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%F=36,\PARA{038<p type="texpara" tag="Body Text" >\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - Spojitos\U{165} funkcie - Cvi\U{10d}enia\dotfill \thepage }}
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\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{Spojitos\v{t} 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}{}{}{M4.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{M4.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O4.tex}}%
%BeginExpansion
\msihyperref{Ot\'{a}zky}{}{}{O4.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. }Zistite, \v{c}i je funkcia $f\left( x\right)
=5x^{2}+2x-6$ spojit\'{a} v bode $a=-3,\;0,\;1.%
\CustomNote{Answer}{%
Je spojit\'{a}}$

\textbf{Pr\'{\i}klad 2. }Zistite, \v{c}i je funkcia $f\left( x\right)
=\left| 6x+3\right| $ spojit\'{a} v bode $a=-\frac{1}{2}.%
\CustomNote{Answer}{%
\[
\lim_{x\longrightarrow -\frac{1}{2}^{-}}\left| 6x+3\right|
=\lim_{x\longrightarrow -\frac{1}{2}^{-}}-6x-3=0. 
\]%
\[
\lim_{x\longrightarrow -\frac{1}{2}^{+}}\left| 6x+3\right|
=\lim_{x\longrightarrow -\frac{1}{2}^{+}}6x+3=0. 
\]%
\[
f\left( -\frac{1}{2}\right) =0\Longrightarrow 
\]%
$\Longrightarrow \lim_{x\longrightarrow -\frac{1}{2}^{-}}\left| 6x+3\right|
=\lim_{x\longrightarrow -\frac{1}{2}^{+}}\left| 6x+3\right| =f\left( -\frac{1%
}{2}\right) .$%
\par
Je spojit\'{a} v bode $a=-\frac{1}{2}$}$

\textbf{Pr\'{\i}klad 3. }Zistite, \v{c}i je funkcia $f\left( x\right) =\sqrt{%
3-x^{2}}$ z\v{l}ava (sprava) spojit\'{a} v bode $a=\sqrt{3}.$%
\CustomNote{Answer}{%
Funkcia $f\left( x\right) =\sqrt{3-x^{2}}$ m\'{a} defini\v{c}n\'{y} obor $%
D\left( f\right) =\left\langle -\sqrt{3},\sqrt{3}\right\rangle .$ M\'{a}me: $%
f\left( \sqrt{3}\right) =0$, $\lim_{x\longrightarrow \sqrt{3}^{-}}f\left(
x\right) =\lim_{x\longrightarrow \sqrt{3}^{-}}\sqrt{3-x^{2}}=0$, teda $f$ je
v bode $\sqrt{3}$ spojit\'{a} z\v{l}ava. Vzh\v{l}adom na defini\v{c}n\'{y}
obor $\lim_{x\longrightarrow \sqrt{3}^{+}}f\left( x\right) $ nem\'{a}
zmysel, teda $f$ je z\v{l}ava spojit\'{a} v bode $a=\sqrt{3}.$}

\textbf{Pr\'{\i}klad 4. }Zistite, \v{c}i je funkcia $f\left( x\right) =\frac{%
1}{4-x^{{}}}$ spojit\'{a} v bodoch $a=-2,\;0,\;4.%
\CustomNote{Answer}{$f$ \ je spojit\'{a} v bodoch $a=-2,\;0.$ V bode $a=4$
nie je definovan\'{a} (teda tam ani nem\^{o}\v{z}e by\v{t} spojit\'{a})}$

\textbf{Pr\'{\i}klad 5. }Zistite, \v{c}i je funkcia $f\left( x\right) =\frac{%
4-x^{2}}{\left| 4x-x^{3}\right| }$ spojit\'{a} v bodoch $a=-2,\;0,\;2.%
\CustomNote{Prob_Solv_Hint}{%
Funkcia 
\[
f\left( x\right) =\left\{ 
\begin{tabular}{ccc}
$\frac{1}{x}$ & pre & $x\in \left( -\infty ,-2\right) $ \\ 
$-\frac{1}{x}$ & pre & $x\in \left( -2,0\right) $ \\ 
$\frac{1}{x}$ & pre & $x\in \left( 0,2\right) $ \\ 
$-\frac{1}{x}$ & pre & $x\in \left( 2,\infty \right) $%
\end{tabular}%
\right. 
\]%
nie je definovan\'{a} v bodoch $a=-2,0,2$, teda v t\'{y}chto bodoch nie je
ani spojit\'{a}.}$

\textbf{Pr\'{\i}klad 6. }Zistite, \v{c}i je funkcia $f\left( x\right) =\frac{%
\sin x}{x}$ spojit\'{a} na intervale $\left( 0,\infty \right) .%
\CustomNote{Answer}{%
Pre ka\v{z}d\'{e} $a\in \left( 0,\infty \right) $ m\'{a}me 
\[
\lim_{x\longrightarrow a}f\left( x\right) =\lim_{x\longrightarrow a}\frac{%
\sin x}{x}=\frac{\sin a}{a}=f\left( a\right) . 
\]%
Funkcia je spojit\'{a} na $\left( 0,\infty \right) $}$

\textbf{Pr\'{\i}klad 7. }Zistite, \v{c}i je funkcia $f\left( x\right) =\frac{%
\sqrt{x-1}}{\sqrt{3-x}}$ spojit\'{a} na intervale $\left\langle 1,3\right) $%
, alebo na $\left( 1,3\right\rangle .%
\CustomNote{Prob_Solv_Hint}{$f=\frac{h}{g}$, kde 
\[
h:\left\langle 1,\infty \right) \longrightarrow \mathbf{R},\,h\left(
x\right) =\sqrt{x-1}, 
\]%
\[
\,g:\left( -\infty ,3\right) \longrightarrow \mathbf{R},\,g\left( x\right) =%
\sqrt{3-x}, 
\]%
s\'{u} spojit\'{e}, potom 
\[
f:\left\langle 1,3\right) \longrightarrow ,\,f\left( x\right) =\frac{\sqrt{%
x-1}}{\sqrt{3-x}} 
\]%
je podiel dvoch spojit\'{y}ch funkci\'{\i}, teda je spojit\'{a}, alebo to uk%
\'{a}\v{z}eme takto: $\forall x\in \left\langle 1,3\right) $ plat\'{\i}: $%
\lim_{x\longrightarrow a}f\left( x\right) =\lim_{x\longrightarrow a}\frac{%
\sqrt{x-1}}{\sqrt{3-x}}=\frac{\sqrt{a-1}}{\sqrt{3-a}}=f\left( a\right) .$}$

\textbf{Pr\'{\i}klad 8.} Zistite, \v{c}i je funkcia $f$ spojit\'{a} v bode $%
a $ a na\v{c}rtnite jej graf ak:

$f\left( x\right) =\left\{ 
\begin{tabular}{cc}
$2\sqrt{x}$ & $x\in \left\langle 0,1\right\rangle $ \\ 
$4-2x$ & $x\in \left( 1,\frac{5}{2}\right) $ \\ 
$2x-7$ & $x\in \left( \frac{5}{2},\infty \right) $%
\end{tabular}%
\right. ,\;a=1,\,\frac{5}{2}.%
\CustomNote{Answer}{%
V bode $a=1$ je spojit\'{a}, v bode $a=\frac{5}{2}$ nie je definovan\'{a},
teda ani spojit\'{a}.}$

\textbf{Pr\'{\i}klad 9.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie $f\left(
x\right) =\frac{\left( x-1\right) ^{2}-1}{x-2}.$ Ak sa d\'{a} n\'{a}jdite tak%
\'{e} roz\v{s}\'{\i}renie funkcie $f,$ ktor\'{e} ozna\v{c}\'{\i}me $F$ s
defini\v{c}n\'{y}m oborom $\mathbf{R}$, aby $F$ bola spojit\'{a}! Nap\'{\i}%
\v{s}te predpis z\'{\i}skanej spojitej funkcie $F$. 
\CustomNote{Prob_Solv_Hint}{$D\left( f\right) =\left( -\infty ,2\right) \cup
\left( 2,\infty \right) .$Funkcia $f$ je spojit\'{a}.%
\[
\lim_{x\longrightarrow 2}f\left( x\right) =\lim_{x\longrightarrow 2}\frac{%
\left( x-1\right) ^{2}-1}{x-2}=\lim_{x\longrightarrow 2}\frac{x\left(
x-2\right) }{x-2}=2. 
\]%
h\v{l}adan\'{a} spojit\'{a} funkcia je dan\'{a}: $F:\mathbf{R}%
\longrightarrow \mathbf{R},\,F\left( x\right) =\left\{ 
\begin{tabular}{ccc}
$f\left( x\right) $ & pre & $x\neq 2$ \\ 
$2$ & pre & $x=2$%
\end{tabular}%
\right. .$}

\textbf{Pr\'{\i}klad 10.} N\'{a}jdite defini\v{c}n\'{y} obor funkcie $%
f\left( x\right) =\frac{x}{x-2}.$ Ak sa d\'{a} n\'{a}jdite tak\'{e} roz\v{s}%
\'{\i}renie funkcie $f,$ ktor\'{e} ozna\v{c}\'{\i}me $F$ s defini\v{c}n\'{y}%
m oborom $\mathbf{R}$, aby $F$ bola spojit\'{a}! Nap\'{\i}\v{s}te predpis z%
\'{\i}skanej spojitej funkcie $F$.%
\CustomNote{Answer}{$D\left( f\right) =\left( -\infty ,2\right) \cup \left(
2,\infty \right) .$Ned\'{a} sa}

\textbf{Pr\'{\i}klad 11. }Ur\v{c}te hodnotu parametra $p$ tak, aby funkcia 
\[
f\left( x\right) =\left\{ 
\begin{tabular}{cc}
$8e^{px}$ & $x<0$ \\ 
$p-3x$ & $x\geq 0$%
\end{tabular}%
\right. , 
\]
bola v bode $a=0$ spojit\'{a}.%
\CustomNote{Answer}{$p=8.$}

\textbf{Pr\'{\i}klad 12. }Ur\v{c}te hodnotu parametra $p$ tak, aby funkcia 
\[
f\left( x\right) =\left\{ 
\begin{tabular}{cc}
$\frac{\sqrt{1+x}-\sqrt{1+x^{2}}}{\sqrt{1+x}-1}$ & $x\neq 0$ \\ 
$p^{2}+2p-2$ & $x=0$%
\end{tabular}%
\right. , 
\]%
bola v bode $a=0$ spojit\'{a}. 
\CustomNote{Answer}{$p=-3\vee p=1$}

\textbf{Pr\'{\i}klad 13. }N\'{a}jdite defini\v{c}n\'{y} obor funkcie $%
f\left( x\right) =\frac{x^{2}+2x-3}{x^{3}+5x^{2}+6x}.$ Ak sa d\'{a} n\'{a}%
jdite tak\'{e} roz\v{s}\'{\i}renie funkcie $f$ v bode $a$, aby bola v bode $%
a $ spojit\'{a}. Nap\'{\i}\v{s}te predpis z\'{\i}skanej spojitej funkcie $F$
, ke\v{d} $a=0,-2,-3.%
\CustomNote{Answer}{%
\[
f\left( x\right) =\frac{x^{2}+2x-3}{x^{3}+5x^{2}+6x}=\frac{\left( x+3\right)
\left( x-1\right) }{x\left( x+2\right) \left( x+3\right) }\Longrightarrow 
\]%
\[
\Longrightarrow D\left( f\right) =\mathbf{R\setminus }\left\{
-3,-2,0\right\} , 
\]%
je spojit\'{a} (podiel). Preto\v{z}e 
\[
\lim_{x\longrightarrow -3}f\left( x\right) =-\frac{4}{3}, 
\]%
\[
\,\lim_{x\longrightarrow -2^{\pm }}f\left( x\right) =\pm \infty , 
\]%
\[
\,\lim_{x\longrightarrow 0^{\pm }}f\left( x\right) =\mp \infty , 
\]%
funkciu $f$ mo\v{z}no dodefinova\v{t} iba v bode $x=-3$ takto: 
\[
F:\mathbf{R\setminus }\left\{ -2,0\right\} \longrightarrow \mathbf{R}, 
\]%
\[
\,F\left( x\right) =\left\{ 
\begin{tabular}{ccc}
$f\left( x\right) $ & pre & $x\in D\left( f\right) $ \\ 
$-\frac{4}{3}$ & pre & $x=-3$%
\end{tabular}%
.\right. 
\]%
}$

\textbf{Pr\'{\i}klad 14. }Zistite, \v{c}i je funkcia $\,f\left( x\right) =%
\frac{1}{x-3}$ spojit\'{a} a ohrani\v{c}en\'{a} na intervale $\left\langle
0,3\right) .%
\CustomNote{Answer}{%
Je spojit\'{a}, ale nie je ohrani\v{c}en\'{a}.}$

\textbf{Pr\'{\i}klad 15. }Zistite, \v{c}i je funkcia $\,f\left( x\right)
=\left| 4x-8\right| $ spojit\'{a} na intervale $\left\langle
-1,4\right\rangle .$ Ak \'{a}no, n\'{a}jdite jej minimum a maximum na danom
intervale. 
\CustomNote{Prob_Solv_Hint}{$f\left( x\right) =\left\{ 
\begin{tabular}{ccc}
$8-4x$ & pre & $x\in \left\langle -1,2\right) $ \\ 
$4x-8$ & pre & $x\in \left\langle 2,4\right\rangle $%
\end{tabular}%
\right. ,$ spojit\'{a}, 
\[
\min_{x\in \left\langle -1,4\right\rangle }f\left( x\right) =0=f\left(
2\right) , 
\]%
\[
\max_{x\in \left\langle -1,4\right\rangle }f\left( x\right) =12=f\left(
-1\right) . 
\]%
}

\textbf{Pr\'{\i}klad 16. }Zistite, \v{c}i je funkcia $\,f\left( x\right) =%
\sqrt{\left| x\right| }$ spojit\'{a} na intervale $\left\langle
-3,2\right\rangle .$ Ak \'{a}no, n\'{a}jdite jej minimum a maximum na danom
intervale. 
\CustomNote{Answer}{%
Je spojit\'{a}, 
\[
\min_{x\in \left\langle -3,2\right\rangle }f\left( x\right) =0, 
\]%
\[
\max_{x\in \left\langle -3,2\right\rangle }f\left( x\right) =\sqrt{3}. 
\]%
}

\textbf{Pr\'{\i}klad 17. }Zistite, \v{c}i je funkcia $f\left( x\right) =%
\frac{1}{x^{2}-6x+8}$ spojit\'{a} na intervale $\left\langle
0,3\right\rangle .$ Ak \'{a}no, n\'{a}jdite jej minimum a maximum na danom
intervale. 
\CustomNote{Answer}{%
Funkcia $f$ nie je definovan\'{a} a teda ani spojit\'{a} v bode $a=2,$ a na
intervale $\left\langle 0,3\right\rangle $ nenadob\'{u}da maximum ani
minimum. $\left( \lim_{x\longrightarrow 2^{\pm }}f\left( x\right) =\mp
\infty \right) .$}

\textbf{Pr\'{\i}klad 18. }Pou\v{z}it\'{\i}m vety o medzihodnote uk\'{a}\v{z}%
te, \v{z}e funkcia $x^{3}+x+1$ m\'{a} na intervale $\left\langle
-1,1\right\rangle $ aspo\v{n} jeden kore\v{n}. 
\CustomNote{Prob_Solv_Hint}{%
Nech $f\left( x\right) =x^{3}+x+1$ na intervale $\left\langle
-1,1\right\rangle .$ Funkcia $f$ je spojit\'{a} na uzavretom intervale a plat%
\'{\i}: $f\left( -1\right) =-1<0<f\left( 1\right) =3.$ Pou\v{z}ijeme vetu o
medzihodnote pre $p=0.$ Pod\v{l}a tejto vety v intervale $\left\langle
-1,1\right\rangle $ existuje aspo\v{n} jeden kore\v{n} danej rovnice.}

\textbf{Pr\'{\i}klad 19. }Pou\v{z}it\'{\i}m vety o medzihodnote uk\'{a}\v{z}%
te, \v{z}e rovnica $x^{3}+\frac{1}{x}=3$ m\'{a} na intervale $\left\langle
1,3\right\rangle $ aspo\v{n} jedno rie\v{s}enie. 
\CustomNote{Answer}{%
M\'{a} na danom intervale aspo\v{n} jeden kore\v{n}.}

\textbf{Pr\'{\i}klad 20. }Zistite, \v{c}i je funkcia $f\left( x\right) =%
\frac{1}{x}$ ohrani\v{c}en\'{a} a \v{c}i m\'{a} minimum a maximum na
intervale $\left\langle 1,\infty \right) .%
\CustomNote{Prob_Solv_Hint}{%
Funkcia $f\left( x\right) =\frac{1}{x}$ je na intervale $\left\langle
1,\infty \right) $ spojit\'{a} a je aj ohrani\v{c}en\'{a}. Plat\'{\i}: 
\[
\max_{x\in \left\langle 1,\infty \right) }f\left( x\right) =f\left( 1\right)
=1, 
\]%
\[
\inf_{x\in \left\langle 1,\infty \right) }f\left( x\right) =0. 
\]%
Funkcia nenadob\'{u}da minimum.}$

\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}{}{}{M4.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{M4.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O4.tex}}%
%BeginExpansion
\msihyperref{Ot\'{a}zky}{}{}{O4.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{Spojitos\v{t} funkcie}

\end{document}
