1 files changed, 1609 insertions, 0 deletions
diff --git a/šola/ana1/prak.lyx b/šola/ana1/prak.lyx
new file mode 100644
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+++ b/šola/ana1/prak.lyx
@@ -0,0 +1,1609 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\DeclareMathOperator{\ctg}{ctg}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children no
+\language slovene
+\language_package default
+\inputencoding auto-legacy
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\float_placement class
+\float_alignment class
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 1cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tablestyle default
+\tracking_changes false
+\output_changes false
+\change_bars false
+\postpone_fragile_content false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\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 $\binom{n}{k}\coloneqq\frac{n!}{k!\left(n-k\right)!}$
+\end_inset
+
+,
+ 
+\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 $z^{n}=r^{n}\left(\cos\left(n\phi\right)+i\sin\left(n\phi\right)\right)$
+\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
+\begin_inset Tabular
+<lyxtabular version="3" rows="8" 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
+Izraz
+\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
+Odvod
+\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
+Izraz
+\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
+Odvod
+\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 $\frac{f}{g}$
+\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 $\frac{f'g-fg'}{g^{2}}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $g\not=0$
+\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 $f\left(g\right)$
+\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 $f'\left(g\right)g'$
+\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 $\tan x$
+\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 $\cos^{-2}x$
+\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 $\cot x$
+\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 $-sin^{-2}x$
+\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 $a^{x}$
+\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 $a^{x}\text{\ensuremath{\ln a}}$
+\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 $x^{x}$
+\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 $x^{x}\left(1+\ln x\right)$
+\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 $\log_{a}x$
+\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 $\frac{1}{x\ln a}$
+\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 $f^{-1}\left(a\right)$
+\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 $\frac{1}{f'\left(f^{-1}\left(a\right)\right)}$
+\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 $\arcsin x$
+\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 $\left(1-x^{2}\right)^{-\frac{1}{2}}$
+\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 $\arccos x$
+\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 $-\left(1-x^{2}\right)^{-\frac{1}{2}}$
+\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 $\arctan x$
+\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 $\frac{1}{1+x^{2}}$
+\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 $\text{arccot\,}x$
+\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 $-\frac{1}{1+x^{2}}$
+\end_inset
+
+
+\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 $x^{n}$
+\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 $nx^{n-1}$
+\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
+
+\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
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f''\left(I\right)>0\Leftrightarrow f$
+\end_inset
+
+ konveksna na 
+\begin_inset Formula $I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f''\left(I\right)<0\Leftrightarrow f$
+\end_inset
+
+ konkavna na 
+\begin_inset Formula $I$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+ab>0\wedge a<b\Leftrightarrow a^{-1}>b^{-1},\quad ab<0\wedge a<b\Leftrightarrow a^{-1}<b^{-1}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\lim_{x\to0}\frac{\sin x}{x}=1\quad\quad\tan\phi=\left|\frac{k_{1}-k_{2}}{1+k_{1}k_{2}}\right|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\lim_{x\to0}x\ln x=0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+f\text{ zv.+odv.@ }\left[a,b\right]\Rightarrow\exists\xi\in\left[a,b\right]\ni:f\left(b\right)-f\left(a\right)=f'\left(\xi\right)\left(b-a\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+T_{f,a,n}\left(x\right)=\sum_{k=0}^{n}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^{k}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f\text{\ensuremath{\in C^{n+1}}}$
+\end_inset
+
+ na odprtem 
+\begin_inset Formula $I\subset\mathbb{R}\Rightarrow\forall a,x\in I\exists c\in\left(\min\left\{ a,x\right\} ,\max\left\{ a,x\right\} \right)\ni:f\left(x\right)-T_{f,a,n}\left(x\right)=R_{f,a,n}\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}$
+\end_inset
+
+
+\begin_inset Formula $\left(x-a\right)^{n+1}.\text{ Posledično velja tudi takale ocena:}$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\exists M>0\forall x\in I:\left|f^{\left(n+1\right)}\right|\leq M\Rightarrow R_{f,a,n}\left(x\right)=\frac{M}{\left(n+1\right)!}\left|x-a\right|^{n+1}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+R=\lim_{n\to\infty}\left|\frac{c_{n}}{c_{n+1}}\right|,\quad R=\lim_{n\to\infty}\frac{1}{\sqrt[n]{\left|c_{n}\right|}}
+\]
+
+\end_inset
+
+
+\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 Paragraph
+Uporabne vrste
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sin x=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{\left(2n+1\right)!}x^{2n+1}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\cos x=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{\left(2n\right)!}x^{2n}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\sinh x=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{\left(2n+1\right)!}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $e^{x}=\sum_{x=0}^{\infty}\frac{x^{n}}{n!}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\left(1+x\right)^{\alpha}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\ln\left(1+x\right)=\sum_{n=1}^{\infty}\left(-1\right)^{n+1}\frac{x^{n}}{n}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $\ln\left(1-x\right)=-\sum_{n=1}^{\infty}\frac{x^{n}}{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Razcep racionalnih
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\frac{p\left(x\right)}{\left(x-a\right)^{3}}=\frac{A}{x-a}+\frac{B}{\left(x-1\right)^{2}}+\frac{C}{\left(x-1\right)^{3}}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\frac{p\left(x\right)}{\left(x-a\right)\left(x-b\right)^{2}}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{\left(x-b\right)^{2}}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\frac{p\left(x\right)}{\left(x-a\right)\left(x^{2}-b\right)}=\frac{A}{x-a}+\frac{Bx-C}{x^{2}-b}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Integrali
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\int\frac{1}{x^{2}+a^{2}}dx=\frac{1}{a}\arctan\frac{x}{a}+C
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{x^{2}-a^{2}}dx=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{a^{2}-x^{2}}dx=\frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right|+C
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{ax+b}dx=\frac{1}{a}\ln\left|ax+b\right|+C
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\left(ax+b\right)^{n}dx=\frac{\left(ax+b\right)^{n+1}}{a\left(n+1\right)}+C
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int f\left(x\right)g'\left(x\right)dx=f\left(x\right)g\left(x\right)-\int f'\left(x\right)g\left(x\right)dx
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{\sin^{2}\left(x\right)}dx=-\ctg\left(x\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{\cos^{2}\left(x\right)}=\tan\left(x\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{\sqrt{a^{2}+x^{2}}}dx=\ln\left|x+\sqrt{x^{2}+a^{2}}\right|
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{1}{\sqrt{x^{2}-a^{2}}}dx=\ln\left|x+\sqrt{x^{2}-a^{2}}\right|
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\sqrt{a^{2}+x^{2}}dx=\frac{1}{2}\left(x\sqrt{a^{2}+x^{2}}+a^{2}\ln\left(\sqrt{a^{2}+x^{2}}+x\right)\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\sqrt{a^{2}-x^{2}}dx=\frac{1}{2}\left(x\sqrt{a^{2}-x^{2}}+a^{2}\arctan\left(\frac{x}{\sqrt{a^{2}-x^{2}}}\right)\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{A}{x-a}dx=A\ln\left|x-a\right|
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{A}{\left(x-a\right)^{n}}dx=\frac{-A}{n-1}\cdot\frac{1}{\left(x-a\right)^{n-1}}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\int\frac{Bx+C}{x^{2}+bx+c}=\frac{B}{2}\ln\left|x^{2}+bx+c\right|+\frac{2C-Bb}{\sqrt{-D}}\arctan\left(\frac{2x+b}{\sqrt{-D}}\right)
+\]
+
+\end_inset
+
+ In velja 
+\begin_inset Formula $D=b^{2}-4c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Pri 
+\begin_inset Formula $\int\sin\left(x\right)^{p}\cos\left(x\right)^{q}dx$
+\end_inset
+
+ lih 
+\begin_inset Formula $q$
+\end_inset
+
+ substituiramo 
+\begin_inset Formula $t=\cos\left(x\right)$
+\end_inset
+
+,
+ lih 
+\begin_inset Formula $p$
+\end_inset
+
+ pa 
+\begin_inset Formula $t=\sin\left(x\right)$
+\end_inset
+
+.
+ Pri sodih nižamo stopnje s formulo dvonega kota.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+https://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document