Initial commit - Task 1

This commit is contained in:
Oystein Kristoffer Tveit 2020-08-26 15:50:56 +02:00
commit 84fd77356a
25 changed files with 5335 additions and 0 deletions

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Task 1/figures/5a.tex Normal file
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\begin{tikzpicture}
\begin{axis}[
xlabel = $x$, ylabel = $y$,
width = 10cm, height=10cm,
axis lines = middle,
grid,
xmin=-1.5, xmax=5.5,
ymin=-2.5, ymax=4.5,
legend pos=outer north east
]
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no marks,
domain = 0:4,
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\addlegendentry{$y=x-1$}
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no marks,
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\addlegendentry{$y=3-x$}
\node[label={90:{(2,\ 1)}},circle,fill,inner sep=2pt] at (axis cs:2,1) {};
\end{axis}
\end{tikzpicture}

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\begin{tikzpicture}
\begin{axis}[
xlabel = $x$, ylabel = $y$,
width = 10cm, height=10cm,
axis lines = middle,
grid,
xmin=-1.5, xmax=5.5,
ymin=-2.5, ymax=4.5,
legend pos=outer north east
]
\addplot+ [
no marks,
domain = 0.5:4.5,
]{abs(x-2)+1};
\addlegendentry{$y=|x-2|+1$}
\node[label={270:{(2,\ 1)}},circle,fill,inner sep=2pt] at (axis cs:2,1) {};
\end{axis}
\end{tikzpicture}

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\usepackage{geometry}
\geometry{
a4paper,
left=30mm,
right=30mm,
top=30mm,
bottom=20mm,
}

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\usepackage{fancyhdr}
\usepackage{titling}
\pagestyle{fancy}
\fancyhf{}
\rhead{\theauthor}
\lhead{\thetitle}
\rfoot{Side \thepage}
\renewcommand{\headrulewidth}{1pt}
\renewcommand{\footrulewidth}{1pt}

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Task 1/lib/math.tex Normal file
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\usepackage{pgfplots}
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{
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%--------------------------%
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%--------------------------%
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\usepackage{titling}
\pretitle{
\begin{flushleft}
\LARGE
}
\posttitle{
\par
\end{flushleft}
}
\preauthor{
\begin{flushleft}
\large
}
\postauthor{
\par
\end{flushleft}
}
\predate{
\begin{flushleft}
\large
}
\postdate{
\par
\end{flushleft}
}
%---------------------------------------%
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coltext=white,
arc=0pt,
boxrule=0pt,
parbox=false,
top=5mm,
bottom=5mm
}
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\vspace*{2mm}
\centering
\includegraphics[scale=1.2]{lib/ntnu_alt_versjon_uten_slagord.pdf}
\end{minipage}
\vspace*{2cm}
}

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\BOOKMARK [1][-]{section.1}{Forberedende oppgaver}{}% 1
\BOOKMARK [2][-]{Item.1}{Oppgave 1}{section.1}% 2
\BOOKMARK [3][-]{Item.2}{a\)}{Item.1}% 3
\BOOKMARK [3][-]{Item.3}{b\)}{Item.1}% 4
\BOOKMARK [3][-]{Item.4}{c\)}{Item.1}% 5
\BOOKMARK [2][-]{Item.5}{Oppgave 2}{section.1}% 6
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\documentclass{article}
\usepackage[norsk]{babel}
\usepackage[utf8]{inputenc}
\usepackage{hyperref}
\usepackage{xcolor}
\usepackage[fleqn]{amsmath}
\usepackage[many]{tcolorbox}
\usepackage{graphicx}
\usepackage{parskip}
\usepackage{gensymb}
\definecolor{ntnublue}{RGB}{0,80,158}
\input{lib/geometry.tex}
\input{lib/header.tex}
\input{lib/math.tex}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=magenta,
urlcolor=blue,
}
\pgfplotsset{compat=newest}
\author{Øystein Tveit}
\title{MA0001 Øving 1}
\input{lib/titling.tex}
\begin{document}
\thispagestyle{plain}
\tittel
\tableofcontents
\newpage
\section{Forberedende oppgaver}
\begin{oppgaver}
\oppg
\input{tasks/1.tex}
\oppg
\input{tasks/2.tex}
\end{oppgaver}
\section{Innleveringsoppgaver}
\begin{oppgaver}
\setoppg{2}
\oppg
\input{tasks/3.tex}
\oppg
\input{tasks/4.tex}
\break
\oppg
\input{tasks/5.tex}
\end{oppgaver}
\end{document}

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\babel@toc {norsk}{}
\contentsline {section}{\numberline {1}Forberedende oppgaver}{2}{section.1}%
\contentsline {subsection}{Oppgave 1}{2}{Item.1}%
\contentsline {subsubsection}{a)}{2}{Item.2}%
\contentsline {subsubsection}{b)}{2}{Item.3}%
\contentsline {subsubsection}{c)}{2}{Item.4}%
\contentsline {subsection}{Oppgave 2}{2}{Item.5}%
\contentsline {section}{\numberline {2}Innleveringsoppgaver}{2}{section.2}%
\contentsline {subsection}{Oppgave 3}{2}{Item.6}%
\contentsline {subsubsection}{a)}{2}{Item.7}%
\contentsline {subsubsection}{b)}{2}{Item.8}%
\contentsline {subsubsection}{c)}{3}{Item.9}%
\contentsline {subsubsection}{d)}{3}{Item.10}%
\contentsline {subsubsection}{e)}{3}{Item.11}%
\contentsline {subsection}{Oppgave 4}{3}{Item.12}%
\contentsline {subsubsection}{a)}{3}{Item.13}%
\contentsline {subsubsection}{b)}{3}{Item.14}%
\contentsline {subsubsection}{c)}{4}{Item.15}%
\contentsline {subsubsection}{d)}{4}{Item.16}%
\contentsline {subsection}{Oppgave 5}{6}{Item.17}%
\contentsline {subsubsection}{a)}{6}{Item.18}%
\contentsline {subsubsection}{b)}{6}{Item.19}%

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\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}
\begin{deloppgaver}
\delo
\[ -135\degree \cdot \frac{\pi}{180\degree} \approx -2.35 \]
\delo
\[ 1,5\pi \cdot \frac{180\degree}{\pi} = 270\deg \]
\delo
\[ sin\left( \frac{-\pi}{2} \right) = -1\]
\end{deloppgaver}

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\begin{deloppgaver}
\delo
\begin{alignat*}{4}
& 1^{-1} &&+ 2^{-1} &&+ 3^{-1} \\[1ex]
& \frac{1}{1} &&+ \frac{1}{2} &&+ \frac{1}{3} \\[2ex]
& \frac{6}{6} &&+ \frac{3}{6} &&+ \frac{2}{6} & \qquad \text{Største felles divisor er 6} \\[2ex]
& \frac{11}{6} \\
\end{alignat*}
\delo
\begin{align*}
& x^{-7} x^{6} \\
& x^{-7+6} \\
& x^{-1} \\
& \frac{1}{x}
\end{align*}
\delo
\begin{align*}
& \frac{ \left ( x^2 y z^3 \right )^2 }{ x^4 y^3 z^5 } \\[2ex]
& \frac{ x^4 y^2 z^6 }{ x^4 y^3 z^5 } \\[2ex]
& x^{4-4} y^{2-3} z^{6-5} \\
& x^0 y^{-1} z^1 \\[2ex]
& \frac{z}{y} \\
\end{align*}
\delo
\begin{align*}
& \frac{ e^{z-x} }{ e^x e^z} \\[2ex]
& \frac{ \cancel{e^z} }{ e^{x+x} \cancel{e^z} } \\[2ex]
& \frac{1}{e^{2x}} \\
\end{align*}
\delo
\begin{align*}
& \frac{ z^2 - y^2 }{ z+y } \\[2ex]
& \frac{ (z+y)(z-y) }{ z+y } && \text{3. Kvadratsetning} \\[2ex]
& \frac{ \cancel{(z+y)}(z-y) }{ \cancel{z+y} } \\[2ex]
& z-y
\end{align*}
\end{deloppgaver}

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\begin{deloppgaver}
\delo
\begin{align*}
3+2x &= 2-x\\
\cancel{3} +2x -\cancel{3} +x &= 2 -\cancel{x} -3 +\cancel{x} \\
3x &= -1\\
x &= -\frac{1}{3}\\
\end{align*}
\delo
\begin{align*}
x^2 + x &= 3 \\
x^2 + x - 3 &= 0 \\
\end{align*}
Vi bruker andregradsformelen:
\begin{equation*}
x = \frac{-b \pm \sqrt{ b^2 - 4ac }}{ 2a }
\end{equation*}
\begin{align*}
x &= \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot -3}}{ 2 \cdot 1} \\[2ex]
x &= \frac{-1 \pm \sqrt{1 - (-12)}}{2} \\[2ex]
x &= -\frac{1}{2} \pm \frac{\sqrt{13}}{2}
\end{align*}
\begin{equation*}
x = -\frac{1}{2} + \frac{\sqrt{13}}{2} \quad \vee \quad x = -\frac{1}{2} - \frac{\sqrt{13}}{2}
\end{equation*}
\delo
\begin{align*}
-x \left( x+2 \right) \left( 5x-4 \right) = 0 \\
\end{align*}
Etter nullfaktorregelen må en av faktorene være 0 for at produktet skal bli 0.
\begin{align*}
-x &= 0 &&\vee& x+2 &= 0 &&\vee& 5x-4 &=0 \\
x &= 0 &&\vee& x &= -2 &&\vee& x &= \frac{4}{5} \\
\end{align*}
\delo
\begin{align*}
\frac{x}{x+1} &= \frac{1}{3} + \frac{x-1}{3} \\[2ex]
\frac{x}{x+1} &= \frac{x}{3} \\[2ex]
\frac{3 \cdot x}{3 \cdot (x+1)} &= \frac{(x+1) \cdot x}{(x+1) \cdot 3} \\[2ex]
\frac{3x - x(x+1)}{3(x+1)} &= 0 \\[2ex]
\frac{x(3 - (x+1))}{3(x+1)} &= 0 \\[2ex]
\frac{x(-x+2)}{3(x+1)} &= 0 \\[2ex]
\end{align*}
Telleren må være 0 for at hele uttrykket skal bli 0, men x-verdien er ikke en gyldig løsning når nevner også blir 0. Vi deler opp telleren etter nullfaktorregelen:
\begin{align*}
x &= 0 &&\vee& -x+2 &= 0 \\
x &= 0 &&\vee& x &= 2 \\
\end{align*}
Om vi løser for nevneren:
\begin{equation*}
3(x+1) = 0 \rightarrow x=-1
\end{equation*}
så ser vi at begge løsningene er gyldige.
\end{deloppgaver}

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\begin{deloppgaver}
\delo
\begin{align*}
y&=3-x \\
y&=x-1
\end{align*}
Ettersom utrykkene er rette linjer, vil de ha samme y-verdi på kun ett punkt:
\begin{align*}
3-x &= x-1 \\
2x &= 4 \\
x = 2
\end{align*}
Vi setter x-verdien i ett av uttrykkene og får at
\begin{align*}
y &= 3-2 = 1
\end{align*}
Linjene møtes i $(2, 1)$
Linjene har stigningstallene 1 og -1, og vi kan skissere dem ut ifra skjæringspunktet.
\begin{graphbox}
\input{figures/5a.tex}
\end{graphbox}
\delo
Ettersom absoluttverdien av et uttrykk ikke kan bli mindre enn null, vil $|x-2|+1$ aldri være mindre enn 1. Dette skjer når
\begin{align*}
|x-2| &= 0 \\
x-2 &= \pm 0 \\
x &= 2
\end{align*}
Dermed vender stigningsfarta fra -1 til 1 i punktet $(2, 1)$.
Vi kan skissere grafen fra dette punktet.
\begin{graphbox}
\input{figures/5b.tex}
\end{graphbox}
\end{deloppgaver}