Initial commit - Task 1
This commit is contained in:
commit
84fd77356a
26
Task 1/figures/5a.tex
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26
Task 1/figures/5a.tex
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\begin{tikzpicture}
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\begin{axis}[
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xlabel = $x$, ylabel = $y$,
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width = 10cm, height=10cm,
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axis lines = middle,
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grid,
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xmin=-1.5, xmax=5.5,
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ymin=-2.5, ymax=4.5,
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legend pos=outer north east
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]
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\addplot+ [
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no marks,
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domain = 0:4,
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]{x-1};
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\addlegendentry{$y=x-1$}
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\addplot+ [
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no marks,
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domain = 0:4,
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]{3-x};
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\addlegendentry{$y=3-x$}
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\node[label={90:{(2,\ 1)}},circle,fill,inner sep=2pt] at (axis cs:2,1) {};
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\end{axis}
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\end{tikzpicture}
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21
Task 1/figures/5b.tex
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21
Task 1/figures/5b.tex
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\begin{tikzpicture}
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\begin{axis}[
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xlabel = $x$, ylabel = $y$,
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width = 10cm, height=10cm,
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axis lines = middle,
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grid,
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xmin=-1.5, xmax=5.5,
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ymin=-2.5, ymax=4.5,
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legend pos=outer north east
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]
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\addplot+ [
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no marks,
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domain = 0.5:4.5,
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]{abs(x-2)+1};
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\addlegendentry{$y=|x-2|+1$}
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\node[label={270:{(2,\ 1)}},circle,fill,inner sep=2pt] at (axis cs:2,1) {};
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\end{axis}
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\end{tikzpicture}
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9
Task 1/lib/geometry.tex
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9
Task 1/lib/geometry.tex
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\usepackage{geometry}
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\geometry{
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a4paper,
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left=30mm,
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right=30mm,
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top=30mm,
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bottom=20mm,
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}
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11
Task 1/lib/header.tex
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11
Task 1/lib/header.tex
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\usepackage{fancyhdr}
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\usepackage{titling}
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\pagestyle{fancy}
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\fancyhf{}
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\rhead{\theauthor}
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\lhead{\thetitle}
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\rfoot{Side \thepage}
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\renewcommand{\headrulewidth}{1pt}
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\renewcommand{\footrulewidth}{1pt}
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60
Task 1/lib/math.tex
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60
Task 1/lib/math.tex
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\usepackage{cancel}
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\usepackage{tikz}
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\usepackage{pgfplots}
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\usepackage{enumitem}
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%--------------------------%
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\renewcommand{\theenumi}{\arabic{enumi}}
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\renewcommand{\labelenumi}{\Large\fbox{\theenumi}}
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\newenvironment{oppgaver}{
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\begin{enumerate}
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}
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{
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\end{enumerate}
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}
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\newcommand{\setoppg}[1]{\setcounter{enumi}{#1}}
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\newcommand{\oppg}{
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\item
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\addcontentsline{toc}{subsection}{Oppgave \theenumi}
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}
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%--------------------------%
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\renewcommand{\theenumii}{\alph{enumii})}
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\renewcommand{\labelenumii}{\textbf{\theenumii}}
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\newenvironment{deloppgaver}{
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\begin{enumerate}
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}{
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\end{enumerate}
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}
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\newcommand{\setdelo}[1]{\setcounter{enumii}{#1}}
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\newcommand{\delo}{
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\vspace*{0.5cm}
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\item
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\addcontentsline{toc}{subsubsection}{\theenumii}
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}
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%--------------------------%
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\newtcolorbox{coloredgraphbox}{
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colframe=ntnublue,
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arc=0pt,
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outer arc=0pt
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colback=white,
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boxrule=3pt
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}
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\newenvironment{graphbox}{
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\begin{center}
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\begin{coloredgraphbox}
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}{
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\end{coloredgraphbox}
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\end{center}
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}
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3261
Task 1/lib/ntnu_alt_versjon_uten_slagord.pdf
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3261
Task 1/lib/ntnu_alt_versjon_uten_slagord.pdf
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File diff suppressed because one or more lines are too long
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Task 1/lib/ntnu_logo-eps-converted-to.pdf
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Task 1/lib/ntnu_logo-eps-converted-to.pdf
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Binary file not shown.
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Task 1/lib/ntnu_logo.eps
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Task 1/lib/ntnu_logo.eps
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%!PS-Adobe-3.0 EPSF-3.0
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%%Creator: cairo 1.17.3 (https://cairographics.org)
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%%CreationDate: Thu Aug 20 19:13:37 2020
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%%Pages: 1
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%%DocumentData: Clean7Bit
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%%LanguageLevel: 2
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%%BoundingBox: 0 0 45 60
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%%EndComments
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%%BeginProlog
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50 dict begin
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/q { gsave } bind def
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/Q { grestore } bind def
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/cm { 6 array astore concat } bind def
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/w { setlinewidth } bind def
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/J { setlinecap } bind def
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/j { setlinejoin } bind def
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/M { setmiterlimit } bind def
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/d { setdash } bind def
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/m { moveto } bind def
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/l { lineto } bind def
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/c { curveto } bind def
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/h { closepath } bind def
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/re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto
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0 exch rlineto 0 rlineto closepath } bind def
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/S { stroke } bind def
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/f { fill } bind def
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/f* { eofill } bind def
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/n { newpath } bind def
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/W { clip } bind def
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/W* { eoclip } bind def
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/BT { } bind def
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/ET { } bind def
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/BDC { mark 3 1 roll /BDC pdfmark } bind def
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/EMC { mark /EMC pdfmark } bind def
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/cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def
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/Tj { show currentpoint cairo_store_point } bind def
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/TJ {
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{
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dup
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type /stringtype eq
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{ show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse
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} forall
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currentpoint cairo_store_point
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} bind def
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/cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore
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cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def
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/Tf { pop /cairo_font exch def /cairo_font_matrix where
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{ pop cairo_selectfont } if } bind def
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/Td { matrix translate cairo_font_matrix matrix concatmatrix dup
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/cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point
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/cairo_font where { pop cairo_selectfont } if } bind def
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/Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def
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cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def
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/g { setgray } bind def
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/rg { setrgbcolor } bind def
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/d1 { setcachedevice } bind def
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/cairo_data_source {
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CairoDataIndex CairoData length lt
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{ CairoData CairoDataIndex get /CairoDataIndex CairoDataIndex 1 add def }
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{ () } ifelse
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} def
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/cairo_flush_ascii85_file { cairo_ascii85_file status { cairo_ascii85_file flushfile } if } def
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/cairo_image { image cairo_flush_ascii85_file } def
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/cairo_imagemask { imagemask cairo_flush_ascii85_file } def
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%%EndProlog
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%%BeginSetup
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%%EndSetup
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%%Page: 1 1
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%%BeginPageSetup
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%%PageBoundingBox: 0 0 45 60
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%%EndPageSetup
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q 0 0 45 60 rectclip
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1 0 0 -1 0 60 cm q
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Q q
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q
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0 0 45 60 re W n
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[ 1 0 0 1 0 0 ] concat
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q
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0.129412 0.298039 0.631373 rg
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11.023 11.023 m 33.078 11.023 l 33.078 33.082 l 11.023 33.082 l h
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33.078 0 m 11.023 0 l 4.934 0 0 4.934 0 11.023 c 0 33.078 l 0 39.168 4.934
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44.102 11.023 44.102 c 33.078 44.102 l 39.168 44.102 44.102 39.168 44.102
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33.078 c 44.102 11.023 l 44.102 4.934 39.168 0 33.078 0 c f
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21.977 12.93 m 16.98 12.93 12.934 16.98 12.934 21.977 c 12.934 26.973 16.98
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31.023 21.977 31.023 c 26.977 31.023 31.023 26.973 31.023 21.977 c 31.023
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16.98 26.977 12.93 21.977 12.93 c f
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0.129412 0.054902 0.0313726 rg
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||||||
|
4.801 49.043 m 9.617 56.047 l 9.617 50.66 l 9.602 49.379 l 8.156 49.379
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l 8.156 49.043 l 11.258 49.043 l 11.258 49.379 l 10.062 49.379 l 10.047
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50.66 l 10.047 59 l 9.828 59.07 l 3.367 49.672 l 3.367 57.219 l 3.379 58.539
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l 4.871 58.539 l 4.871 58.875 l 1.656 58.875 l 1.656 58.539 l 2.938 58.539
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l 2.949 57.219 l 2.949 50.66 l 2.938 49.379 l 1.656 49.379 l 1.656 49.043
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l h
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4.801 49.043 m f
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20.867 49.043 m 20.867 52.191 l 20.449 52.191 l 20.406 51.703 20.352 51.094
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19.918 50.367 c 19.418 49.52 18.738 49.477 18.066 49.449 c 17.289 49.449
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l 17.273 50.699 l 17.273 57.121 l 17.289 58.539 l 18.793 58.539 l 18.793
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58.875 l 14.227 58.875 l 14.227 58.539 l 15.73 58.539 l 15.742 57.121 l
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15.742 50.699 l 15.73 49.449 l 15.09 49.449 l 14.281 49.461 13.668 49.461
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13.082 50.395 c 12.625 51.133 12.582 51.719 12.555 52.191 c 12.152 52.191
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l 12.152 49.043 l 20.867 49.043 l f
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24.82 49.043 m 29.637 56.047 l 29.637 50.66 l 29.621 49.379 l 28.176 49.379
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l 28.176 49.043 l 31.281 49.043 l 31.281 49.379 l 30.082 49.379 l 30.066
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50.66 l 30.066 59 l 29.844 59.07 l 23.387 49.672 l 23.387 57.219 l 23.402
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58.539 l 24.891 58.539 l 24.891 58.875 l 21.676 58.875 l 21.676 58.539
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l 22.953 58.539 l 22.969 57.219 l 22.969 50.66 l 22.953 49.379 l 21.676
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49.379 l 21.676 49.043 l h
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24.82 49.043 m f
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36.684 49.043 m 36.684 49.379 l 35.305 49.379 l 35.293 50.562 l 35.293
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55.879 l 35.305 57.258 35.336 58.566 37.926 58.566 c 40.543 58.566 40.707
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57.426 40.75 56.188 c 40.75 50.562 l 40.738 49.379 l 39.332 49.379 l 39.332
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49.043 l 42.406 49.043 l 42.406 49.379 l 41.195 49.379 l 41.184 50.562
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l 41.184 55.629 l 41.168 57.453 41.152 59.012 37.547 59.012 c 37.031 59.012
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35.809 58.957 35.027 58.637 c 33.789 58.109 33.762 57.023 33.746 55.688
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c 33.746 50.562 l 33.734 49.379 l 32.539 49.379 l 32.539 49.043 l 36.684
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49.043 l f
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Q
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Q
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Q Q
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showpage
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%%Trailer
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end
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%%EOF
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61
Task 1/lib/titling.tex
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61
Task 1/lib/titling.tex
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\usepackage{titling}
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\pretitle{
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\begin{flushleft}
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\LARGE
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}
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\posttitle{
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\par
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\end{flushleft}
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}
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\preauthor{
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\begin{flushleft}
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\large
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}
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\postauthor{
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\par
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\end{flushleft}
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}
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\predate{
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\begin{flushleft}
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\large
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}
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\postdate{
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\par
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\end{flushleft}
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}
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%---------------------------------------%
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\newtcolorbox{headerbox}{
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% tikznode,
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% tcbox width=auto limited,
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% capture=hbox,
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enhanced,
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colback=ntnublue,
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coltext=white,
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arc=0pt,
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boxrule=0pt,
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parbox=false,
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top=5mm,
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bottom=5mm
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|
}
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\newcommand{\tittel}{
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||||||
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\begin{minipage}{0.80\textwidth}
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\begin{headerbox}
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\Huge\thetitle
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\LARGE\theauthor
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\end{headerbox}
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\end{minipage}
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\hfill
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\begin{minipage}{0.10\textwidth}
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\vspace*{2mm}
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||||||
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\centering
|
||||||
|
\includegraphics[scale=1.2]{lib/ntnu_alt_versjon_uten_slagord.pdf}
|
||||||
|
\end{minipage}
|
||||||
|
\vspace*{2cm}
|
||||||
|
}
|
||||||
42
Task 1/main.aux
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42
Task 1/main.aux
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\relax
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\providecommand\hyper@newdestlabel[2]{}
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\@nameuse{bbl@beforestart}
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\catcode `"\active
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||||||
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\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
|
||||||
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\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
|
||||||
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\global\let\oldcontentsline\contentsline
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||||||
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\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
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\global\let\oldnewlabel\newlabel
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\gdef\newlabel#1#2{\newlabelxx{#1}#2}
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\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
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\AtEndDocument{\ifx\hyper@anchor\@undefined
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\let\contentsline\oldcontentsline
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\let\newlabel\oldnewlabel
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|
\fi}
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|
\fi}
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\global\let\hyper@last\relax
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\gdef\HyperFirstAtBeginDocument#1{#1}
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\providecommand\HyField@AuxAddToFields[1]{}
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\providecommand\HyField@AuxAddToCoFields[2]{}
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\babel@aux{norsk}{}
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\@writefile{toc}{\contentsline {section}{\numberline {1}Forberedende oppgaver}{2}{section.1}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{Oppgave 1}{2}{Item.1}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{a)}{2}{Item.2}\protected@file@percent }
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224
Task 1/main.fdb_latexmk
Normal file
224
Task 1/main.fdb_latexmk
Normal file
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||||||
|
\oppg
|
||||||
|
\input{tasks/3.tex}
|
||||||
|
|
||||||
|
\oppg
|
||||||
|
\input{tasks/4.tex}
|
||||||
|
|
||||||
|
\break
|
||||||
|
|
||||||
|
\oppg
|
||||||
|
\input{tasks/5.tex}
|
||||||
|
|
||||||
|
\end{oppgaver}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\end{document}
|
||||||
22
Task 1/main.toc
Normal file
22
Task 1/main.toc
Normal file
@ -0,0 +1,22 @@
|
|||||||
|
\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}%
|
||||||
14
Task 1/tasks/1.tex
Normal file
14
Task 1/tasks/1.tex
Normal file
@ -0,0 +1,14 @@
|
|||||||
|
|
||||||
|
\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}
|
||||||
0
Task 1/tasks/2.tex
Normal file
0
Task 1/tasks/2.tex
Normal file
43
Task 1/tasks/3.tex
Normal file
43
Task 1/tasks/3.tex
Normal file
@ -0,0 +1,43 @@
|
|||||||
|
\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}
|
||||||
20
Task 1/tasks/3.tex.aux
Normal file
20
Task 1/tasks/3.tex.aux
Normal file
@ -0,0 +1,20 @@
|
|||||||
|
\relax
|
||||||
|
\@setckpt{tasks/3.tex}{
|
||||||
|
\setcounter{page}{3}
|
||||||
|
\setcounter{equation}{0}
|
||||||
|
\setcounter{enumi}{3}
|
||||||
|
\setcounter{enumii}{0}
|
||||||
|
\setcounter{enumiii}{0}
|
||||||
|
\setcounter{enumiv}{0}
|
||||||
|
\setcounter{footnote}{0}
|
||||||
|
\setcounter{mpfootnote}{0}
|
||||||
|
\setcounter{part}{0}
|
||||||
|
\setcounter{section}{0}
|
||||||
|
\setcounter{subsection}{0}
|
||||||
|
\setcounter{subsubsection}{0}
|
||||||
|
\setcounter{paragraph}{0}
|
||||||
|
\setcounter{subparagraph}{0}
|
||||||
|
\setcounter{figure}{0}
|
||||||
|
\setcounter{table}{0}
|
||||||
|
\setcounter{parentequation}{0}
|
||||||
|
}
|
||||||
70
Task 1/tasks/4.tex
Normal file
70
Task 1/tasks/4.tex
Normal file
@ -0,0 +1,70 @@
|
|||||||
|
\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}
|
||||||
20
Task 1/tasks/4.tex.aux
Normal file
20
Task 1/tasks/4.tex.aux
Normal file
@ -0,0 +1,20 @@
|
|||||||
|
\relax
|
||||||
|
\@setckpt{tasks/4.tex}{
|
||||||
|
\setcounter{page}{4}
|
||||||
|
\setcounter{equation}{0}
|
||||||
|
\setcounter{enumi}{4}
|
||||||
|
\setcounter{enumii}{0}
|
||||||
|
\setcounter{enumiii}{0}
|
||||||
|
\setcounter{enumiv}{0}
|
||||||
|
\setcounter{footnote}{0}
|
||||||
|
\setcounter{mpfootnote}{0}
|
||||||
|
\setcounter{part}{0}
|
||||||
|
\setcounter{section}{0}
|
||||||
|
\setcounter{subsection}{0}
|
||||||
|
\setcounter{subsubsection}{0}
|
||||||
|
\setcounter{paragraph}{0}
|
||||||
|
\setcounter{subparagraph}{0}
|
||||||
|
\setcounter{figure}{0}
|
||||||
|
\setcounter{table}{0}
|
||||||
|
\setcounter{parentequation}{0}
|
||||||
|
}
|
||||||
47
Task 1/tasks/5.tex
Normal file
47
Task 1/tasks/5.tex
Normal file
@ -0,0 +1,47 @@
|
|||||||
|
\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}
|
||||||
Loading…
Reference in New Issue
Block a user