Add Exerise 2

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Oystein Kristoffer Tveit 2020-09-14 09:33:21 +02:00
parent 7a888fa983
commit 0d8128a777
9 changed files with 233 additions and 0 deletions

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Exercise 2/figures/3a.tex Normal file
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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}

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Exercise 2/main.pdf Normal file

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Exercise 2/main.tex Normal file
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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}

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Exercise 2/tasks/1.tex Normal file
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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}

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Exercise 2/tasks/2.tex Normal file
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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}

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Exercise 2/tasks/3.tex Normal file
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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}

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Exercise 2/tasks/4.tex Normal file
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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*}

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Exercise 2/tasks/5.tex Normal file
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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*}

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lib/lib.tex Normal file
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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}