From 5ddc2e50dcfe5a1e36bb5a04008e29a33a661921 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= Date: Mon, 18 Nov 2024 00:18:24 +0100 Subject: kombi do konca 5. predavanja --- "\305\241ola/ana3/prak.lyx" | 1240 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1240 insertions(+) create mode 100644 "\305\241ola/ana3/prak.lyx" (limited to 'šola/ana3/prak.lyx') diff --git "a/\305\241ola/ana3/prak.lyx" "b/\305\241ola/ana3/prak.lyx" new file mode 100644 index 0000000..6675894 --- /dev/null +++ "b/\305\241ola/ana3/prak.lyx" @@ -0,0 +1,1240 @@ +#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 $\arctan a\pm\arctan b=\frac{a\pm b}{1\mp ab}$ +\end_inset + + +\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 $\euler^{1/k}\coloneqq\lim_{n\to\infty}\left(1+\frac{1}{nk}\right)^{n}$ +\end_inset + + +\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 + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sin$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\cos$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\tan$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $30^{\circ}=\frac{\pi}{6}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{3}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{3}}{3}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $45^{\circ}=\frac{\pi}{4}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{2}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{2}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +1 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $60^{\circ}=\frac{\pi}{3}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{3}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sqrt{3}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\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 + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout +Izraz +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Odvod +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Izraz +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +Odvod +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{f}{g}$ +\end_inset + + +\end_layout + +\end_inset + + +\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 + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $f\left(g\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $f'\left(g\right)g'$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\tan x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\cos^{-2}x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\cot x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $-sin^{-2}x$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $a^{x}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $a^{x}\text{\ensuremath{\ln a}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $x^{x}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $x^{x}\left(1+\ln x\right)$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\log_{a}x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{x\ln a}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $f^{-1}\left(a\right)$ +\end_inset + + +\end_layout + +\end_inset + + +\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 + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\arcsin x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\left(1-x^{2}\right)^{-\frac{1}{2}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\arccos x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $-\left(1-x^{2}\right)^{-\frac{1}{2}}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\arctan x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{1+x^{2}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\text{arccot\,}x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $-\frac{1}{1+x^{2}}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $x^{n}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $nx^{n-1}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + + + +\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 ab^{-1},\quad ab<0\wedge a