%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %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 17:49:42} %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 - Funkcie - Cvi\U{10d}enia\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{Funkcie} \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{G1.tex}}% %BeginExpansion \msihyperref{Index}{}{}{G1.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Cvi\v{c}enia} \textbf{Pr\'{\i}klad 1. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie $f\left( x\right) =\frac{\sqrt{x+4}-2}{\sin 2x}+\log \left( 1-x^{2}\right) .% \CustomNote{Prob_Solv_Hint}{% Sk\'{u}majte defini\v{c}n\'{y} obor podielu a potom\ s\'{u}\v{c}tu funkci% \'{\i}. \par V\'{y}sledok: $D\left( f\right) =\left( -1,0\right) \cup \left( 0,1\right) $} $ \textbf{Pr\'{\i}klad 2.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie $f\left( x\right) =\ln \left( 1-\log \left( x^{2}-5x+16\right) \right) .% \CustomNote{Answer}{$D\left( f\right) =\left( 2,3\right) .$}$ \textbf{Pr\'{\i}klad 3.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie $f\left( x\right) =\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{\sqrt{1+x}}}.% \CustomNote{Answer}{% Funkcia nie je nikde definovan\'{a}}$ \textbf{Pr\'{\i}klad 4.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie $f\left( x\right) =\left( x^{2}+x+1\right) ^{-\frac{3}{2}}.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R}$}$ \textbf{Pr\'{\i}klad 5.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie $f\left( x\right) =\ln \frac{x-5}{x^{2}-10x+24}-\sqrt[3]{x+5}.% \CustomNote{Answer}{$D\left( f\right) =\left( 4,5\right) \cup \left( 6,\infty \right) $}$ \textbf{Pr\'{\i}klad 6.} Dan\'{a} je funkcia $f\left( x\right) =\log \left( \frac{x^{2}-2}{x}\right) .$ N\'{a}jdite a) defini\v{c}n\'{y} obor funkcie, b) v\v{s}etky re\'{a}lne \v{c}\'{\i}sla, pre ktor\'{e} je $f\left( x\right) >0.% \CustomNote{Answer}{$D\left( f\right) =\left( -\sqrt{2},0\right) \cup \left( \sqrt{2},\infty \right) .$% \par Ak $x\in \left( -1,0\right) \cup \left( 2,\infty \right) ,$ potom $f\left( x\right) >0.$}$ \textbf{Pr\'{\i}klad 7.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =\frac{x% }{\log \left( 1-x\right) }.% \CustomNote{Prob_Solv_Hint}{% Sk\'{u}majte symetriu defini\v{c}n\'{e}ho oboru. V\'{y}sledok: $D\left( f\right) =\left( -\infty ,0\right) \cup \left( 0,1\right) ,\,f$ ani p\'{a}% rna ani nep\'{a}rna.}$ \textbf{Pr\'{\i}klad 8. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =x\left[ \log \left( x+1\right) -\log x\right] .% \CustomNote{Prob_Solv_Hint}{% Sk\'{u}majte symetriu defini\v{c}n\'{e}ho oboru. V\'{y}sledok: $D\left( f\right) =\left( 0,\infty \right) ,$\ $f$\ ani p\'{a}rna ani nep\'{a}rna.}$ \textbf{Pr\'{\i}klad 9. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =1-% \sqrt{2\cos 2x}.% \CustomNote{Answer}{$D\left( f\right) =\cup _{k\in \mathbf{Z}}\left\langle -% \frac{\pi }{4}+k\pi ,\frac{\pi }{4}+k\pi \right\rangle ,$ p\'{a}rna}$ \textbf{Pr\'{\i}klad 10. }Pre funkciu $f\left( x\right) =x^{3}-3x^{2}+2x$ n% \'{a}jdite defini\v{c}n\'{y} obor, nulov\'{e} body (body, v ktor\'{y}ch je $% f\left( x\right) =0$), v\v{s}etky re\'{a}lne \v{c}\'{\i}sla, pre ktor\'{e} je $f\left( x\right) >0$ a $f\left( x\right) <0.% \CustomNote{Prob_Solv_Hint}{% Rozlo\v{z}te funkciu $f\left( x\right) =x\left( x-a\right) \left( x-b\right) .$}$ \textbf{Pr\'{\i}klad 11. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =2^{-x^{2}}.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R,}$ p\'{a}rna}$ \textbf{Pr\'{\i}klad 12. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =\frac{% a^{x}+a^{-x}}{2},\,a>0.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R,}$ p\'{a}rna}$ \textbf{Pr\'{\i}klad 13. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =\frac{% a^{x}-a^{-x}}{2},\,a>0.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R,}$ nep\'{a}rna funkcia}$ \textbf{Pr\'{\i}klad 14. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie a zistite, \v{c}i je p\'{a}rna, alebo nep\'{a}rna ak $f\left( x\right) =\ln \frac{1-x}{1+x}.% \CustomNote{Answer}{$D\left( f\right) =\left( -1,1\right) ,$ nep\'{a}rna}$ \textbf{Pr\'{\i}klad 15. }Pre funkciu $f\left( x\right) =\left| x\right| $ n% \'{a}jdite defini\v{c}n\'{y} obor, podmno\v{z}iny defini\v{c}n\'{e}ho oboru, na ktor\'{y}ch je funkcia rast\'{u}ca alebo klesaj\'{u}ca, zistite \v{c}i je ohrani\v{c}en\'{a} a na\v{c}rtnite jej graf.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R,}$ na $\left( -\infty ,0\right) $ je klesaj\'{u}ca, na $\left( 0,\infty \right) $ je rast\'{u}ca, zdola ohrani\v{c}en\'{a} $\min_{x\in \mathbf{R}}f\left( x\right) =f\left( 0\right) =0,$ nie je zhora ohrani\v{c}en\'{a}. \par Graf $\left| x\right| $\FRAME{dtbpFX}{3in}{2in}{0in}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 3in;height 2in;depth 0in;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "-5";xmax "5";xviewmin "-5.2";xviewmax "5.204";yviewmin "-0.094965386250000";yviewmax "5.101899307725";plottype 4;numpoints 49;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};function \TEXUX{$\left| x\right| $};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'GJLB4Q03.wmf';tempfile-properties "PR";}}% } \textbf{Pr\'{\i}klad 16. }Pre funkciu $f\left( x\right) =\left| x\right| -x$ n\'{a}jdite defini\v{c}n\'{y} obor, podmno\v{z}iny defini\v{c}n\'{e}ho oboru, na ktor\'{y}ch je funkcia rast\'{u}ca alebo klesaj\'{u}ca, zistite \v{c}i je ohrani\v{c}en\'{a} a na\v{c}rtnite jej graf.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R},\,$% \par $f\left( x\right) =\left| x\right| -x=\left\{ \begin{tabular}{ccc} $-2x$ & pre & $x<0$ \\ $0$ & pre & $x\geq 0$% \end{tabular}% \right. ,$% \par na $\left( -\infty ,0\right) $ je klesaj\'{u}ca, na $\left( 0,\infty \right) $ je kon\v{s}tantn\'{a}, zdola ohrani\v{c}en\'{a} $\min_{x\in \mathbf{R}% }f\left( x\right) =0.$ Graf $\left| x\right| -x$\FRAME{dtbpFX}{3in}{2in}{0pt% }{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height 2in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "-5";xmax "5";xviewmin "-5.2";xviewmax "5.204";yviewmin "-0.2";yviewmax "10.204";plottype 4;numpoints 49;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};function \TEXUX{$\left| x\right| -x$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'GJLB4Q04.wmf';tempfile-properties "PR";}}% } \textbf{Pr\'{\i}klad 17. }Pre funkciu $f\left( x\right) =1-\cos x$ ur\v{c}te defini\v{c}n\'{y} obor, podmno\v{z}iny defini\v{c}n\'{e}ho oboru, na ktor% \'{y}ch je funkcia rast\'{u}ca alebo klesaj\'{u}ca, zistite \v{c}i je ohrani% \v{c}en\'{a}, n\'{a}jdite jej supremum, infimum, maximum, minimum a na\v{c}% rtnite jej graf.% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R,}$ na intervaloch $\left( 2k\pi ,\pi +2k\pi \right) $ je rast\'{u}ca, na intervaloch $\left( -\pi +2k\pi ,2k\pi \right) $ je klesaj\'{u}ca, $\min_{x\in \mathbf{R}}f\left( x\right) =f\left( k\pi \right) =0,\,$% \par $\max_{x\in \mathbf{R}}f\left( x\right) =f\left( \pi +k\pi \right) =2$. Graf $f\left( x\right) =1-\cos x$\FRAME{dtbpFX}{3in}{2in}{0pt}{}{}{Plot}{\special% {language "Scientific Word";type "MAPLEPLOT";width 3in;height 2in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "-5";xmax "5";xviewmin "-5.2";xviewmax "5.204";yviewmin "-0.039985310443582";yviewmax "2.04068422093533";plottype 4;numpoints 49;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};function \TEXUX{$1-\cos x$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'GJLB4Q05.wmf';tempfile-properties "PR";}}% } \textbf{Pr\'{\i}klad 18. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie, obor funk\v{c}n\'{y}ch hodn\^{o}t. N\'{a}jdite inverzn\'{u} funkciu, ak $f\left( x\right) =\sqrt[3]{x^{2}+1}.$% \CustomNote{Answer}{$D\left( f\right) =\mathbf{R,\,}H\left( f\right) =\left\langle 1,\infty \right) ,$ zdola ohrani\v{c}en\'{a} $\min_{x\in \mathbf{R}}f\left( x\right) =f\left( 0\right) =1.$ Nie je prost\'{a} preto nem\'{a} inverzn\'{u} funkciu.} \textbf{Pr\'{\i}klad 19. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie, obor funk\v{c}n\'{y}ch hodn\^{o}t. N\'{a}jdite inverzn\'{u} funkciu. Na\v{c}% rtnite graf funkcie aj inverznej funkcie, ak $f\left( x\right) =-4+3\sqrt{x}.% \CustomNote{Answer}{$D\left( f\right) =\left\langle 0,\infty \right) \mathbf{% ,\,}H\left( f\right) =\left\langle -4,\infty \right) ,$ je prost\'{a} $% f^{-1}:\left\langle -4,\infty \right) \longrightarrow \left\langle 0,\infty \right) ,\,f^{-1}\left( x\right) =\left( \frac{x+4}{3}\right) ^{2}.$}$ \textbf{Pr\'{\i}klad 20. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie, obor funk\v{c}n\'{y}ch hodn\^{o}t. N\'{a}jdite inverzn\'{u} funkciu. Na\v{c}% rtnite graf funkcie aj inverznej funkcie, ak $f\left( x\right) =1+\ln \left( x+2\right) .% \CustomNote{Answer}{$D\left( f\right) =\left( -2,\infty \right) \mathbf{,\,}% H\left( f\right) =\mathbf{R},$ je prost\'{a} $f^{-1}:\mathbf{R}% \longrightarrow \left( -2,\infty \right) ,\,f^{-1}\left( x\right) =-2+e^{x-1}.$}$ \begin{center} \begin{tabular}{|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{M2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.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{Funkcie} \end{document}