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Feat: fin des exercices sur la tangente pour les 1ST
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Bertrand Benjamin 2022-11-14 11:06:22 +01:00
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1ST/03_Nombre_derive_et_tangente/exercises.tex
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@ -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 [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)};
\addplot[tangent at/.list={0.29,0.645,0.795},domain=0:5,samples=20, color=red, very thick]{(x-3)^2 + 1};
\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 [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)};
\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};
\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 {(-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)};
\addplot [black, mark=*, very thick, only marks] coordinates {(-2,-3) (-1,-5) (0,0) (1.5,5) (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 [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)};
\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}
\begin{tikzpicture}
\begin{axis}[
axis lines = center,
grid= both,
xlabel = {$x$},
xtick distance=1,
ylabel = {$f(x)$},
ytick distance=1,
]
\addplot[domain=-2:2,samples=50, color=red, very thick]{sin(deg(x*pi/2))*2};
\addplot [black, mark=*, very thick, only marks] coordinates {(-1,-2)};
\addplot [black, mark=*, very thick, only marks] coordinates {(1.5,2*sin(deg(1.5*pi/2)))};
\addplot [black, mark=*, very thick, only marks] coordinates {(0,0)};
\end{axis}
\pgfkeys{tikz/.cd}
\tikzset{tangent/.style={black,thick},
tangent at/.style={postaction={decorate,decoration={markings,
mark=at position #1 with {\fill[tangent] (0,0) circle (3pt);}
}}},
}
\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{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}

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1ST/03_Nombre_derive_et_tangente/plan_de_travail.pdf
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10
1ST/03_Nombre_derive_et_tangente/plan_de_travail.tex
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@ -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}

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1ST/03_Nombre_derive_et_tangente/solutions.pdf
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7
1ST/03_Nombre_derive_et_tangente/solutions.tex
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@ -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}