%% 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 17:57:44} %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 - Deriv\U{e1}cia 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{Diferenci\'{a}lny po\v{c}et funkci\'{\i} jednej re\'{a}lnej premennej} \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}{}{}{M5.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.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. }Vypo\v{c}\'{\i}tajte deriv\'{a}cie funkci\'{\i}: \[ f_{1}\left( x\right) =\sqrt{\sin \left( \frac{2x}{3}\right) }% ,\;\,f_{2}\left( x\right) =4^{3x},\;\,f_{3}\left( x\right) =\ln \frac{5+4x}{% 3+7x},\;f_{4}\left( x\right) =x10^{-x},\;\,f_{5}\left( x\right) =\ln \sin 2x.% \CustomNote{Answer}{% \[ f_{1}^{\prime }\left( x\right) =\frac{\cos \left( \frac{2x}{3}\right) }{3% \sqrt{\sin \left( \frac{2x}{3}\right) }}, \]% \[ \,f_{2}^{\prime }\left( x\right) =3.\ln 4.4^{3x}, \]% $\,$% \[ f_{3}^{\prime }\left( x\right) =-\frac{23}{\left( 3+7x\right) \left( 5+4x\right) }, \]% \[ f_{4}^{\prime }\left( x\right) =10^{-x}\left( 1-x\ln 10\right) , \]% \[ \,f_{5}^{\prime }\left( x\right) =2\func{cotg}2x. \]% } \]% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textbf{Pr\'{\i}klad 2. }Vypo\v{c}\'{\i}tajte deriv\'{a}cie funkci\'{\i}: \[ f_{1}\left( x\right) =\arcsin \sqrt{x},\;\,f_{2}\left( x\right) =x\arcsin x+% \sqrt{1-x^{2}},\;\,f_{3}\left( x\right) =\limfunc{arctg}\left( x-\sqrt{% 1+x^{2}}\right) , \]% \[ \;f_{4}\left( x\right) =x^{x},\;\,f_{5}\left( x\right) =x^{\sin x}.% \CustomNote{Answer}{% \[ f_{1}^{\prime }\left( x\right) =\frac{1}{2\sqrt{x}\sqrt{1-x}}, \]% \[ \,f_{2}^{\prime }\left( x\right) =\arcsin x, \]% \[ \,f_{3}^{\prime }\left( x\right) =\frac{1}{2\left( 1+x^{2}\right) }, \]% \[ f_{4}^{\prime }\left( x\right) =x^{x}\left( 1+\ln x\right) , \]% \[ \,f_{5}\left( x\right) =x^{\sin x}\left( \cos x\ln x+\frac{\sin x}{x}\right) . \]% } \] \textbf{Pr\'{\i}klad 3. }Zistite, \v{c}i je funkcia \[ f\left( x\right) =\left\{ \begin{tabular}{cc} $x\sin \frac{1}{x},$ & $x\neq 0$ \\ $0$ & $x=0$% \end{tabular}% ,\right. \]% diferencovate\v{l}n\'{a} v bodoch $a=0,\;\frac{2}{\pi }.% \CustomNote{Prob_Solv_Hint}{% V bode $a=0$ plat\'{\i}: $\lim_{x\longrightarrow 0}\frac{f\left( x\right) -f\left( 0\right) }{x-0}=\lim_{x\longrightarrow 0}\frac{x\sin \frac{1}{x}}{x}% =\lim_{x\longrightarrow 0}\sin \frac{1}{x}\nexists .$% \par V bode $a=\frac{2}{\pi }$ nemus\'{\i}me deriv\'{a}ciu po\v{c}\'{\i}ta\v{t} z defin\'{\i}cie, ale sta\v{c}\'{\i} pre $x\neq 0$\ n\'{a}js\v{t} : $f^{\prime }\left( x\right) =\sin \frac{1}{x}-\frac{1}{x}\cos \frac{1}{x},\;f^{\prime }\left( \frac{2}{\pi }\right) =1.$}$ \textbf{Pr\'{\i}klad 4.} Zistite, \v{c}i je funkcia \[ f\left( x\right) =\left\{ \begin{tabular}{cc} $x\limfunc{arctg}\frac{1}{x},$ & $x\neq 0$ \\ $0,$ & $x=0$% \end{tabular}% ,\right. \]% v bode $a=0$ a) spojit\'{a}, b) diferencovate\v{l}n\'{a}.% \CustomNote{Answer}{% a) je spojit\'{a} v bode $a=0.$% \par b) nie je diferencovate\v{l}n\'{a} v bode $a=0.$} \textbf{Pr\'{\i}klad 5.} Pre funkciu $f\left( x\right) =\left| 2x-6\right| $ n\'{a}jdite $f^{\prime }.$ V bodoch, v ktor\'{y}ch deriv\'{a}cia neexistuje vypo\v{c}\'{\i}tajte deriv\'{a}ciu sprava a deriv\'{a}ciu z\v{l}ava. Na\v{c}% rtnite grafy $f$ a $f^{\prime }.% \CustomNote{Prob_Solv_Hint}{$f\left( x\right) =\left| 2x-6\right| =\left\{ \begin{tabular}{ccc} $6-2x$ & pre & $x<3$ \\ $2x-6$ & pre & $x\geq 3$% \end{tabular}% ,\text{ }\right. $% \par $f^{\prime }\left( x\right) =\left\{ \begin{tabular}{ccc} $-2$ & pre & $x<3$ \\ $2$ & pre & $x>3$% \end{tabular}% \,\ \ \ \right. ,$% \par $\left. \begin{tabular}{c} $f_{+}^{\prime }\left( 3\right) =\lim_{x\longrightarrow 3^{+}}\frac{2x-6}{x-3% }=2$ \\ $f_{-}^{\prime }\left( 3\right) =\lim_{x\longrightarrow 3^{-}}\frac{6-2x}{x-3% }=-2$% \end{tabular}% \right\} \Longrightarrow f^{\prime }\left( 3\right) \nexists .$}$ \textbf{Pr\'{\i}klad 6.} Pre funkciu $f\left( x\right) =\sqrt{\left| x-1\right| }$ n\'{a}jdite $f^{\prime }.$ V bodoch, v ktor\'{y}ch deriv\'{a}% cia neexistuje vypo\v{c}\'{\i}tajte deriv\'{a}ciu sprava a deriv\'{a}ciu z% \v{l}ava. Na\v{c}rtnite grafy $f$ a $f^{\prime }.% \CustomNote{Answer}{$f^{\prime }\left( x\right) =\left\{ \begin{tabular}{ccc} $\frac{-1}{2\sqrt{1-x}}$ & pre & $x<1$ \\ $\frac{1}{2\sqrt{x-1}}$ & pre & $x>1$% \end{tabular}% \,\ \ \ \right. ,f^{\prime }\left( 1\right) \nexists .$% \par {}}$ \textbf{Pr\'{\i}klad 7.} Pre funkciu $f\left( x\right) =\left| x^{2}-x-2\right| $ n\'{a}jdite $f^{\prime }.$ V bodoch, v ktor\'{y}ch deriv% \'{a}cia neexistuje vypo\v{c}\'{\i}tajte deriv\'{a}ciu sprava a deriv\'{a}% ciu z\v{l}ava. Na\v{c}rtnite grafy $f$ a $f^{\prime }.% \CustomNote{Answer}{$f^{\prime }\left( x\right) =\left\{ \begin{tabular}{ccc} $-2x+1$ & pre & $x\in \left( -1,2\right) $ \\ $2x-1$ & pre & $x\notin \left\langle -1,2\right\rangle $% \end{tabular}% \ \right. ,$% \par $f^{\prime }\left( -1\right) ,\,f^{\prime }\left( 2\right) \nexists .$}$ \textbf{Pr\'{\i}klad 8. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f\left( x\right) =e^{-x}\cos 2x$ v bode $A=\left( 0,?\right) .% \CustomNote{Answer}{$A=\left( 0,1\right) ,\,t:x+y-1=0,\,n:x-y+1=0$}$ \textbf{Pr\'{\i}klad 9. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f\left( x\right) =e^{1-x^{2}}$ v priese\v{c}n\'{\i}ku s priamkou $y=1.% \CustomNote{Answer}{\'{U}loha m\'{a} dve rie\v{s}enia: \par v bode $\ T_{1}=\left( 1,1\right) :$ $\,t_{1}:2x+y-3=0,\,n_{1}:x-2y+1=0,\,$% \par v bode $\ T_{2}=\left( -1,1\right) :$ $\,t_{2}:2x-y+3=0,\,n_{2}:x+2y-1=0.$}$ \textbf{Pr\'{\i}klad 10. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f\left( x\right) =\ln \left( x+1\right) $ v bode $A=\left( 0,?\right) .% \CustomNote{Answer}{$A=\left( 0,0\right) ,\,t:x-y=0,\,n:x+y=0.$}$ \textbf{Pr\'{\i}klad 11. }Na\v{c}rtnite graf funkcie $f\left( x\right) =\arccos 3x$ a n\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f$ v jeho priese\v{c}n\'{\i}ku s osou $o_{y}.% \CustomNote{Answer}{$\,t:3x+y-\frac{\pi }{2}=0,\,n:x-3y+\frac{3\pi }{2}=0.$}$ \textbf{Pr\'{\i}klad 12. }Ku grafu funkcie $f\left( x\right) =x\ln x$ n\'{a}% jdite rovnicu norm\'{a}ly, ktor\'{a} je rovnobe\v{z}n\'{a} s priamkou $% p:2x-2y+3=0$. \CustomNote{Answer}{$\,n:y-x+3e^{-2}=0.$} \textbf{Pr\'{\i}klad 13. }N\'{a}jdite uhol, pod ktor\'{y}m sa pret\'{\i}naj% \'{u} grafy funkci\'{\i} $f\left( x\right) =\ln x$ a $g\left( x\right) =\ln ^{2}x.% \CustomNote{Prob_Solv_Hint}{% N\'{a}vod: najsk\^{o}r ur\v{c}te priese\v{c}n\'{\i}k funkci\'{\i} $f$ a $g$, potom pou\v{z}ite vz\v{t}ah $\;tg\varphi =\left| \frac{k_{2}-k_{1}}{% 1+k_{1}k_{2}}\right| ,$ kde $k_{1},k_{2}$ s\'{u} smernice doty\v{c}n\'{\i}c ku grafom funkcie $f$ resp. $g$ v ich priese\v{c}n\'{\i}ku. V\'{y}sledok: $% \varphi _{1}=\frac{\pi }{4},\,\varphi _{2}=\limfunc{arctg}\frac{e}{e^{2}+2}.$% }$ \textbf{Pr\'{\i}klad 14. }N\'{a}jdite uhol, pod ktor\'{y}m sa pret\'{\i}naj% \'{u} grafy funkci\'{\i} $f:\left\langle 0,\frac{\pi }{2}\right\rangle \longrightarrow \mathbf{R},\,f\left( x\right) =\sin x$ a $g:\left\langle 0,% \frac{\pi }{2}\right\rangle \longrightarrow \mathbf{R},\,g\left( x\right) =\cos x.$ \CustomNote{Answer}{$\varphi =\limfunc{arctg}2\sqrt{2}.$} \textbf{Pr\'{\i}klad 15. }V induk\v{c}nej cievke pretek\'{a} pr\'{u}d $i$, pre ktor\'{y} plat\'{\i} $i=15\sin ^{5}3t$, kde pr\'{u}d $i$ je dan\'{y} v amp\'{e}roch a \v{c}as $t$ v sekund\'{a}ch. Vypo\v{c}\'{\i}tajte indukovan% \'{e} elektromotorick\'{e} nap\"{a}tie $e_{i}=-L\frac{di}{dt}$ v \v{c}ase $t=% \frac{2\pi }{9}\unit{s}$, ak $L=0,03\unit{H}.% \CustomNote{Answer}{$1,9\unit{V}$}$ \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}{}{}{M5.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{M5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.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}