From af582621e63f51b58b65bd255112ac67b763d845 Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Mon, 14 Nov 2022 11:06:22 +0100 Subject: [PATCH] Feat: fin des exercices sur la tangente pour les 1ST --- .../exercises.tex | 299 +++++++++++++++--- .../plan_de_travail.pdf | Bin 50106 -> 51968 bytes .../plan_de_travail.tex | 10 +- .../solutions.pdf | Bin 25538 -> 37956 bytes .../solutions.tex | 7 +- 5 files changed, 271 insertions(+), 45 deletions(-) diff --git a/1ST/03_Nombre_derive_et_tangente/exercises.tex b/1ST/03_Nombre_derive_et_tangente/exercises.tex index 13864c2..a7b40d5 100644 --- a/1ST/03_Nombre_derive_et_tangente/exercises.tex +++ b/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 [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} diff --git a/1ST/03_Nombre_derive_et_tangente/plan_de_travail.pdf b/1ST/03_Nombre_derive_et_tangente/plan_de_travail.pdf index 3eccc3bcc3c15c8aeec708155140aad4649ace77..9f62e8e86fc8c4e1c067c8ecc6bb22e2fcbe4944 100644 GIT binary patch delta 41742 zcmbSxQ*fYN6y(grPA2xmwvCB7v2EM@z24J( z?swXPL41-xYC^z48Ce;b;`ItpfJ?3MsKX|t%_o%;h_s{p=N}O;CCAPPJDK~dnA#k* zoj0I0Y%d=u6mk0E9p>VdzrdwdcvSM9$??$2;VDW7f^{HxJ-zqFbx3<0SZs>~VBYQI z%~;+;Rn}ELZ(mLi$LHB>mSBITI3=C$j?>AeeE&lBca-wQtaysB=(uxw25vQ`4zZ8Z z9N?>;i*HmnL3KG(*6Ts>qHgX^`cb*@ZkL1Z?H6^!rcAKz{Ra3dP&XV|a17vk#Vg-g z@F{#xpERqiou+WlA}`0&Vduaw^*@O^Dx7RN7L`fK|@&v(^KaC4hqos?Fh2|{*y4n_c z8vRCg4UuF0BsKHUH*~*g-#ht}f*Ab_nt~K{%Medhke<8Tn5rlcFi0!$smYBaC+)v{Jn@d0b{(1g$S-};je-x zCrvKW#j*#HVF#68PH()p!|)sP5frWZXY2N#0g9LS6M>*6S<&QylfuKqVl$xZ;P7m) z0UR*!G&D3aPvkHAdfnAiL)<-$R;d-kZ!DVOn)59*jtZ8k%`}xk3z;j<%IQjQwcZ?) z-{I=jDkH3>a>H+d`;wed5`%(Mr~z;gyEI8HqeGPmFdS_$)+t>x5H 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