Feat: fin des exercices sur la tangente pour les 1ST

1ST/03_Nombre_derive_et_tangente/exercises.tex

@@ -31,7 +31,6 @@
         Rapport (vertical sur horizontal) & & & & \\
         \hline
     \end{tabular}
-
 \end{annexe}
 
 
@@ -170,6 +169,12 @@
 \begin{exercise}[subtitle={Tracer des tangentes}, step={2}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}]
     Tracer les tangentes aux points marqués sur les graphiques
 
+    \pgfkeys{tikz/.cd}
+    \tikzset{tangent/.style={black,thick},
+        tangent at/.style={postaction={decorate,decoration={markings,
+                    mark=at position #1 with {\fill[tangent] (axis direction cs:0,0) circle (2pt);}
+        }}},
+    }
     \begin{minipage}{0.5\linewidth}
         \begin{tikzpicture}
             \begin{axis}[
@@ -182,10 +187,7 @@
                 ymin = 0,
                 ymax = 11,
                 ]
-                \addplot[domain=0:5,samples=20, color=red, very thick]{(x-3)^2 + 1};
+                \addplot[tangent at/.list={0.29,0.645,0.795},domain=0:5,samples=20, color=red, very thick]{(x-3)^2 + 1};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(3,1)};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(1,5)};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(4,2)};
             \end{axis}
         \end{tikzpicture}
     \end{minipage}
@@ -198,18 +200,56 @@
                 xtick distance=1,
                 ylabel = {$f(x)$},
                 ytick distance=1,
-                ymin = -6,
-                ymax = 6,
                 ]
-                \addplot[domain=-2:2,samples=50, color=red, very thick]{sin(deg(x*pi/2))*5};
+                \addplot[domain=-2:2,samples=50, color=red, very thick,tangent at/.list={0.25,0.5,0.865}]{sin(deg(x*pi/2))*5};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5*sin(deg(1.5*pi/2)))};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)};
             \end{axis}
         \end{tikzpicture}
     \end{minipage}
 \end{exercise}
 
+\begin{solution}
+    \pgfkeys{tikz/.cd,
+        tangent length/.store in=\TangentLength,
+        tangent length=30mm
+    }
+    \tikzset{tangent/.style={black,thick},
+        tangent at/.style={postaction={decorate,decoration={markings,
+                    mark=at position #1 with {\draw[tangent] (axis direction cs:-\TangentLength,0) -- (axis direction cs:\TangentLength,0);
+        \fill[tangent] (axis direction cs:0,0) circle (2pt);}}}},
+    }
+    \begin{minipage}{0.5\linewidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                axis lines = center,
+                grid= both,
+                xlabel = {$x$},
+                xtick distance=1,
+                ylabel = {$f(x)$},
+                ytick distance=1,
+                ymin = 0,
+                ymax = 11,
+                ]
+                \addplot[tangent at/.list={0.29,0.64,0.795},domain=0:5,samples=20, color=red, very thick]{(x-3)^2 + 1};
+            \end{axis}
+        \end{tikzpicture}
+    \end{minipage}
+    \begin{minipage}{0.5\linewidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                axis lines = center,
+                grid= both,
+                xlabel = {$x$},
+                xtick distance=1,
+                ylabel = {$f(x)$},
+                ytick distance=1,
+                ymin=-6, ymax=6,
+                ]
+                \addplot[domain=-2:2,samples=50, color=red, very thick,tangent at/.list={0.24715,0.5,0.865}]{sin(deg(x*pi/2))*5};
+            \end{axis}
+        \end{tikzpicture}
+    \end{minipage}
+\end{solution}
+
 
 \begin{exercise}[subtitle={Tracer une courbe}, step={2}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}]
     \begin{multicols}{2}
@@ -227,11 +267,7 @@
                         ymin = -6,
                         ymax = 6,
                         ]
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3)};
+                        \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3) (-1,-5) (0,0) (1.5,5) (2,2)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(2,2)};
                     \end{axis}
                 \end{tikzpicture}
 
@@ -248,15 +284,11 @@
                         ymin = -6,
                         ymax = 6,
                         ]
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3)};
+                        \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3) (-1,-5) (0,0) (1.5,5) (2,2)};
                         \addplot [mark=, very thick] coordinates {(-2,-3) (-1.8, -3.5)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-5)};
                         \addplot [mark=, very thick] coordinates {(-1.2,-5) (-0.8, -5)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)};
                         \addplot [mark=, very thick] coordinates {(-0.2,0) (0.2, 0)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,5)};
                         \addplot [mark=, very thick] coordinates {(1.3, 4.8) (1.7, 5.2)};
-                        \addplot [black, mark=*, very thick, only marks] coordinates {(2,2)};
                         \addplot [mark=, very thick] coordinates {(1.8, 2) (2, 2)};
                     \end{axis}
                 \end{tikzpicture}
@@ -319,13 +351,118 @@
     \end{enumerate}
 \end{exercise}
 
+\begin{solution}
+    \begin{multicols}{2}
+        \begin{enumerate}
+            \item
+
+                \begin{tikzpicture}
+                    \begin{axis}[
+                        axis lines = center,
+                        grid= both,
+                        xlabel = {$x$},
+                        xtick distance=1,
+                        ylabel = {$f(x)$},
+                        ytick distance=1,
+                        ymin = -6,
+                        ymax = 6,
+                        ]
+                        \addplot [black, mark=*, very thick] coordinates {(-2,-3) (-1,-5) (0,0) (1.5,5) (2,2)};
+                    \end{axis}
+                \end{tikzpicture}
+
+            \item
+
+                \begin{tikzpicture}
+                    \begin{axis}[
+                        axis lines = center,
+                        grid= both,
+                        xlabel = {$x$},
+                        xtick distance=1,
+                        ylabel = {$f(x)$},
+                        ytick distance=1,
+                        ymin = -6,
+                        ymax = 6,
+                        ]
+                        \addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3) (-1,-5) (0,0) (1.5,5) (2,2)};
+                        \addplot [mark=, very thick] coordinates {(-2,-3) (-1.8, -3.5)};
+                        \addplot [mark=, very thick] coordinates {(-1.2,-5) (-0.8, -5)};
+                        \addplot [mark=, very thick] coordinates {(-0.2,0) (0.2, 0)};
+                        \addplot [mark=, very thick] coordinates {(1.3, 4.8) (1.7, 5.2)};
+                        \addplot [mark=, very thick] coordinates {(1.8, 2) (2, 2)};
+                    \end{axis}
+                \end{tikzpicture}
+        \end{enumerate}
+    \end{multicols}
+    \begin{enumerate}
+        \setcounter{enumi}{2}
+        \item
+            Tracer une courbe qui respecte les points et les tangentes représentées dans les graphiques suivants.
+
+            \pgfkeys{tikz/.cd,
+                tangent length/.store in=\TangentLength,
+                tangent length=7mm
+            }
+            \tikzset{tangent/.style={black,thick},
+                tangent at/.style={postaction={decorate,decoration={markings,
+                            mark=at position #1 with {\draw[tangent] (-\TangentLength,0) -- (\TangentLength,0);
+                \fill[tangent] (0,0) circle (2pt);}}}},
+            }
+
+            \begin{tikzpicture}[scale=1]
+                % Axes
+                \draw [-latex] (-0.5,0) -- (8,0) node [above] {$x$};
+                \draw [-latex] (0,-0.5) -- (0,4) node [right] {$y$};
+                % Origin
+                \node at (0,0) [below left] {$0$};
+                % Points
+                \coordinate (start) at (0,-0.8);
+                \coordinate (c1) at (3,3);
+                \coordinate (c2) at (5,1.5);
+                \coordinate (c3) at (6,4);
+                \coordinate (end) at (8,2);
+                % show the points
+                % \foreach \n in {start,c1,c2,c3,end} \fill [black] (\n)
+                %     circle (2pt) node [below] {};
+                % join the coordinates
+                \draw [tangent at/.list={0.15,0.3,...,1}] (start) to[out=70,in=180] (c1) to[out=0,in=180]
+                (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);
+            \end{tikzpicture}
+            \hfill
+            \begin{tikzpicture}[scale=1]
+                % Axes
+                \draw [-latex] (-4,0) -- (4,0) node [above] {$x$};
+                \draw [-latex] (0,-3) -- (0,3) node [right] {$y$};
+                % Origin
+                \node at (0,0) [below left] {$0$};
+                % Points
+                \coordinate (start) at (-4,-1);
+                \coordinate (c1) at (-2,3);
+                \coordinate (c2) at (0,1);
+                \coordinate (c3) at (2,-2);
+                \coordinate (end) at (4,0);
+                % show the points
+                % \foreach \n in {start,c1,c2,c3,end} \fill [black] (\n)
+                %     circle (2pt) node [below] {};
+                % join the coordinates
+                \draw [tangent at/.list={0.2,0.4,...,1}] (start) to[out=70,in=180] (c1) to[out=0,in=180]
+                (c2) to[out=0,in=180] (c3) to[out=0,in=150] (end);
+            \end{tikzpicture}
+    \end{enumerate}
+\end{solution}
+
 % Nombre dérivé et tangente
 
 
 \begin{exercise}[subtitle={Lire le nombre dérivé}, step={3}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}]
     Sur les courbes suivantes, tracer les tangentes aux points puis lire graphiquement le nombre dérivé.
-    Tracer les tangentes aux points marqués sur les graphiques
 
+    \pgfkeys{tikz/.cd}
+    \tikzset{tangent/.style={black,thick},
+        tangent at/.style={postaction={decorate,decoration={markings,
+                    mark=at position #1 with {\fill[tangent] (axis direction cs:0,0) circle (2pt);}
+        }}},
+    }
     \begin{minipage}{0.5\linewidth}
         \begin{tikzpicture}
             \begin{axis}[
@@ -335,33 +472,115 @@
                 xtick distance=1,
                 ylabel = {$f(x)$},
                 ytick distance=1,
+                ymin=-2,
                 ]
-                \addplot[domain=-4:2,samples=20, color=red, very thick]{0.1*(x+1)^3 + 1};
+                \addplot[tangent at/.list={0,0.29,0.78},domain=-4:4,samples=20, color=red, very thick]{0.1*(x+1)^3 + 1};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(1,1)};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(1,5)};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(4,2)};
             \end{axis}
         \end{tikzpicture}
     \end{minipage}
     \begin{minipage}{0.5\linewidth}
-        \begin{tikzpicture}
+        \pgfkeys{tikz/.cd}
-            \begin{axis}[
+        \tikzset{tangent/.style={black,thick},
-                axis lines = center,
+            tangent at/.style={postaction={decorate,decoration={markings,
-                grid= both,
+                        mark=at position #1 with {\fill[tangent] (0,0) circle (3pt);}
-                xlabel = {$x$},
+            }}},
-                xtick distance=1,
+        }
-                ylabel = {$f(x)$},
+        \begin{tikzpicture}[xscale=0.5,yscale=0.8]
-                ytick distance=1,
+                % Axes
-                ]
+                \draw [-latex, thick] (-0.5,0) -- (15,0) node [above] {$x$};
-                \addplot[domain=-2:2,samples=50, color=red, very thick]{sin(deg(x*pi/2))*2};
+                \draw [-latex, thick] (0,-3.5) -- (0,5) node [left] {$f(x)$};
-                \addplot [black, mark=*, very thick, only marks] coordinates {(-1,-2)};
+                \draw [very thin] (0,-3) grid (15,5);
-                \addplot [black, mark=*, very thick, only marks] coordinates {(1.5,2*sin(deg(1.5*pi/2)))};
+                % Origin
-                \addplot [black, mark=*, very thick, only marks] coordinates {(0,0)};
+                \node at (0,0) [below left] {$0$};
-            \end{axis}
+                % Points
+                \coordinate (start) at (0,-3);
+                \coordinate (c1) at (3,0);
+                \coordinate (c2) at (7,4);
+                \coordinate (c3) at (11,1);
+                \coordinate (end) at (15,5);
+
+                \draw [tangent at/.list={0.21,0.49,0.72,1}] (start) to [out=0,in=225] (c1) to[out=45,in=180] (c2) to[out=0,in=180] (c3) to[out=0,in=225] (end);
         \end{tikzpicture}
     \end{minipage}
 
 \end{exercise}
 
+\begin{solution}
+    \pgfkeys{tikz/.cd}
+    \tikzset{tangent/.style={black,thick},
+        tangent at/.style={postaction={decorate,decoration={markings,
+                    mark=at position #1 with {\fill[tangent] (axis direction cs:0,0) circle (2pt);}
+        }}},
+    }
+    \begin{minipage}{0.5\linewidth}
+        \begin{tikzpicture}
+            \begin{axis}[
+                axis lines = center,
+                grid= both,
+                xlabel = {$x$},
+                xtick distance=1,
+                ylabel = {$f(x)$},
+                ytick distance=1,
+                ymin=-2,
+                ]
+                \addplot[tangent at/.list={0,0.29,0.78},domain=-4:4,samples=20, color=red, very thick]{0.1*(x+1)^3 + 1};
+            \end{axis}
+        \end{tikzpicture}
+    \end{minipage}
+    \begin{minipage}{0.5\linewidth}
+        \pgfkeys{tikz/.cd}
+        \tikzset{tangent/.style={black,thick},
+            tangent at/.style={postaction={decorate,decoration={markings,
+                        mark=at position #1 with {\fill[tangent] (axis direction cs:0,0) circle (2pt);}
+            }}},
+        }
+        \begin{tikzpicture}[xscale=0.5,yscale=0.8]
+            % Axes
+            \draw [-latex, thick] (-0.5,0) -- (15,0) node [above] {$x$};
+            \draw [-latex, thick] (0,-3.5) -- (0,5) node [left] {$f(x)$};
+            \draw [very thin] (0,-3) grid (15,5);
+            % Origin
+            \node at (0,0) [below left] {$0$};
+            % Points
+            \coordinate (start) at (0,-3);
+            \coordinate (c1) at (3,0);
+            \coordinate (c2) at (7,4);
+            \coordinate (c3) at (11,1);
+            \coordinate (end) at (15,5);
+
+            \draw [tangent at/.list={0.21,0.49,0.72,1}] (start) to [out=0,in=225] (c1) to[out=45,in=180] (c2) to[out=0,in=180] (c3) to[out=0,in=225] (end);
+        \end{tikzpicture}
+    \end{minipage}
+
+\end{solution}
+
 \begin{exercise}[subtitle={Tracer la courbe avec les nombres dérivés}, step={3}, origin={ma tête}, topics={ Nombre dérivé et tangente }, tags={ Dérivation }, mode={\trainMode}]
+    Pour chacun des tableaux ci-dessous, placer les points, puis tracer les tangentes et enfin tracer une courbe qui respecte les points et les tangentes.
+    \begin{multicols}{2}
+        \begin{enumerate}
+            \item
+                \begin{tabular}{|c|*{5}{c|}}
+                    \hline
+                    $x$ & -2 & -1 & 0 & 1 & 2\\
+                    \hline
+                    $f(x)$ & 3 & 1 & -1 & -3 & 1\\
+                    \hline
+                    $f'(x)$ & 0 & -1 & -1 & 0 & 0\\
+                    \hline
+                \end{tabular}
+            \item
+                \begin{tabular}{|c|*{5}{c|}}
+                    \hline
+                    $x$ & -2 & -1 & 0 & 1 & 2\\
+                    \hline
+                    $g(x)$ & 0 & 2 & 4 & 0 & -3\\
+                    \hline
+                    $g'(x)$ & 2 & 1 & 0 & -2 & 0\\
+                    \hline
+                \end{tabular}
+        \end{enumerate}
+    \end{multicols}
+    \begin{enumerate}
+        \item (*) Que peut-on des points où le nombre dérivé est nul?
+    \end{enumerate}
 \end{exercise}

1ST/03_Nombre_derive_et_tangente/plan_de_travail.tex

@@ -2,6 +2,7 @@
 \usepackage{myXsim}
 \usepackage{pgfplots}
 \usetikzlibrary{decorations.markings}
+\pgfplotsset{compat=1.18}
 
 \author{Benjamin Bertrand}
 \title{Nombre dérivé et tangente - Plan de travail}
@@ -28,7 +29,6 @@ Savoir-faire de la séquence
     \item Faire le lien entre le taux de variations et la pente de la droite passant par les points
     \item Construire une tangente à une courbe en un point
     \item Interpréter géométriquement le nombre dérivé comme coefficient directeur de la tangente.
-    \item Déterminer l’équation réduite de la tangente à une courbe en un point.
 \end{itemize}
 
 \bigskip
@@ -45,12 +45,16 @@ Savoir-faire de la séquence
 
 \listsectionexercises
 
-\pagebreak
+\bigskip
 
 \input{exercises.tex}
 \printcollection{banque}
 
-\clearpage
+\bigskip
+\hline
+\bigskip
+
+
 \printannexes
 
 \end{document}

1ST/03_Nombre_derive_et_tangente/solutions.tex

@@ -1,5 +1,8 @@
-\documentclass[a4paper,10pt]{article}
+\documentclass[a4paper,12pt]{article}
 \usepackage{myXsim}
+\usepackage{pgfplots}
+\usetikzlibrary{decorations.markings}
+\pgfplotsset{compat=1.18}
 
 \usetikzlibrary{shapes.geometric}
 
@@ -23,6 +26,6 @@
 
 \input{exercises.tex}
 %\printcollection{banque}
-\printsolutions{exercises}
+%\printsolutions{exercises}
 
 \end{document}