%% 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 15:48:07} %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 - Diferenci\U{e1}lny po\U{10d}et funkci\U{ed} jednej re\U{e1}lnej premennej - Deriv\U{e1}cia funkcie\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{Diferenci\'{a}lny po\v{c}et funkci\'{\i} jednej re\'{a}lnej premennej} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M5.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M52.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M52.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} Deriv\'{a}cia funkcie v bode je \v{d}al\v{s}\'{\i}m d\^{o}le\v{z}it\'{y}m pojmom matematickej anal\'{y}zy. Jeho znalos\v{t} m\'{a} aj ve\v{l}k\'{y} praktick\'{y} v\'{y}znam. Pomocou deriv\'{a}ci\'{\i} funkcie sa daj\'{u} zisti\v{t} \v{d}al\v{s}ie vlastnosti funkci\'{\i}. Pojem deriv\'{a}cie funkcie sa \v{c}asto vyskytuje v aplik\'{a}ci\'{a}ch najm\"{a} vo fyzik\'{a}% lnych a technick\'{y}ch ved\'{a}ch, kde s\'{u} niektor\'{e} ich pojmy definovan\'{e} priamo pomocou deriv\'{a}cie. \paragraph{Ciele} Po pre\v{s}tudovan\'{\i} tejto \v{c}asti by ste mali by\v{t} schopn\'{\i}: \begin{itemize} \item rozumie\v{t} pojmu deriv\'{a}cia funkcie v bode, \item ch\'{a}pa\v{t} geometrick\'{y} v\'{y}znam deriv\'{a}cie, \item rozumie\v{t} vz\v{t}ahu medzi diferencovate\v{l}nos\v{t}ou funkcie v bode a jej spojitos\v{t}ou, \item ovl\'{a}da\v{t} pravidl\'{a} derivovania, \item vedie\v{t} vypo\v{c}\'{\i}ta\v{t} deriv\'{a}cie r\^{o}znych funkci% \'{\i}, \item vypo\v{c}\'{\i}ta\v{t} diferenci\'{a}l funkcie, \item n\'{a}js\v{t} rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie v bode, \item vypo\v{c}\'{\i}ta\v{t} deriv\'{a}cie vy\v{s}\v{s}\'{\i}ch r\'{a}dov. \end{itemize} \paragraph{Po\v{z}adovan\'{e} vedomosti} \begin{itemize} \item znalos\v{t} vlastnost\'{\i} \ spojit\'{y}ch funkci\'{\i}, \item znalos\v{t} limity funkcie a viet o limit\'{a}ch, \item znalos\v{t} spojitosti funkci\'{\i}. \end{itemize} \subsection{Deriv\'{a}cia funkcie} Teraz sa budeme zaobera\v{t} deriv\'{a}ciou funkcie a jej pou\v{z}it\'{\i}m. K pojmu deriv\'{a}cie funkcie ved\'{u} hlavne nasleduj\'{u}ce typy \'{u}loh. \begin{example} Pozorujeme hmotn\'{y} bod v \v{c}asovom intervale $I$ pri jeho pohybe po priamke, ktor\'{a} je \v{c}\'{\i}selnou osou $o_{t}$. Funkcia $% f:I\longrightarrow \mathbf{R}$, ktor\'{a} ka\v{z}d\'{e}mu $t\in I$ jednozna% \v{c}ne prirad\'{\i} hodnotu $f(t)$ - bod, v ktorom sa nach\'{a}dza hmotn% \'{y} bod na \v{c}\'{\i}selnej osi $o_{t}$ v okamihu $t$, popisuje pohyb tohto bodu. \v{C}\'{\i}slo $\frac{f\left( t\right) -f\left( t_{0}\right) }{% t-t_{0}}$ naz\'{y}vame \emph{priemernou r\'{y}chlos\v{t}ou} bodu pohybuj\'{u}% ceho sa v \v{c}asovom intervale $\left\langle t_{0},t\right\rangle $. \v{C}% \'{\i}slo $\lim_{t\longrightarrow t_{0}}\frac{f\left( t\right) -f\left( t_{0}\right) }{t-t_{0}}=f\,$\/$\,^{\prime }\left( t_{0}\right) $ sa naz\'{y}% va \emph{okam\v{z}itou r\'{y}chlos\v{t}ou} pohybuj\'{u}ceho sa bodu v okamihu $t_{0}$. \end{example} \begin{example} V \v{c}asovom intervale $I$ pozorujeme elektrick\'{y} n\'{a}boj pretekaj\'{u}% ci prierezom vodi\v{c}a. Funkcia $f:I\longrightarrow \mathbf{R}$, ktor\'{a} ka\v{z}d\'{e}mu $t\in I$ jednozna\v{c}ne prirad\'{\i} hodnotu $f(t)$ ud\'{a}% va ve\v{l}kos\v{t} elektrick\'{e}ho n\'{a}boja, ktor\'{y} pretiekol prierezom vodi\v{c}a od za\v{c}iato\v{c}n\'{e}ho okamihu $t_{0}$ po okamih $% t $, popisuje pozorovan\'{y} jav. \v{C}\'{\i}slo $\frac{f\left( t\right) -f\left( t_{0}\right) }{t-t_{0}}$ naz\'{y}vame \emph{priemernou intenzitou} elektrick\'{e}ho pr\'{u}du na \v{c}asovom intervale$\left\langle t_{0},t\right\rangle $. \v{C}\'{\i}slo $\lim_{t\longrightarrow t_{0}}\frac{% f\left( t\right) -f\left( t_{0}\right) }{t-t_{0}}=f\,$\/$\,^{\prime }\left( t_{0}\right) $ sa naz\'{y}va \emph{intenzitou elektrick\'{e}ho pr\'{u}du} v okamihu $t_{0}$. \end{example} \begin{example} Nech $I$ je interval a nech $f:I\longrightarrow \mathbf{R}$. Nech $x_{0}\in I $. \v{C}\'{\i}slo $\frac{f\left( x\right) -f\left( x_{0}\right) }{x-x_{0}}$ je \emph{smernica se\v{c}nice} grafu funkcie $f:I\longrightarrow \mathbf{R}$ prech\'{a}dzaj\'{u}cej bodmi $(x,f(x))$, $(x_{0},f(x_{0}))$. \v{C}\'{\i}slo $% \lim_{x\longrightarrow x_{0}}\frac{f\left( x\right) -f\left( x_{0}\right) }{% x-x_{0}}=f\,$\/$\,^{\prime }\left( x_{0}\right) $ je \emph{smernica doty\v{c}% nice} ku grafu funkcie $f$ v bode $(x_{0},f(x_{0}))$. \end{example} Predch\'{a}dzaj\'{u}ce pr\'{\i}klady a cel\'{y} rad in\'{y}ch fyzik\'{a}% lnych a praktick\'{y}ch \'{u}loh n\'{a}s ved\'{u} k limite: \[ \lim_{x\longrightarrow x_{0}}\frac{f\left( x\right) -f\left( x_{0}\right) }{% x-x_{0}}, \]% s ktorou sa teraz budeme \v{c}astej\v{s}ie streta\v{t} a podrobnej\v{s}ie zaobera\v{t}. \begin{definition} \label{1}Nech $f:A\longrightarrow \mathbf{R}$ je definovan\'{a} v okol\'{\i} bodu $a\in A.$ Ak existuje vlastn\'{a} limita \[ \lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}=f\,% \text{\/}\,^{\prime }\left( a\right) , \]% tak \v{c}\'{\i}slo $f\,$\/$\,^{\prime }\left( a\right) $ naz\'{y}vame \emph{% deriv\'{a}ciou} funkcie $f:A\longrightarrow \mathbf{R}$ v bode $a$ a hovor% \'{\i}me, \v{z}e funkcia $f$ je v bode $a$ diferencovate\v{l}n\'{a}. \end{definition} \begin{description} \item[Pozn\'{a}mka] \v{C}asto budeme namiesto limity z predch\'{a}dzaj\'{u}% cej defin\'{\i}cie pou\v{z}\'{\i}va\v{t} ekvivalentn\'{u} defin\'{\i}ciu deriv\'{a}cie: $\lim_{h\longrightarrow 0}\frac{f\left( a+h\right) -f\left( a\right) }{h}=f\,$\/$\,^{\prime }\left( a\right) .$T\'{u}to defin\'{\i}ciu dostaneme, ak v defin\'{\i}cii deriv\'{a}cie polo\v{z}\'{\i}me $x-a=h.$ \end{description} Deriv\'{a}ciu funkcie $f$ ozna\v{c}ujeme: $f\,$\/$\,^{\prime },Df,\frac{df}{% dx}$ (Leibnizovo ozna\v{c}enie). My budeme naj\v{c}astej\v{s}ie pou\v{z}% \'{\i}va\v{t} prv\'{e} ozna\v{c}enie. \begin{quote} %TCIMACRO{% %\hyperref{\textbf{Geometrick\'{y} v\'{y}znam deriv\'{a}cie}}{}{}{Derivacia\Derivacia.html} }% %BeginExpansion \msihyperref{\textbf{Geometrick\'{y} v\'{y}znam deriv\'{a}cie}}{}{}{Derivacia\Derivacia.html} %EndExpansion - anim\'{a}cia zobrazuj\'{u}ca zmenu se\v{c}nice ku grafu funkcie so smernicou $\frac{f\left( x\right) -f\left( a\right) }{x-a}$ na doty\v{c}nicu ku grafu funkcie so smernicou $f\,$\/$\,^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}$ pre $x\longrightarrow a$. \end{quote} \begin{example} Nech $f:\mathbf{R}\longrightarrow \mathbf{R},\,f\left( x\right) =\frac{1}{4}% x^{2}+1$. N\'{a}jdite $f\,$\/$\,^{\prime }(-2),\,f\,$\/$\,^{\prime }(4)$. \end{example} \begin{solution} Po\v{c}\'{\i}tajme limity:% \[ f\,\/\,^{\prime }\left( -2\right) =\lim_{x\longrightarrow -2}\frac{f\left( x\right) -f\left( -2\right) }{x-\left( -2\right) }=\lim_{x\longrightarrow -2}% \frac{\frac{1}{4}x^{2}+1-2}{x+2}= \]% \[ =\frac{1}{4}\lim_{x\longrightarrow -2}\frac{x^{2}-4}{x+2}=\frac{1}{4}% \lim_{x\longrightarrow -2}\frac{\left( x+2\right) \left( x-2\right) }{x+2}% =-1. \]% \[ f\,\/\,^{\prime }\left( 4\right) =\lim_{x\longrightarrow 4}\frac{f\left( x\right) -f\left( 4\right) }{x-4}=\lim_{x\longrightarrow 4}\frac{\frac{1}{4}% x^{2}+1-5}{x-4}= \]% \[ =\frac{1}{4}\lim_{x\longrightarrow 4}\frac{x^{2}-16}{x-4}=\frac{1}{4}% \lim_{x\longrightarrow 4}\frac{\left( x+4\right) \left( x-4\right) }{x-4}% =2.\square \] \end{solution} \begin{example} Nech $I$ je otvoren\'{y} interval, $k\in \mathbf{R,\,}a\in A$ je \v{l}ubovo% \v{l}n\'{y} bod. Nech $f:I\longrightarrow \mathbf{R,\,}f\left( x\right) =k$. Potom $f\,$\/$\,^{\prime }(a)=0$. \end{example} \begin{solution} Nech $a\in I$ je \v{l}ubovo\v{l}n\'{y} bod.$\ $% \[ f\,\/\,^{\prime }(a)=\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}=\lim_{x\longrightarrow a}\frac{k-k}{x-a}=0. \]% Preto\v{z}e bod $a$ bol \v{l}ubovo\v{l}n\'{y}, tak $f\,$\/$\,^{\prime }(a)=0\,\forall a\in I$. $\square $ \end{solution} \begin{example} Nech $f:\mathbf{R}\longrightarrow \mathbf{R},\,f\left( x\right) =x$. N\'{a}% jdite $f\,$\/$\,^{\prime }(a)$. \end{example} \begin{solution} Nech $a\in \mathbf{R}$ je \v{l}ubovo\v{l}n\'{y} bod, potom \[ f\,\/\,^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}=\lim_{x\longrightarrow a}\frac{x-a}{x-a}% =1.\square \] \end{solution} \begin{example} Nech $f:\mathbf{R}\longrightarrow \mathbf{R},\,f\left( x\right) =x^{2}$. Uk% \'{a}\v{z}te, \v{z}e $f\,$\/$\,^{\prime }(a)=2a,\,\forall a\in \mathbf{R}$. \end{example} \begin{solution} Nech $a\in \mathbf{R}$ je \v{l}ubovo\v{l}n\'{y} bod, potom \[ f\,\/\,^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a}=\lim_{x\longrightarrow a}\frac{x^{2}-a^{2}% }{x-a}=\lim_{x\longrightarrow a}\frac{\left( x-a\right) \left( x+a\right) }{% x-a}=2a.\square \] \end{solution} Ak pre funkciu $f:A\longrightarrow \mathbf{R,\,}a\in A$ existuje $f\,$\/$% \,^{\prime }(a)$, potom \label{2}\emph{rovnica doty\v{c}nice} ku grafu funkcie $f$ v bode $(a,f(a))$ m\'{a} tvar \[ y-f(a)=f\,\/\,^{\prime }(a)(x-a),\text{ alebo }y=f(a)+f\,\/\,^{\prime }(a)(x-a) \]% ak pre funkciu $f:A\longrightarrow \mathbf{R,\,}a\in A$ existuje $f\,$\/$% \,^{\prime }(a)\neq 0$, potom \label{3}\emph{rovnica norm\'{a}ly } ku grafu funkcie $f$ v bode $(a,f(a))$ m\'{a} tvar \[ y-f(a)=-\frac{1}{f\,\/\,^{\prime }\left( a\right) }(x-a),\text{ alebo }% y=f(a)-\frac{1}{f\,\/\,^{\prime }\left( a\right) }(x-a). \]% Norm\'{a}la ku grafu funkcie je priamka kolm\'{a} na doty\v{c}nicu ku grafu funkcie v dotykovom bode. \begin{definition} \label{4}Nech $f:A\longrightarrow \mathbf{R}$ je definovan\'{a} v \v{l}avom (pravom) okol\'{\i} bodu $a\in A.$ Ak existuje vlastn\'{a} limita \[ \lim_{x\longrightarrow a^{-}}\frac{f\left( x\right) -f\left( a\right) }{x-a}% =f\,\/_{-}^{\prime }\left( a\right) \;\left( \lim_{x\longrightarrow a^{+}}% \frac{f\left( x\right) -f\left( a\right) }{x-a}=f\,\/_{+}^{\prime }\left( a\right) \right) , \]% tak \v{c}\'{\i}slo $f\,\/_{-}^{\prime }\left( a\right) $ $\left( f\,\/_{+}^{\prime }\left( a\right) \right) $\ \ naz\'{y}vame \emph{deriv\'{a}% ciou z\v{l}ava (sprava)} funkcie $f:A\longrightarrow \mathbf{R}$ v bode $a$. \end{definition} $f\,\/_{-}^{\prime }\left( a\right) ,$ $f\,\/_{+}^{\prime }\left( a\right) $ naz\'{y}vame \emph{jednostrann\'{y}mi deriv\'{a}ciami} funkcie $f$ v bode $a$% . \begin{theorem} Funkcia $f:A\longrightarrow \mathbf{R}$ definovan\'{a} v okol\'{\i} bodu $% a\in A$ m\'{a} v bode $a$ deriv\'{a}ciu $f\,$\/$\,^{\prime }\left( a\right) $ pr\'{a}ve vtedy, ak m\'{a} v bode $a$ deriv\'{a}ciu z\v{l}ava $% f\,\/_{-}^{\prime }\left( a\right) $ aj deriv\'{a}ciu sprava $% f\,\/_{+}^{\prime }\left( a\right) $\ a plat\'{\i} $f\,\/_{-}^{\prime }\left( a\right) =$ $f\,\/_{+}^{\prime }\left( a\right) .$ Potom $f\,$\/$% \,^{\prime }(a)=f\,\/_{-}^{\prime }\left( a\right) =f\,\/_{+}^{\prime }\left( a\right) .$ \end{theorem} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M5.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M52.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M52.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C5.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{Diferenci\'{a}lny po\v{c}et funkci\'{\i} jednej re\'{a}lnej premennej} \end{document}