%% 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 15:53:11} %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 - Vlastnosti diferencovate\U{13e}n\U{fd}ch funkci\U{ed}\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}{}{}{M61.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{M61.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M63.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{M63.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{Vlastnosti diferencovate\v{l}n\'{y}ch funkci\'{\i}} \begin{theorem} \label{3}(Rolleho veta) Nech $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je spojit\'{a} funkcia. Nech $f$ je diferencovate% \v{l}n\'{a} na $(a,b)$ a nech $f(a)=f(b)$. Potom existuje \v{c}\'{\i}slo $% c\in (a,b)$, tak\'{e} \v{z}e $f\,$\/$\,^{\prime }(c)=0$. \end{theorem} $% \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO621.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO621.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1}$ Geometrick\'{y} v\'{y}znam Rolleho vety: $f:\left\langle -2,3\right\rangle \longrightarrow \mathbf{R,}$ spojit\'{a}, diferencovate\v{l}n\'{a} na $% \left( -2,3\right) ,$\thinspace $f(-2)=f(3),$ potom existuje $c\in (-2,3)$, tak\'{e} \v{z}e $f\,$\/$\,^{\prime }(c)=0$. Body $c_{1},\,c_{2},\,c_{3}$ s% \'{u} vyzna\v{c}en\'{e} \v{c}erven\'{y}m \v{s}tvor\v{c}ekom na osi $% o_{x}.x^{4}-5x^{2}+4$\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-2.5";xmax "2.5";xviewmin "-2.505";xviewmax "2.505";yviewmin "-2.2640625";yviewmax "11.8265625";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x^{4}-5x^{2}+4$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";function \TEXUX{$4$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";rangeset"X";function \TEXUX{$\frac{-9}{4}$};linecolor "maroon";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-2.5,2.5";num-x-gridlines 100;curveColor "[flat::RGB:0x00800000]";curveStyle "Line";function \TEXUX{$\left[ \MATRIX{2,1}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,}\CELL{-\sqrt{\frac{5}{2}}}% \CELL{0}\right] $};linecolor "black";linestyle 1;pointplot TRUE;pointstyle "circle";linethickness 2;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Point";function \TEXUX{$\left[ \MATRIX{2,1}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,}\CELL{0}\CELL{4}\right] $};linecolor "black";linestyle 1;pointplot TRUE;pointstyle "circle";linethickness 2;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Point";function \TEXUX{$\left[ \MATRIX{2,1}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,}\CELL{\sqrt{\frac{5}{2}}}% \CELL{0}\right] $};linecolor "black";linestyle 1;pointplot TRUE;pointstyle "circle";linethickness 2;lineAttributes "Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Point";valid_file "T";tempfilename 'HTRIFY0R.wmf';tempfile-properties "XPR";}} \begin{example} Preverte platnos\v{t} Rolleho vety pre funkciu $f\left( x\right) =x^{3}+4x^{2}-7x-10$ na intervale $\left\langle -1,2\right\rangle .$ \begin{solution} Funkcia $f$ je spojit\'{a} na intervale $\left\langle -1,2\right\rangle $ jej deriv\'{a}cia je $f\,$\/$\,^{\prime }\left( x\right) =3x^{2}+8x-7$. Deriv% \'{a}cia existuje $\forall x\in \mathbf{R}$, \v{c}o znamen\'{a}, \v{z}e $f$ je diferencovate\v{l}n\'{a} na $\left( -1,2\right) .$ Plat\'{\i} $f\left( -1\right) =0=f\left( 2\right) .$ Predpoklady Rolleho vety s\'{u} splnen\'{e}% , teda existuje \v{c}\'{\i}slo $c\in (-1,2)$, tak\'{e} \v{z}e $f\,$\/$% \,^{\prime }(c)=0.$ N\'{a}jdeme toto \v{c}\'{\i}slo: \v{c}\'{\i}slo $c$ sp% \'{l}\v{n}a rovnicu $3x^{2}+8x-7=0\Longrightarrow c_{1,2}=\frac{-8\pm \sqrt{% 64+84}}{6}=\frac{-4\pm \sqrt{37}}{3}$. \v{L}ahko vidie\v{t}, \v{z}e h\v{l}% adan\'{e} $c=$ $\frac{-4+\sqrt{37}}{3}\in (-1,2).\square $ \end{solution} \end{example} \begin{theorem} \label{5}(Cauchyho veta) Nech $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$, $g:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ s\'{u} spojit\'{e} funkcie. Nech $f,\,g$ s\'{u} diferencovate\v{l}n\'{e} na $(a,b)$ a $g^{\prime }(x)\neq 0$ pre ka\v{z}d% \'{e} $x\in (a,b)$. Potom existuje $c\in (a,b)$ tak\'{e}, \v{z}e \[ \frac{f\left( b\right) -f\left( a\right) }{g\left( b\right) -g\left( a\right) }=\frac{f\,\/\,^{\prime }\left( c\right) }{g^{\prime }\left( c\right) }. \] \end{theorem} $% \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small D\^{o}kaz}}{}{}{DO622.tex} }% %BeginExpansion \msihyperref{{\small D\^{o}kaz}}{}{}{DO622.tex} %EndExpansion \\ \hline \end{tabular}% \label{2}$ \begin{example} Zistite, \v{c}i Cauchyho veta plat\'{\i} pre funkcie $f\left( x\right) =x^{3} $ a $g\left( x\right) =x^{2}+1$ na intervale $\left\langle 1,2\right\rangle . $ \end{example} \begin{solution} Funkcie $f,g$ s\'{u} spojit\'{e} na intervale $\left\langle 1,2\right\rangle $ ich deriv\'{a}cie s\'{u} $f\,$\/$\,^{\prime }\left( x\right) =3x^{2},\,g^{\prime }\left( x\right) =2x$. Deriv\'{a}cie existuj\'{u} $% \forall x\in \mathbf{R}$, \v{c}o znamen\'{a}, \v{z}e $f$ \ a $g$\ s\'{u} diferencovate\v{l}n\'{e} na $\left( 1,2\right) $ a $g^{\prime }(x)\neq 0$ pre ka\v{z}d\'{e} $x\in (1,2)$. Predpoklady Cauchyho vety s\'{u} splnen\'{e}% , teda existuje \v{c}\'{\i}slo $c\in (1,2)$, tak\'{e} \v{z}e $\frac{% f\,\/\,^{\prime }\left( c\right) }{g^{\prime }\left( c\right) }=\frac{% f\left( 2\right) -f\left( 1\right) }{g\left( 2\right) -g\left( 1\right) }=% \frac{8-1}{5-2}=\frac{7}{3}.$ N\'{a}jdeme \v{c}\'{\i}slo $c$ sp\'{l}\v{n}a rovnicu $\frac{3c^{2}}{2c}=\frac{7}{3}\Longrightarrow c=\frac{14}{9}\in \left( 1,2\right) $ $.\square $ \end{solution} \begin{theorem} \label{7}(Lagrangeova veta) Nech $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je spojit\'{a} funkcia, diferencovate\v{l}n\'{a} na $(a,b)$. Potom existuje $c\in (a,b)$ tak\'{e}, \v{z}e \[ f\,\/\,^{\prime }(c)=\frac{f\left( b\right) -f\left( a\right) }{b-a}% ,\;\left( f(b)-f(a)=f\,\/\,^{\prime }(c)(b-a)\right) . \] \end{theorem} $% \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{{\small D\^{o}kaz}}{}{}{DO623.tex} }% %BeginExpansion \msihyperref{{\small D\^{o}kaz}}{}{}{DO623.tex} %EndExpansion \\ \hline \end{tabular}% \label{8}$ \begin{description} \item[Pozn\'{a}mka] Nech platia predpoklady Lagrangeovej vety. Potom ak v jej tvrden\'{\i} navz\'{a}jom zamen\'{\i}me $a$ a $b,$ tvrdenie zostane v platnosti, t.j.\ existuje tak\'{e} $c\in (a,b)$, \v{z}e% \[ f\,\/\,^{\prime }(c)=\frac{f\left( a\right) -f\left( b\right) }{a-b}% ,\;\left( f(a)-f(b)=f\,\/\,^{\prime }(c)(a-b)\right) . \] \end{description} Geometrick\'{y} v\'{y}znam Lagrangeovej vety: $f:\left\langle -2,2\right\rangle \longrightarrow \mathbf{R,}$ spojit\'{a}, diferencovate% \v{l}n\'{a} na $\left( -2,2\right) ,$ potom existuje $c\in (-2,2)$, tak\'{e} \v{z}e $f\,$\/$\,^{\prime }(c)=\frac{f\left( 2\right) -f\left( -2\right) }{% 2-\left( -2\right) },\,$teda doty\v{c}nica ku grafu funkcie v bode $\left( c,f\left( c\right) \right) $ je rovnobe\v{z}n\'{a} so se\v{c}nicou, prech% \'{a}dzaj\'{u}cou bodmi $\left( -2,f\left( -2\right) \right) ,\,\left( 2,f\left( 2\right) \right) $. Body $c_{1},\,c_{2}\,$ s\'{u} vyzna\v{c}en\'{e} \v{c}erven\'{y}m \v{s}tvor\v{c}ekom na osi $o_{x}.$\FRAME{dtbpFX}{4.4996in}{% 3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-2";xmax "2";xviewmin "-2.004";xviewmax "2.004";yviewmin "-7.09735983854936";yviewmax "11.0973598385494";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$x^{3}-x+2$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "-2,2";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";function \TEXUX{$3x-\frac{8}{3\sqrt{3}}+\frac{2}{\sqrt{3}}+2+\frac{6}{% \sqrt{3}}$};linecolor "green";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2,2";num-x-gridlines 100;curveColor "[flat::RGB:0x0000ff00]";curveStyle "Line";function \TEXUX{$3x+\frac{8}{3\sqrt{3}}-\frac{2}{\sqrt{3}}+2-\frac{6}{% \sqrt{3}}$};linecolor "green";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2,2";num-x-gridlines 100;curveColor "[flat::RGB:0x0000ff00]";curveStyle "Line";function \TEXUX{$3x+2$};linecolor "green";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-2,2";num-x-gridlines 100;curveColor "[flat::RGB:0x0000ff00]";curveStyle "Line";function \TEXUX{$\left[ \MATRIX{2,1}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,}\CELL{-\frac{2\sqrt{3}}{3}}% \CELL{0}\right] $};linecolor "blue";linestyle 1;pointplot TRUE;pointstyle "circle";linethickness 2;lineAttributes "Solid";curveColor "[flat::RGB:0x00000080]";curveStyle "Point";function \TEXUX{$\left[ \MATRIX{2,1}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,}\CELL{\frac{2\sqrt{3}}{3}}% \CELL{0}\right] $};linecolor "blue";linestyle 1;pointplot TRUE;pointstyle "circle";linethickness 2;lineAttributes "Solid";curveColor "[flat::RGB:0x00000080]";curveStyle "Point";valid_file "T";tempfilename 'HTRKW20T.wmf';tempfile-properties "XPR";}} \begin{example} Pomocou Lagrangeovej vety dok\'{a}\v{z}te$\label{4}$ nerovnicu: $\frac{b-a}{b% }\leq \ln \frac{b}{a}\leq \frac{b-a}{a}$ pre $0