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diff --git a/mat/domace_naloge/29/dokument.tex b/mat/domace_naloge/29/dokument.tex new file mode 100644 index 0000000..690a382 --- /dev/null +++ b/mat/domace_naloge/29/dokument.tex @@ -0,0 +1,221 @@ +% !TeX encoding = UTF-8 +% !TeX spellcheck = sl_SI +% do-vimlatex-onwrite +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{etoolbox} +\usepackage[hidelinks]{hyperref} +\usepackage{xurl} % da breaka url +\usepackage[a4paper, total={7in, 10in}]{geometry} +\usepackage{graphicx} +\usepackage{hologo} +\usepackage{amssymb} +\usepackage{calculator} +\usepackage{pgfplots} +\usepackage[inline]{enumitem} +\usepackage{siunitx} +\usepackage{multicol} +\usepackage{tabularcalc} +\usepackage{amsmath} +\usepackage{float} +\usepackage{tasks} +\usepackage{filecontents} +\usepackage{textcomp} +\usepackage{tkz-euclide} +\usepackage{dirtytalk} +\usepackage{csquotes} +\usepackage{listings} +\usepackage{datetime} % [ddMMyyyy] +\usepackage{ccicons} +\usepackage{chemformula} +\usepackage{gensymb} +\usepackage[normalem]{ulem} +\usepackage[slovene]{babel} +\usepackage{cancel} +\usepackage{tabularx} +\usepackage{xcolor} +\usepackage{colortbl} +\usepackage{exsheets} +\usepackage{harpoon} +\usepackage{readarray} +\usepackage{forloop} +\usepackage[nomessages]{fp} +\usepackage{multirow} +\newcolumntype{Y}{>{\centering\arraybackslash}X} +% \usepackage{multienum} % weird with labels +\usetikzlibrary{calc} %% not really needed............. idk. +%\usetikzlibrary{external} +% \usetkzobj{all} % tkz-euclide > 3.02 tega ne potrebuje več (: +%\tikzexternalize +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction,per-mode=fraction} % per-mode=symbol +\newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}} +\newcommand{\functionSamples}{100} % fix to fancier value upon release, keep low during development +\newcommand{\razhroscevanje}{1} +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codered}{rgb}{1,0,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} +\renewcommand{\dateseparator}{. } +\settimeformat{hhmmsstime} +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} +\lstset{style=mystyle} +\def\@maketitle{% + \newpage + \null + \vskip 2em% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% <------ + \@author% <------ Authors + \end{tabular}\par}% <------ + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em} +%opening +\newcommand{\snovdn}{Koreni poljubnih stopenj } +\newcommand{\predmdn}{mat} +\newcommand{\predmkaj}{domace\_naloge} +\newcommand{\stevilkadn}{29} +\newcommand{\cm}[1]{\SI{#1}{\centi\meter}} +\newcommand{\kmh}[1]{\SI{#1}{\kilo\meter\per\hour}} +\makeatletter +\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname} +\newcommand{\@xslalph}[1]{% + \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i% + \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}% + \or t\or u\or v\or z\or \v{z} + \else\@ctrerr\fi% +} +\AddEnumerateCounter{\xslalph}{\@xslalph}{m} +\makeatother +\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma +\newcommand*\textfrac[2]{ + \frac{\text{#1}}{\text{#2}} +} +\NewTasks[counter-format=\xslalph*),label-format=\bfseries]{primerTasks}(3) +\newcommand\vektor{\overrightarrow} +\newcommand{\iic}{I\textsuperscript{2}C } +\title{% + \snovdn --- \stevilkadn. domača naloga + \\ + \large Matematika, Gimnazija Bežigrad} +\author{\begin{tabular}{rl} + \textbf{Profesor:} & prof. Vilko Domajnko \\ + \textbf{Avtor:} & Anton Luka Šijanec, 2. a +% \textbf{Avtor:} & Anton Luka Šijanec \\ & Member 2 \\ & Member 3 +\end{tabular}} +\newcommand\hcancel[2][black]{\setbox0=\hbox{$0#2$}% +\rlap{\raisebox{.45\ht0}{\textcolor{#1}{\rule{\wd0}{1pt}}}}#2} +\everymath{\displaystyle} % https://tex.stackexchange.com/a/32847/212260 +\begin{document} +\maketitle +\begin{abstract} +Ta dokument vsebuje navodila in rešitve domače naloge snovi \textit{\snovdn}pri matematiki, ki sem jih spisal sam. +Kjer je bilo potrebno izbrati neke poljubne naloge, sem jih vedno izbiral naključno. +\end{abstract} +\paragraph{Navodilo naloge} \textbf{listi}: \textit{Koreni poljubnih stopenj} / 1, ... 10 (izberi tri sode in tri lihe, povsod po en primer) +%\tableofcontents +\begin{enumerate}[label=\textbf{\arabic*.}] + \item Izračunaj brez pomoči kalkulatorja: + \begin{enumerate}[label=\textbf{\xslalph*})] + \setcounter{enumii}{1} + \item $$\sqrt[3]{12,5\cdot10^{-5}}=\sqrt[3]{\frac{25}{2}\cdot\frac{1}{10^5}}=\sqrt[3]{\frac{1}{8000}}=\frac{1}{20}$$ + \end{enumerate} + \setcounter{enumi}{2} + \item Izračunaj in poenostavi: + \begin{enumerate}[label=\textbf{\xslalph*})] + \setcounter{enumii}{4} + \item $$a\cdot\frac{\left(\sqrt[4]{a}+\sqrt[4]{b}\right)^2+\left(\sqrt[4]{a}-\sqrt[4]{b}\right)^2}{a+\sqrt{ab}} + =a\cdot\frac{\sqrt{a}+2\sqrt[4]{ab}+\sqrt{b}+\sqrt{a}-2\sqrt[4]{ab}+\sqrt{b}}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)} + =a\cdot\frac{2\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}=$$$$=\frac{a2}{\sqrt{2}}=2\sqrt{a}$$ + \end{enumerate} + \item Poenostavi: + \begin{enumerate}[label=\textbf{\xslalph*})] + \setcounter{enumii}{7} + \item $$\frac{\sqrt[3]{x^2\cdot\sqrt[4]{xy^3}}}{\sqrt{x\sqrt{xy}}} + =\frac{\sqrt[3]{\sqrt[4]{x^2^4\cdot xy^3}}}{\sqrt{\sqrt{x^2\cdot xy}}} + =\frac{\sqrt[3]{\sqrt[4]{x^9y^3}}}{\sqrt{\sqrt{x^3y}}} + =\frac{\sqrt[12]{\left(x^3y\right)^3}}{\sqrt{\sqrt{x^3y}}} + =\frac{\sqrt[4]{x^3y}}{\sqrt[4]{x^3y}} + =1$$ + \end{enumerate} + \setcounter{enumi}{7} + \item Zaprta kvadratna škatla z merami $\SI{6}{\deci\meter}\times\SI{9}{\deci\meter}\times\SI{30}{\deci\meter}$ ima enako površino kot kocka. Izračunaj dolžino kockinega roba (do milimetra natančno). + $$ + \sqrt{ + \frac{ + 2\left(\SI{30}{\deci\meter}\cdot\SI{6}{\deci\meter}\right)+ + 2\left(\SI{6}{\deci\meter}\cdot\SI{9}{\deci\meter}\right)+ + 2\left(\SI{9}{\deci\meter}\cdot\SI{30}{\deci\meter}\right) + }{6} + } + =\sqrt{\frac{\SI{1008}{\deci\meter\squared}}{6}}=\sqrt{\SI{168}{\deci\meter\squared}}=\SI{12,961481}{\deci\meter} + $$ + \item \begin{enumerate}[label=\textbf{\xslalph*})] + \setcounter{enumii}{1} + \item Za koliko se mora povečati stranica kocke, da se bo njena prostornina s tem povečala za četrtino (prejšnje vrednosti)? + $$ + \left(n+x\right)^2=x^2\cdot\frac{5}{4} \rightarrow n+x=\frac{\sqrt{5}x}{2} \rightarrow n=\frac{\sqrt{5}x}{2}-x + $$ + \end{enumerate} + \item \begin{enumerate}[label=\textbf{\xslalph*})] + \setcounter{enumii}{1} + \item V veliki steklenici je 10 litrov vina. Iz nje izlijemo $x$ litrov vina in nato dolijemo $x$ litrov vode. Zatem ponovno izlijemo iz steklenice $x$ litrov te mešanice in za \say{nadomestek} spet dolijemo $x$ litrov vode. In zatem še enkrat tako. Tedaj je v sodu 5 litrov vina in 5 litrov vode. Izračuaj $x$. \small{\textit{Moj dedek v orehovi lupinici.}} + \def\de{\left(1-x\right)-x\left(\frac{1-x}{1}\right)} + $$ + \left(\de\right)-x\frac{\de}{1}=\frac{1}{2}=1-x-x\left(1-x\right)-x\left(1-x-x\left(1-x\right)\right)=-x^3+3x^2-3x+1= + $$ + $$ + =-\left(x^3-1\right)+3x\left(x-1\right)=-\left(x-1\right)\cdot\left(x^2+x+1\right)+3x\left(x-1\right) + =-\left(x-1\right)\cdot\left(x^2+x+1-3x\right)= + $$ + $$ + -\left(x-1\right)\cdot\left(x^2-2x+1\right)=-\left(x-1\right)\cdot\left(x-1\right)^2=-\left(x-1\right)^3=\frac{1}{2} + \rightarrow \left(x-1\right)^3=\frac{-1}{2} \rightarrow x=-\frac{\sqrt[3]{4}}{2}+1 + $$ + Sedaj pa $x$ za pravilno vrednost še pomnožimo z 10 in dobimo $2,05299$, torej je, ker smo prej namesto \SI{10}{\liter} uporabili 1, rezultat \SI{2,05299}{\liter}. + \end{enumerate} +\end{enumerate} +\section{Zaključek} +Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in +\hologo{LaTeX}\footnote{Za izdelavo dokumenta potrebujete \texttt{TeXLive 2020}.} +izvorna koda, zgodovina sprememb in prejšnje različice, je na voljo v mojem šolskem Git repozitoriju na +\url{https://git.sijanec.eu/sijanec/sola-gimb-2} v mapi +\href{https://git.sijanec/sola-gimb-2/src/branch/master/\predmdn/\predmkaj/\stevilkadn/}{/\predmdn/\predmkaj/\stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/\predmdn/\predmkaj/\stevilkadn/dokument.pdf} in/ali \url{https://git.sijanec.eu/sijanec/sola-gimb-2/raw/branch/master/\predmdn/\predmkaj/\stevilkadn/dokument.pdf}. +\if\razhroscevanje1 +\vfill +\section*{Razhroščevalne informacije} +Te informacije so generirane, ker je omogočeno razhroščevanje. Pred objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroscevanje} na 0 v začetku dokumenta. + +Grafi imajo natančnost \functionSamples\space točk na graf. + +Konec generiranja dokumenta: \today\ ob \currenttime\footnote{To ne nakazuje dejanskega časa, ko je bil dokument napisan, temveč čas, ko je bi dokument generiran v PDF/DVI obliko. Isto velja za datum v glavi dokumenta. Če berete direktno iz LaTeX datoteke, bo to vedno današnji datum.}%\input|"date -Ins" + +Dokument se je generiral R0qK1KR2 \SI{}{\second}. +\fi +% \item $$$$ aaasecgeninsaaa R0qK1KR2 +\end{document} |