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4 changed files with 205 additions and 7 deletions
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@ -2,7 +2,16 @@
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\usepackage{mathtools}
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\usepackage{amssymb}
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\usepackage{pgfplots}
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\usepgfplotslibrary{external}
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\tikzexternalize
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\tikzset{%
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external/system call={pdflatex \tikzexternalcheckshellescape --halt-on-error --interaction=batchmode --output-directory=./build --jobname "\image" "\texsource"},
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/pgf/images/include external/.code={%
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\includegraphics{build/#1}%
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},
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}
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\title{Analyse 1 - TD 1 \\ NON CORRIGÉ}
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\author{Timéo Pochin}
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@ -144,8 +153,8 @@
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -5, xmax = 5,
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ymin = -5, ymax = 5,
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xmin = -4.9, xmax = 4.9,
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ymin = -4.9, ymax = 4.9,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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@ -155,20 +164,209 @@
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth,
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height = \textwidth
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height = \textwidth,
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]
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\addplot[
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color = black
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]{-(1/3)*x+(8/3)};
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]{-(1/3)*x+(8/3)} node[above right,pos=0.83] {$-\frac{1}{3}x+\frac{8}{3}$};
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\addplot[
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color = black
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]{-(5/3)*x+(4/3)};
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]{-(5/3)*x+(4/3)} node[above right,pos=0.85] {$-\frac{5}{3}x+\frac{4}{3}$};
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\addplot[
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color = black
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]{(1/3)*x-(8/3)};
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]{(1/3)*x-(8/3)} node[above left,pos=0.95] {$\frac{1}{3}x-\frac{8}{3}$};
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\addplot[
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color = black
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]{(5/3)*x-(4/3)};
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]{(5/3)*x-(4/3)} node[above left,pos=0.8] {$\frac{5}{3}x-\frac{4}{3}$};
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\end{axis}
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\end{tikzpicture}
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\section*{Exercice 3}
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\subsection*{a)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -4.9, xmax = 4.9,
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ymin = -0.9, ymax = 8.9,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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color = black,
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samples = 1000
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]{abs(x)+abs(2*x-4)};
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\addplot[
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ultra thin,
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color = red,
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samples = 1000
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]{abs(x)};
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\addplot[
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ultra thin,
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color = blue,
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samples = 1000
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]{abs(2*x-4)};
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\end{axis}
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\end{tikzpicture}
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\subsection*{b)}
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L’ensemble $f(\mathbb{R})$ est égal à $\left[2,+\infty\right[$.
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%
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\\
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La fonction $f$ est minorée et elle n’est pas majorée.
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\subsection*{c)}
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Les antécédents par $f$ de $3$ sont $1$ et $\frac{7}{3}$.
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\\
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$1$ n’a pas d’antécédents par $f$.
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\\
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L’antécédent par $f$ de $2$ est $2$.
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\section*{Exercice 4}
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\subsection*{a)}
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\subsubsection*{i)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-1:2
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]{-(0.5*x^2-x-1)};
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\end{axis}
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\end{tikzpicture}
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\subsubsection*{ii)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-2:1
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]{0.5*(-x)^2-(-x)-1};
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\end{axis}
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\end{tikzpicture}
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\subsubsection*{iii)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x+1};
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\end{axis}
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\end{tikzpicture}
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\subsubsection*{iv)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-3:0
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]{0.5*(x+2)^2-(x+2)-1};
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\end{axis}
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\end{tikzpicture}
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