1 files changed, 658 insertions, 0 deletions

diff --git a/šola/ana1/kolokvij2.lyx b/šola/ana1/kolokvij2.lyx
index 486a401..a057288 100644
--- a/šola/ana1/kolokvij2.lyx
+++ b/šola/ana1/kolokvij2.lyx
@@ -158,6 +158,580 @@ begin{multicols}{2}
 \end_inset
 
 
+\begin_inset Formula $\log_{a}1=0$
+\end_inset
+
+, 
+\begin_inset Formula $\log_{a}a=1$
+\end_inset
+
+, 
+\begin_inset Formula $\log_{a}a^{x}=x$
+\end_inset
+
+, 
+\begin_inset Formula $a^{\log_{a}x}=x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\log_{a}x^{n}=n\log_{a}x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $D=b^{2}-4ac$
+\end_inset
+
+, 
+\begin_inset Formula $x_{1,2}=\frac{-b\pm\sqrt{D}}{2a}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $zw=\left(ac-bd\right)+\left(ad+bc\right)i$
+\end_inset
+
+, 
+\begin_inset Formula $\vert zw\vert=\vert z\vert\vert w\vert$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\arg\left(zw\right)=\arg z+\arg w$
+\end_inset
+
+ (kot)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $z\overline{z}=a^{2}-\left(bi\right)^{2}=a^{2}+b^{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\cos\phi+i\sin\phi\right)$
+\end_inset
+
+
+\begin_inset Formula $\left(\cos\psi+i\sin\psi\right)=\cos\left(\phi+\psi\right)+i\sin\left(\phi+\psi\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $z^{2}=a^{2}+2abi-b^{2}$
+\end_inset
+
+, 
+\begin_inset Formula $z^{3}=a^{3}-3ab^{2}+\left(3a^{2}b-b^{3}\right)i$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $(a+b)^{n}=\sum_{k=0}^{n}{n \choose k}ab^{n-k}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $z^{n}=r^{3}\left(\cos\left(3\phi\right)+i\sin\left(3\phi\right)\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\phi=\arctan\frac{\Im z}{\Re z}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Odprta množica ne vsebuje robnih točk.
+ Zaprta vsebuje vse.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sin\left(x\pm y\right)=\sin x\cdot\cos y\pm\sin y\cdot\cos x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\cos\left(x\pm y\right)=\cos x\cdot\cos y\mp\sin y\cdot\sin x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tan\left(x\pm y\right)=\frac{\tan x\pm\tan y}{1\text{\ensuremath{\mp\tan}x\ensuremath{\cdot\tan y}}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a_{n}$
+\end_inset
+
+je konv.
+ 
+\begin_inset Formula $\Longleftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\forall\varepsilon>0:\exists n_{0}\ni:\forall n,m:n_{0}<n<m\wedge\vert a_{n}-a_{m}\vert<\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\euler^{1/k}\coloneqq\lim_{n\to\infty}\left(1+\frac{1}{nk}\right)^{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vrsta je konv., če je konv.
+ njeno zap.
+ delnih vsot.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $s_{n}=\begin{cases}
+\frac{1-q^{n+1}}{1-q}; & q\not=1\\
+n+1; & q=1
+\end{cases}$
+\end_inset
+
+.
+ Geom.
+ vrsta konv.
+ 
+\begin_inset Formula $\Longleftrightarrow q\in\left(-1,1\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Primerjalni krit.
+\series default
+: 
+\begin_inset Formula $\sum_{1}^{\infty}a_{k}$
+\end_inset
+
+ konv.
+ 
+\begin_inset Formula $\wedge$
+\end_inset
+
+ 
+\begin_inset Formula $b_{k}\leq a_{k}$
+\end_inset
+
+za 
+\begin_inset Formula $k>n_{0}$
+\end_inset
+
+ 
+\begin_inset Formula $\wedge$
+\end_inset
+
+ vrsti sta navzdol omejeni 
+\begin_inset Formula $\Longrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\sum_{1}^{\infty}b_{k}$
+\end_inset
+
+ konv.
+ 
+\begin_inset Formula $\sum_{1}^{\infty}a_{k}$
+\end_inset
+
+ rečemo 
+\shape italic
+majoranta
+\shape default
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Kvocientni
+\series default
+: 
+\begin_inset Formula $a_{k}>0$
+\end_inset
+
+, 
+\begin_inset Formula $D_{n}\coloneqq\frac{a_{n}+1}{a_{n}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\forall n<n_{0}:D_{n}\in\left(0,1\right)\Longrightarrow\sum_{1}^{\infty}a_{k}<\infty$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\forall n<n_{0}:D_{n}\geq1\Longrightarrow\sum_{1}^{\infty}a_{k}=\infty$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Če 
+\begin_inset Formula $\exists D\coloneqq\lim_{n\to\infty}D_{n}$
+\end_inset
+
+: 
+\begin_inset Formula $\vert D\vert<1\Longrightarrow$
+\end_inset
+
+konv., 
+\begin_inset Formula $\vert D\vert>1\Longrightarrow div.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Korenski
+\series default
+: Kot Kvocientni, le da 
+\begin_inset Formula $D_{n}\coloneqq\sqrt[n]{a_{n}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Leibnizov
+\series default
+: 
+\begin_inset Formula $a_{n}\to0\Longrightarrow\sum_{1}^{\infty}\left(\left(-1\right)^{k}a_{k}\right)<\infty$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Absolutna konvergenca 
+\begin_inset Formula $\left(\sum_{1}^{\infty}\vert a_{n}\vert<\infty\right)$
+\end_inset
+
+ 
+\begin_inset Formula $\Longrightarrow$
+\end_inset
+
+ konvergenca
+\end_layout
+
+\begin_layout Standard
+Pri konv.
+ po točkah je 
+\begin_inset Formula $n_{0}$
+\end_inset
+
+ odvisen od 
+\begin_inset Formula $x$
+\end_inset
+
+, pri enakomerni ni.
+\end_layout
+
+\begin_layout Standard
+Potenčna vrsta: 
+\begin_inset Formula $\sum_{j=1}^{\infty}b_{j}x^{j}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $R^{-1}=\limsup_{k\to\infty}\sqrt[k]{\vert b_{k}\vert}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\vert x\vert<R\Longrightarrow$
+\end_inset
+
+abs.
+ konv., 
+\begin_inset Formula $\vert x\vert>R\Longrightarrow$
+\end_inset
+
+divergira
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\lim_{x\to a}\left(\alpha f\left(x\right)\right)=\alpha\lim_{x\to a}f\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="4">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sin$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\cos$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\tan$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $30^{\circ}=\frac{\pi}{6}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{1}{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sqrt{3}}{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sqrt{3}}{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $45^{\circ}=\frac{\pi}{4}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sqrt{2}}{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sqrt{2}}{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $60^{\circ}=\frac{\pi}{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\sqrt{3}}{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{1}{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sqrt{3}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Krožnica: 
+\begin_inset Formula $\left(x-p\right)^{2}+\left(y-q\right)^{2}=r^{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Elipsa: 
+\begin_inset Formula $\frac{\left(x-p\right)^{2}}{a^{2}}+\frac{\left(y-q\right)^{2}}{b^{2}}=1$
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -670,6 +1244,90 @@ R=\lim_{n\to\infty}\left|\frac{c_{n}}{c_{n+1}}\right|,\quad R=\lim_{n\to\infty}\
 \end_layout
 
 \begin_layout Standard
+Zvezna 
+\begin_inset Formula $\text{f}$
+\end_inset
+
+ na zaprtem intervalu 
+\begin_inset Formula $\left[a,b\right]$
+\end_inset
+
+ doseže 
+\begin_inset Formula $\inf$
+\end_inset
+
+ in 
+\begin_inset Formula $\sup$
+\end_inset
+
+, je omejena in doseže vse funkcijske vrednosti na 
+\begin_inset Formula $\left[f\left(a\right),f\left(b\right)\right]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:I\to\mathbb{R}$
+\end_inset
+
+ je enakomerno zvezna na 
+\begin_inset Formula $I$
+\end_inset
+
+, če 
+\begin_inset Formula $\forall\varepsilon>0\exists\delta_{\left(\varepsilon\right)}>0\ni:\forall x,y\in I:\left|x-y\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(y\right)\right|<\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:I\to\mathbb{R}$
+\end_inset
+
+ je zvezna na 
+\begin_inset Formula $I$
+\end_inset
+
+, če 
+\begin_inset Formula $\forall\varepsilon>0\forall x\in I\exists\delta_{\left(x,\varepsilon\right)}>0\ni:\forall x,y\in I:\left|x-y\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(y\right)\right|<\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Zvezna 
+\begin_inset Formula $f$
+\end_inset
+
+ na kompaktni množici je enakomerno zvezna.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+f'\left(x\right)=\lim_{x\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\sinh x=\frac{e^{x}-e^{-x}}{2},\quad\cosh x=\frac{e^{x}+e^{-x}}{2}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 \begin_inset ERT
 status open