Add Exerise 2

Exercise 2/figures/3a.tex

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+\begin{tikzpicture}
+
+  % Frame
+  \draw [thick, <->] (0,-4) -- (0,0) -- (9,0);
+  
+  % Vertical lines
+  \draw (3,0) node[anchor=south] {1.1} -- (3,-4) ;
+  \draw (6,0) node[anchor=south] {4.5} -- (6,-4) ;
+  
+  % Upper horizontal lines
+  \draw [dashed] (0,-1) node [anchor=east] {$|4.5-c|$} -- (3,-1) node [midway, above, sloped] {$-c$} -- (6,-1);
+  \draw [dashed] (0,-2) node [anchor=east] {$|c-1.1|$} -- (3,-2) node [midway, above, sloped] {$-c$};
+  \draw (6,-1) -- (9,-1) node [midway, above, sloped] {$c$};
+  \draw (3,-2) --(6,-2) node [midway, above, sloped] {$c$} -- (9,-2) ;
+  
+  % Points
+  \node[circle,fill,inner sep=1.5pt] at (6,-1) {};
+  \node[circle,fill,inner sep=1.5pt] at (3,-2) {};
+  \node[circle,fill,inner sep=1.5pt] at (3,-3) {};
+  \node[circle,fill,inner sep=1.5pt] at (6,-3) {};
+  
+  % Lower horizontal line
+  \draw [dashed] (0,-3) node [anchor=east] {$|4.5-c| + |c-1.1|$} -- (3,-3) node [midway, above, sloped] {$-2c$};
+  \draw (3,-3)  -- (6,-3) node [midway, above, sloped] {$0$};
+  \draw (6,-3) -- (9,-3) node [midway, above, sloped] {$2c$};
+  
+\end{tikzpicture}

Exercise 2/main.tex

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+\documentclass{article}
+
+\author{Øystein Tveit}
+\title{MA0001 Øving 2}
+
+\input{../lib/lib.tex}
+
+\begin{document}
+  
+  \thispagestyle{plain}
+  \tittel
+  \tableofcontents
+
+  \newpage
+
+  \section{Forberedende oppgaver}
+  \begin{oppgaver}
+    
+    \oppg
+    \input{tasks/1.tex}
+
+  \end{oppgaver}
+
+  % \newpage
+
+  \section{Innleveringsoppgaver}
+  \begin{oppgaver}
+    \setoppg{1}
+
+    \oppg
+    \input{tasks/2.tex}
+
+    \oppg
+    \input{tasks/3.tex}
+
+    \oppg
+    \input{tasks/4.tex}
+
+    \oppg
+    \input{tasks/5.tex}
+    
+  \end{oppgaver}
+
+\end{document}

Exercise 2/tasks/1.tex

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+\hfill
+
+\begin{tabular}{ l l r }
+  a) \addcontentsline{toc}{subsubsection}{a)} & $\sqrt{-2}$ & udefinert \\
+  b) \addcontentsline{toc}{subsubsection}{b)} & $\sqrt{2}$ & definert \\
+  c) \addcontentsline{toc}{subsubsection}{c)} & $sin\left(-400\right)$ & definert \\
+  d) \addcontentsline{toc}{subsubsection}{d)} & $e^{-3}$ & definert \\
+  e) \addcontentsline{toc}{subsubsection}{e)} & $log_3\left(-9\right)$ & udefinert \\
+  f) \addcontentsline{toc}{subsubsection}{f)} & $log_{-3}\left(9\right)$ & udefinert \\
+  g) \addcontentsline{toc}{subsubsection}{g)} & $log_{-3}\left(-9\right)$ & udefinert \\
+  h) \addcontentsline{toc}{subsubsection}{h)} & $log_3\left(9\right)$ & definert
+\end{tabular}

Exercise 2/tasks/2.tex

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+\begin{deloppgaver}
+
+  \delo
+    \begin{align*}
+      |5-2x|          &<          3 \\
+      -3 < 5-2x       &<          3 \\
+      -3 < 5-2x \quad &\vee \quad 5-2x < 3 \\
+      -8 < 2x   \quad &\vee \quad -2x < -2 \\
+      8 > 2x    \quad &\vee \quad 2x > 2 \\
+      4 > x     \quad &\vee \quad x > 1 \\
+      x               &\in        \left(1,4\right)
+    \end{align*}
+
+  \delo
+    \begin{align*}
+      |x^2 - 3|              &<          6 \\
+      -6 < x^2-3       \quad &\vee \quad x^2-3 < 6 \\
+      -3 < x^2         \quad &\vee \quad x*2 < 9 \\
+      \pm\sqrt{-3} < x \quad &\vee \quad x < \pm 3 \\
+    \end{align*}
+
+    Ettersom $\sqrt{-3}$ er et imaginert tall, er dette ikke et valid skjæringspunkt. Vi tar den ikke med i beregningen.
+
+    \[ x \in (-3,3) \]
+    
+\end{deloppgaver}

Exercise 2/tasks/3.tex

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+\begin{deloppgaver}
+  \delo \label{delo:3a}
+    \begin{align*}
+      |a-b| &= |a-c| +|c-b|
+    \end{align*}
+    
+    Vi substituerer $a=-4.5$ og $b=1.1$
+
+    \begin{align*}
+      | -4.5 - 1.1 | &= | -4.5 - c | +| c - 1.1 | \\
+      5.6 &= | -4.5 - c | + | c - 1.1 |
+    \end{align*}
+
+    Om vi ser på stigningstallene til leddene på høyre side, ser vi at
+
+    \begin{graphbox}
+      \input{figures/3a.tex}
+    \end{graphbox}
+
+    stigningstallet til uttrykket på høyre side er $0$ mellom $c=1.1$ og $c=4.5$.
+
+    Det betyr at 
+
+    \begin{alignat*}{3}
+      & | -4.5 - c | + | c - 1.1 | &&= | -4.5 | + | - 1.1 | && \qquad c \in \left[1.1, 4.5\right] \\
+      &                            &&= 5.6 && \qquad c \in [1.1, 4.5] \\
+    \end{alignat*}
+
+    Alle $c$-verdier mellom $1.1$ og $4.5$ er reelle tall som oppfyller
+    \[ |a-b| = |a-c| + |c-b|, \qquad a=-4.5,\ b=1.1 \] 
+  
+  \delo
+    Fra oppgave \ref{delo:3a} vet vi at $c$ i
+
+    \[ |a-c| + |c-b|\] 
+
+    synker med $-2c$ før $c=1.1$ og øker med $2c$ etter $c=4.5$.
+
+    Ut ifra det kan vi konkludere med at 
+    
+    \[ |a-b| < |a-c| + |c-b|, \qquad a=-4.5,\ b=1.1 \qquad c \in \left(-\infty,1.1\right) \cup \left(4.5,\infty\right)  \] 
+
+\end{deloppgaver}

Exercise 2/tasks/4.tex

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+\hfill
+
+La $ a \neq b $
+
+Stigningstallet $m$ til en rett linje som krysser $(a,b)$ og $(b,a)$ vil være
+
+\[ \frac{\Delta y}{\Delta x}\]
+
+hvor
+
+\begin{align*}
+  \Delta y &= a - b \\
+  \Delta x &= b - a
+\end{align*}
+
+Herifra bruker vi ettpunktsformelen og ett av punktene $(a,b)$
+
+\begin{align*}
+  y - y_1 &= m (x - x_1) \\[2ex]
+  y - b   &= \frac{a - b}{b - a}\left(x - a\right) \\[2ex]
+  y       &= \frac{a - b}{b - a}\left(x - a\right) + b
+\end{align*}
+

Exercise 2/tasks/5.tex

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+\begin{align*}
+  5x + 3y &= -4 \\[1ex]
+  y &= -\frac{5}{3}x - \frac{4}{3}
+\end{align*}
+
+Ettersom
+
+\[ l_1 \perp l_2 \quad \Leftrightarrow \quad m_1 m_2 = -1 \]
+
+vil stigningstallet til den vinkelrette linja være
+
+\begin{align*}
+  -\frac{5}{3} m_2 &= -1 \\[1ex]
+  m_2 &= \frac{-1}{\left(\frac{-5}{3}\right)} \\[1ex]
+      &= \frac{3}{5}
+\end{align*}
+
+Og med ettpunktsformelen og punktet $(0,4)$ vil linja være
+
+\begin{align*}
+  y - y_1 &= m (x - x_1) \\[1ex]
+  y - 4 &= \frac{3}{5}(x - 0) \\[1ex]
+  y &=  \frac{3}{5}x + 4
+\end{align*}
+

lib/lib.tex

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+\usepackage[norsk]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{hyperref}
+\usepackage{xcolor}
+\usepackage[fleqn]{amsmath}
+\usepackage[many]{tcolorbox}
+\usepackage{graphicx}
+\usepackage{textcomp}
+\usepackage{gensymb}
+\usepackage{float}
+
+\definecolor{ntnublue}{RGB}{0,80,158}
+
+\input{../lib/geometry.tex}
+\input{../lib/header.tex}
+\input{../lib/math.tex}
+\usetikzlibrary{angles, quotes}
+
+\hypersetup{
+    colorlinks=true,
+    linkcolor=blue,
+    filecolor=magenta,
+    urlcolor=blue,
+}
+
+\pgfplotsset{compat=newest}
+
+\author{Øystein Tveit}
+\title{MA0001 Øving 2}
+
+\input{../lib/titling.tex}
+
+\setlength{\parindent}{0cm}