From 0d8128a77764e875bf9ea8ebfca909b6ab9d0bdc Mon Sep 17 00:00:00 2001 From: h7x4 Date: Mon, 14 Sep 2020 09:33:21 +0200 Subject: [PATCH] Add Exerise 2 --- Exercise 2/figures/3a.tex | 27 +++++++++++++++++++++++ Exercise 2/main.pdf | Bin 0 -> 834224 bytes Exercise 2/main.tex | 44 ++++++++++++++++++++++++++++++++++++++ Exercise 2/tasks/1.tex | 12 +++++++++++ Exercise 2/tasks/2.tex | 26 ++++++++++++++++++++++ Exercise 2/tasks/3.tex | 43 +++++++++++++++++++++++++++++++++++++ Exercise 2/tasks/4.tex | 23 ++++++++++++++++++++ Exercise 2/tasks/5.tex | 25 ++++++++++++++++++++++ lib/lib.tex | 33 ++++++++++++++++++++++++++++ 9 files changed, 233 insertions(+) create mode 100644 Exercise 2/figures/3a.tex create mode 100644 Exercise 2/main.pdf create mode 100644 Exercise 2/main.tex create mode 100644 Exercise 2/tasks/1.tex create mode 100644 Exercise 2/tasks/2.tex create mode 100644 Exercise 2/tasks/3.tex create mode 100644 Exercise 2/tasks/4.tex create mode 100644 Exercise 2/tasks/5.tex create mode 100644 lib/lib.tex diff --git a/Exercise 2/figures/3a.tex b/Exercise 2/figures/3a.tex new file mode 100644 index 0000000..5a39279 --- /dev/null +++ b/Exercise 2/figures/3a.tex @@ -0,0 +1,27 @@ +\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 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+\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} \ No newline at end of file diff --git a/Exercise 2/tasks/1.tex b/Exercise 2/tasks/1.tex new file mode 100644 index 0000000..03309b8 --- /dev/null +++ b/Exercise 2/tasks/1.tex @@ -0,0 +1,12 @@ +\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} \ No newline at end of file diff --git a/Exercise 2/tasks/2.tex b/Exercise 2/tasks/2.tex new file mode 100644 index 0000000..1c650a4 --- /dev/null +++ b/Exercise 2/tasks/2.tex @@ -0,0 +1,26 @@ +\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} \ No newline at end of file diff --git a/Exercise 2/tasks/3.tex b/Exercise 2/tasks/3.tex new file mode 100644 index 0000000..46bf2a5 --- /dev/null +++ b/Exercise 2/tasks/3.tex @@ -0,0 +1,43 @@ +\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} \ No newline at end of file diff --git a/Exercise 2/tasks/4.tex b/Exercise 2/tasks/4.tex new file mode 100644 index 0000000..d8e37bf --- /dev/null +++ b/Exercise 2/tasks/4.tex @@ -0,0 +1,23 @@ +\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*} + diff --git a/Exercise 2/tasks/5.tex b/Exercise 2/tasks/5.tex new file mode 100644 index 0000000..d092236 --- /dev/null +++ b/Exercise 2/tasks/5.tex @@ -0,0 +1,25 @@ +\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*} + diff --git a/lib/lib.tex b/lib/lib.tex new file mode 100644 index 0000000..d5627f3 --- /dev/null +++ b/lib/lib.tex @@ -0,0 +1,33 @@ +\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} \ No newline at end of file