updated

analyse_1_td_1.tex

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