbf{R}.$ Preto\v{z}e $f\in C^{1},\,\Phi $ je diferencovate\v{l}n\'{a}, pod\v{l}a re% \v{t}azov\'{e}ho pravidla m\'{a}me \[ \Phi ^{\prime }(t)=Df_{\mathbf{a}+t\mathbf{h}}(\mathbf{h})=\nabla f(\mathbf{a% }+t\mathbf{h})\cdot \mathbf{h}=\sum_{i=1}^{n}\frac{\partial f\left( \mathbf{a% }+t\mathbf{h}\right) }{\partial x_{i}}h_{i}. \]% Preto\v{z}e $f\in C^{2},\,\frac{\partial f}{\partial x_{i}}$ s\'{u} $C^{1}$ funkcie, teda diferencovate\v{l}n\'{e} a my m\^{o}\v{z}eme pokra\v{c}ova\v{t} podobn\'{y}m sp\^{o}sobom: \[ \Phi ^{\prime \prime }(t)=\frac{d}{dt}\left( \sum_{i=1}^{n}\frac{\partial f\left( \mathbf{a}+t\mathbf{h}\right) }{\partial x_{i}}h_{i}\right) =\sum_{i=1}^{n}\left( \sum_{j=1}^{n}\frac{\partial ^{2}f\left( \mathbf{a}+t% \mathbf{h}\right) }{\partial x_{i}\partial x_{j}}h_{j}\right) h_{i}. \]% Tento proces bude pokra\v{c}ova\v{t} ak namiesto $i,j,\dots $ budeme uva\v{z}% ova\v{t} $i_{1},i_{2},\dots ,i_{q}$ a dostaneme: \[ \Phi ^{\left( q\right) }(t)=\sum_{i_{1}=1}^{n}\sum_{i_{2}=1}^{n}\dots \sum_{i_{q}=1}^{n}\frac{\partial ^{q}f\left( \mathbf{a}+t\mathbf{h}\right) }{% \partial x_{i_{1}}\partial x_{i_{2}}\dots \partial x_{i_{q}}}% h_{i_{1}}h_{i_{2}}\dots h_{i_{q}},\,1\leq q\leq p. \]% Potom pod\v{l}a \hyperref{vety.}{}{}{Ma42.tex#13} \[ \Phi (1)=\Phi (0)+\Phi ^{\prime }(0)+\frac{\Phi ^{\prime \prime }(0)}{2!}% +\dots +\frac{\Phi ^{\left( p-1\right) }(0)}{(p-1)!}+E_{p}(\Theta ),\,0<\Theta <1.\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma42.tex#2}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}