%% 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 16:28:09} %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 - Zbierka 2\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{Zbierka \'{u}loh} \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Funkcia - deriv\'{a}cia a jej pou\v{z}itie.} \textbf{Pr\'{\i}klad 1. }Definujte deriv\'{a}ciu funkcie v bode $a\ $a pomocou defin\'{\i}cie n\'{a}jdite deriv\'{a}ciu funkcie $f\left( x\right) =% \sqrt{4x+1}$ v bode $a=2.\;% \CustomNote{Answer}{$f\,\,^{\prime }\left( a\right) =\lim_{x\longrightarrow a}\frac{f\left( x\right) -f\left( a\right) }{x-a},\,f\,\,^{\prime }\left( 2\right) =\frac{2}{3}.$}$ \textbf{Pr\'{\i}klad 2. }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}. $% \CustomNote{Answer}{% nie je diferencovate\v{l}n\'{a} v bode \ $0.$}$ \textbf{Pr\'{\i}klad 3. }Zistite, \v{c}i je funkcia $\ $% \[ f\left( x\right) =\left\{ \begin{tabular}{cc} $x\func{arctg}\frac{1}{x}$ & $x\neq 0$ \\ $0$ & $x=0$% \end{tabular}% \right. \]% a. spojit\'{a} v bode $a=0,$ b. diferencovate\v{l}n\'{a} v bode $a=0$. $\ \CustomNote{Answer}{% a. je spojit\'{a} v bode $a=0,\,$% \par b. nie je diferencovate\v{l}n\'{a} v bode $0.$}$ \textbf{Pr\'{\i}klad 4. }Vypo\v{c}\'{\i}tajte deriv\'{a}ciu funkcie $f\left( x\right) =2x+\left| x-1\right| .$ V bodoch, v ktor\'{y}ch deriv\'{a}cia nee$% x $istuje vypo\v{c}\'{\i}tajte deriv\'{a}ciu z\v{l}ava a deriv\'{a}ciu sprava. $% \CustomNote{Answer}{$f\,\,^{\prime }\left( x\right) =\left\{ \begin{tabular}{cc} $1$ & $x<1$ \\ $3$ & $x>1$% \end{tabular}% \right. ,\,$% \par $f\,\,^{\prime -}\left( 1\right) =1,\,f\,\,^{\prime +}\left( 1\right) =3.$}$ \textbf{Pr\'{\i}klad 5. }Vypo\v{c}\'{\i}tajte deriv\'{a}ciu funkcie $% f:\left( -\infty ,-5\right) \cup \left( -1,1\right) \cup \left( 1,\infty \right) \longrightarrow \mathbf{R},\,f\left( x\right) =\log _{2}\left( \left| \frac{10}{x-1}\right| -\left| -\frac{5}{x+2}\right| \right) ,\,$v bode $a=-6.$ $% \CustomNote{Answer}{$-\frac{17}{28\log 2}.$}$ V pr\'{\i}kladoch 6 - 9 n\'{a}jdite $f\,\,^{\prime }$ a na\v{c}rtnite grafy \ $f$ a $f\,\,^{\prime },\,$ke\v{d} \textbf{Pr\'{\i}klad 6. }$f\left( x\right) =\left| 2x-6\right| .\;% \CustomNote{Answer}{$f\,\,^{\prime }\left( x\right) =\left\{ \begin{tabular}{cc} $-2$ & $x<3$ \\ $2$ & $x>3$% \end{tabular}% \right. ,\,f\,\,^{\prime }\left( 3\right) $ \ neexistuje.}$ \textbf{Pr\'{\i}klad 7. }$f\left( x\right) =\left| x^{2}-x-2\right| .$ $% \CustomNote{Answer}{$f\,\,^{\prime }\left( x\right) =\left\{ \begin{tabular}{cc} $-2x+1$ & $x\in \left( -1,2\right) $ \\ $2x-1$ & $x\notin \left\langle -1,2\right\rangle $% \end{tabular}% \right. ,\,$% \par $f\,\,^{\prime }\left( -1\right) ,\,f\,\,^{\prime }\left( 2\right) $ neexistuj\'{u}$.$}$ \textbf{Pr\'{\i}klad 8. }$f\left( x\right) =\sqrt{\left| x-1\right| }.$ $% \CustomNote{Answer}{$f\,\,^{\prime }\left( x\right) =\left\{ \begin{tabular}{cc} $\frac{1}{2\sqrt{x-1}}$ & $x>1$ \\ $\frac{-1}{2\sqrt{1-x}}$ & $x<1$% \end{tabular}% \right. ,$% \par $f\,\,^{\prime }\left( 1\right) $ neexistuje$.$}$ \textbf{Pr\'{\i}klad 9. }$f\left( x\right) $ =$\ln \left| 5-x\right| .$ $% \CustomNote{Answer}{$f\,\,^{\prime }\left( x\right) =\frac{1}{x-5},\,D\left( f\right) =\left( -\infty ,5\right) \cup \left( 5,\infty \right) .$}$ V pr\'{\i}kladoch 10, 11 n\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f$ v bode $A$ \textbf{Pr\'{\i}klad 10. }$f\left( x\right) =\frac{3x-4}{2x-3},\,A=\left( 2,?\right) .$ $% \CustomNote{Answer}{$A=\left( 2,2\right) ,\,t:x+y-4=0,\,n:x-y=0.$}$ \textbf{Pr\'{\i}klad 11. }$f\left( x\right) =e^{-x}\cos \left( 2x\right) ,\,A=\left( 0,?\right) .% \CustomNote{Answer}{$A=\left( 0,1\right) ,\,t:x+y-1=0,\,n:x-y+1=0.$}$ \textbf{Pr\'{\i}klad 12. }Funkcia $f$ je dan\'{a} predpisom $f\left( x\right) =\frac{x+9}{x+5}.$ N\'{a}jdite rovnicu doty\v{c}nice ku grafu funkcie $f,\,$ktor\'{a} prech\'{a}dza za\v{c}iatkom s\'{u}radnicovej s\'{u}% stavy. $% \CustomNote{Answer}{$T_{1}=\left( -15,\,\frac{3}{5}\right) ,\,T_{2}=\left( -3,3\right) ,\,$% \par $t_{1}:x+25y=0,\,t_{2}:x+y=0.$}$ \textbf{Pr\'{\i}klad 13. }Funkcia $f$ dan\'{a} predpisom $f\left( x\right) =\arccos \left( 3x\right) .$ a. Na\v{c}rtnite graf funkcie $f$. b. N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f,\,$v 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 14. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f:f\left( x\right) =e^{1-x^{2}},\,$ktor\'{a} prech\'{a}dza priese\v{c}n\'{\i}kom grafu funkcie s priamkou $y=1.$ $% \CustomNote{Answer}{$t_{1}:2x+y-3=0,\,n_{1}^{{}}:x-2y+1=0,\,$% \par $t_{2}:2x-y+3=0,\,n_{2}:x+2y-1=0.$}$ \textbf{Pr\'{\i}klad 15. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f:f\left( x\right) =x^{2}-2x+3,\,$ak doty\v{c}nica $t$ je rovnobe\v{z}n\'{a} s priamkou $p:3x-y+5=0.$ $% \CustomNote{Answer}{$t:12x-4y-13=0,\,n:4x+12y-61=0.$}$ \textbf{Pr\'{\i}klad 16. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f:f\left( x\right) =x^{2}-5x+5,\,$ak doty\v{c}nica $t$ je rovnobe\v{z}n\'{a} s priamkou $p:3x-y+2=0.$ $% \CustomNote{Answer}{$t:3x-y-11=0,\,n:x+3y-7=0.$}$ \textbf{Pr\'{\i}klad 17. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f:f\left( x\right) =\frac{1}{x},\,$ak doty\v{c}nica $t$ je rovnobe\v{z}n\'{a} s priamkou $p:\frac{1}{9}x+y-1=0.$ $% \CustomNote{Answer}{$t_{1}:x+9y-6=0,\,n_{1}:27x-3y-80=0,\,$% \par $t_{2}:x+9y+6=0,\,n_{2}:27x-3y+80=0.$}$ \textbf{Pr\'{\i}klad 18. }N\'{a}jdite rovnicu norm\'{a}ly ku grafu funkcie $% f:f\left( x\right) =x\ln x,\,$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 19. }Zistite, v ktorom bode je doty\v{c}nica ku grafu funkcie $f:f\left( x\right) =\frac{\ln x}{x}$ rovnobe\v{z}n\'{a} s osou $% o_{x}.$ $% \CustomNote{Answer}{$\left( e,\,\frac{1}{e}\right) .$}$ \textbf{Pr\'{\i}klad 20. }N\'{a}jdite rovnicu doty\v{c}nice a norm\'{a}ly ku grafu funkcie $f:f\left( x\right) =\ln x,\,$ak doty\v{c}nica je kolm\'{a} na priamku $p:x+2y-2=0.$ $% \CustomNote{Answer}{$t:y-2x+1+\ln 2=0,$% \par $\,n:4y+2x-1+4\ln 2=0.$}$ \textbf{Pr\'{\i}klad 21. }N\'{a}jdite rovnice doty\v{c}n\'{\i}c k hyperbole $% 7x^{2}-2y^{2}=14,\,$ktor\'{e} s\'{u} kolm\'{e} na priamku $p:2x+4y-3=0.$ $% \CustomNote{Answer}{$t_{1}:2x-y-1=0,\,t_{2}:2x-y+1=0.$}$ \textbf{Pr\'{\i}klad 22. }Pre parabolu $f\left( x\right) =x^{2}-2x+3$ n\'{a}% jdite rovnicu norm\'{a}ly, ktor\'{a} je kolm\'{a} na spojnicu vrcholu paraboly s bodom $A=\left( 2,0\right) .$ $% \CustomNote{Answer}{$n:2y-x-6=0.$}$ \textbf{Pr\'{\i}klad 23. }Zistite pod ak\'{y}m uhlom sa pret\'{\i}na graf funkcie $f\left( x\right) =\sin \left( 2x\right) $ s kladn\'{y}m smerom osi $% o_{x}.$(N\'{a}vod: $\limfunc{tg}\varphi =\left| \frac{k_{2}-k_{1}}{% 1+k_{1}k_{2}}\right| $). $% \CustomNote{Answer}{$\varphi _{1}=\func{arctg}2,\,\varphi _{2}=\pi -\func{% arctg}2.$}$ \textbf{Pr\'{\i}klad 24. }Zistite pod ak\'{y}m uhlom sa pret\'{\i}naj\'{u} krivky $x^{2}+y^{2}-8=0$ a $y^{2}=2x$ v prvom kvadrante. $% \CustomNote{Answer}{$\varphi =\func{arctg}3.$}$ \textbf{Pr\'{\i}klad 25. }N\'{a}jdite uhol, pod ktor\'{y}m sa pret\'{\i}naj% \'{u} grafy funkci\'{\i} $f:\left( 0,\,\infty \right) \longrightarrow \mathbf{R},\,f\left( x\right) =\ln x,\,g:\left( 0,\,\infty \right) \longrightarrow \mathbf{R},\,f\left( x\right) =\ln ^{2}x.$ $% \CustomNote{Answer}{$\varphi _{1}=\frac{\pi }{4},\,\varphi _{2}=\func{arctg}% \frac{e}{e^{2}+2}.$}$ \textbf{Pr\'{\i}klad 26. }Vypo\v{c}\'{\i}tajte uhol, pod ktor\'{y}m sa pret% \'{\i}naj\'{u} grafy funkci\'{\i} $f:\left\langle 0,\frac{\pi }{2}% \right\rangle \longrightarrow \mathbf{R},\,f\left( x\right) =\sin x,\,g:\left\langle 0,\frac{\pi }{2}\right\rangle \longrightarrow \mathbf{R}% ,\,g\left( x\right) =\cos x.$ $% \CustomNote{Answer}{$\varphi =\func{arctg}\left( 2\sqrt{2}\right) .$}$ V pr\'{\i}kladoch 27 - 29 sk\'{u}majte monot\'{o}nnos\v{t} funkcie $f$\ a jej extr\'{e}my, ak existuj\'{u} ak \textbf{Pr\'{\i}klad 27. }$f\left( x\right) =x^{\frac{1}{x}}.\,% \CustomNote{Answer}{% rastie na $\left( 0,e\right) ,\,$% \par kles\'{a} na $\left( e,\infty \right) ,\,\max_{x\in \left( 0,\infty \right) }f\left( x\right) =f\left( e\right) =e^{\frac{1}{e}}.$}$ \textbf{Pr\'{\i}klad 28. }$f\left( x\right) =\frac{x}{\ln x}.\,% \CustomNote{Answer}{% rastie na $\left( e,\infty \right) ,\,$kles\'{a} na $\left( 0,1\right) $ a $% \left( 1,e\right) ,\,\min_{x\in \left( 0,\infty \right) }f\left( x\right) =f\left( e\right) =e.$}$ \textbf{Pr\'{\i}klad 29. }$f\left( x\right) =x-\ln \left( 1+x^{2}\right) .\,% \CustomNote{Answer}{% rastie na $\mathbf{R},\,$nem\'{a} extr\'{e}m.}$ \textbf{Pr\'{\i}klad 30. }Ur\v{c}te absol\'{u}tne extr\'{e}my funkcie $% f\left( x\right) =x^{3}-6x^{2}+9x-4$ na intervale $\left\langle 0,2\right\rangle .$ $% \CustomNote{Answer}{$\min_{\left\langle 0,2\right\rangle }f\left( x\right) =f\left( 0\right) =-4,\,$% \par $\max_{\left\langle 0,2\right\rangle }f\left( x\right) =f\left( 1\right) =0.$% }$ \textbf{Pr\'{\i}klad 31. }N\'{a}jdite absol\'{u}tne extr\'{e}my funkcie $% f:\left\langle -\frac{\pi }{2},\frac{\pi }{2}\right\rangle \longrightarrow \mathbf{R},\,f\left( x\right) =\cos \left( 2x\right) -2x.$ $% \CustomNote{Answer}{$\min_{\left\langle -\frac{\pi }{2},\frac{\pi }{2}% \right\rangle }f\left( x\right) =f\left( \frac{\pi }{2}\right) =-1-\pi ,\,$% \par $\max_{\left\langle -\frac{\pi }{2},\frac{\pi }{2}\right\rangle }f\left( x\right) =f\left( -\frac{\pi }{2}\right) =-1+\pi .$}$ \textbf{Pr\'{\i}klad 32. }N\'{a}jdite rovnice asympt\^{o}t grafu funkcie $% f:f\left( x\right) =\sqrt[3]{2x^{2}-x^{3}}.$ $% \CustomNote{Answer}{% m\'{a} asymptoty v bodoch $-\infty ,\,\infty $ s rovnicou\ $y=-x+\frac{2}{3}$% }$ \textbf{Pr\'{\i}klad 33. }Do rota\v{c}n\'{e}ho ku\v{z}e\v{l}a s polomerom $% r=4$ a v\'{y}\v{s}kou $v=6$\ vp\'{\i}\v{s}te valec s najv\"{a}\v{c}\v{s}% \'{\i}m objemom. $% \CustomNote{Answer}{$R=\frac{8}{3},\,h=2.$}$ \textbf{Pr\'{\i}klad 34. }Do elipsy s polosami $a,\,b$ je vp\'{\i}san\'{y} obd\'{l}\v{z}nik so stranami rovnobe\v{z}n\'{y}mi s osami elipsy. Ak\'{e} rozmery m\'{a} obd\'{l}\v{z}nik s najv\"{a}\v{c}\v{s}\'{\i}m obsahom? $% \CustomNote{Answer}{$a\sqrt{2},\,b\sqrt{2}$}$ \textbf{Pr\'{\i}klad 35. }Potrebujeme oploti\v{t} obd\'{l}\v{z}nikov\'{y} pozemok, priliehaj\'{u}ci jednou stranou k domu. Pozemok m\'{a} dan\'{u} v% \'{y}meru (plo\v{s}n\'{y} obsah) $P.$ Ak\'{y} m\'{a} by\v{t} pomer d\'{l}% \v{z}ok jeho str\'{a}n, aby sme na stavbu plotu spotrebovali najmenej materi% \'{a}lu? $% \CustomNote{Answer}{$1:2$}$ \textbf{Pr\'{\i}klad 36. }Kus dr\^{o}tu s d\'{l}\v{z}kou $a$ m\'{a}me rozdeli% \v{t} na dve \v{c}asti $a_{1},\,a_{2},$ z ktor\'{y}ch prv\'{a} sa zohne do tvaru \v{s}tvorca, druh\'{a} do tvaru kruhu. Ak\'{e} ve\v{l}k\'{e} maj\'{u} by\v{t} \v{c}asti $a_{1},\,a_{2},$ aby s\'{u}\v{c}et obsahu \v{s}tvorca a obsahu kruhu bol najmen\v{s}\'{\i}? $% \CustomNote{Answer}{$a_{1}=\frac{4}{\pi +4}a,\,a_{2}=\frac{\pi }{\pi +4}a$}$ \textbf{Pr\'{\i}klad 37. }N\'{a}jdite tak\'{e} kladn\'{e} \v{c}\'{\i}slo $x,$ aby s\'{u}\v{c}et tohto \v{c}\'{\i}sla a jeho prevr\'{a}tenej hodnoty bol najmen\v{s}\'{\i}. $% \CustomNote{Answer}{$x=1.$}$ \textbf{Pr\'{\i}klad 38. }N\'{a}jdite dve kladn\'{e} \v{c}\'{\i}sla tak, aby ich s\'{u}\v{c}et bol $80$ a s\'{u}\v{c}et ich \v{s}tvorcov minim\'{a}lny. $% \CustomNote{Answer}{$40,40.$}$ \textbf{Pr\'{\i}klad 39. }Dan\'{e} s\'{u} dve kladn\'{e} \v{c}\'{\i}sla, ktor% \'{y}ch s\'{u}\v{c}et sa rovn\'{a} \v{c}\'{\i}slu $a.$ N\'{a}jdite najmen% \v{s}iu hodnotu s\'{u}\v{c}tu ich n-t\'{y}ch mocn\'{\i}n. $% \CustomNote{Answer}{$\frac{a}{2},\,\frac{a}{2},\,s_{min}=2\left( \frac{a}{2}% \right) ^{n}.$}$ \textbf{Pr\'{\i}klad 40. }\v{C}islo $10$ rozde\v{l}te na dve \v{c}asti tak, aby s\'{u}\v{c}et ich druh\'{y}ch mocn\'{\i}n bol najmen\v{s}\'{\i}. $% \CustomNote{Answer}{$5,5.$}$ \textbf{Pr\'{\i}klad 41. }Ur\v{c}te dve kladn\'{e} \v{c}\'{\i}sla, ktor\'{y}% ch rozdiel je $100$ tak, aby rozdiel medzi \v{s}tvorcom v\"{a}\v{c}\v{s}ieho a p\"{a}\v{t}n\'{a}sobkom \v{s}tvorca men\v{s}ieho bol maxim\'{a}lny. $% \CustomNote{Answer}{$125,25.$}$ \textbf{Pr\'{\i}klad 42. }Priamo\v{c}iary pohyb telesa je ur\v{c}en\'{y} rovnicou $s=2t^{3}-15t^{2}+36t+2,$ kde $s$ je dr\'{a}ha vyjadren\'{a} v metroch, $t$ je \v{c}as vyjadren\'{y} v sekund\'{a}ch. Ur\v{c}te \v{c}as $t,$ v ktorom je r\'{y}chlos\v{t} nulov\'{a}. $% \CustomNote{Answer}{$t_{1}=2\unit{s},t_{2}=3\unit{s}.$}$ \textbf{Pr\'{\i}klad 43. }Kmitav\'{y} pohyb hmotn\'{e}ho bodu popisuje funkcia $f:f\left( t\right) =l\cos \left( \omega t\right) ,\,(l,\,\omega \in \mathbf{R},\,l\neq 0,\,\omega \neq 0),$ kde $f\left( t\right) $ ozna\v{c}uje v\'{y}chylku. Vypo\v{c}\'{\i}tajte r\'{y}chlos\v{t} a zr\'{y}chlenie pohybu v \v{c}ase, ke\v{d} $f\left( t\right) =1.$ $% \CustomNote{Answer}{$v=0,\,a=-l\omega ^{2}.$}$ \textbf{Pr\'{\i}klad 44. }Na ak\'{y} vonkaj\v{s}\'{\i} odpor $R_{e}$ mus% \'{\i}me zapoji\v{t} galvanick\'{y} \v{c}l\'{a}nok s nap\"{a}t\'{\i}m $U$ a vn\'{u}torn\'{y}m odporom $R_{1},$ aby jeho v\'{y}kon bol \v{c}o najv\"{a}% \v{c}\v{s}\'{\i}. $% \CustomNote{Answer}{$R_{e}=R_{1}.$}$ \textbf{Pr\'{\i}klad 45. }Mno\v{z}stvo elektrick\'{e}ho n\'{a}boja, ktor\'{e} prech\'{a}dza vodi\v{c}om sa men\'{\i} pod\v{l}a funkcie $Q:Q\left( t\right) =3t^{2}+2t+2$ ($Q$ v $\unit{C},\,t$ v $\unit{s}$). Vypo\v{c}\'{\i}tajte intenzitu elektrick\'{e}ho pr\'{u}du v \v{c}ase $t_{0}=0,\,1,\,5\unit{s}.$ Zistite, kedy sa intenzita bude rovna\v{t} $20\unit{A}.$ $% \CustomNote{Answer}{$Q^{\prime }\left( 0\right) =2\unit{A},\,Q^{\prime }\left( 1\right) =8\unit{A},\,\,$% \par $Q^{\prime }\left( 5\right) =32\unit{A},\,t=3\unit{s}.$}$ V pr\'{\i}kladoch 46 - 69 vypo\v{c}\'{\i}tajte limity (aj s pou\v{z}it\'{\i}% m L' Hospitalovho pravidla) \textbf{Pr\'{\i}klad 46. }$\lim_{x\longrightarrow \infty }\frac{x-\cos x}{% x+\sin x}.$ $% \CustomNote{Answer}{$1,$ pozor v tomto pr\'{\i}pade nem\^{o}\v{z}eme pou\v{z}% i\v{t} L'Hospitalovo pravidlo, preto\v{z}e limita $\lim_{x\longrightarrow \infty }\frac{1+\sin x}{1-\cos x}$ neexiatuje.}$ \textbf{Pr\'{\i}klad 47. }$\lim_{x\longrightarrow 0}\frac{\arcsin \left( 2x\right) -2\arcsin x}{x^{3}}.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 48. }$\lim_{x\longrightarrow 0}\frac{\sin x-x}{\arcsin x-x}.$ $% \CustomNote{Answer}{$-1.$}$ \textbf{Pr\'{\i}klad 49. }$\lim_{x\longrightarrow 0}\frac{\limfunc{tg}x-x}{% x-\sin x}.$ $% \CustomNote{Answer}{$2.$}$ \textbf{Pr\'{\i}klad 50. }$\lim_{x\longrightarrow 1}\frac{1-\limfunc{tg}% \left( \frac{\pi }{4}x\right) }{1-x^{2}}.$ $% \CustomNote{Answer}{$\frac{\pi }{4}.$}$ \textbf{Pr\'{\i}klad 51. }$\lim_{x\longrightarrow \infty }\frac{\frac{\pi }{2% }-\func{arctg}x}{\ln \sqrt{\frac{x-1}{x+1}}}.$ $% \CustomNote{Answer}{$-1.$}$ \textbf{Pr\'{\i}klad 52. }$\lim_{x\longrightarrow 0}\frac{e^{\limfunc{tg}% x}-e^{x}}{\limfunc{tg}x-x}.$ $% \CustomNote{Prob_Solv_Hint}{$1,\,$L'Hospitalovo pravidlo treba pou\v{z}i\v{t} viackr\'{a}t}$ \textbf{Pr\'{\i}klad 53. }$\lim_{x\longrightarrow 0}\frac{x-\func{arctg}x}{% x^{3}}.$ $% \CustomNote{Answer}{$\frac{1}{3}.$}$ \textbf{Pr\'{\i}klad 54. }$\lim_{x\longrightarrow 0^{+}}\frac{\ln \left( \sin \left( ax\right) \right) }{\ln \left( \sin \left( bx\right) \right) }% ,\,a>0,\,b>0.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 55. }$\lim_{x\longrightarrow 0^{+}}\left( e^{x}-1\right) \func{cotg}x.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 56. }$\lim_{x\longrightarrow a}\left[ \arcsin \left( x-a\right) \func{cotg}\left( x-a\right) \right] .$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 57. }$\lim_{x\longrightarrow 2}\frac{x^{2}-4}{x^{2}}% \limfunc{tg}\left( \frac{\pi x}{4}\right) .$ $% \CustomNote{Answer}{$-\frac{4}{\pi }.$}$ \textbf{Pr\'{\i}klad 58. }$\lim_{x\longrightarrow 0^{+}}\left( \limfunc{tg}% x\right) ^{\sin x}.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 59. }$\lim_{x\longrightarrow 0}x^{\sin x}.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 60. }$\lim_{x\longrightarrow \infty }x^{\frac{1}{x}}.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 61. }$\lim_{x\longrightarrow 0^{+}}\left( \frac{1}{x}% \right) ^{\limfunc{tg}x}.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 62. }$\lim_{x\longrightarrow 0^{+}}\left( \ln \frac{1}{x% }\right) ^{x}.$ $% \CustomNote{Answer}{$1.$}$ \textbf{Pr\'{\i}klad 63. }$\lim_{x\longrightarrow 0^{+}}\left( e^{2x}+x\right) ^{\frac{1}{x}}.$ $% \CustomNote{Answer}{$e^{3}.$}$ \textbf{Pr\'{\i}klad 64. }$\lim_{x\longrightarrow a}\left( \frac{\sin x}{% \sin a}\right) ^{\func{cotg}\left( x-a\right) }.$ $% \CustomNote{Answer}{$e^{\func{cotg}a}.$}$ \textbf{Pr\'{\i}klad 65. }$\lim_{x\longrightarrow \infty }\left( \frac{2}{% \pi }\func{arctg}x\right) ^{x}.$ $% \CustomNote{Answer}{$e^{-\frac{2}{\pi }}.$}$ \textbf{Pr\'{\i}klad 66. }$\lim_{x\longrightarrow \frac{\pi }{4}}\left( \limfunc{tg}x\right) ^{\limfunc{tg}2x}.$ $% \CustomNote{Answer}{$e^{-1}.$}$ \textbf{Pr\'{\i}klad 67. }$\lim_{x\longrightarrow \infty }\left( \cos \frac{1% }{x}\right) ^{x^{2}}.$ $% \CustomNote{Answer}{$e^{-\frac{1}{2}}.$}$ \textbf{Pr\'{\i}klad 68. }$\lim_{x\longrightarrow 1^{+}}\left( \frac{x}{x-1}-% \frac{1}{\ln x}\right) .$ $% \CustomNote{Answer}{$\frac{1}{2}.$}$ \textbf{Pr\'{\i}klad 69. }$\lim_{x\longrightarrow 1^{-}}\left( \frac{x}{x-1}-% \frac{1}{\ln x}\right) .$ $% \CustomNote{Answer}{$\frac{1}{2}.$}$ \textbf{Pr\'{\i}klad 70. }$\lim_{x\longrightarrow 1^{+}}\left( \frac{1}{\ln x% }-\frac{1}{x-1}\right) .$ $% \CustomNote{Answer}{$\frac{1}{2}.$}$ \textbf{Pr\'{\i}klad 71. }$\lim_{x\longrightarrow 1^{-}}\left( \frac{1}{\ln x% }-\frac{1}{x-1}\right) .$ $% \CustomNote{Answer}{$\frac{1}{2}.$}$ V pr\'{\i}kladoch 72 - 75 vy\v{s}etrite spojitos\v{t} funkcie $f$\ na $% D\left( f\right) $\ ak: \textbf{Pr\'{\i}klad 72. }$f\left( x\right) =\left\{ \begin{tabular}{cc} $\frac{x}{\sin x}$ & $x\in \left\langle -\frac{\pi }{2},0\right) $ \\ $0$ & $x=0$ \\ $x^{2}\ln x$ & $x\in \left( 0,1\right\rangle $% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% funkcia $f$\ \ nie je spojit\'{a} v bode $0,\,$v ostatn\'{y}ch bodoch $% D\left( f\right) $ je spojit\'{a}.}$ \textbf{Pr\'{\i}klad 73. }$f\left( x\right) =\left\{ \begin{tabular}{cc} $x^{2}\ln x$ & $x\in \left( 0,1\right\rangle $ \\ $\frac{1}{\ln x}-\frac{1}{x-1}$ & $x\in \left( 1,2\right\rangle $% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% funkcia $f$\ \ nie je spojit\'{a} v bode $1,\,$v ostatn\'{y}ch bodoch $% D\left( f\right) $ je spojit\'{a}.}$ \textbf{Pr\'{\i}klad 74. }$f\left( x\right) =\left\{ \begin{tabular}{cc} $\frac{\limfunc{tg}3x}{\limfunc{tg}5x}$ & $x\in \left( \frac{\pi }{3},\frac{% \pi }{2}\right) $ \\ $1$ & $x=\frac{\pi }{2}$ \\ $\limfunc{tg}x-\frac{1}{\cos x}$ & $x\in \left( \frac{\pi }{2},\pi \right\rangle $% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% funkcia $f$\ \ nie je spojit\'{a} v bode $\frac{\pi }{2},\,$v ostatn\'{y}ch bodoch $D\left( f\right) $ je spojit\'{a}.}$ \textbf{Pr\'{\i}klad 75. }$f\left( x\right) =\left\{ \begin{tabular}{cc} $\left( x-1\right) \ln \left( 1-x\right) $ & $x\in \left( 0,1\right) $ \\ $0$ & $x=1$ \\ $x^{\frac{1}{1-x}}$ & $x\in \left( 1,\infty \right) $% \end{tabular}% \right. .$ $% \CustomNote{Answer}{% funkcia $f$\ \ nie je spojit\'{a} v bode $1,\,$v ostatn\'{y}ch bodoch $% D\left( f\right) $ je spojit\'{a}.}$ \textbf{Pr\'{\i}klad 76. }Vy\v{s}etrite spojitos\v{t} funkcie $% f:\left\langle -\frac{\pi }{2},\frac{\pi }{2}\right\rangle \longrightarrow \mathbf{R},\,f\left( x\right) =\left\{ \begin{tabular}{cc} $\func{cotg}x-\frac{1}{x}$ & $x\neq 0$ \\ $0$ & $x=0$% \end{tabular}% .\text{ }% \CustomNote{Answer}{% Funkcia $f$ je spojit\'{a} na $D\left( f\right) =\left\langle -\frac{\pi }{2}% ,\frac{\pi }{2}\right\rangle .$}\right. $ \textbf{Pr\'{\i}klad 77. }Zistite, \v{c}i je funkcia $f:\mathbf{R}% \longrightarrow \mathbf{R},\,f\left( x\right) =\left\{ \begin{tabular}{cc} $\frac{1}{x}-\frac{1}{e^{x}-1}$ & $x\neq 0$ \\ $\frac{1}{2}$ & $x=0$% \end{tabular}% \text{ spojit\'{a} a vypo\v{c}\'{\i}tajte }f\,\,^{\prime }\left( 0\right) .% \text{ }\right. \CustomNote{Answer}{% Funkcia $f$ je spojit\'{a}, $f\,\,^{\prime }\left( 0\right) =-\frac{1}{12}.$} $ \begin{center} \begin{tabular}{|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Zbierka}{}{}{Z.tex}}% %BeginExpansion \msihyperref{Zbierka}{}{}{Z.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za I} \section{Zbierka \'{u}loh} \end{document}