%% 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 15:53:50} %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 - Aplik\U{e1}cie diferenci\U{e1}lneho po\U{10d}tu - Taylorova veta\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{Aplik\'{a}cie diferenci\'{a}lneho po\v{c}tu} \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}{}{}{M6.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M6.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M62.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M62.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M64.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M64.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O6.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O6.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C6.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C6.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Taylorova veta} Predpokladajme, \v{z}e by sme chceli aproximova\v{t} hodnotu funkcie $f(x)$ v bl\'{\i}zkosti bodu $a$ a pozn\'{a}me hodnotu $f(a)$. Aproxim\'{a}ciu u% \v{z} dok\'{a}\v{z}eme urobi\v{t} pomocou diferenci\'{a}lu (t.j. znalosti hodnoty $f\,$\/$\,^{\prime }(a)$). Ak v\v{s}ak chceme e\v{s}te presnej\v{s}% iu aproxim\'{a}ciu je potrebn\'{e} namiesto aproxim\'{a}cie line\'{a}rnou funkciou hodnotu $f(x)$ aproximova\v{t} polyn\'{o}mom. \begin{theorem} \label{2}(Taylorova veta) Nech $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je n-kr\'{a}t spojite diferencovate\v{l}n\'{a} funkcia. Nech $f$ je (n+1)-kr\'{a}t diferencovate\v{l}n\'{a} na otvorenom intervale $(a,b)$. Potom existuje $c\in (a,b)$ tak\'{e}, \v{z}e \[ f(b)=f(a)+f\,\/\,^{\prime }(a)(b-a)+\frac{f\,\/\,^{\prime \prime }\left( a\right) }{2!}\left( b-a\right) ^{2}+...+\frac{f^{\left( n\right) }\left( a\right) }{n!}\left( b-a\right) ^{n}+\frac{f^{\left( n+1\right) }\left( c\right) }{\left( n+1\right) !}(b-a)^{n+1}. \] \end{theorem} $% \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small D\^{o}kaz}}{}{}{DO631.tex} }% %BeginExpansion \msihyperref{{\small D\^{o}kaz}}{}{}{DO631.tex} %EndExpansion \\ \hline \end{tabular}% \label{1}$ \begin{description} \item[Pozn\'{a}mka] a) Taylorova veta je \v{c}asto prezentovan\'{a} v tvare, ke\v{d} namiesto $b$ p\'{\i}\v{s}eme $x$ a namiesto $a$ p\'{\i}\v{s}eme $% x_{0}$. b) Taylorovu vetu pou\v{z}\'{\i}vame hlavne v pr\'{\i}pade, ke\v{d} $% f^{\left( n+1\right) }:A_{n+1}\longrightarrow \mathbf{R}$ je spojit\'{a} na $% O_{\delta }^{\circ }\left( x_{0}\right) $. Potom $\forall x\in O_{\delta }^{\circ }\left( x_{0}\right) $ m\'{a}me \[ f(x)=f(x_{0})+f\,\/\,^{\prime }(x_{0})(x-x_{0})+\frac{f\,\/\,^{\prime \prime }\left( x_{0}\right) }{2!}\left( x-x_{0}\right) ^{2}+...+\frac{f^{\left( n\right) }\left( x_{0}\right) }{n!}\left( x-x_{0}\right) ^{n}+\frac{% f^{\left( n+1\right) }\left( c\right) }{\left( n+1\right) !}(x-x_{0})^{n+1}, \]% kde $c\in O_{\delta }^{\circ }\left( x_{0}\right) $ z\'{a}vis\'{\i} nielen od $x,\,x_{0}$ ale aj od $n\in \mathbf{N}$. c) Polyn\'{o}m \[ \label{3}T_{n}:\mathbf{R}\longrightarrow \mathbf{R},\,T_{n}\left( x\right) =f(x_{0})+f\,\/\,^{\prime }(x_{0})(x-x_{0})+\frac{f\,\/\,^{\prime \prime }\left( x_{0}\right) }{2!}\left( x-x_{0}\right) ^{2}+...+\frac{f^{\left( n\right) }\left( x_{0}\right) }{n!}\left( x-x_{0}\right) ^{n} \]% \textsl{naz\'{y}vame n-t\'{y} Taylorov polyn\'{o}m funkcie} $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ v bode $x_{0}$ a$\ $% \[ r_{n}:O_{\delta }^{\circ }\left( x_{0}\right) \longrightarrow \mathbf{R}% ,\,r_{n}\left( x\right) =\frac{f^{\left( n+1\right) }\left( c\right) }{% \left( n+1\right) !}(x-x_{0})^{n+1} \]% naz\'{y}vame \textsl{zvy\v{s}ok po n-tom Taylorovom polyn\'{o}me funkcie (Lagrangeov tvar zvy\v{s}ku)} $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ v bode $x_{0}$. \end{description} \begin{example} Aproximujte hodnotu $\sin \frac{7\pi }{36}$ s chybou men\v{s}ou ako $0,00001$% . \end{example} \begin{solution} Nech $f(x)=\sin x$, $x=\frac{7\pi }{36}$, $x_{0}=0$. Vieme, \v{z}e $\forall c $ a $\,\forall n$ plat\'{\i} $\left| f^{\left( n+1\right) }\left( c\right) \right| =\left| \sin c\right| \leq 1$, alebo $\left| f^{\left( n+1\right) }\left( c\right) \right| =\left| \cos c\right| \leq 1$ potom m\'{a}me odhad pre $R_{n}\left( \frac{7\pi }{36}\right) $:% \[ \left| r_{n}\left( \frac{7\pi }{36}\right) \right| =\left| \frac{f^{\left( n+1\right) }\left( c\right) }{\left( n+1\right) !}\right| \left| \frac{7\pi }{36}\right| ^{n+1}\leq \frac{1}{\left( n+1\right) !}\left( \frac{7\pi }{36}% \right) ^{n+1}. \]% M\'{a} plati\v{t} $\left| r_{n}\left( \frac{7\pi }{36}\right) \right| \leq 0,00001$. T\'{a}to nerovnica je splnen\'{a} pre $n\geq 6$, teda \[ \sin \frac{7\pi }{36}\approx T_{6}\left( \frac{7\pi }{36}\right) =\left( \frac{7\pi }{36}\right) -\frac{1}{3!}\left( \frac{7\pi }{36}\right) ^{3}+% \frac{1}{5!}\left( \frac{7\pi }{36}\right) ^{5}=0,573583.\square \] \end{solution} \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}{}{}{M6.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M6.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M62.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M62.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M64.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M64.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O6.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O6.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C6.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C6.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Aplik\'{a}cie diferenci\'{a}lneho po\v{c}tu} \end{document}