\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Monday, March 24, 2003 13:19:10} %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{Vektor gradientu, vety o strednej hodnote} \subsection{Veta o strednej hodnote.} \textbf{D\^{o}kaz: }Nech $\mathbf{\,}\Phi :\left\langle 0,1\right\rangle \longrightarrow \mathbf{R},\,\Phi (t)=f(\mathbf{a}+t\mathbf{h}).$ Potom $% \Phi =f\mathbf{\circ g}$ kde $\mathbf{g}:\left\langle 0,1\right\rangle \longrightarrow \mathbf{R}^{n},\mathbf{g}(t)=\mathbf{a}+t\mathbf{h}.$ Potom $% \mathbf{g}$ je diferencovate\v{l}n\'{a} na $(0,1),$ pri\v{c}om $D\mathbf{g}% _{t}=\mathbf{h}.$ Potom \[ \Phi ^{\prime }(t)=\frac{d\Phi }{dt}=Df_{\mathbf{g}(t)}D\mathbf{g}_{t}=Df_{% \mathbf{a}+t\mathbf{h}}(\mathbf{h})=\mathbf{\,}\nabla f(\mathbf{a}+t\mathbf{h% })\cdot \mathbf{h.} \]% Potom z vety o strednej hodnote pre jednu premenn\'{u} dostaneme: \[ \Phi (1)-\Phi (0)=\Phi ^{\prime }(\theta )(1-0),\text{\textbf{\thinspace\ \ }% kde \ }\mathbf{\,}\theta \in (0,1). \]% To znamen\'{a}, \v{z}e \[ f(\mathbf{a}+\mathbf{h})-f(\mathbf{a})=Df_{\mathbf{a}+\theta \mathbf{h}}(% \mathbf{h}).\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{Ma34.tex#3}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \end{document}