input courbes;input geo; verbatimtex %&latex \documentclass{polymaths} \begin{document} etex color vert_e, turquoise, orange, vert_fonce, rose, vert_mer, bleu_ciel, or, rouge_v,bleu_m,bleu,bleu_f; vert_e:=(0,0.790002,0.340007); turquoise:=(0.250999,0.878399,0.815699); orange:=(0.589999,0.269997,0.080004); vert_fonce:=(0,1.4*0.392193,0); rose:=(1.0, 0.752907, 0.796106); bleu_ciel:=(1.2*0.529405,1.2*0.807794,1);%.2*0.921598); or:=(1,0.843104,0); rouge_v:=(0.829997,0.099994,0.119999); bleu_m:=(0.7*0.529405,0.7*0.807794,0.7);%*0.921598); bleu_f:=(0.211762,0.3231176,0.3686392); bleu:=(0.529405,0.807794,1); beginfig(1) repere(17cm,5cm,0.5cm,2.5cm,4cm,0.3125cm); %trace.grille(0.5,0.5pt,0.88white); %trace.grille(1,0.7pt,0.8white); marque.unites(1mm); rtrace.axes(0.5pt); vardef fx(expr t)=t enddef; vardef fy(expr t)=t**2-1/t enddef; rtrace.courbe(0.01,6,1500,1.5pt,vert_fonce); draw rpoint(0,-5)--rpoint(100,420) dashed evenly; dotlabel.lrt(btex $A$ etex, rpoint(2,3.5)); decoupe.repere; etiquette.axes; etiquette.unites; endfig; beginfig(2) repere(10cm,5cm,1cm,1.5cm,5cm,2.5cm); %trace.grille(0.5,0.5pt,0.88white); %trace.grille(1,0.7pt,0.8white); marque.unites(1mm); rtrace.axes(0.5pt); vardef fx(expr t)=t enddef; vardef fy(expr t)=cos(t) enddef; rtrace.courbe(0,2,1500,1.5pt,bleu_m); dotlabel.top(btex etex,rpoint(1,0.540)); draw rpoint(1,0)--rpoint(1,0.540) dashed evenly; draw rpoint(0,0.540)--rpoint(1,0.540) dashed evenly; label.lft(btex cos 1 etex, rpoint(0,0.540)); decoupe.repere; etiquette.axes; etiquette.unites; endfig; beginfig(3) repere(10cm,3.5cm,1cm,0.5cm,5cm,2.5cm); %trace.grille(0.5,0.5pt,0.88white); %trace.grille(1,0.7pt,0.8white); marque.unites(1mm); rtrace.axes(0.5pt); vardef fx(expr t)=t enddef; vardef fy(expr t)=cos(t) enddef; rtrace.courbe(0,1,1500,1.5pt,vert_fonce); vardef fy(expr t)=cos(t-1) enddef; rtrace.courbe(1,2,1500,1.5pt,vert_fonce); dotlabel.top(btex etex,rpoint(1,0.540)); draw rpoint(1,0)--rpoint(1,0.540) dashed evenly; draw rpoint(0,0.540)--rpoint(1,0.540) dashed evenly; label.lft(btex cos 1 etex, rpoint(0,0.540)); draw halfcircle scaled 0.2cm rotated 270 shifted(rpoint(0.97,1)) ; decoupe.repere; etiquette.axes; etiquette.unites; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Visualisation de la def des lim %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1.5cm; uy:=1.5cm; xmin := -0.2 ; xmax := 5; ymin := -0.2 ; ymax := 3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; vardef f(expr x) =1/2*x+1/2 enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,g(i)*uy) .. endfor (b*ux,g(b)*uy) enddef; beginfig(4); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Courbe et point central avec pointillés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% axes; label.llft(btex $0$ etex,(0,0)); draw tracee(f,-.2,4.5,.01) ; label.rt(btex $ y=f(x)$ etex,(4.5*ux,f(4.5)*uy)); pair A; pair h,b,g,d; h:=(0,.1*uy); b:=(0,-.1*uy); g:=(-.1*ux,0); d:=(.1*ux,0); A:=(3*ux,f(3)*uy) ; draw A withpen pencircle scaled 2.5bp; pair B,C; B:=A xscaled 0; C:=A yscaled 0; draw A--B dashed evenly; draw A--C dashed evenly; draw (B shifted (-.05*ux,0))--(B shifted (.05*ux,0)) withpen pencircle scaled 2 bp; draw (C shifted (0,-.05*uy))--(C shifted (0,.05*uy)) withpen pencircle scaled 2 bp; label.bot(btex $a$ etex,C shifted 1.2 b); label.lft(btex $\ell$ etex,B shifted g); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Les intervalles et l'axe des y %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair M,N; % points de l'axe des y pair S,T; % points de l'axe des x pair p,q; % shift d'epsilon numeric n; % epsilon n:=.5; p:=(0,n*uy); q:=(0,-n*uy); N:= B shifted p; M:= B shifted q; draw N--(N shifted d)--(N shifted (d+b)) withpen pencircle scaled 2 bp withcolor bleu_m; draw N--(N shifted g)--(N shifted (g+b)) withpen pencircle scaled 2 bp withcolor bleu_m ; draw M--(M shifted d)--(M shifted (d+h)) withpen pencircle scaled 2 bp withcolor bleu_m; draw M--(M shifted g)--(M shifted (g+h)) withpen pencircle scaled 2 bp withcolor bleu_m ; label.lft(btex $\ell\!-\!\varepsilon$ etex, M shifted g); label.lft(btex $\ell\!+\!\varepsilon$ etex, N shifted g); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Les crochets sur l'axe des x %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S:=((3-(2*n))*ux,0); T:=((3+(2*n))*ux,0); draw S--(S shifted h)--(S shifted (h+d)) withpen pencircle scaled 2 bp withcolor 0.6white; draw S--(S shifted b)--(S shifted (b+d)) withpen pencircle scaled 2 bp withcolor 0.6white ; draw T--(T shifted h)--(T shifted (h+g)) withpen pencircle scaled 2 bp withcolor 0.6white; draw T--(T shifted b)--(T shifted (b+g)) withpen pencircle scaled 2 bp withcolor 0.6white ; label.bot(btex $a\!+\!\alpha$ etex,T shifted b); label.bot(btex $a\!-\!\alpha$ etex,S shifted b); %%%%%%%%%%%%%%%%%%%%%%%% % Le gras %%%%%%%%%%%%%%%%%%%%%%%% pair Q,R; % les points de la courbe path fleched, flecheb; fleched=(h+g)--(0,0)--(b+g);%flèche vers la droite flecheb=(h+g)--(0,0)--(h+d);%flèche vers le bas Q:=((3-(2*n))*ux,f(3-(2*n))*uy); R:=((3+(2*n))*ux,f(3+(2*n))*uy); draw S--T withpen pencircle scaled 2bp withcolor 0.6white; draw M--N withpen pencircle scaled 2bp withcolor bleu_m; draw (S shifted 1.5 h)--Q dashed evenly withpen pencircle scaled 1.2bp; draw (T shifted 1.5 h)--R dashed evenly withpen pencircle scaled 1.2bp; draw (N shifted 1.5 d)--R dashed evenly withpen pencircle scaled 1.2bp; draw (M shifted 1.5 d)--Q dashed evenly withpen pencircle scaled 1.2bp; draw fleched shifted (N+(1*ux,0)) withpen pencircle scaled 1.2bp; draw fleched shifted (M+(1*ux,0)) withpen pencircle scaled 1.2bp; draw flecheb shifted (S+(0,0.7*uy)) withpen pencircle scaled 1.2bp; draw flecheb shifted (T+(0,0.7*uy)) withpen pencircle scaled 1.2bp; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure limite infinie en finie 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=4cm; uy:=1.5cm; xmin := 0 ; xmax := 1.1; ymin := -.1 ; ymax := 3.3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax))shifted (.08*ux,0); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(5); axes; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La courbe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =1/(sqrt x) enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; draw tracee(g,.1,1,.01) withcolor vert_fonce ; label.urt(btex $ y=1/x$ etex,(.5*ux,f(.5)*uy)shifted (.05*ux,.35*uy)); %%%%%%%%%%%%%%%%%%%%%%%%% % Le point M %%%%%%%%%%%%%%%%%%%%%%%%% draw(ux,uy) withpen pencircle scaled 3bp; label.urt(btex Depart etex, (ux,uy)); %%%%%%%%%%%%%%%%%%%%%%%%% % Tirets %%%%%%%%%%%%%%%%%%%%%%%%% draw ((-.05*ux,uy)--(.05*ux,uy))shifted (.08*ux,0); label.lft(btex $1$ etex,(-.05*ux,uy)); draw (ux,-.1*uy)--(ux,.1*uy); label.bot(btex $1$ etex,(ux,-.1*uy)); %%%%%%%%%%%%%%%%%%%%%%%% % Flèches %%%%%%%%%%%%%%%%%%%%%%%% path p,q,r; p:=(-.04*ux,-.1*uy)--(0,0)--(.04*ux,-.1*uy); q:=p rotated 15; r:=p rotated 60; draw q shifted (.15*ux,f(.15)*uy) withcolor vert_fonce; draw r shifted (.5*ux,f(.5)*uy) withcolor vert_fonce; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Limite finie en l'infini : 1-1/x %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=.5cm; uy:=3cm; xmin := -.5 ; xmax := 12; ymin := -.1 ; ymax := 1.5; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; vardef f(expr x) =x/(x+1) enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; beginfig(6); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La courbe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% label.llft(btex $0$ etex,(-.05*ux,-.1*uy)); label.rt(btex $y=\displaystyle \mathbf{1-\frac{1}{x}}$ etex,(1.8*ux,.5*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % L'asymptote et les horizontales %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path z; numeric e; e:=.15; z:= (0,(1+e)*uy)--(11.5*ux,(1+e)*uy)--(11.5*ux,(1-e)*uy)--(0,(1-e)*uy)--cycle; fill z withcolor bleu_ciel; draw(0,(1+e)*uy)--(11.5*ux,(1+e)*uy) withcolor bleu_m; label.rt(btex $y=1+\varepsilon$ etex,(11.5*ux,(1+e)*uy)); label.lft(btex $1+\varepsilon$ etex,(-.1*ux,(1+e)*uy)); draw(0,(1-e)*uy)--(11.5*ux,(1-e)*uy)withcolor bleu_m; label.rt(btex $y=1-\varepsilon$ etex,(11.5*ux,(1-e)*uy)); label.lft(btex $1-\varepsilon$ etex,(-.1*ux,(1-e)*uy)); draw (-.1*ux,uy)--(.1*ux,uy)withcolor bleu_m; draw (.1*ux,uy)--(11.5*ux,uy) dashed evenly withcolor bleu_m; label.lft(btex $1$ etex,(-.1*ux,uy)); label.bot(btex $ 1/\varepsilon$ etex,((1/e-1)*ux,0)) ; draw ((1/e-1)*ux,0)--((1/e-1)*ux,(1-e)*uy) dashed evenly; draw tracee(f,0,11.5,.008) withpen pencircle scaled 1.5bp withcolor red; draw ((1/e-1)*ux,(1-e)*uy) withpen pencircle scaled 3bp; axes; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % lim infinie en infini : racine carrée %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax; ux:=1cm; uy:=1cm; xmin := -0.5; xmax := 8; ymin := xmin; ymax := xmax-1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(7) %%%%%%%%%%%%%%%%%%%%%%%%%% %Axe %%%%%%%%%%%%%%%%%%%%%%%%%% label.llft(btex $0$ etex,(0,0)); %%%%%%%%%%%%%%%%%%%%%%% %Fonction, graphe %%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =2.3*sqrt(x) enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %label.rt(btex $y=2\sqrt{x}$ etex,(xmax*ux,(f(xmax))*uy)); %%%%%%%%%%%%%%%%%%%%%%%% %Points sur le graphe %%%%%%%%%%%%%%%%%%%%%%% pair A[],M[],O[]; path z; u:=.7;%facteur d'échelle %a:=2;b:=5;c:=9;%numeros des points particuliers aa:=2;bb:=5;cc:=9;%numeros des points particuliers for i=1 upto 12: A[i]:=(i*u*ux,0);% A comme abscisse M[i]:=(i*u*ux,f(i*u)*uy); %M comme courbe !! O[i]:=(0,f(i*u)*uy); %O comme ordonnée endfor ; z:=O[bb]--O[bb]shifted (xmax*ux,0)--O[bb]shifted (xmax*ux,0)shifted(0,(ymax-f(5*u))*uy)--O[bb]shifted(0,(ymax-f(5*u))*uy)--cycle; fill z withcolor bleu_ciel; axes; dotlabel.lrt(btex Buzz etex, M[cc]); for i=1 upto 10: draw M[i] withpen pencircle scaled 6bp ; endfor ; for i=aa,bb,cc: draw A[i]--M[i]--O[i] dashed evenly; endfor; label.bot(btex premier essai etex,A[aa]); label.bot(btex 2ème essai etex,A[bb]); label.bot(btex 3ème essai etex,A[cc]); label.lft(btex trop court etex,O[aa] shifted (0,.1*uy)); %label.lft(btex c'est juste etex,O[bb] shifted (0,.1*uy)); label.lft(btex vers l'infini et au-delà etex,O[cc] shifted (0,.2*uy)); %%%%%%%%%%%%%%%%%%% %Tracé des droites %%%%%%%%%%%%%%%%%% pair C,D; C=O[bb]; %Point seuil D=1/3(O[cc]+2O[cc-1]); %Point seuil pickup pencircle scaled 2bp; draw C--C shifted (xmax*ux,0) withcolor bleu_m; %draw D--D shifted (xmax*ux,0); label.lft(btex Seuil etex,C shifted (0,-.1*uy)); %label.lft(btex $A_2$ etex,D shifted (0,-.1*uy)); draw tracee(g,0,xmax,.1)withcolor red; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % comparaison de limites %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax; ux:=1cm; uy:=0.5cm; xmin := -0.5; xmax := 8; ymin := xmin; ymax := xmax-1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(8) %%%%%%%%%%%%%%%%%%%%%%%%%% %Axe %%%%%%%%%%%%%%%%%%%%%%%%%% label.llft(btex $0$ etex,(0,0)); %%%%%%%%%%%%%%%%%%%%%%% %Fonction, graphe %%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =2.3*sqrt(x) enddef; vardef h(expr x)=4*sqrt(x)-1.7 enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; vardef traceh (suffix gg)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,h(i)*uy) .. endfor (b*ux,h(b)*uy) enddef; %label.rt(btex $y=2\sqrt{x}$ etex,(xmax*ux,(f(xmax))*uy)); %%%%%%%%%%%%%%%%%%%%%%%% %Points sur le graphe %%%%%%%%%%%%%%%%%%%%%%% pair A[],M[],O[],OO[],MM[]; path z; u:=.7;%facteur d'échelle %a:=2;b:=5;c:=9;%numeros des points particuliers aa:=1.3;bb:=5;cc:=9;%numeros des points particuliers for i=1 upto 12: A[i]:=(i*u*ux,0);% A comme abscisse M[i]:=(i*u*ux,f(i*u)*uy); %M comme courbe !! O[i]:=(0,f(i*u)*uy); %O comme ordonnée OO[i]:=(0,h(i*u)*uy); MM[i]:=(i*u*ux,h(i*u)*uy); endfor ; z:=O[bb]--O[bb]shifted (xmax*ux,0)--O[bb]shifted (xmax*ux,0)shifted(0,(ymax-f(5*u))*uy)--O[bb]shifted(0,(ymax-f(5*u))*uy)--cycle; fill z withcolor bleu_ciel; axes; %dotlabel.lrt(btex Buzz etex, M[cc]); draw tracee(g,0,xmax,.1)withcolor bleu_f withpen pencircle scaled 1.5bp; draw traceh(gg,0,4.73,.1)withcolor 0.6white withpen pencircle scaled 1.5bp; draw M[bb] withpen pencircle scaled 6bp ; draw (aa*u*ux,f(aa*u)*uy) withpen pencircle scaled 6bp ; draw MM[bb] withpen pencircle scaled 6bp; draw A[bb]--M[bb]--O[bb] dashed evenly; draw (aa*u*ux,0)-- (aa*u*ux,f(aa*u)*uy)--(0,f(aa*u)*uy) dashed evenly; draw M[bb]--MM[bb]--OO[bb] dashed evenly; label.bot(btex $m$ etex,(aa*u*ux,0)); label.bot(btex $M$ etex,A[bb]); %label.bot(btex 3ème essai etex,A[cc]); label.lft(btex intersection éventuelle etex, (0,f(aa*u)*uy) shifted (0,.1*uy)); %label.lft(btex c'est juste etex,O[bb] shifted (0,.1*uy)); label.lft(btex déjà au-dessus du seuil etex,OO[bb] shifted (0,.2*uy)); label.lrt(btex $y=f(x)$ etex,M[cc]) withcolor bleu_f; label.lrt(btex $y=g(x)$ etex,(4.73*ux,h(4.73)*uy))withcolor 0.6white; %%%%%%%%%%%%%%%%%%% %Tracé des droites %%%%%%%%%%%%%%%%%% pair C,D; C=O[bb]; %Point seuil D=1/3(O[cc]+2O[cc-1]); %Point seuil pickup pencircle scaled 2bp; draw C--C shifted (xmax*ux,0) withcolor bleu_m; %draw D--D shifted (xmax*ux,0); label.lft(btex Seuil etex,C shifted (0,-.1*uy)); %label.lft(btex $A_2$ etex,D shifted (0,-.1*uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Gendarmes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=.5cm; uy:=2.5cm; xmin := -.5 ; xmax := 12; ymin := -.1 ; ymax := 1.5; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; vardef f(expr x) =1-1/(x+1) enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; vardef fp(expr x) =1+(sin(x))/(x*sqrt(x)) enddef; vardef tracep (suffix gp)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,fp(i)*uy) .. endfor (b*ux,fp(b)*uy) enddef; vardef fs(expr x) =1+1/(x-1) enddef; vardef traces (suffix gs)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,fs(i)*uy) .. endfor (b*ux,fs(b)*uy) enddef; beginfig(9) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La courbe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% label.llft(btex $0$ etex,(-.05*ux,-.1*uy)); %label.rt(btex $y=\displaystyle \mathbf{1-\frac{1}{x}}$ etex,(1.8*ux,.5*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % L'asymptote et les horizontales %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path z; numeric e; e:=.15; z:= (0,(1+e)*uy)--(11.5*ux,(1+e)*uy)--(11.5*ux,(1-e)*uy)--(0,(1-e)*uy)--cycle; fill z withcolor bleu_ciel; draw(0,(1+e)*uy)--(11.5*ux,(1+e)*uy) withcolor bleu_m; label.rt(btex $y=h(x)$ etex,(11.5*ux,(1+e)*uy))withcolor bleu_f; label.lft(btex $\ell+\varepsilon$ etex,(-.1*ux,(1+e)*uy)); draw(0,(1-e)*uy)--(11.5*ux,(1-e)*uy)withcolor bleu_m; label.rt(btex $y=g(x)$ etex,(11.5*ux,(1-e)*uy))withcolor 0.6white; label.lft(btex $\ell-\varepsilon$ etex,(-.1*ux,(1-e)*uy)); label.rt(btex $y=f(x)$ etex,(11.5*ux,1*uy))withcolor bleu; draw (-.1*ux,uy)--(.1*ux,uy)withcolor bleu_m; draw (.1*ux,uy)--(11.5*ux,uy) dashed evenly withcolor bleu_m; label.lft(btex $\ell$ etex,(-.1*ux,uy)); label.bot(btex $ A_g$ etex,((1/e-1)*ux,0)) ; draw ((1/e-1)*ux,0)--((1/e-1)*ux,(1-e)*uy) dashed evenly; draw ((1/e+1)*ux,0)--((1/e+1)*ux,(1+e)*uy) dashed evenly; label.bot(btex $ A_h$ etex,((1/e+1)*ux,0)) ; dotlabel.bot(btex etex ,((1/e-1)*ux,(1-e)*uy)); dotlabel.top(btex etex ,((1/e+1)*ux,(1+e)*uy)); draw tracee(f,0,11.5,.008) withpen pencircle scaled 1.5bp withcolor 0.6white; draw tracep(fp,1.3,11.5,.008) withpen pencircle scaled 1.5bp withcolor bleu; draw traces(fs,2.5,11.5,.008) withpen pencircle scaled 1.5bp withcolor bleu_f; draw ((1/e-1)*ux,(1-e)*uy) withpen pencircle scaled 3bp; axes; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% ASYMPTOTE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(11) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=0.4cm; uy:=2cm; xmin := -1; xmax := 20; ymin := -.7; ymax := 1.5; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax) ; % axe des y %label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x %label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; vardef f(expr x) =1/sqrt(x)+0.125*x-1 enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; vardef fd(expr x) =0.125*x-1 enddef; vardef traced (suffix gd)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,fd(i)*uy) .. endfor (b*ux,fd(b)*uy) enddef; axes; label.llft(btex $0$ etex,(0,-.01*uy)); pair h,b,d,g; % shift des tirets b:=(0,-.05*uy); h:=(0,.05*uy); d:=(0.1*ux,0); g:=(-0.1*ux,0); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Le deuxième point et l'écart %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path p,T; pair D,E,F,ee, EO,FO; %draw D withpen pencircle scaled 3bp; p:=tracee(f,0.2,20,0.01); draw p withpen pencircle scaled 2bp withcolor bleu_f; D:=(14*ux,f(14)*uy); T:= traced(fd,3,20,0.01); draw T dashed evenly withpen pencircle scaled 1.5bp withcolor 0.6white; E:=D yscaled 0; EO:=D xscaled 0; path r,rr; r:=D--E; F:=(T intersectionpoint r); FO:= F xscaled 0; rr:=D--F; ee:=(14*ux,f(14)*uy); label.lrt(btex $\mathbf{e(x)}$ etex, ee)withcolor bleu_m; drawarrow D--F withpen pencircle scaled 1.5bp withcolor bleu_m; drawarrow F--D withpen pencircle scaled 1.5bp withcolor bleu_m; draw F--E dashed withdots withcolor bleu_m withpen pencircle scaled 1.5bp ; draw F--FO dashed withdots withcolor bleu_m withpen pencircle scaled 1.5bp ; draw D--EO dashed withdots withcolor bleu_m withpen pencircle scaled 1.5bp ; label.lft(btex $f(x)$ etex,EO shifted g)withcolor bleu_m; draw (EO shifted g)--(EO shifted d)withcolor bleu_m; label.lft(btex $ax+b$ etex,FO shifted g)withcolor bleu_m; draw (FO shifted g)--(FO shifted d)withcolor bleu_m; label.bot(btex $x$ etex,E shifted b)withcolor bleu_m; draw (E shifted h)--(E shifted b)withcolor bleu_m; label.ulft(btex $y=f(x)$ etex, (18*ux,f(18)*uy))withcolor bleu_f; label.lrt(btex $y=ax+b$ etex, (18*ux,fd(18)*uy))withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù %%%%%%% Cercle Trigo %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(12); numeric u; u=2.5cm ; drawarrow (-1.2*u,0)--(1.2*u,0); drawarrow (0,-1.2*u)--(0,1.2*u); draw fullcircle scaled 5cm; draw (0,0)--(u,(sind(45)/cosd(45))*u)withcolor 0.4white withpen pencircle scaled 1.5bp ; %draw (0,sind(45)*u)--(cosd(45)*u,sind(45)*u) dashed evenly; draw (cosd(45)*u,0)--(cosd(45)*u,sind(45)*u) dashed evenly withcolor bleu_f withpen pencircle scaled 1.5bp ; draw (u,-1.3*u)--(u,1.3*u) withcolor bleu_m withpen pencircle scaled 1.5bp ;draw((cosd45)*u,sind(45)*u)--(u,0); drawarrow (0.22*u,0)..(0.22*cosd(45)*u,0.22*sind(45)*u); dotlabel.llft(btex $0$ etex, (0,0)); dotlabel.lrt(btex $I$ etex, (u,0)); %dotlabel.ulft(btex $J$ etex, (0,u)); %dotlabel.lft(btex $\sin x$ etex, (0,sind(45)*u)); dotlabel.bot(btex $C$ etex, (cosd(45)*u,0)); dotlabel.rt(btex $T$ etex , (u,(sind(45)/cosd(45))*u)); dotlabel.top(btex $ $ etex,(cosd(45)*u,sind(45)*u));label.top(btex $M$ etex,(cosd(45)*u,sind(45)*1.1*u)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DIAGRAMME %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(13); % picture(16.600000,-14.087500)(34.700000,2.850000) x = 1.000000cm; y = -1.000000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 path p; p = (22.300000x,4.675000y)..(19.475000x,12.150000y)..(16.650000x,4.675000y)..(19.475000x,-2.800000y)..cycle; fill p withcolor bleu_ciel; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (22.300000x,4.675000y)..(19.475000x,12.150000y)..(16.650000x,4.675000y)..(19.475000x,-2.800000y)..cycle withpen pencircle scaled 1.2bp withcolor bleu_f; path p; p = (34.650000x,4.850000y)..(31.975000x,12.400000y)..(29.300000x,4.850000y)..(31.975000x,-2.700000y)..cycle; fill p withcolor bleu_ciel; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (34.650000x,4.850000y)..(31.975000x,12.400000y)..(29.300000x,4.850000y)..(31.975000x,-2.700000y)..cycle withpen pencircle scaled 1.2bp withcolor bleu_f; % set_linewidth 0.100000 %metapost_arc % center->x = 25.531695% center->y = -15.961931% width = 35.080988% height = 35.080988% angle1 = 251.789043% angle2 = 289.008028% set_line_color 0.000000, 0.000000, 0.000000 draw (20.050000x,0.700000y)..(25.653701x,1.578139y)..(31.244645x,0.622132y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (31.595419x,0.489533y)--(31.216119x,0.900181y)--(31.244645x,0.622132y)--(31.039321x,0.432481y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (31.595419x,0.489533y)--(31.216119x,0.900181y)--(31.244645x,0.622132y)--(31.039321x,0.432481y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 %metapost_arc % center->x = 37.262705% center->y = -15.504852% width = 52.331576% height = 52.331576% angle1 = 227.543481% angle2 = 258.221659% set_line_color 0.000000, 0.000000, 0.000000 draw (19.600000x,3.800000y)..(25.335928x,7.784640y)..(31.921587x,10.110006y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (32.290124x,10.179331y)--(31.752525x,10.332588y)--(31.921587x,10.110006y)--(31.844959x,9.841206y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (32.290124x,10.179331y)--(31.752525x,10.332588y)--(31.921587x,10.110006y)--(31.844959x,9.841206y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 %metapost_arc % center->x = 15.959226% center->y = -15.193976% width = 48.438383% height = 48.438383% angle1 = 278.645818% angle2 = 310.494626% set_line_color 0.000000, 0.000000, 0.000000 draw (19.600000x,8.750000y)..(26.029764x,6.832224y)..(31.686605x,3.223917y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (31.966540x,2.974393y)--(31.759643x,3.493714y)--(31.686605x,3.223917y)--(31.426945x,3.120468y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (31.966540x,2.974393y)--(31.759643x,3.493714y)--(31.686605x,3.223917y)--(31.426945x,3.120468y)--cycle withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(19.250000x,0.800000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(19.000000x,4.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(19.050000x,9.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(32.100000x,0.550000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(32.350000x,3.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(32.500000x,7.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \texttt{x} etex scaled 2.67988,(32.550000x,10.150000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex D\'EPART etex scaled 3,(17.500000x,13.700000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex ARRIV\'EE etex scaled 3,(30.950000x,13.700000y)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Visualisation PHYSIQUE de la def des lim %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1.25cm; uy:=1.15cm; xmin := -0.2 ; xmax := 5; ymin := -0.2 ; ymax := 3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $T$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; vardef f(expr x) =1/2*x+1/2 enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,g(i)*uy) .. endfor (b*ux,g(b)*uy) enddef; beginfig(14); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Courbe et point central avec pointillés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% axes; label.llft(btex $0$ etex,(0,0)); draw tracee(f,-.2,4.5,.01) ; label.rt(btex $ y=p(\mathit{T})$ etex,(4.5*ux,f(4.5)*uy)); pair A; pair h,b,g,d; h:=(0,.1*uy); b:=(0,-.1*uy); g:=(-.1*ux,0); d:=(.1*ux,0); A:=(3*ux,f(3)*uy) ; draw A withpen pencircle scaled 2.5bp; pair B,C; B:=A xscaled 0; C:=A yscaled 0; draw A--B dashed evenly; draw A--C dashed evenly; draw (B shifted (-.05*ux,0))--(B shifted (.05*ux,0)) withpen pencircle scaled 2 bp; draw (C shifted (0,-.05*uy))--(C shifted (0,.05*uy)) withpen pencircle scaled 2 bp; label.bot(btex $\textit{T}_0$ etex,C shifted 1.2 b); label.lft(btex $p_0$ etex,B shifted g); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Les intervalles et l'axe des y %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair M,N; % points de l'axe des y pair S,T; % points de l'axe des x pair p,q; % shift d'epsilon numeric n; % epsilon n:=.5; p:=(0,n*uy); q:=(0,-n*uy); N:= B shifted p; M:= B shifted q; draw N--(N shifted d)--(N shifted (d+b)) withpen pencircle scaled 2 bp withcolor bleu_m; draw N--(N shifted g)--(N shifted (g+b)) withpen pencircle scaled 2 bp withcolor bleu_m ; draw M--(M shifted d)--(M shifted (d+h)) withpen pencircle scaled 2 bp withcolor bleu_m; draw M--(M shifted g)--(M shifted (g+h)) withpen pencircle scaled 2 bp withcolor bleu_m ; label.lft(btex $p_0\!-\!\varepsilon$ etex, M shifted g); label.lft(btex $p_0\!+\!\varepsilon$ etex, N shifted g); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Les crochets sur l'axe des x %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S:=((3-(2*n))*ux,0); T:=((3+(2*n))*ux,0); draw S--(S shifted h)--(S shifted (h+d)) withpen pencircle scaled 2 bp withcolor 0.6white; draw S--(S shifted b)--(S shifted (b+d)) withpen pencircle scaled 2 bp withcolor 0.6white ; draw T--(T shifted h)--(T shifted (h+g)) withpen pencircle scaled 2 bp withcolor 0.6white; draw T--(T shifted b)--(T shifted (b+g)) withpen pencircle scaled 2 bp withcolor 0.6white ; label.bot(btex $\textit{T}_0\!+\!\varepsilon V/R$ etex,T shifted b); label.bot(btex $\textit{T}_0\!-\!\varepsilon V/R$ etex,S shifted b); %%%%%%%%%%%%%%%%%%%%%%%% % Le gras %%%%%%%%%%%%%%%%%%%%%%%% pair Q,R; % les points de la courbe path fleched, flecheb; fleched=(h+g)--(0,0)--(b+g);%flèche vers la droite flecheb=(h+g)--(0,0)--(h+d);%flèche vers le bas Q:=((3-(2*n))*ux,f(3-(2*n))*uy); R:=((3+(2*n))*ux,f(3+(2*n))*uy); draw S--T withpen pencircle scaled 2bp withcolor 0.6white; draw M--N withpen pencircle scaled 2bp withcolor bleu_m; draw (S shifted 1.5 h)--Q dashed evenly withpen pencircle scaled 1.2bp; draw (T shifted 1.5 h)--R dashed evenly withpen pencircle scaled 1.2bp; draw (N shifted 1.5 d)--R dashed evenly withpen pencircle scaled 1.2bp; draw (M shifted 1.5 d)--Q dashed evenly withpen pencircle scaled 1.2bp; draw fleched shifted (N+(1*ux,0)) withpen pencircle scaled 1.2bp; draw fleched shifted (M+(1*ux,0)) withpen pencircle scaled 1.2bp; draw flecheb shifted (S+(0,0.7*uy)) withpen pencircle scaled 1.2bp; draw flecheb shifted (T+(0,0.7*uy)) withpen pencircle scaled 1.2bp; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % LIMITE INFINIE EN a NEWTON %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=9cm; uy:=0.8cm; xmin := 0 ; xmax := 1.1; ymin := -.1 ; ymax := 4.7; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric w; %shift de l'axe vertical w:=.04*ux; vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax))shifted (w,0); % axe des y label.lrt(btex $d$ etex,(xmax*ux,0)); % label de l'axe des x label.lft(btex $f$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(15); numeric a; % abscisses vardef f(expr x) =1/(sqrt x) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; a:=.3; path z; %zone rouge z:= (0,f(a)*uy)--(xmax*ux,f(a)*uy)--(xmax*ux,ymax*uy)--(0,ymax*uy)--cycle; fill z withcolor bleu_ciel; axes; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La courbe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% draw trace(g,.05,1,.01) withcolor bleu_f withpen pencircle scaled 1.5bp; %label.urt(btex $y=\frac{1}{\sqrt x}$ etex,(.5*ux,f(.5)*uy)shifted (.07*ux,.55*uy)); label.llft(btex $0$ etex,(-.05*ux,-.1*uy)shifted(w,0)); %%%%%%%%%%%%%%%%%%%%%%%%% % Le point M %%%%%%%%%%%%%%%%%%%%%%%%% draw(ux,uy) withpen pencircle scaled 3bp; %label.urt(btex M etex, (ux,uy)); %%%%%%%%%%%%%%%%%%%%%%%%% % Tirets %%%%%%%%%%%%%%%%%%%%%%%%% %draw ((-.03*ux,uy)--(.03*ux,uy))shifted (w,0); %label.lft(btex $1$ etex,(-.05*ux,uy)shifted(w,0)); %draw (ux,-.1*uy)--(ux,.1*uy); %label.bot(btex $1$ etex,(ux,-.1*uy)); %%%%%%%%%%%%%%%%%%%%%%%% % Flèches %%%%%%%%%%%%%%%%%%%%%%%% path p,q,r,s; p:=(-.02*ux,-.1*uy)--(0,0)--(.02*ux,-.1*uy); q:=p rotated 15; r:=p rotated 60; s:=p rotated 5; draw q shifted (.15*ux,f(.15)*uy) withcolor bleu_f; draw r shifted (.5*ux,f(.5)*uy) withcolor bleu_f; draw s shifted (.07*ux,f(.07)*uy) withcolor bleu_f; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Les points N et NA %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric b; % abscisses b:=.06; draw (b*ux,f(b)*uy) withpen pencircle scaled 3bp; %label.urt(btex $N$ etex,(b*ux,f(b)*uy)); draw (a*ux,f(a)*uy) withpen pencircle scaled 3bp; %label.urt(btex $F_{A}$ etex,(a*ux,f(a)*uy)) withcolor red; draw (a*ux,-.1*uy)--(a*ux,.1*uy); label.bot(btex $\displaystyle \sqrt{\ofr{Gmm'}{A}}$ etex,(a*ux,-.1*uy)shifted(.02*ux,0)); draw ((-.03*ux,f(a)*uy)--(.03*ux,f(a)*uy))shifted(w,0) withcolor bleu_m dashed withdots withpen pencircle scaled 1.5bp; label.lft(btex $A$ etex,(-.05*ux,f(a)*uy)shifted(w,0)) withcolor bleu_m; draw (w,f(a)*uy)--(a*ux,f(a)*uy)--(a*ux,0) dashed withdots withcolor bleu_m withpen pencircle scaled 1.5bp; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Limite finie en l'infini : HUILE MOTEUR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=.5cm; uy:=2.5cm; xmin := -.5 ; xmax := 12; ymin := -.1 ; ymax := 1.5; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $t$ etex,(xmax*ux,0)); % label de l'axe des x label.rt(btex $\othertheta$ etex,(0,ymax*uy)); % label de l'axe des y enddef; vardef f(expr x) =x/(x+1) enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; beginfig(16); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La courbe %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% label.llft(btex $0$ etex,(-.05*ux,-.1*uy)); %label.rt(btex $y=\displaystyle \mathbf{1-\frac{1}{x}}$ etex,(1.8*ux,.5*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % L'asymptote et les horizontales %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path z; numeric e; e:=.15; z:= (0,(1+e)*uy)--(11.5*ux,(1+e)*uy)--(11.5*ux,(1-e)*uy)--(0,(1-e)*uy)--cycle; fill z withcolor bleu_ciel; draw(0,(1+e)*uy)--(11.5*ux,(1+e)*uy) withcolor bleu_m; %label.rt(btex $y=1+\varepsilon$ etex,(11.5*ux,(1+e)*uy)); label.lft(btex $\tau+\varepsilon$ etex,(-.1*ux,(1+e)*uy)); draw(0,(1-e)*uy)--(11.5*ux,(1-e)*uy)withcolor bleu_m; %label.rt(btex $y=1-\varepsilon$ etex,(11.5*ux,(1-e)*uy)); label.lft(btex $\tau-\varepsilon$ etex,(-.1*ux,(1-e)*uy)); draw (-.1*ux,uy)--(.1*ux,uy)withcolor bleu_m; draw (.1*ux,uy)--(11.5*ux,uy) dashed evenly withcolor bleu_m; label.lft(btex $1$ etex,(-.1*ux,uy)); label.bot(btex seuil etex,((1/e-1)*ux,0)) ; draw ((1/e-1)*ux,0)--((1/e-1)*ux,(1-e)*uy) dashed evenly; draw tracee(f,0,11.5,.008) withpen pencircle scaled 1.5bp withcolor bleu_f; draw ((1/e-1)*ux,(1-e)*uy) withpen pencircle scaled 3bp; axes; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =1/(sqrt x) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % lim infinie en infini : Energie cinétique %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax; ux:=1cm; uy:=0.9cm; xmin := -0.5; xmax := 8; ymin := xmin; ymax := xmax-4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $v$ etex,(xmax*ux,0)); % label de l'axe des x label.lft(btex $\ER_c$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(17) %%%%%%%%%%%%%%%%%%%%%%%%%% %Axe %%%%%%%%%%%%%%%%%%%%%%%%%% label.llft(btex $0$ etex,(0,0)); %%%%%%%%%%%%%%%%%%%%%%% %Fonction, graphe %%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =.2*x*x enddef; vardef tracee (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %label.rt(btex $y=2\sqrt{x}$ etex,(xmax*ux,(f(xmax))*uy)); %%%%%%%%%%%%%%%%%%%%%%%% %Points sur le graphe %%%%%%%%%%%%%%%%%%%%%%% pair A[],M[],O[]; path z; u:=.7;%facteur d'échelle %a:=2;b:=5;c:=9;%numeros des points particuliers aa:=2;bb:=5;cc:=9;%numeros des points particuliers for i=1 upto 12: A[i]:=(i*u*ux,0);% A comme abscisse M[i]:=(i*u*ux,f(i*u)*uy); %M comme courbe !! O[i]:=(0,f(i*u)*uy); %O comme ordonnée endfor ; z:=O[bb]--O[bb]shifted (xmax*ux,0)--O[bb]shifted (xmax*ux,0)shifted(0,(ymax-f(5*u))*uy)--O[bb]shifted(0,(ymax-f(5*u))*uy)--cycle; fill z withcolor bleu_ciel; axes; %dotlabel.lrt(btex Buzz etex, M[cc]); %for i=1 upto 10: % draw M[i] withpen pencircle scaled 6bp ; %endfor ; %for i=aa,bb,cc: draw A[bb]--M[bb]--O[bb] dashed evenly withcolor bleu_m; %endfor; %label.bot(btex premier essai etex,A[aa]); label.bot(btex $v_0$ etex,A[bb]); %label.bot(btex 3ème essai etex,A[cc]); %label.lft(btex trop court etex,O[aa] shifted (0,.1*uy)); %label.lft(btex c'est juste etex,O[bb] shifted (0,.1*uy)); %label.lft(btex vers l'infini et au-delà etex,O[cc] shifted (0,.2*uy)); %%%%%%%%%%%%%%%%%%% %Tracé des droites %%%%%%%%%%%%%%%%%% pair C,D; C=O[bb]; %Point seuil D=1/3(O[cc]+2O[cc-1]); %Point seuil pickup pencircle scaled 2bp; draw C--C shifted (xmax*ux,0) withcolor bleu_m; %draw D--D shifted (xmax*ux,0); label.lft(btex $\ER_{\text{seuil}}$ etex,C shifted (0,-.1*uy)); %label.lft(btex $A_2$ etex,D shifted (0,-.1*uy)); draw tracee(g,0,4.472,.1)withcolor bleu_f; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% ARBRE ASYMPTOTIQUE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Metapost TeX macro % Title: Diagramme1.dia % Creator: Dia v0.93 % CreationDate: Thu Sep 8 18:03:17 2005 % For: moi beginfig(18); % picture(2.800000,-31.722785)(34.168750,-15.731250) x = 0.4500000cm; y = -0.6500000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (5.150000x,20.100000y)--(12.710566x,16.496103y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (13.049076x,16.334745y)--(12.705301x,16.775562y)--(12.710566x,16.496103y)--(12.490158x,16.324216y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (13.049076x,16.334745y)--(12.705301x,16.775562y)--(12.710566x,16.496103y)--(12.490158x,16.324216y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (5.000000x,20.086638y)--(12.764874x,23.981728y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (13.100065x,24.149870y)--(12.541049x,24.149141y)--(12.764874x,23.981728y)--(12.765238x,23.702220y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (13.100065x,24.149870y)--(12.541049x,24.149141y)--(12.764874x,23.981728y)--(12.765238x,23.702220y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (15.450000x,24.000000y)--(19.530353x,20.503401y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (19.815104x,20.259388y)--(19.598112x,20.774572y)--(19.530353x,20.503401y)--(19.272761x,20.394905y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (19.815104x,20.259388y)--(19.598112x,20.774572y)--(19.530353x,20.503401y)--(19.272761x,20.394905y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (15.400000x,24.050000y)--(19.833137x,27.916654y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (20.115743x,28.163148y)--(19.574607x,28.022894y)--(19.833137x,27.916654y)--(19.903264x,27.646086y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (20.115743x,28.163148y)--(19.574607x,28.022894y)--(19.833137x,27.916654y)--(19.903264x,27.646086y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (23.600000x,28.100000y)--(31.690877x,25.248451y) withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (32.044554x,25.123801y)--(31.656085x,25.525786y)--(31.690877x,25.248451y)--(31.489885x,25.054216y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (32.044554x,25.123801y)--(31.656085x,25.525786y)--(31.690877x,25.248451y)--(31.489885x,25.054216y)--cycle withpen pencircle scaled 0.1000x; % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (23.800000x,28.100000y)--(31.793624x,31.067231y) withpen pencircle scaled 0.1000x; drawarrow (23.800000x,28.100000y)--(31.793624x,((31.067231+25.24845)/2)*y) withpen pencircle scaled 0.2x; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (32.145185x,31.197730y)--(31.589438x,31.258105y)--(31.793624x,31.067231y)--(31.763437x,30.789358y)--cycle; fill p withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (32.145185x,31.197730y)--(31.589438x,31.258105y)--(31.793624x,31.067231y)--(31.763437x,30.789358y)--cycle withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $\infty$ ou pas de limite : pas d'asymptote ($x\mapsto x^2$) etex scaled 1.6,(13.450000x,16.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $a$ etex scaled 1.6,(13.950000x,24.250000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $a=0$ : direction asymptotique horizontale ($x\mapsto \ln x$) etex scaled 1.6,(20.500000x,20.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $a\neq 0$ etex scaled 1.6,(20.550000x,28.186638y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $\displaystyle\lim_{x\to+\infty}\big(f(x)-ax)=\infty$ ($x\mapsto x+\sqrt{x}$) etex scaled 1.6,(32.700000x,25.336638y)); label.urt(btex branche parabolique dans la direction $y=ax$ etex scaled 1.6,(32.700000x,26.336638y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $\displaystyle\lim_{x\to+\infty}\big(f(x)-ax)=b$ ($x\mapsto x+\ofr{1}{x}$) etex scaled 1.67988,(32.800000x,31.286638y)); label.urt(btex asymptote d'équation $y=ax+b$ etex scaled 1.6,(32.800000x,32.286638y)); label.urt(btex $\big(f(x)-ax\big)$ n'admet pas de limite ($x\mapsto x+\sin x$) etex scaled 1.6,(32.3x,((31.067231+25.24845)/2-0.25)*y)); label.urt(btex direction asymptotique d'équation $y=ax$ etex scaled 1.6,(32.3x,((31.067231+25.24845)/2+0.75)*y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $\displaystyle\lim_{x\to+\infty}\frac{f(x)}{x}$ etex scaled 1.6,(0.1x,21y)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% TVI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(19); repere(10cm,4cm,1cm,1cm,1cm,0.4cm); %trace.grille(0.5,0.5pt,0.88white); %trace.grille(1,0.7pt,0.8white); %marque.unites(1mm); rtrace.axes(0.5pt); draw rpoint(2,1) withpen pencircle scaled 5pt; draw rpoint(6,4) withpen pencircle scaled 5pt; draw rpoint(2,1)..rpoint(5,3)..rpoint(6,4)withpen pencircle scaled 1.5pt withcolor bleu_m; draw rpoint(0,1)--rpoint(2,1)--rpoint(2,0)dashed evenly; draw rpoint(0,4)--rpoint(6,4)--rpoint(6,0)dashed evenly; draw rpoint(5,3)--rpoint(5,0) dashed evenly withcolor bleu; draw rpoint(0,3)--rpoint(10,3) withcolor bleu; label.lft(btex $f(b)$ etex,rpoint(0,4)); label.lft(btex $f(a)$ etex,rpoint(0,1)); label.bot(btex $a$ etex,rpoint(2,0)); label.bot(btex $b$ etex,rpoint(6,0)); label.llft(btex $0$ etex, rpoint(0,0)); label.lft(btex $k$ etex, rpoint(0,3)); label.bot(btex $c$ etex, rpoint(5,0)); decoupe.repere; etiquette.axes; %etiquette.unites; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% TVI multiple %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(20); repere(11cm,4cm,1cm,1cm,1cm,0.4cm); %trace.grille(0.5,0.5pt,0.88white); %trace.grille(1,0.7pt,0.8white); %marque.unites(1mm); rtrace.axes(0.5pt); draw rpoint(2,0.5) withpen pencircle scaled 5pt; %draw rpoint(4,1.7) withpen pencircle scaled 5pt; draw rpoint(3,2) withpen pencircle scaled 5pt; draw rpoint(6,2) withpen pencircle scaled 5pt; draw rpoint(8,2) withpen pencircle scaled 5pt; draw rpoint(9,4) withpen pencircle scaled 5pt; draw rpoint(2,0.5)..rpoint(3,2)..rpoint(4,3.7)..rpoint(6,2)..rpoint(7,1.5)..rpoint(8,2)..rpoint(9,4)withpen pencircle scaled 1.5pt withcolor bleu_m; draw rpoint(0,0.5)--rpoint(2,0.5)--rpoint(2,0)dashed evenly; draw rpoint(0,4)--rpoint(9,4)--rpoint(9,0)dashed evenly; draw rpoint(3,2)--rpoint(3,0) dashed evenly withcolor bleu; draw rpoint(6,2)--rpoint(6,0) dashed evenly withcolor bleu; draw rpoint(8,2)--rpoint(8,0) dashed evenly withcolor bleu; draw rpoint(0,2)--rpoint(10,2) withcolor bleu; %draw rpoint(4,0)--rpoint(4,1.7) dashed evenly; label.lft(btex $f(b)$ etex,rpoint(0,4)); label.lft(btex $f(a)$ etex,rpoint(0,0.5)); label.bot(btex $a$ etex,rpoint(2,0)); label.bot(btex $b$ etex,rpoint(9,0)); label.llft(btex $0$ etex, rpoint(0,0)); label.lft(btex $k$ etex, rpoint(0,2)); label.bot(btex $c_1$ etex, rpoint(3,0)); label.bot(btex $c_2$ etex, rpoint(6,0)); label.bot(btex $c_3$ etex, rpoint(8,0)); %label.bot(btex $m$ etex, rpoint(4,0)); decoupe.repere; etiquette.axes; %etiquette.unites; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TANGENTE qui tourne %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1.3cm; uy:=1.3cm; xmin := -.5; xmax := 4.5; ymin := -.5; ymax := 3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(21); axes; label.llft(btex $0$ etex,(-.1*ux,-.1*uy)); %%%%%%%%%%%%%%% % Courbes %%%%%%%%%%%%%%% path p,q; p:=(0,0){dir10}..(3.5*ux,3*uy); q:=(.2*ux,.5*uy)--(4.5*ux,1.5*uy); draw p withpen pencircle scaled 1.2bp withcolor bleu_m ; draw q withcolor green; label.rt(btex $\Gamma$ etex,(3.5*ux,3.2*uy)); %label.rt(btex D etex,(4.5*ux,2.1*uy)); %%%%%%%%%%%%%%%%%%%%%%%%% % Point d'intersection %%%%%%%%%%%%%%%%%%%%%%%% pair A; A:=(p intersectionpoint q); draw A withpen pencircle scaled 3.5bp; label.ulft(btex A etex,A shifted (.1*ux,.1*uy)) ; %%%%%%%%%%%%%%%%%%%%%%%%% % Verticale %%%%%%%%%%%%%%%%%%%%%%%%% pair B; B:=A yscaled0; draw A--B dashed evenly ; draw (B shifted (0,.1*uy))--(B shifted (0,-.1*uy)); label.bot(btex $a$ etex,B shifted (0,-.1*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Droites tournées %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% draw q rotatedaround(A,9)withcolor green; draw q rotatedaround(A,18)withcolor green; draw q rotatedaround(A,27) withpen pencircle scaled 1.5bp withcolor vert_e; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La flèche %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair C; % point de départ de la flèche pair D; % point d'arrivée de la flèche C=(4.7*ux,1.7*uy); D=C rotatedaround(A,20); drawarrow C{dir(80+angle (C))}..{dir(100+angle (D))}D dashed evenly withpen pencircle scaled 1.5bp withcolor green; %label.urt(btex \mg$n\!\to\!\infty$ etex,C rotated 20) withpen pencircle scaled 1.5bp; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % LEIBNIZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1.3cm; uy:=1.3cm; xmin := -.5; xmax := 4.5; ymin := -.5; ymax := 3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(22); axes; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)); %%%%%%%%%%%%%%% % Courbes %%%%%%%%%%%%%%% path p,q; p:=(0,0){dir10}..(3.5*ux,3*uy); q:=(.2*ux,.5*uy)--(4.5*ux,1.5*uy); draw p withpen pencircle scaled 1.2bp withcolor bleu_m; label.rt(btex $\Gamma$ etex,(3.5*ux,3.2*uy)); %%%%%%%%%%%%%%%%%%%%%%%%% % Point d'intersection %%%%%%%%%%%%%%%%%%%%%%%% pair A; A:=(p intersectionpoint q); draw A withpen pencircle scaled 3.5bp; label.ulft(btex A etex,A shifted (.1*ux,.1*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%% % Tangente %%%%%%%%%%%%%%%%%%%%%%%%%% draw q rotatedaround(A,27) withpen pencircle scaled 1.5bp withcolor bleu; %%%%%%%%%%%%%%%%%%%%%%%%% % Verticale %%%%%%%%%%%%%%%%%%%%%%%%% pair B, BB; B:=A yscaled0; BB:=A xscaled 0; draw A--B dashed evenly; draw A--BB dashed evenly; draw (B shifted (0,.1*uy))--(B shifted (0,-.1*uy)); label.bot(btex $a$ etex,B shifted (0,-.17*uy)); draw (BB shifted (.1*ux,0))--(BB shifted (-.1*ux,0)); label.lft(btex $f(a)$ etex,BB shifted (-.17*ux,0)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Droite tournée %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path r; r:=q rotatedaround(A,38); draw r withcolor bleu_f; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mh et sa verticale %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair C; C:=(3.25*ux,2.4*uy); draw C withpen pencircle scaled 3.5bp; label.ulft(btex $M_{h}$ etex,C shifted (-.1*ux,.1*uy)); pair D,DD; D:=C yscaled0; DD:=C xscaled0; draw C--D dashed evenly; draw C--DD dashed evenly; draw (D shifted (0,.1*uy))--(D shifted (0,-.1*uy)); label.bot(btex $a+h$ etex,D shifted (0,-.1*uy)); draw (DD shifted (.1*ux,0))--(DD shifted (-.1*ux,0)); label.lft(btex $f(a+h)$ etex,DD shifted (-.1*ux,0)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La flèche %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair E; % point de départ de la flèche pair F; % point d'arrivée de la flèche E=(3.5*ux,2.7*uy); F=E rotatedaround(A,-10); drawarrow E{dir(-60+angle (E))}..{dir(-110+angle (F))}F dashed evenly withpen pencircle scaled 1.5bp withcolor bleu_f; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % triangle caractéristique %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path pp,qq; pair G,AA; AA:=A shifted(10*ux,0); pp:=BB--AA; qq:=C--D; G:=(pp intersectionpoint qq); drawdblarrow G shifted(0.2*ux,0)--C shifted(0.2*ux,0) withcolor 0.6white; drawdblarrow G shifted(0,-0.2*uy)--A shifted(0,-0.2*uy) withcolor 0.6white; label.rt(btex $\Delta y$ etex,((C+G)/2)shifted(0.3*ux,0)) withcolor 0.6white; label.bot(btex $\Delta x$ etex,((A+G)/2)shifted(0,-0.3*uy))withcolor 0.6white; draw A--G dashed evenly; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure mouette %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %beginfig(23) %repere(12cm,4cm,4cm,3.5cm,4cm,0.5cm); %trace.grille(0.5,0.5pt,0.88white); %trace.grille(1,0.7pt,0.8white); %marque.unites(1mm); %rtrace.axes(0.5pt); %vardef fx(expr t)=t enddef; %vardef fy(expr t)=t*(t-2) enddef; %rtrace.courbe(-1,0,500,1.5pt,bleu_m); %draw rpoint(0,-5)--rpoint(100,420) dashed evenly; %vardef fy(expr t)=t*(2-t) enddef; %rtrace.courbe(0,1.8,500,1.5pt,bleu_m); %dotlabel.urt(btex $y=|x(x-2)|$ etex, rpoint(1.8,0.36)); %decoupe.repere; %etiquette.axes; %etiquette.unites; beginfig(23) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=2.7cm; uy:=1.75cm; xmin := -0.8; xmax := 1.4; ymin := -.5; ymax := 1.5; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Fonction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =abs(x*(x-2)) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; axes; label.llft(btex $0$ etex,(-.05*ux,-.15*uy)); %%%%%%%%%%%%%%% % Courbe %%%%%%%%%%%%%%% draw trace(f,-.5,1.3,.01)withpen pencircle scaled 1.3bp withcolor bleu_m; %%%%%%%%%%%%%%%%% % Labels %%%%%%%%%%%%%%%%% pair h,b,g,d; h:=(0,.1*uy); b:=(0,-.1*uy); g:=(-.1*ux,0); d:=(.1*ux,0); draw ((ux,0) shifted h)--((ux,0) shifted b); %draw ((-ux,0) shifted h)--((-ux,0) shifted b); label.bot(btex $1$ etex,(ux,0) shifted b); %label.bot(btex $-1$ etex,(-ux,0) shifted b); label.ulft(btex $1$ etex,(0,uy) shifted h+.5g); label.urt(btex $y=\vert x(x-2)\vert$ etex,(1.3*ux,1*uy)); drawarrow (0,0)--(0.5*ux,1*uy) dashed evenly withcolor bleu_f; drawarrow (0,0)--(-0.5*ux,1*uy) dashed evenly withcolor bleu_f; endfig; beginfig(241) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=2.5cm; uy:=1.8cm; xmin := -0.2; xmax := 3.1; ymin := -.25; ymax := 1.8; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Fonction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =sqrt(x-0.5) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; axes; label.llft(btex $0$ etex,(-.05*ux,-.15*uy)); %%%%%%%%%%%%%%% % Courbe %%%%%%%%%%%%%%% draw trace(f,.5,3,.01)withpen pencircle scaled 1.3bp withcolor bleu_m; %%%%%%%%%%%%%%%%% % Labels %%%%%%%%%%%%%%%%% pair h,b,g,d; h:=(0,.1*uy); b:=(0,-.1*uy); g:=(-.1*ux,0); d:=(.1*ux,0); draw ((ux,0) shifted h)--((ux,0) shifted b); %draw ((-ux,0) shifted h)--((-ux,0) shifted b); label.bot(btex $1$ etex,(ux,0) shifted b); %label.bot(btex $-1$ etex,(-ux,0) shifted b); label.ulft(btex $1$ etex,(0,uy) shifted h+.5g); label.urt(btex $\displaystyle y=\sqrt{x-\frac{1}{2}}$ etex,(3*ux,1.8*uy)); drawarrow (0.5*ux,0)--(0.5*ux,1*uy) dashed evenly withcolor bleu_f; %drawarrow (0,0)--(-0.5*ux,1*uy) dashed evenly withcolor red; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % approximation affine %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1cm; uy:=0.9cm; xmin := 1.1; xmax := 7; ymin := -.3; ymax := 4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax)) shifted (1.4*ux,0) ; % axe des y %label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x %label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(24); axes; label.llft(btex $0$ etex,(1.3*ux,-.1*uy)); pair h,b; % shift des tirets b:=(0,-.1*uy); h:=(0,.1*uy); %%%%%%%%%%%%%%% % Courbe %%%%%%%%%%%%%%% path p; p:=(1.6ux,1.2*uy)..(2.5*ux,uy)..(5.5*ux,4*uy); draw p withpen pencircle scaled 1.3bp withcolor bleu_m; label.rt(btex $\mathcal{C}_f$ etex,(5.5*ux,4*uy)shifted 2h); %%%%%%%%%%%%%%%%%% % Tangente %%%%%%%%%%%%%%%%%% numeric e; %paramètre du point A pair M,A; e:=1.3; A:=point e of p; M:=direction e of p; path T; T:=(A shifted -1.5M)--(A shifted 1.5M); draw T withcolor bleu; label.rt(btex $\mathcal{T}$ etex,A shifted M+(.7*ux,0)+4h) withcolor bleu; %%%%%%%%%%%%%%%%%%%%% % Verticales %%%%%%%%%%%%%%%%%%%%% draw A withpen pencircle scaled 3bp withcolor bleu_m; label.top(btex $A$ etex, A shifted h)withcolor bleu_m; pair B; B:= A yscaled 0; draw A--B dashed evenly withcolor 0.6white; label.bot(btex $a$ etex,B shifted b) withcolor 0.6white; draw (B shifted b)--(B shifted h) withcolor 0.6white; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Le deuxième point et l'écart %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair D,E,F; D:=point 1.7 of p; %draw D withpen pencircle scaled 3bp; E:=D yscaled 0; path r; r:=D--E; F:=(T intersectionpoint r); drawarrow D--F withpen pencircle scaled 1.5bp withcolor 0.6white; drawarrow F--D withpen pencircle scaled 1.5bp withcolor 0.6white; label.rt(btex $e(x)$ etex,(4*ux,3.4*uy)) withcolor 0.6white; draw F--E dashed evenly withcolor 0.6white; label.bot(btex $x$ etex,E shifted b)withcolor 0.6white; draw (E shifted h)--(E shifted b) withcolor bleu_f; %%%%%%%%%%%%%%%%%%%%%%%%% % La flèche %%%%%%%%%%%%%%%%%%%%%%%%% pair G; % point de départ de la flèche pair H; % point d'arrivée de la flèche G=(4.7*ux,3.1*uy); H=G rotated -10; drawarrow G{dir(250+angle (G))}..{dir(-20-angle (H))}H withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %fonction décroissante et pentes (25) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, W; ux:=1cm; uy:=0.8cm; xmin := -.5 ; xmax := 5.5; ymin := -.5 ; ymax := 4 ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(25); axes; %%%%%%%%%%%%%%%%%%%% % Shifts %%%%%%%%%%%%%%%%%%%% pair h,g; h:=(0,.1*uy); g:=(-.1*ux,0); %%%%%%%%%%%%%%%%%%%% % Courbe %%%%%%%%%%%%%%%%%%%% path N; N:=(.5*ux,3.5*uy){dir-70}..(5*ux,.5*uy){dir-10}; draw N withpen pencircle scaled 1.3bp withcolor bleu_f; label.urt(btex $y=f(x)$ etex,(5*ux,.5*uy)) withcolor bleu_f; pair A,B; A:=point 0.2 of N ; B:=point 0.7 of N ; draw A withpen pencircle scaled 3bp withcolor 0.6white; draw B withpen pencircle scaled 3bp withcolor 0.6white; pair M,P; M:=direction 0.2 of N; P:=direction 0.7 of N; drawarrow (A--(A shifted .6M)) withpen pencircle scaled 1.3bp withcolor 0.6white; drawarrow (A--(A shifted -.6M))withpen pencircle scaled 1.3bp withcolor 0.6white; drawarrow (B--(B shifted .6P))withpen pencircle scaled 1.3bp withcolor 0.6white; drawarrow (B--(B shifted -.6P))withpen pencircle scaled 1.3bp withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Figure 26 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=.005cm; uy:=.004cm; xmin := -280 ; xmax :=900; ymin := -350; ymax :=900; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric p,q; % shift des axes p:=200; q:=120; vardef axes = drawarrow ((ux*xmin,0) -- (ux*xmax,0))shifted (0,p*uy) ; % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax))shifted (q*ux,0); % axe des y label(btex $x$ etex,(950*ux,200*uy)); % label de l'axe des x label(btex $y$ etex,(120*ux,950*uy)); % label de l'axe des y enddef; beginfig(26); %%%%%%%%%%%%%%%%%%%%%%%% %Courbe %%%%%%%%%%%%%%%%%%%%%%%% axes; vardef f(expr x) =x-50*cosd(x) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,g(i)*uy) .. endfor (b*ux,g(b)*uy) enddef; path P; P:=trace(f,-280,830,10) ; draw P withcolor bleu_m; label.rt(btex $y=x-\cos x$ etex,(830*ux,f(830)*uy)) withcolor bleu_m; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Tangentes horizontales %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair A,B,C,D,a,b,c,d; pair T; A:=(-110*ux,0) ; B:=(250*ux,0); C:=(610*ux,0); a:=(-110*ux,f(-110)*uy); b:=(250*ux,f(250)*uy); c:=(610*ux,f(610)*uy); T:=(80*ux,0); drawarrow (a--(a shifted T))withcolor bleu_f; drawarrow (a--(a shifted -T))withcolor bleu_f; drawarrow (b--(b shifted T))withcolor bleu_f; drawarrow (b--(b shifted -T))withcolor bleu_f; drawarrow (c--(c shifted T))withcolor bleu_f; drawarrow (c--(c shifted -T))withcolor bleu_f; pair d,e,f; d:=(a yscaled 0)shifted(0,p*uy); draw (a--d) dashed evenly withcolor bleu_f; e:=(b yscaled 0)shifted(0,p*uy); draw (b--e) dashed evenly withcolor bleu_f; f:=(c yscaled 0)shifted(0,p*uy); draw (c--f) dashed evenly withcolor bleu_f; draw (d shifted (0,20*uy))--(d shifted (0,-20*uy)) withcolor bleu_f; label.top(btex \small{$\displaystyle\frac{-3\pi}{2}$} etex,d shifted (0,20*uy)) withcolor bleu_f; draw (e shifted (0,20*uy))--(e shifted (0,-20*uy)) withcolor bleu_f; label.bot(btex \small{$\displaystyle\frac{\pi}{2}$} etex,e shifted (0,-20*uy)) withcolor bleu_f; draw (f shifted (0,20*uy))--(f shifted (0,-20*uy)) withcolor bleu_f; label.bot(btex \small{$\displaystyle\frac{5\pi}{2}$} etex,f shifted (0,-20*uy)) withcolor bleu_f; label.ulft(btex \small{$0$} etex,((q-20)*ux,(p+20)*uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% les dents de la mer %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(27); drawdblarrow (0,0)--(0,3cm) dashed evenly withcolor bleu_f ; drawdblarrow (0,3cm)--(7cm,3cm) dashed evenly withcolor bleu_f; draw (0,0)--(2cm,3cm); mark_rt_angle((0,0),(0,3cm),(7cm,3cm)); label.llft(btex $A$ etex,(0,0)); label.ulft(btex $H$ etex,(0,3cm)); dotlabel.lrt(btex $M(x)$ etex,(2cm,3cm)); label.top(btex $B$ etex,(7cm,3cm)); label.lft(btex $18$m etex,(0,1.5cm)) withcolor bleu_f; label.top(btex $30$m etex,(3.5cm,3cm)) withcolor bleu_f; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% La baleine %%%%%%%%%%%%%%%%%%%%%%%% beginfig(28) repere(9cm,3.5cm,0.5cm,0.5cm,0.75cm,0.777cm); draw rpoint(0,0){dir0}..rpoint(4,1)..rpoint(5.5,3.1754){dir30}..tension1.3..rpoint(11,0) withpen pencircle scaled 1.5bp; draw rpoint(0,0)--rpoint(8,4.6188) withpen pencircle scaled 0.5bp; draw rpoint(5.5,0)--rpoint(5.5,3.1754) dashed withdots scaled 0.5; label.bot(btex $x_0$ etex, rpoint(5.5,0)); label.lrt(btex $0$ etex, rpoint(0,0)); label.bot(btex $1$ etex, rpoint(11,0)); label.ulft(btex $M_0$ etex, rpoint(5.5,3.1754)); label.urt(btex $y=f(x)$ etex, rpoint(9,3)); rtrace.axes(0.5pt); decoupe.repere; etiquette.axes; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Différentielle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1.3cm; uy:=1.3cm; xmin := -.5; xmax := 4.5; ymin := -.5; ymax := 3; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(29); axes; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)); %%%%%%%%%%%%%%% % Courbes %%%%%%%%%%%%%%% path p,q; p:=(0,0){dir10}..(3.5*ux,3*uy); q:=(.2*ux,.5*uy)--(4.5*ux,1.5*uy); draw p withpen pencircle scaled 1.2bp withcolor bleu_m; label.rt(btex $\Gamma$ etex,(3.5*ux,3.2*uy)); %%%%%%%%%%%%%%%%%%%%%%%%% % Point d'intersection %%%%%%%%%%%%%%%%%%%%%%%% pair A; A:=(p intersectionpoint q); draw A withpen pencircle scaled 3.5bp; label.ulft(btex A etex,A shifted (.1*ux,.1*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%% % Tangente %%%%%%%%%%%%%%%%%%%%%%%%%% path qqq; qqq:=q rotatedaround(A,27); draw q rotatedaround(A,27) withpen pencircle scaled 1.5bp withcolor vert_e; %%%%%%%%%%%%%%%%%%%%%%%%% % Verticale %%%%%%%%%%%%%%%%%%%%%%%%% pair B, BB; B:=A yscaled0; BB:=A xscaled 0; draw A--B dashed evenly; draw A--BB dashed evenly; draw (B shifted (0,.1*uy))--(B shifted (0,-.1*uy)); label.bot(btex $a$ etex,B shifted (0,-.17*uy)); draw (BB shifted (.1*ux,0))--(BB shifted (-.1*ux,0)); label.lft(btex $f(a)$ etex,BB shifted (-.17*ux,0)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Droite tournée %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path r; r:=q rotatedaround(A,38); %draw r withcolor green; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mh et sa verticale %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair C; C:=(3.25*ux,2.4*uy); draw C withpen pencircle scaled 3.5bp; label.ulft(btex $M'$ etex,C shifted (-.1*ux,.1*uy)); pair D,DD; D:=C yscaled0; DD:=C xscaled0; draw C--D dashed evenly; draw C--DD dashed evenly; draw (D shifted (0,.1*uy))--(D shifted (0,-.1*uy)); label.bot(btex $a+$d$x$ etex,D shifted (0,-.1*uy)); draw (DD shifted (.1*ux,0))--(DD shifted (-.1*ux,0)); label.lft(btex $f(a+$d$x)$ etex,DD shifted (-.1*ux,0)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % La flèche %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair E; % point de départ de la flèche pair F; % point d'arrivée de la flèche E=(3.5*ux,2.7*uy); F=E rotatedaround(A,-10); %drawarrow E{dir(-60+angle (E))}..{dir(-110+angle (F))}F dashed evenly withpen pencircle scaled 1.5bp withcolor green; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % triangle caractéristique %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% path pp,qq; pair G,AA,H,HH; AA:=A shifted(10*ux,0); pp:=BB--AA; qq:=C--D; G:=(pp intersectionpoint qq); H:=(qqq intersectionpoint qq); HH:=H shifted(10*ux,0); draw H withpen pencircle scaled 3.5bp; draw G withpen pencircle scaled 3.5bp; drawarrow G shifted(0.5*ux,0)--H shifted(0.5*ux,0) withcolor 0.6white; drawarrow A shifted(0,-0.2*uy)--G shifted(0,-0.2*uy) withcolor 0.6white; label.rt(btex d$f$ etex,((H+G)/2)shifted(0.5*ux,0)) withcolor 0.6white; label.bot(btex d$x$ etex,((A+G)/2)shifted(0,-0.3*uy))withcolor 0.6white; label.lrt(btex $T$ etex, (H shifted(0.1*ux,0))) withcolor 0.6white; label.lrt(btex $G$ etex, (G shifted(0.1*ux,0))) withcolor 0.6white; draw A--G shifted(0.95*ux,0) dashed evenly; draw H--H shifted(0.5*ux,0) dashed evenly; draw C--C shifted(0.95*ux,0) dashed evenly; drawarrow G shifted(0.95*ux,0)--C shifted(0.95*ux,0) withcolor 0.6white; label.rt(btex $\Delta f$ etex,((C+G)/2)shifted(0.95*ux,0)) withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% ROLLE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef droite(expr a,b,t) = (t[a,b])--(t[b,a]) enddef; path c,cartouche; c=(2.8cm,2.7cm)..(3.5cm,3.2cm)..(7.5cm,3.2cm)..(7.8cm,2.6cm); cartouche = (4cm,2mm)--(9cm,2mm)--(9cm,8mm)--(4cm,8mm)--cycle; pair A,M; A = point 1 of c; M = point 2 of c; vardef tangente(expr t,x) = pair X,Y; X := point (t-.05) of c; Y := point (t+.05) of c; droite(X,Y,x) enddef; beginfig(30); % grille(1cm,0,10cm,0,7cm); repere(10cm,5cm,2cm,2cm,1cm,1cm); rtrace.axes(0.5pt); %marque.unites(0.1); %% lectures sur la grille numeric xa,ya,xm,ym,xc,yc; xa = 1.5; ya = 1.2 ; xm = 5.5 ; ym = 1.2; xc = 3.8 ; yc =1.95; pair AA,MM,CC; AA = (xa,ya) ; MM = (xm,ym) ; CC = (xc,yc); projection.axes(AA,0.5,1.7); projection.axes(MM,0.5,1.7); draw (CC--(xc,0)) en_place dashed evenly scaled 1.7 withpen pencircle scaled 0.5; label(btex $a$ etex,rpoint(xa,-0.2)); label(btex $b$ etex,rpoint(xm,-0.2)); label(btex $c$ etex,rpoint(xc,-0.2)); label.lft(btex $f(b) = f(a)$ etex, rpoint(0,ym)); %% fin des lectures draw c withpen pencircle scaled 1.5pt withcolor bleu_f; draw droite(A,M,1.2); draw tangente(1.53325,6) withpen pencircle scaled 1.5pt withcolor 0.6white; dotlabel.lrt(btex $A$ etex ,A); dotlabel.llft(btex $B$ etex ,M); label.lrt(btex $\CR_f$ etex ,point 3 of c); %% Cartouche %fill cartouche withcolor .8white; %draw cartouche; %label.rt(btex \textbf{Th. de Rolle} etex xscaled 1.15,(4cm,5mm)); %% fin du cartouche decoupe.repere; etiquette.axes; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% TAF %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pair A,M; path c,cartouche; c=(2.3cm,1cm)..(4.5cm,4cm)..(9cm,3cm); cartouche = (4cm,2mm)--(9cm,2mm)--(9cm,8mm)--(4cm,8mm)--cycle; A = point 0.6 of c; M = point 1.5 of c; beginfig(31); %grille(1cm,0,10cm,0,7cm); repere(10cm,5cm,2cm,2cm,1cm,1cm); rtrace.axes(0.5pt); %marque.unites(0.1); %% lectures sur la grille numeric xa,ya,xm,ym,xc,yc; xa = 1.25; ya = 1.1 ; xm = 4.9 ; ym = 2.25; xc = 2.9 ; yc =2.2; pair AA,MM,CC; AA = (xa,ya) ; MM = (xm,ym) ; CC = (xc,yc); % projection.axes(AA,0.5,1.7); %projection.axes(MM,0.5,1.7); path ff,ab; ff:=rpoint(xc,yc)--rpoint(xc,0); ab:=(AA--MM) en_place; pair P,m; m:=(ff intersectionpoint c); P:=(ff intersectionpoint ab); draw (CC--(xc,0)) en_place dashed evenly scaled 1.7 withpen pencircle scaled 0.5; draw rpoint(xa,ya)--rpoint(xa,-0.2) dashed evenly; draw rpoint(xm,ym)--rpoint(xm,-0.2) dashed evenly; label(btex $a$ etex,rpoint(xa,-0.2)); label(btex $b$ etex,rpoint(xm,-0.2)); dotlabel.ulft(btex $M$ etex, m); dotlabel.lrt(btex $P$ etex, P); label(btex $x$ etex,rpoint(xc,-0.2)); %label.lft(btex $f(a)$ etex, rpoint(0,ya)); %label.lft(btex $f(b)$ etex, rpoint(0,ym)); %% fin des lectures draw c withpen pencircle scaled 1.5pt withcolor bleu_f; draw droite(A,M,1.2); draw tangente(1.09,6) withpen pencircle scaled 1.5pt withcolor 0.6white; dotlabel.lrt(btex $A$ etex ,A); % dotlabel.lrt(btex $B$ etex scaled 1.5,M); dotlabel.lrt(btex etex,M); label.bot(btex $B$ etex ,rpoint(5.15,2.2)); label.bot(btex $\CR_f$ etex ,point 2 of c); %% Cartouche %fill cartouche withcolor .8white; %draw cartouche; %label.rt(btex \textbf{Th. des accroissements finis} etex %xscaled 0.95,(4cm,5mm)); %% fin du cartouche decoupe.repere; etiquette.axes; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù %%%%%%%% COMPLEXES COMPLEXES COMPLEXES %%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% AXE DES REELS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(32); drawarrow (-5cm,0)--(5cm,0) withcolor bleu_m withpen pencircle scaled 1.5bp; draw (-3cm,-0.2cm)--(-3cm,0.2cm); draw (-1.414cm,-0.2cm)--(-1.414cm,0.2cm); draw (0cm,-0.2cm)--(0cm,0.2cm); draw (0.66666666cm,-0.2cm)--(0.66666666cm,0.2cm); draw (3.14cm,-0.2cm)--(3.14cm,0.2cm); label.bot(btex $3$ etex,(-3cm,-0.2cm)); label.bot(btex $-\sqrt{2}$ etex,(-1.414cm,-0.2cm)); label.bot(btex $0$ etex,(0cm,-0.2cm)); label.bot(btex $\frac{2}{3}$ etex,(0.66666666cm,-0.2cm)); label.bot(btex $\pi$ etex,(3.14cm,-0.2cm)); label.bot(btex $\mathbb{R}$ etex,(5cm,-0.2cm))withcolor bleu_m withpen pencircle scaled 1.5bp; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% FAUX ELEMENT NEUTRE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(33); drawarrow (-5cm,0)--(5cm,0) withcolor bleu_m withpen pencircle scaled 1.5bp; drawarrow (0,-1.5cm)--(0,2.5cm) dashed evenly withcolor 0.6white withpen pencircle scaled 1.5bp ; label.urt(btex $(1,1)$ etex, (1cm,1cm))withcolor 0.6white dashed evenly; dotlabel.urt(btex $ $ etex, (1cm,1cm)); %draw (-3cm,-0.2cm)--(-3cm,0.2cm); draw (1cm,-0.2cm)--(1cm,0.2cm); draw (-0.2cm,1cm)--(0.2cm,1cm) withcolor 0.6white; draw (0cm,-0.2cm)--(0cm,0.2cm); %draw (0.66666666cm,-0.2cm)--(0.66666666cm,0.2cm); %draw (3.14cm,-0.2cm)--(3.14cm,0.2cm); draw (0,1cm)--(1cm,1cm)--(1cm,0) withcolor 0.6white dashed withdots withpen pencircle scaled 1.5bp; %label.bot(btex $3$ etex,(-3cm,-0.2cm)); label.bot(btex $1$ etex,(1cm,-0.2cm))withcolor bleu_m; label.lft(btex $1$ etex,(-0.2cm,1cm)) withcolor 0.6white dashed evenly; label.llft(btex $0$ etex,(-0.2cm,-0.2cm))withcolor bleu_m; %label.bot(btex $\frac{2}{3}$ etex,(0.66666666cm,-0.2cm)); %label.bot(btex $\pi$ etex,(3.14cm,-0.2cm)); label.bot(btex $\mathbb{R}$ etex,(5cm,-0.2cm))withcolor bleu_m withpen pencircle scaled 1.5bp; label.urt(btex $(1,0)$ etex,(1cm,0.1cm)) withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% REPRESENTATION COMPLEXE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(34); drawarrow (-5cm,0)--(5cm,0) withcolor bleu_m withpen pencircle scaled 1.5bp; drawarrow (0,-1.5cm)--(0,2.5cm) withcolor bleu_f withpen pencircle scaled 1.5bp ; label.urt(btex \boldmath $M(a,b)$\unboldmath etex, (3cm,1.5cm))withcolor 0.6white dashed evenly; draw (3cm,-0.2cm)--(3cm,0.2cm)withcolor 0.6white; draw (-0.2cm,1.5cm)--(0.2cm,1.5cm) withcolor 0.6white; dotlabel.urt(btex $ $ etex, (3cm,1.5cm)); %draw (0cm,-0.2cm)--(0cm,0.2cm); draw (0,1.5cm)--(3cm,1.5cm)--(3cm,0) withcolor 0.6white dashed withdots withpen pencircle scaled 1.5bp; drawarrow (0,0)--(1cm,0) withcolor bleu_m withpen pencircle scaled 3bp; drawarrow (0,0)--(0,1cm) withcolor bleu_f withpen pencircle scaled 3bp; label.bot(btex $\overrightarrow{e_1}$ etex,(0.5cm,-0.2cm))withcolor bleu_m; label.lft(btex $\overrightarrow{e_2}$ etex,(-0.2cm,0.5cm)) withcolor bleu_f dashed evenly; drawarrow (0,0)--(3cm,1.5cm) withcolor 0.6white withpen pencircle scaled 1.5bp; label.bot(btex $a$ etex,(3cm,-0.2cm))withcolor bleu_m; label.lft(btex $b$ etex,(-0.2cm,1.5cm)) withcolor bleu_f dashed evenly; label.llft(btex $0$ etex,(-0.2cm,-0.2cm)); label.bot(btex \textbf{axe réel} etex,(5cm,-0.2cm))withcolor bleu_m withpen pencircle scaled 1.5bp; label.lft(btex \textbf{axe imaginaire} etex,(0cm,2.5cm))withcolor bleu_f withpen pencircle scaled 1.5bp; label.ulft(btex \boldmath$\overrightarrow{u}$\unboldmath etex, (1.5cm,0.75cm))withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% CONJUGUE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(35); drawarrow (-5cm,0)--(5cm,0) withcolor bleu_m withpen pencircle scaled 1.5bp; drawarrow (0,-1.5cm)--(0,2.5cm) withcolor bleu_f withpen pencircle scaled 1.5bp ; label.urt(btex \boldmath $M(z)$\unboldmath etex, (3cm,1.5cm))withcolor 0.6white dashed evenly; dotlabel.urt(btex $ $ etex, (3cm,1.5cm)); label.urt(btex \boldmath $M(\overline{z})$\unboldmath etex, (3cm,-1.5cm))withcolor 0.6white dashed evenly; dotlabel.urt(btex $ $ etex, (3cm,-1.5cm)); draw(3cm,1.5cm)--(3cm,-1.5cm) withcolor 0.6white dashed evenly; %draw (3cm,-0.2cm)--(3cm,0.2cm)withcolor rose; %draw (-0.2cm,1.5cm)--(0.2cm,1.5cm) withcolor rose; %draw (0cm,-0.2cm)--(0cm,0.2cm); %draw (0,1.5cm)--(3cm,1.5cm)--(3cm,0) withcolor rose dashed withdots withpen pencircle scaled 1.5bp; drawarrow (0,0)--(1cm,0) withcolor bleu_m withpen pencircle scaled 3bp; drawarrow (0,0)--(0,1cm) withcolor bleu_f withpen pencircle scaled 3bp; label.bot(btex $\overrightarrow{e_1}$ etex,(0.5cm,-0.2cm))withcolor bleu_m; label.lft(btex $\overrightarrow{e_2}$ etex,(-0.2cm,0.5cm)) withcolor bleu_f dashed evenly; drawarrow (0,0)--(3cm,1.5cm) withcolor 0.6white withpen pencircle scaled 1.5bp; drawarrow (0,0)--(3cm,-1.5cm) withcolor 0.6white withpen pencircle scaled 1.5bp; %label.bot(btex $a$ etex,(3cm,-0.2cm))withcolor bleu_m; %label.lft(btex $b$ etex,(-0.2cm,1.5cm)) withcolor vert_fonce dashed evenly; label.llft(btex $0$ etex,(-0.2cm,-0.2cm)); label.bot(btex \textbf{axe réel} etex,(5cm,-0.2cm))withcolor bleu_m withpen pencircle scaled 1.5bp; %label.lft(btex \textbf{axe imaginaire} etex,(0cm,2.5cm))withcolor vert_fonce withpen pencircle scaled 1.5bp; %label.ulft(btex \boldmath$\overrightarrow{u}$\unboldmath etex, (1.5cm,0.75cm))withcolor rose; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% MODULE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(36); drawarrow (-2cm,0)--(7cm,0) withcolor bleu_m withpen pencircle scaled 1.5bp; drawarrow (0,-1.5cm)--(0,2.5cm) withcolor bleu_f withpen pencircle scaled 1.5bp ; label.urt(btex \boldmath $M(a+ib)$\unboldmath etex, (4cm,2cm))withcolor 0.6white dashed evenly; dotlabel.urt(btex $ $ etex, (4cm,2cm)); draw (4cm,-0.2cm)--(4cm,0.2cm)withcolor 0.6white; draw (-0.2cm,2cm)--(0.2cm,2cm) withcolor 0.6white; %draw (0cm,-0.2cm)--(0cm,0.2cm); draw (0,2cm)--(4cm,2cm)--(4cm,0) withcolor 0.6white dashed withdots withpen pencircle scaled 1.5bp; drawarrow (0,0)--(1cm,0) withcolor bleu_m withpen pencircle scaled 3bp; drawarrow (0,0)--(0,1cm) withcolor bleu_f withpen pencircle scaled 3bp; label.bot(btex $\overrightarrow{e_1}$ etex,(0.5cm,-0.2cm))withcolor bleu_m; label.lft(btex $\overrightarrow{e_2}$ etex,(-0.2cm,0.5cm)) withcolor bleu_f dashed evenly; drawarrow (0,0)--(4cm,2cm) withcolor 0.6white withpen pencircle scaled 1.5bp; label.bot(btex $a$ etex,(4cm,-0.2cm))withcolor bleu_m; label.lft(btex $b$ etex,(-0.2cm,2cm)) withcolor bleu_f dashed evenly; label.llft(btex $0$ etex,(-0.2cm,-0.2cm)); label.bot(btex \textbf{axe réel} etex,(5cm,-0.2cm))withcolor bleu_m withpen pencircle scaled 1.5bp; label.lft(btex \textbf{axe imaginaire} etex,(0cm,2.5cm))withcolor bleu_f withpen pencircle scaled 1.5bp; label.ulft(btex \boldmath$\sqrt{a^2+b^2}$\unboldmath etex, (2cm,0.85cm))withcolor 0.6white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% COORDONNEES POLAIRES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(37); pair b,bb; b:=(4cm,0); bb:=b rotated 30; drawarrow (-2cm,0)--(7cm,0) withcolor bleu_m withpen pencircle scaled 1.5bp; drawarrow (0,-1.5cm)--(0,2.5cm) withcolor bleu_f withpen pencircle scaled 1.5bp ; label.urt(btex \boldmath $M(z)$\unboldmath etex, bb)withcolor 0.6white dashed evenly; dotlabel.urt(btex $ $ etex, bb); pair bx,by; %projetés bx:= bb yscaled 0; by:= bb xscaled 0; draw by--bb--bx withcolor 0.6white dashed withdots withpen pencircle scaled 1.5bp; drawarrow (0,0)--(1cm,0) withcolor bleu_m withpen pencircle scaled 3bp; drawarrow (0,0)--(0,1cm) withcolor bleu_f withpen pencircle scaled 3bp; label.bot(btex $\overrightarrow{e_1}$ etex,(0.5cm,-0.2cm))withcolor bleu_m; label.lft(btex $\overrightarrow{e_2}$ etex,(-0.2cm,0.5cm)) withcolor bleu_f dashed evenly; drawarrow (0,0)--bb withcolor 0.6white withpen pencircle scaled 1.5bp; label.bot(btex $r\,\cos\theta$ etex,bx)withcolor bleu_m; label.lft(btex $r\,\sin\theta$ etex,by) withcolor bleu_f dashed evenly; label.llft(btex $0$ etex,(-0.2cm,-0.2cm)); %label.bot(btex \textbf{axe réel} etex,(5cm,-0.2cm))withcolor bleu_m withpen pencircle scaled 1.5bp; %label.lft(btex \textbf{axe imaginaire} etex,(0cm,2.5cm))withcolor vert_fonce withpen pencircle scaled 1.5bp; label.ulft(btex \boldmath$r$\unboldmath etex, 0.5*bb)withcolor 0.6white; %%%%%%%%%%%%%%%%%%%%%%% %Tracé de la flèche arrondie %%%%%%%%%%%%%%%%%%%%%% %pair b; %axe des x pair C; % point de départ de la flèche pair D; % point d'arrivée de la flèche C=1/3(origin+b); D=C rotated 30; drawarrow C{dir(90+angle (C))}..{dir(90+angle (D))}D; label.rt(btex $\theta$ etex,C rotated 15); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% QUADRILATERES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %@AUTEUR: Claudy Diguet numeric u; u = 1cm; beginfig(38); draw (1.5,0)*u--(2,0)*u withpen pencircle scaled 1.5pt; draw (2,7)*u--(2,-7)*u withpen pencircle scaled 1.5pt; drawarrow (2,7)*u--(16.3,7)*u withpen pencircle scaled 1.5pt; drawarrow (2,5)*u--(6.5,5)*u withpen pencircle scaled 1.5pt; drawarrow (2,3)*u--(6.5,3)*u withpen pencircle scaled 1.5pt; drawarrow (2,1)*u--(6.5,1)*u withpen pencircle scaled 1.5pt; drawarrow (2,-5)*u--(6.5,-5)*u withpen pencircle scaled 1.5pt; drawarrow (2,-3)*u--(6.5,-3)*u withpen pencircle scaled 1.5pt; drawarrow (2,-1)*u--(6.5,-1)*u withpen pencircle scaled 1.5pt; drawarrow (2,-7)*u--(16.3,-7)*u withpen pencircle scaled 1.5pt; draw (6.5,5)*u--(6.5,-5)*u withpen pencircle scaled 1.5pt; draw (6.5,0)*u--(7,0)*u withpen pencircle scaled 1.5pt; draw (11,0)*u--(11.5,0)*u withpen pencircle scaled 1.5pt; draw (11.5,5)*u--(11.5,-5)*u withpen pencircle scaled 1.5pt; drawarrow (11.5,5)*u--(16.3,5)*u withpen pencircle scaled 1.5pt; drawarrow (11.5,3)*u--(16.3,3)*u withpen pencircle scaled 1.5pt; draw (16.3,7)*u--(16.3,3)*u withpen pencircle scaled 1.5pt; draw (16.3,5)*u--(16.8,5)*u withpen pencircle scaled 1.5pt; draw (19.2,5)*u--(20,5)*u withpen pencircle scaled 1.5pt; draw (20,5)*u--(20,1)*u withpen pencircle scaled 1.5pt; drawarrow (20,3)*u--(24.5,3)*u withpen pencircle scaled 1.5pt; drawarrow (20,1)*u--(24.5,1)*u withpen pencircle scaled 1.5pt; drawarrow (11.5,-5)*u--(16.3,-5)*u withpen pencircle scaled 1.5pt; drawarrow (11.5,-3)*u--(16.3,-3)*u withpen pencircle scaled 1.5pt; draw (16.3,-7)*u--(16.3,-3)*u withpen pencircle scaled 1.5pt; draw (16.3,-5)*u--(16.8,-5)*u withpen pencircle scaled 1.5pt; draw (19.2,-5)*u--(20,-5)*u withpen pencircle scaled 1.5pt; draw (20,-5)*u--(20,-1)*u withpen pencircle scaled 1.5pt; drawarrow (20,-3)*u--(24.5,-3)*u withpen pencircle scaled 1.5pt; drawarrow (20,-1)*u--(24.5,-1)*u withpen pencircle scaled 1.5pt; draw (24.5,3)*u--(24.5,-3)*u withpen pencircle scaled 1.5pt; draw (24.5,0)*u--(25,0)*u withpen pencircle scaled 1.5pt; label.top(btex $ qui~a~ses~diagonales~qui $ etex,(4.25,5)*u); label.bot(btex $ se~coupent~en~leur~milieu $ etex,(4.25,5)*u); label.top(btex $ qui~a~tous~ses~c\hat{o}t\acute{e}s~oppos\acute{e}s $ etex,(4.25,3)*u); label.bot(btex $ parall\grave{e}les $ etex,(4.25,3)*u); label.top(btex $ qui~a~tous~ses~c\hat{o}t\acute{e}s~oppos\acute{e}s $ etex,(4.25,1)*u); label.bot(btex $ de~m\hat{e}me~longueur $ etex,(4.25,1)*u); label.top(btex $ qui~a~\underline{deux}~c\hat{o}t\acute{e}s~parall\grave{e}les $ etex,(4.25,-1)*u); label.bot(btex $ et~de~m\hat{e}me~longueur $ etex,(4.25,-1)*u); label.top(btex $ qui~a~tous~ses~angles $ etex,(4.25,-3)*u); label.bot(btex $ oppos\acute{e}s~de~m\hat{e}me~mesure $ etex,(4.25,-3)*u); label.top(btex $ qui~a~ses~angles $ etex,(4.25,-5)*u); label.bot(btex $ cons\acute{e}cutifs~suppl\acute{e}mentaires $ etex,(4.3,-5)*u); label.top(btex $ qui~a~3~angles~droits $ etex,(9,7)*u); label.top(btex $ qui~a~un~angle $ etex,(13.9,5)*u); label.bot(btex $ droit $ etex,(13.9,5)*u); label.top(btex $ qui~a~ses~diagonales $ etex,(13.9,3)*u); label.bot(btex $ de~m\hat{e}me~longueur $ etex,(13.9,3)*u); label.top(btex $ qui~a~4~c\hat{o}t\acute{e}s~de~m\hat{e}me~longueur $ etex,(9,-7)*u); label.top(btex $ qui~a~deux~c\hat{o}t\acute{e}s~cons\acute{e}cutifs $ etex,(13.9,-3)*u); label.bot(btex $ de~m\hat{e}me~longueur $ etex,(13.9,-3)*u); label.top(btex $ qui~a~ses~diagonales $ etex,(13.9,-5)*u); label.bot(btex $ perpendiculaires $ etex,(13.9,-5)*u); label.top(btex $ qui~a~deux~c\hat{o}t\acute{e}s~cons\acute{e}cutifs $ etex,(22.25,3)*u); label.bot(btex $ de~m\hat{e}me~longueur $ etex,(22.25,3)*u); label.top(btex $ qui~a~ses~diagonales $ etex,(22.25,1)*u); label.bot(btex $ perpendiculaires $ etex,(22.25,1)*u); label.top(btex $ qui~a~un~angle $ etex,(22.25,-3)*u); label.bot(btex $ droit $ etex,(22.25,-3)*u); label.top(btex $ qui~a~ses~diagonales $ etex,(22.25,-1)*u); label.bot(btex $ de~m\hat{e}me~longueur $ etex,(22.25,-1)*u); label(btex $ Quadrilat\grave{e}re $ etex,(0,0)*u); draw (1.5,-1)*u--(1.5,1)*u withpen pencircle scaled 1.5pt; draw (-1.5,1.5)*u--(1.5,1)*u withpen pencircle scaled 1.5pt; draw (-1.5,1.5)*u--(-2,-2)*u withpen pencircle scaled 1.5pt; draw (-2,-2)*u--(1.5,-1)*u withpen pencircle scaled 1.5pt; label(btex $ Parall\acute{e}logramme $ etex,(9,0)*u); draw (7.25,0.7)*u--(11.25,0.7)*u withpen pencircle scaled 1.5pt; draw (6.75,-0.7)*u--(10.75,-0.7)*u withpen pencircle scaled 1.5pt; draw (6.75,-0.7)*u--(7.25,0.7)*u withpen pencircle scaled 1.5pt; draw (11.25,0.7)*u--(10.75,-0.7)*u withpen pencircle scaled 1.5pt; label(btex $ Rectangle $ etex,(18,5)*u); draw (16.8,5.7)*u--(19.2,5.7)*u--(19.2,4.3)*u--(16.8,4.3)*u--cycle withpen pencircle scaled 1.5pt; label(btex $ Losange $ etex,(18,-5)*u); draw (16.8,-5)*u--(18,-6)*u--(19.2,-5)*u--(18,-4)*u--cycle withpen pencircle scaled 1.5pt; label(btex $ Carr\acute{e} $ etex,(26.5,0)*u); draw (25,1.5)*u--(28,1.5)*u--(28,-1.5)*u--(25,-1.5)*u--cycle withpen pencircle scaled 1.5pt; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% MULTIPLICATION PAR i %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(39); drawarrow (-5cm,0)--(5cm,0); drawarrow(0,-4cm)--(0,4cm); drawarrow(0,0)--(3cm,2cm) withpen pencircle scaled 2bp; drawarrow(0,0)--(0,1cm); drawarrow(0,0)--(1cm,0); drawarrow(0,0)--(-3cm,-2cm) withpen pencircle scaled 2bp dashed evenly withcolor bleu_m; drawarrow(0,0)--(-2cm,3cm) withpen pencircle scaled 2bp dashed withdots scaled 0.5 withcolor red; % % %label.rt(btex Imaginaires etex,(0,4cm)); %label.top(btex Réels etex,(5cm,0cm)); label.llft(btex 0 etex,(0,-0.2cm)); label.urt(btex $M(3\ ,\ 2)$ etex,(3cm,2cm)); label.bot(btex $\overrightarrow{e_1}$ etex,(0.5cm,0)); label.ulft(btex $\overrightarrow{e_2}$ etex,(0,0.7cm)); label.llft(btex $M'(-3\ ,\ -2)$ etex,(-3cm,-2cm)); label.top(btex $M''(-2\ ,\ 3)$ etex,(-2cm,3cm)); draw fullcircle scaled 7.212cm dashed withdots; mark_rt_angle((3,2),(0,0),(-2,3)) endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ROTATIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(40) pair b,bb,bbb; b=(7cm,0);bb=b rotated 15;bbb=bb rotated 30; drawarrow(0,0)--(4cm,0)withpen pencircle scaled 1.3bp; drawarrow(0,0)--bb; drawarrow(0,0)--bbb withcolor bleu_m; label.rt(btex $M$ etex,bb); label.ulft(btex $M'$ etex, bbb) withcolor bleu_m; label.ulft(btex $C$ etex, origin); label.bot(btex $\overrightarrow{u}$ etex, (2cm,0)); %%%%%%%%%%%%%%%%%%%%%%% %Tracé arc de cercle %%%%%%%%%%%%%%%%%%%%%%25 %pair bb; % point de départ %pair bbb; % point d'arrivée draw bb{dir(90+angle (bb))}..{dir(90+angle (bbb))}bbb dashed evenly withcolor bleu_m ; %%%%%%%%%%%%%%%%%%%%%%% %Tracé de la flèche arrondie %%%%%%%%%%%%%%%%%%%%%% pair C; % point de départ de la flèche pair D; % point d'arrivée de la flèche C=1/3(origin+bb); D=C rotated 30; drawarrow C{dir(90+angle (C))}..{dir(90+angle (D))}D withcolor bleu_m; label.urt(btex $\alpha$ etex,C rotated 15) withcolor bleu_m; pair CC; % point de départ de la flèche pair DD; % point d'arrivée de la flèche CC=1/4(origin+b); DD=CC rotated 15; drawarrow CC{dir(90+angle (CC))}..{dir(90+angle (DD))}DD; label.urt(btex $\theta$ etex,CC rotated 7); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T V I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=4cm; uy:=4cm; xmin := -.5; xmax := 1.15; ymin := -.15; ymax := 1.15; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(41); axes; label.llft(btex $0$ etex,(-.02*ux,-.02*uy)); draw(0,0)--(1.2*ux,1.2*uy) withcolor 0.55*white; draw(0,uy)--(ux,uy)--(ux,0) withcolor 0.45*white; draw(0,0.4*uy)..(0.3*ux,0.5*uy)..(0.6*ux,0.1*uy)..(ux,0.3*uy) withpen pencircle scaled 1.5bp withcolor bleu_m; draw(0,0.4*uy) withpen pencircle scaled 6bp; draw(ux,0.3*uy) withpen pencircle scaled 6bp; label.bot(btex $1$ etex,(ux,0) shifted b); %label.bot(btex $-1$ etex,(-ux,0) shifted b); label.ulft(btex $1$ etex,(0,uy) shifted h+.5g); label.ulft(btex $y=x$ etex,(1.15*ux,1.15*uy)) withcolor 0.55*white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% P A T A T E S % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(42); x = 0.5000000cm; y = -.5000000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 path p; p = (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; path p; p = (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 14.200876% center->y = 25.647498% width = 33.300684% height = 33.300684% angle1 = 51.668255% angle2 = 97.769316% set_line_color 0.000000, 0.000000, 0.000000 draw (24.527647x,12.586423y)..(18.589187x,9.585848y)..(11.950000x,9.150000y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.814482x,12.827981y)--(24.270996x,12.697127y)--(24.527647x,12.586423y)--(24.593074x,12.314680y)--cycle; fill p withcolor bleu withpen pencircle scaled 0.1000x; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.814482x,12.827981y)--(24.270996x,12.697127y)--(24.527647x,12.586423y)--(24.593074x,12.314680y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 26.953504% center->y = -7.413825% width = 47.582265% height = 47.582265% angle1 = 230.903064% angle2 = 263.027989% set_line_color 0.000000, 0.000000, 0.000000 draw (11.950000x,11.050000y)..(17.644393x,14.480430y)..(24.065630x,16.201385y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 24.653780% center->y = -2.780425% width = 43.473579% height = 43.473579% angle1 = 234.883850% angle2 = 268.443102% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,15.000000y)..(17.815438x,17.852682y)..(24.063199x,18.948341y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 10.241431% center->y = -5.396278% width = 46.947990% height = 46.947990% angle1 = 274.663619% angle2 = 306.997863% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,18.000000y)..(18.588983x,16.543340y)..(24.367735x,13.351415y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,13.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,16.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.100000x,19.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.000000x,9.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,11.150000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.150000x,15.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,18.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{D\'EPART} etex scaled 2 ,(9.750000x,21.250000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ARRIV\'EE} etex scaled 2,(23.850000x,21.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(24.850000x,9.400000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(24.950000x,7.650000y)); endfig; beginfig(43) x = .5000000cm; y = -.5000000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 path p; p = (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle; fill p withcolor bleu_ciel;% (1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; path p; p = (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 14.200876% center->y = 25.647498% width = 33.300684% height = 33.300684% angle1 = 51.668255% angle2 = 97.769316% set_line_color 0.000000, 0.000000, 0.000000 draw (24.527647x,12.586423y)..(18.589187x,9.585848y)..(11.950000x,9.150000y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.814482x,12.827981y)--(24.270996x,12.697127y)--(24.527647x,12.586423y)--(24.593074x,12.314680y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.814482x,12.827981y)--(24.270996x,12.697127y)--(24.527647x,12.586423y)--(24.593074x,12.314680y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 26.953504% center->y = -7.413825% width = 47.582265% height = 47.582265% angle1 = 230.903064% angle2 = 263.027989% set_line_color 0.000000, 0.000000, 0.000000 draw (11.950000x,11.050000y)..(17.644393x,14.480430y)..(24.065630x,16.201385y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 24.653780% center->y = -2.780425% width = 43.473579% height = 43.473579% angle1 = 234.883850% angle2 = 268.443102% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,15.000000y)..(17.815438x,17.852682y)..(24.063199x,18.948341y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 10.241431% center->y = -5.396278% width = 46.947990% height = 46.947990% angle1 = 274.663619% angle2 = 306.997863% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,18.000000y)..(18.588983x,16.543340y)..(24.367735x,13.351415y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X}etex ,(25.050000x,13.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,16.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.100000x,19.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.000000x,9.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,11.150000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.150000x,15.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,18.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ D\'EPART} etex scaled 2 ,(9.750000x,21.250000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ ARRIV\'EE} etex scaled 2,(23.850000x,21.350000y)); endfig; beginfig(44); x = .5000000cm; y = -.5000000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 path p; p = (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; path p; p = (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 17.864070% center->y = 22.677709% width = 29.527961% height = 29.527961% angle1 = 64.726803% angle2 = 113.614082% set_line_color 0.000000, 0.000000, 0.000000 draw (24.167329x,9.326901y)..(18.077823x,7.915275y)..(11.950000x,9.150000y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.500629x,9.498761y)--(23.941655x,9.491815y)--(24.167329x,9.326901y)--(24.170802x,9.047414y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.500629x,9.498761y)--(23.941655x,9.491815y)--(24.167329x,9.326901y)--(24.170802x,9.047414y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 26.953504% center->y = -7.413825% width = 47.582265% height = 47.582265% angle1 = 230.903064% angle2 = 263.027989% set_line_color 0.000000, 0.000000, 0.000000 draw (11.950000x,11.050000y)..(17.644393x,14.480430y)..(24.065630x,16.201385y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 24.653780% center->y = -2.780425% width = 43.473579% height = 43.473579% angle1 = 234.883850% angle2 = 268.443102% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,15.000000y)..(17.815438x,17.852682y)..(24.063199x,18.948341y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 10.241431% center->y = -5.396278% width = 46.947990% height = 46.947990% angle1 = 274.663619% angle2 = 306.997863% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,18.000000y)..(18.588983x,16.543340y)..(24.367735x,13.351415y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,13.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,16.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.100000x,19.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.000000x,9.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,11.150000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.150000x,15.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,18.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ D\'EPART} etex scaled 2 ,(9.750000x,21.250000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ ARRIV\'EE} etex scaled 2,(23.850000x,21.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(24.850000x,9.400000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(24.950000x,7.750000y)); endfig; beginfig(45); x = .5000000cm; y = -.5000000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 path p; p = (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (14.350000x,13.050000y)..(11.100000x,20.000000y)..(7.850000x,13.050000y)..(11.100000x,6.100000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; path p; p = (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle; fill p withcolor bleu_ciel;%(1.000000,1.000000,1.000000); % set_linewidth 0.100000 % set_line_color 0.000000, 0.000000, 0.000000 draw (28.350000x,13.050000y)..(25.100000x,20.100000y)..(21.850000x,13.050000y)..(25.100000x,6.000000y)..cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 17.864070% center->y = 22.677709% width = 29.527961% height = 29.527961% angle1 = 64.726803% angle2 = 113.614082% set_line_color 0.000000, 0.000000, 0.000000 draw (24.167329x,9.326901y)..(18.077823x,7.915275y)..(11.950000x,9.150000y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.500629x,9.498761y)--(23.941655x,9.491815y)--(24.167329x,9.326901y)--(24.170802x,9.047414y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.500629x,9.498761y)--(23.941655x,9.491815y)--(24.167329x,9.326901y)--(24.170802x,9.047414y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 26.953504% center->y = -7.413825% width = 47.582265% height = 47.582265% angle1 = 230.903064% angle2 = 263.027989% set_line_color 0.000000, 0.000000, 0.000000 draw (11.950000x,11.050000y)..(17.644393x,14.480430y)..(24.065630x,16.201385y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438756x,16.238835y)--(23.916289x,16.437652y)--(24.065630x,16.201385y)--(23.966221x,15.940152y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 24.653780% center->y = -2.780425% width = 43.473579% height = 43.473579% angle1 = 234.883850% angle2 = 268.443102% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,15.000000y)..(17.815438x,17.852682y)..(24.063199x,18.948341y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.438197x,18.949619y)--(23.937348x,19.197913y)--(24.063199x,18.948341y)--(23.939052x,18.697916y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 10.241431% center->y = -5.396278% width = 46.947990% height = 46.947990% angle1 = 274.663619% angle2 = 306.997863% set_line_color 0.000000, 0.000000, 0.000000 draw (12.150000x,18.000000y)..(18.588983x,16.543340y)..(24.367735x,13.351415y) withpen pencircle scaled 0.2000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (24.662206x,13.119226y)--(24.424371x,13.625126y)--(24.367735x,13.351415y)--(24.114785x,13.232498y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,13.000000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.050000x,16.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(25.100000x,19.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.000000x,9.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,11.150000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.150000x,15.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(11.050000x,18.200000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ D\'EPART} etex scaled 2 ,(9.750000x,21.250000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{ ARRIV\'EE} etex scaled 2 ,(23.850000x,21.350000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{X} etex ,(24.850000x,9.400000y)); endfig; beginfig(46); x = .5000000cm; y = -.5000000cm; % set_line_color 0.000000, 0.000000, 0.000000 % set_line_color 1.000000, 1.000000, 1.000000 % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{\textit{x}} etex scaled 2.67988,(8.550000x,15.100000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{\textit{y}} etex scaled 2.67988,(25.400000x,15.100000y)); % set_linewidth 0.100000 %metapost_arc % center->x = 17.577207% center->y = 23.947740% width = 24.097172% height = 24.097172% angle1 = 53.638661% angle2 = 128.359080% set_line_color 0.000000, 0.000000, 0.000000 draw (24.720520x,14.245086y)..(17.367168x,11.900985y)..(10.100000x,14.500000y) withpen pencircle scaled 0.3000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (25.012845x,14.479971y)--(24.466488x,14.361675y)--(24.720520x,14.245086y)--(24.779668x,13.971908y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu; % set_line_color 0.000000, 0.000000, 0.000000 draw (25.012845x,14.479971y)--(24.466488x,14.361675y)--(24.720520x,14.245086y)--(24.779668x,13.971908y)--cycle withpen pencircle scaled 0.1000x withcolor bleu; % set_linewidth 0.100000 %metapost_arc % center->x = 17.591311% center->y = 5.673607% width = 24.752792% height = 24.752792% angle1 = 235.258220% angle2 = 307.060239% set_line_color 0.000000, 0.000000, 0.000000 draw (10.538264x,15.843647y)..(17.841698x,18.047470y)..(25.050000x,15.550000y) withpen pencircle scaled 0.3000x withcolor bleu_f; % set_line_color 0.000000, 0.000000, 0.000000 % fill_polygon path p; p = (10.239172x,15.617441y)--(10.788765x,15.719654y)--(10.538264x,15.843647y)--(10.487158x,16.118443y)--cycle; fill p withpen pencircle scaled 0.1000x withcolor bleu_f; % set_line_color 0.000000, 0.000000, 0.000000 draw (10.239172x,15.617441y)--(10.788765x,15.719654y)--(10.538264x,15.843647y)--(10.487158x,16.118443y)--cycle withpen pencircle scaled 0.1000x withcolor bleu_f; % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex \textbf{\textit{f}} etex scaled 2.67988,(17.050000x,11.050000y)); % set_line_color 0.000000, 0.000000, 0.000000 label.urt(btex $f^{-1}$ etex scaled 2.67988,(17.000000x,21.050000y)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % R E C I P R O Q U E % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(47); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% G R A P H I Q U E f-1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax)) ; % axe des y %label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x %label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=0.8cm; uy:=ux; xmin:=-3.5; xmax:=4.5; ymin:=xmin; ymax:=xmax; coefficient:=1; % coefficient d'échelle %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions de la fonction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) :=-2.5/x enddef; vardef trace (suffix F)(expr a,b,inc) := save i; numeric i; for i:=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Tracé des différentes courbes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric pas, seuil, seuilbis, abscisse; pas:=0.009; seuil:=3.75; seuilbis:=1.25; abscisse:=seuil-0.5; xA:=abscisse; yA:=f(abscisse); xB:=f(abscisse); yB:=abscisse; xD:=xmin/2; yD:=xD; xE:=0.75*seuil; yE:=xE; pair A, B, C, D, E, Ax, Ay, Bx, By; A:=(xA*ux,yA*uy); B:=(xB*ux,yB*uy); D:=(xD*ux,yD*uy); E:=(xE*ux,yE*uy); Ax:=A yscaled 0; Ay:=A xscaled 0; Bx:=B yscaled 0; By:=B xscaled 0; C=whatever[A,B]=whatever[D,E]; draw trace(F,f(seuilbis),f(seuil),pas); % tracé de la courbe de f draw trace(F,seuilbis,seuil,pas); % tracé de la courbe de f % segment((seuilbis*ux,0), -90, (seuil*ux,0), 90, (0,0), 1.4bp, 0.2); % segment((0,f(seuilbis)*uy), 0, (0,f(seuil)*uy), 180, (0,0), 1bp, 0.2); % label.top(btex $D$ etex, 2/3[(seuilbis*ux,0),(seuil*ux,0)]); % label.lft(btex $D'$ etex, 1/2[(0,f(seuilbis)*uy),(0,f(seuil)*uy)]); % segment((f(seuilbis)*ux,0), -90, (f(seuil)*ux,0), 90, (0,0), 1.4bp, 0.2); % segment((0,seuilbis*uy), 0, (0,seuil*uy), 180, (0,0), 1bp, 0.2); % label.bot(btex $D'$ etex, 1/2[(f(seuilbis)*ux,0),(f(seuil)*ux,0)]); % label.rt(btex $D$ etex, 2/3[(0,seuilbis*uy),(0,seuil*uy)]); % draw ((seuilbis*ux,f(seuilbis)*uy)--(seuilbis*ux,0)) dashed evenly; % draw ((seuil*ux,f(seuil)*uy)--(seuil*ux,0)) dashed evenly; % draw ((seuilbis*ux,f(seuilbis)*uy)--(0,f(seuilbis)*uy)) dashed evenly; % draw ((seuil*ux,f(seuil)*uy)--(0,f(seuil)*uy)) dashed evenly; % draw ((f(seuilbis)*ux,seuilbis*uy)--(f(seuilbis)*ux,0)) dashed evenly; % draw ((f(seuil)*ux,seuil*uy)--(f(seuil)*ux,0)) dashed evenly; % draw ((f(seuilbis)*ux,seuilbis*uy)--(0,seuilbis*uy)) dashed evenly; % draw ((f(seuil)*ux,seuil*uy)--(0,seuil*uy)) dashed evenly; draw (A--Ax) dashed evenly withcolor bleu; draw ((0,0)--Ax) withpen pencircle scaled 2bp withcolor bleu; draw (A--Ay) dashed evenly withcolor bleu_f; draw ((0,0)--Ay) withpen pencircle scaled 2bp withcolor bleu_f; draw (B--Bx) dashed evenly withcolor bleu_f; draw ((0,0)--Bx) withpen pencircle scaled 2bp withcolor bleu_f; draw (B--By) dashed evenly withcolor bleu; draw ((0,0)--By) withpen pencircle scaled 2bp withcolor bleu; draw (D--E) withcolor 0.55white dashed withdots withpen pencircle scaled 2.5bp; draw (A--B) withpen pencircle scaled 1bp; label(btex $\Vert$ etex, 2/3[A,C]); label(btex $\Vert$ etex, 2/3[B,C]); label.lrt(btex $\Gamma_f$ etex, (seuilbis*ux,f(seuilbis)*uy)); label.ulft(btex $\Gamma_{f^{-1}}$ etex, (f(seuilbis)*ux,seuilbis*uy)); draw (((1,0)--(1,1)--(0,1)) zscaled (5*unitvector(E-D)) shifted C); dotlabel.lrt(btex $(x,y)$ etex, A); dotlabel.ulft(btex $(y,x)$ etex, B); label.top(btex $x$ etex, Ax); label.llft(btex $y$ etex, Ay); label.urt(btex $x$ etex, By); label.ulft(btex $y$ etex, Bx); label.urt(btex $y=x$ etex, E) withcolor 0.55white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% pas monotone %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(48); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=0.45cm; uy:=ux; xmin:=-1; xmax:=9; ymin:=xmin; ymax:=9; coefficient:=1; % coefficient d'échelle %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions de la fonction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric h, debut, fin; h:=1.5; xA:=1*h; yA:=2*h; xE:=2.875*h; yE:=5*h; xD:=4.125*h; yD:=yE; xB:=3.5*h; yB:=5.25*h; xC:=4.5*h; yC:=4.75*h; pair A, D, E, B, C; A:=(xA*ux,yA*uy); D:=(xD*ux,yD*uy); E:=(xE*ux,yE*uy); B:=(xB*ux,yB*uy); C:=(xC*ux,yC*uy); draw (A .. E .. B .. D .. C) withpen pencircle scaled 1.3bp withcolor bleu_f; draw D withpen pencircle scaled 3bp withcolor 0.6white; draw E withpen pencircle scaled 3bp withcolor 0.6white; draw ((xD*ux,0)--D--(0,yD*uy)) dashed evenly withcolor 0.6white; draw (E--(xE*ux,0)) dashed evenly withcolor 0.6white; %dotlabel.llft(btex $A$ etex,A); label.bot(btex $x_2$ etex,(xD*ux,0)); label.bot(btex $x_1$ etex,(xE*ux,0)); label.lft(btex $y_0$ etex,(0,yE*uy)); label.rt(btex $y=f(x)$ etex,C) withcolor bleu_f; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% R E C I D es T A N G E N T E S %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(49); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=0.8cm; uy:=0.9ux; xmin:=-3.5; xmax:=4.5; ymin:=xmin; ymax:=xmax; coefficient:=1; % coefficient d'échelle % k:=-2.5; % intervient dans la fonction homothétique de x--> 1/x %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; label(btex $y=x$ etex,(3.3*ux,3.3*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions de la fonction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) :=(-2.5)/x enddef; vardef trace (suffix F)(expr a,b,inc) := save i; numeric i; for i:=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Tracé des différentes courbes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric pas, seuil, seuilbis, abscisse; pas:=0.009; seuil:=3.75; seuilbis:=0.75; abscisse:=seuilbis+1.5; xA:=abscisse; yA:=f(abscisse); xB:=f(abscisse); yB:=abscisse; xD:=xmin/2; yD:=xD; xE:=0.75*seuil; yE:=xE; xG:=seuilbis-0.25; yG:=2.5*(xG-xA)/(xA*xA)+yA; xH:=seuil-0.25; yH:=2.5*(xH-xA)/(xA*xA)+yA; xG':=yG; yG':=xG; xH':=yH; yH':=xH; pair A, B, C, D, E, G, H, G', H'; A:=(xA*ux,yA*uy); B:=(xB*ux,yB*uy); D:=(xD*ux,yD*uy); E:=(xE*ux,yE*uy); G:=(xG*ux,yG*uy); H:=(xH*ux,yH*uy); G':=(xG'*ux,yG'*uy); H':=(xH'*ux,yH'*uy); C=whatever[A,B]=whatever[D,E]; draw trace(F,f(seuilbis),f(seuil),pas) withpen pencircle scaled 1.3bp withcolor 0.55white; % tracé de la courbe de f draw trace(F,seuilbis,seuil,pas) withpen pencircle scaled 1.3bp withcolor bleu_m; % tracé de la courbe de f draw ((0,yA*uy)--A--(xA*ux,0)) dashed evenly withcolor bleu_f; draw ((0,yB*uy)--B--(xB*ux,0)) dashed evenly withcolor bleu_f; draw D--E withcolor bleu_f dashed withdots withpen pencircle scaled 2.5bp; draw G--H withcolor bleu_m; draw G'--H' withcolor 0.55white; draw A--B withpen pencircle scaled 1bp; label(btex $\Vert$ etex, 1/2[A,C]) withcolor bleu_f; label(btex $\Vert$ etex, 1/2[B,C]) withcolor bleu_f; label.lrt(btex $\Gamma_f$ etex, (seuil*ux,f(seuil)*uy)); label.llft(btex $\Gamma_{f^{-1}}$ etex, (f(seuil)*ux,seuil*uy)); label.urt(btex $\mathcal{T}$ etex, H); label.urt(btex $\mathcal{T'}$ etex, H'); draw (((1,0)--(1,1)--(0,1)) zscaled (5*unitvector(D-E)) shifted C); dotlabel.top(btex $x_0$ etex, (xA*ux,0)); dotlabel.lft(btex $y_0$ etex, (0,yA*uy)); dotlabel.bot(btex $y_0$ etex, (xB*ux,0)); dotlabel.rt(btex $x_0$ etex, (0,yB*uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% F o n c t i o n p u i s s a n c e p a i r e %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(50); vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax)) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=1cm; uy:=0.5*ux; xmin:=-4.5; xmax:=4.5; ymin:=-0.5; ymax:=8; coefficient:=1; % coefficient d'échelle %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; vardef f(expr x) =0.3*x*x enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; draw (trace(g,-4.5,0,.10)) dashed evenly withcolor bleu_m; draw (trace(g,0,4.5,.10)) withcolor bleu_m; endfig; beginfig(51); vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax)) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=1cm; uy:=0.5ux; xmin:=-.5; xmax:=9; ymin:=-0.5; ymax:=9; coefficient:=1; % coefficient d'échelle %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; vardef f(expr x) =0.3*x*x enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; % draw (trace(g,-4.5,0,.10)) dashed evenly withcolor bleu_m; draw (trace(g,0,5,.10)) withcolor bleu_m withpen pencircle scaled 2bp; vardef f(expr x) =sqrt(x/0.3)enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %draw (trace(g,-4.5,0,.10)) dashed evenly withcolor bleu_m; draw (trace(g,0,8,.10)) withcolor vert_e withpen pencircle scaled 2bp; draw ((0,0)--(8*ux,8*uy)) dashed evenly withcolor 0.5*bleu withpen pencircle scaled 2bp; label.urt(btex $y=x^{2n}$ etex scaled 1.5, (5*ux,0.3*25*uy)) withcolor bleu_m; label.urt(btex $y=\sqrt[2n]{x}$ etex scaled 1.5, (8*ux,f(8)*uy)) withcolor vert_e; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Puissances IMPAIRES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(52); vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax)) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=1cm; uy:=0.5ux; xmin:=-4.5; xmax:=4.5; ymin:=-9; ymax:=9; coefficient:=1; % coefficient d'échelle %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; vardef f(expr x) =0.3*x*x*x enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; % draw (trace(g,-4.5,0,.10)) dashed evenly withcolor bleu_m; draw (trace(g,-3,3,.10)) withcolor bleu_m withpen pencircle scaled 2bp; vardef f(expr x) =(x/0.3)**0.333 enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %draw (trace(g,-4.5,0,.10)) dashed evenly withcolor bleu_m; draw (trace(g,0.01,4.5,.10)) withcolor vert_e withpen pencircle scaled 2bp; draw( (trace(g,0.01,4.5,.10))scaled -1) withcolor vert_e withpen pencircle scaled 2bp; draw ((-4*ux,-4*uy)--(4*ux,4*uy)) dashed evenly withcolor 0.5*bleu withpen pencircle scaled 2bp; label.urt(btex $y=x^{2n+1}$ etex scaled 1.5, (3*ux,0.3*27*uy)) withcolor bleu_m; label.urt(btex $y=\sqrt[2n+1]{x}$ etex scaled 1.5, (4.5*ux,f(4.5)*uy)) withcolor vert_e; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% A R C S I N A R C C O S A R C T A N %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef sin(expr x) = sind(x/Pi*180) enddef; vardef cos(expr x) = cosd(x/Pi*180) enddef; vardef tan(expr x) = sin(x)/cos(x) enddef; beginfig(53); pair O,A[],B[],P,Q,M[],T[],X[],Y[]; path p[], axex,axey, droite; picture figure; numeric u,rayon,pi; u=3cm; rayon=1.5u; pi=3.1419; O=(0,0); A1 = (-1.1*pi*u/2,0); A2 =(1.1*pi*u/2,0); axex = A1--A2; drawarrow axex withpen pencircle scaled 1.2bp; label.urt(btex $x$ etex, A2); B1 = (0,-2u); B2 =(0,2u); axey = B1--B2; drawarrow axey withpen pencircle scaled 1.2bp; label.ulft(btex $y$ etex, B2); p6 = (-pi*u/2,u*sind(-90))%{dir(cosd(i))} for i=-89 upto 90: ..(i*pi*u/180,u*sind(i)){dir(cosd(i))} endfor; draw p6 withpen pencircle scaled 2bp withcolor vert_e; droite=(-pi*u/2,-pi*u/2)--(pi*u/2,pi*u/2); draw droite withcolor red dashed withdots withpen pencircle scaled 2bp; p7 = p6reflectedabout ((-pi*u/2,-pi*u/2),(pi*u/2,pi*u/2)); draw p7 withcolor bleu_m withpen pencircle scaled 3bp; %graduation de l'axe y dotlabel.ulft(btex $0$ etex ,(0,0)); dotlabel.lft(btex $1$ etex,(0,1u)); dotlabel.ulft(btex $\dfrac{\pi}{2}$ etex ,(0,pi*u/2)); dotlabel.lft(btex $-1$ etex ,(0,-1u)); dotlabel.llft(btex$-\dfrac{\pi}{2}$ etex ,(0,-pi*u/2)); %graduation de l'axe x dotlabel.bot(btex $0$ etex ,(0,0)); dotlabel.lrt(btex $1$ etex,(1u,0)); dotlabel.bot(btex $\dfrac{\pi}{2}$ etex ,(pi*u/2,0)); dotlabel.llft(btex $-1$ etex ,(-1u,0)); dotlabel.bot(btex $-\dfrac{\pi}{2}$ etex ,(-pi*u/2,0)); P = ((2.5*pi/6)*u, sind((2.5*180)/6)*u); label.bot(btex $\sin x$etex, P); Q = P reflectedabout ((0,0),(pi*u,pi*u)); label.lft(btex$\arcsin x$ etex, Q); draw (-u, -pi*u/2)--(u,-pi*u/2)--(u,pi*u/2)--(-u,pi*u/2)--cycle withcolor bleu; endfig; beginfig(54); pair O,A[],B[],P,Q,M[],T[],X[],Y[]; path p[], axex, axey, droite; picture figure; numeric u,rayon,pi; u=3cm; rayon=1.5u;pi=3.1419; O=(0,0); label.llft(btex $O$ etex, O); A1 = (-1.1u,0); A2 = (1.1*pi*u,0); axex = A1--A2; drawarrow axex withpen pencircle scaled 1.2bp; label.urt(btex $x$ etex, A2); B1 = (0,-1.2u); B2 = (0,1.1*pi*u); axey = B1--B2; drawarrow axey withpen pencircle scaled 1.2bp; label.ulft(btex $y$ etex, B2); p6 = (0*u,cosd(0)*u){dir(-sind(0))} for i=1 upto 180: ..(i*(pi/180)*u,u*cosd(i)){dir(-sind(i))} endfor; draw p6 withpen pencircle scaled 2bp withcolor vert_e; droite =(0,0)--(pi*u,pi*u); draw droite withcolor red dashed withdots withpen pencircle scaled 2bp; p7= p6 reflectedabout ((0,0),(pi*u,pi*u)); draw p7 withcolor bleu_m withpen pencircle scaled 3bp ; %Graduation axe y dotlabel.lft(btex $-1$ etex ,(0,-1u)); dotlabel.lft(btex $-0,5$ etex ,(0,-.5u)); dotlabel.ulft(btex $0$ etex ,(0,0)); dotlabel.lft(btex $0,5$ etex ,(0,0.5u)); dotlabel.lft(btex $1$ etex ,(0,1u)); dotlabel.urt(btex $\dfrac{\pi}{2}$ etex ,(0,pi*u/2)); dotlabel.ulft(btex $\pi$ etex ,(0,pi*u)); %Graduation axe x dotlabel.bot(btex $-1$ etex, (-1u,0)); dotlabel.bot(btex $1$ etex, (1u,0)); dotlabel.bot(btex $\dfrac{\pi}{2}$ etex, (pi*u/2,0)); dotlabel.bot(btex $\pi$ etex, (pi*u,0)); P = ((2.5*pi/6)*u, cosd((2.5*180)/6)*u); label.urt(btex $\cos x$ etex, P); Q = P reflectedabout ((0,0),(pi*u,pi*u)); label.urt(btex $\arccos x$ etex, Q); draw (-u, 0)--(-u,pi*u)--(u,pi*u)--(u,0) withcolor bleu; endfig; beginfig(55); pair O,A[],B[],P,Q,M[],T[],X[],Y[]; path p[], axex, axey, droite; picture figure; numeric u,rayon,pi; u=2cm; rayon=1.5u;pi=3.1419; O=(0,0); label.llft(btex $O$ etex, O); A1 = (-1.1*pi*u,0); A2 = (1.1*pi*u,0); axex = A1--A2; drawarrow axex withpen pencircle scaled 1.2bp; draw axex shifted (0,pi*u/2)withcolor bleu; draw axex shifted (0,-pi*u/2)withcolor bleu; label.urt(btex $x$ etex, A2); B1 = (0,-1.1*pi*u); B2 = (0,1.1*pi*u); axey = B1--B2; drawarrow axey withpen pencircle scaled 1.2bp; draw axey shifted (pi*u/2,0) withcolor bleu; draw axey shifted (-pi*u/2,0)withcolor bleu; label.ulft(btex $y$ etex, B2); p6 = (-75*(pi/180)*u,(sind(-75)/cosd(-75))*u)%{dir(-sind(-89))} for i=-74 upto 75: ..(i*(pi/180)*u,(sind(i)/cosd(i))*u)%{dir(-sind(i))} endfor; draw p6 withpen pencircle scaled 2bp withcolor vert_e; droite=(-pi*u,-pi*u)--(pi*u,pi*u); draw droite withcolor red dashed withdots withpen pencircle scaled 2bp; p7= p6 reflectedabout((-pi*u,-pi*u),(pi*u,pi*u)); draw p7 withcolor bleu_m withpen pencircle scaled 3bp ; %Graduation axe y dotlabel.llft(btex $-\pi$ etex ,(0,-pi*u)); dotlabel.lft(btex $-3$ etex ,(0,-3u)); dotlabel.llft(btex $-2$ etex ,(0,-2u)); dotlabel.lft(btex $-1$ etex ,(0,-1u)); dotlabel.llft(btex $-\dfrac{\pi}{2}$ etex ,(0,-pi*u/2)); dotlabel.lft(btex $-0,5$ etex ,(0,-.5u)); dotlabel.ulft(btex $0$ etex ,(0,0)); dotlabel.lft(btex $1$ etex ,(0,1u)); dotlabel.ulft(btex $\dfrac{\pi}{2}$ etex ,(0,pi*u/2)); dotlabel.ulft(btex $2$ etex ,(0,2u)); dotlabel.lft(btex $3$ etex ,(0,3u)); dotlabel.ulft(btex $\pi$ etex ,(0,pi*u)); %Graduation axe x dotlabel.llft(btex $-\pi$ etex, (-pi*u,0)); dotlabel.bot(btex $-3$ etex, (-3u,0)); dotlabel.llft(btex $-2$ etex, (-2u,0)); dotlabel.llft(btex $-\dfrac{\pi}{2}$ etex, (-pi*u/2,0)); dotlabel.bot(btex $-1$ etex, (-1u,0)); dotlabel.bot(btex $1$ etex, (1u,0)); dotlabel.llft(btex $\dfrac{\pi}{2}$ etex, (pi*u/2,0)); dotlabel.bot(btex $2$ etex, (2u,0)); dotlabel.bot(btex $3$ etex, (3u,0)); dotlabel.lrt(btex $\pi$ etex, (pi*u,0)); P = ((pi/4)*u, (sind(180/4)/cosd(180/4))*u); label.ulft(btex $\tan x$ etex, P); Q = P reflectedabout ((0,0),(pi*u,pi*u)); label.lrt(btex $\arctan x$ etex, Q); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% %% C E R C L E C O N G R U E N C E %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(56); draw fullcircle scaled 4cm; path p[]; pair A[];numeric n[]; p0:= (1.75cm,0)--(2.25cm,0); draw p0 withpen pencircle scaled 3bp; A0:=(2.25cm,0); for i=1 upto 4 : p[i]:=( p[i-1] )rotated 72; draw p[i] withpen pencircle scaled 3bp ; A[i]:=(A[i-1])rotated 72; endfor; label.rt(btex $0\ 5\ 10\ \cdots$ etex, A0) withcolor bleu_m; label.rt(btex $1\ 6\ 11\ \cdots$ etex,A[1]) withcolor bleu; label.lft(btex $\cdots\ 12\ 7\ 2$ etex,A[2]) withcolor 0.7white; label.lft(btex $\cdots\ 13\ 8\ 3$ etex,A[3]) withcolor bleu_f; label.rt(btex $4\ 9\ 14\ \cdots$ etex,A[4]) withcolor 0.5white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % Q C M C O M p L E X e s % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(57); numeric u; u:=1cm; pair o,a,b,c,d; o:=origin; a:=(sqrt(3)*u,-u); b:=a rotated 120; c:=b rotated 120; d:=a scaled -0.5; drawarrow o--a withcolor bleu_f withpen pencircle scaled 1.5bp; drawarrow o--b withcolor bleu_f withpen pencircle scaled 1.5bp; drawarrow o--c withcolor bleu_f withpen pencircle scaled 1.5bp; drawarrow o--d withcolor bleu_f withpen pencircle scaled 1.5bp; drawarrow d--b dashed evenly withcolor bleu_f withpen pencircle scaled 1.5bp; drawarrow d--a dashed evenly withcolor bleu_f withpen pencircle scaled 1.5bp ; label.bot(btex $\Omega$ etex,o shifted(0,-0.1*u)); label.rt(btex $A$ etex,a); label.top(btex $B$ etex,b); label.lft(btex $C$ etex,c); label.lft(btex $D$ etex, d); label.bot(btex $\overrightarrow{u}$ etex, o shifted (1.5*u,-2.2*u)); label.rt(btex $\overrightarrow{v}$ etex, o shifted (u,-1.5*u)); vardef axes = drawarrow (ux*xmin,0)shifted (u,-2*u) -- (ux*xmax,0)shifted (u,-2*u); % axe des x drawarrow (0,uy*ymin)shifted (u,-2*u) -- (0,uy*ymax)shifted (u,-2*u) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)shifted (u,-2*u)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)shifted (u,-2*u)); % label de l'axe des y enddef; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)shifted (u,-2*u)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=1cm; uy:=ux; xmin:=-4.5; xmax:=4.5; ymin:=-.5; ymax:=4.5; coefficient:=1; % coefficient d'échelle %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; drawarrow o shifted (u,-2*u)--o shifted(2*u,-2*u) withpen pencircle scaled 1.5bp withcolor 0.2white; drawarrow o shifted (u,-2*u)--o shifted(u,-u) withpen pencircle scaled 1.5bp withcolor 0.2white; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % L I M I T E S U I T E I N F N I E % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=1cm; uy:=0.8*ux; xmin:=-.5; xmax:=7.5; ymin:=-.5; ymax:=6.5; coefficient:=1; % coefficient d'échelle vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0); % axe des x drawarrow ((0,uy*ymin) -- (0,uy*ymax)) ; % axe des y label.lrt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; label.llft(btex $0$ etex,(-.1*ux,-.15*uy)); beginfig (59); numeric aa,bb,cc; %%%%%%%%%%%%%%%%%%%%%%%%%% %Axe %%%%%%%%%%%%%%%%%%%%%%%%%% axes; label.llft(btex $0$ etex,(0,0)); %%%%%%%%%%%%%%%%%%%%%%% %Fonction, graphe %%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =2.3*sqrt(x) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; draw trace(g,0,xmax,.1) withcolor 0.6white; %label.rt(btex $y=2\sqrt{x}$ etex,(xmax*ux,(f(xmax))*uy)); %%%%%%%%%%%%%%%%%%%%%%%% %Points sur le graphe %%%%%%%%%%%%%%%%%%%%%%% pair A[],M[],O[]; u:=.7;%facteur d'échelle %a:=2;b:=5;c:=9;%numeros des points particuliers aa:=2;bb:=5;cc:=9;%numeros des points particuliers for i=1 upto 12: A[i]:=(i*u*ux,0);% A comme abscisse M[i]:=(i*u*ux,f(i*u)*uy); %M comme courbe !! O[i]:=(0,f(i*u)*uy); %O comme ordonnée endfor ; for i=1 upto 10: draw M[i] withpen pencircle scaled 4bp withcolor 0.3white; endfor ; for i=aa,bb,cc: draw A[i]--M[i]--O[i] dashed evenly; endfor; label.bot(btex $n$ etex,A[aa]); label.bot(btex $N_1$ etex,A[bb]); label.bot(btex $N_2$ etex,A[cc]); label.lft(btex $u_n$ etex,O[aa] shifted (0,.1*uy)); label.lft(btex $u_{N_1}$ etex,O[bb] shifted (0,.1*uy)); label.lft(btex $u_{N_2}$ etex,O[cc] shifted (0,.2*uy)); %%%%%%%%%%%%%%%%%%% %Tracé des droites %%%%%%%%%%%%%%%%%% pair C,D; C=1/3(O[bb]+2O[bb-1]); %Point seuil D=1/3(O[cc]+2O[cc-1]); %Point seuil pickup pencircle scaled 2bp; draw C--C shifted (xmax*ux,0) withcolor bleu_m; draw D--D shifted (xmax*ux,0) withcolor bleu_m; label.lft(btex $A_1$ etex,C shifted (0,-.1*uy)); label.lft(btex $A_2$ etex,D shifted (0,-.1*uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % S U I T E S C O N V E R G E N T E S % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=1.25cm; uy:=1cm; xmin := -0.5; xmax := 8; ymin := xmin; ymax := xmax-1; beginfig(60); %%%%%%%%%%%%%%%%%%%%%%%%%% %Axe %%%%%%%%%%%%%%%%%%%%%%%%%% drawarrow (ux*xmin,0) -- ((1.3*xmax)*ux,0) ;% axe des x label.rt(btex \large{$\bbr$} etex,(1.3*xmax*ux,0)); %%%%%%%%%%%%%%%%%%%%%%%%%% %Points v(n) %%%%%%%%%%%%%%%%%%%%%%%%%% pair v[]; pair b; %shift vers le bas pour les labels for i=3 upto 26: v[i]=((i/(i+1)-3/4)*6*(xmax-.5)*ux,0); endfor b=(0,-.3*uy); for i=3 upto 24: draw v[i] withpen pencircle scaled 4bp; endfor for i=18 upto 23: draw v[i]shifted (v[17]-v[21]) withpen pencircle scaled 4bp; endfor label.bot(btex $u_0$ etex, v[3] shifted b); label.bot(btex $u_1$ etex, v[4] shifted b); %label.bot(btex $u_{N_1}$ etex, v[12] shifted b); %label.bot(btex $u_{N_2}$ etex, v[19] shifted b); %label.bot(btex $\cdots$ etex, 1/2(v[10]+v[16]) shifted b); %label.bot(btex $\cdots$ etex, 1/2(v[16]+v[19]) shifted b); %label.bot(btex $\cdots$ etex, v[20] shifted b+(.2*ux,0)); %%%%%%%%%%%%%%%%%%%%%%%%%% %Points seuils %%%%%%%%%%%%%%%%%%%%%%%%%% pair y[]; %points seuils pair h,hh; %shift vers le haut pour les labels path r,rr; %tiret vertical y[1]=(2v[8]+v[9])/3; y[2]=(v[15]+v[16])/2; y[3]=v[20]+(0.1*ux,0) ; y[4]=v[26]+(0.1*ux,0) ; h=(0,.3*uy);hh=(0,uy); r=(0,0.15*uy)--(0,-0.15*uy); rr=(0,0.85*uy)--(0,-0.85*uy); draw r shifted y[1] withpen pencircle scaled 1.5bp withcolor bleu_m; draw r shifted y[2] withpen pencircle scaled 1.5bp withcolor bleu_m; draw r shifted y[4] withpen pencircle scaled 1.5bp withcolor bleu_m; draw rr shifted y[3] withpen pencircle scaled 4bp withcolor bleu_m; label.top(btex $\ell-1$ etex, y[1] shifted h) withcolor bleu_m; label.top(btex $\ell-0,1$ etex, y[2] shifted h) withcolor bleu_m; label.top(btex $\ell+0,1$ etex, y[4] shifted h) withcolor bleu_m; label.top(btex \LARGE{\boldmath$\ell$\unboldmath} etex, y[3] shifted hh) withcolor bleu; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% % % S U I T E S R E C C o N V E S C A L I E R % %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=2cm; uy:=2cm; xmin :=-1.3 ; xmax :=2.3; ymin := -.2; ymax :=2; pair d,h; d:=(.1*ux,0); h:=(0,.05*uy); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.rt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(61); axes; vardef f(expr x) = sqrt(1+x) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,g(i)*uy) .. endfor (b*ux,g(b)*uy) enddef; path P,Q; P:=trace(f,-1,2,.01) ; Q:=(-.2*ux,-.2*uy)--(2*ux,2*uy); draw P withpen pencircle scaled 1.3bp; draw Q withcolor bleu; label.rt(btex $\displaystyle y=\sqrt{1+x}$ etex,(2*ux,f(2)*uy)); label.rt(btex $y=x$ etex,(2*ux,2*uy)); draw (-.5*ux,0)--(-.5*ux,f(-.5)*uy)--(f(-.5)*ux,f(-.5)*uy)--(f(-.5)*ux,f(f(-.5))*uy)--(f(f(-.5))*ux,f(f(-.5))*uy)--(f(f(-.5))*ux,f(f(f(-.5)))*uy)withcolor bleu_m; %(-.5*ux,0)--(-.5*ux,f(-.5)*uy) drawarrow (-.5*ux,0)--1/2[(-.5*ux,0),(-.5*ux,f(-.5)*uy)]withcolor bleu_m; draw 1/2[(-.5*ux,0),(-.5*ux,f(-.5)*uy)]--(-.5*ux,f(-.5)*uy)withcolor bleu_m; %(-.5*ux,f(-.5)*uy)--(f(-.5)*ux,f(-.5)*uy) drawarrow (-.5*ux,f(-.5)*uy)--1/2[(-.5*ux,f(-.5)*uy),(f(-.5)*ux,f(-.5)*uy)]withcolor bleu_m; draw 1/2[(-.5*ux,f(-.5)*uy),(f(-.5)*ux,f(-.5)*uy)]--(f(-.5)*ux,f(-.5)*uy)withcolor bleu_m; %(f(-.5)*ux,f(-.5)*uy)--(f(-.5)*ux,f(f(-.5))*uy) drawarrow (f(-.5)*ux,f(-.5)*uy)--1/2[(f(-.5)*ux,f(-.5)*uy),(f(-.5)*ux,f(f(-.5))*uy)]withcolor bleu_m; draw 1/2[(f(-.5)*ux,f(-.5)*uy),(f(-.5)*ux,f(f(-.5))*uy)]--(f(-.5)*ux,f(f(-.5))*uy)withcolor bleu_m; %(f(-.5)*ux,f(f(-.5))*uy)--(f(f(-.5))*ux,f(f(-.5))*uy) drawarrow (f(-.5)*ux,f(f(-.5))*uy)--1/2[(f(-.5)*ux,f(f(-.5))*uy),(f(f(-.5))*ux,f(f(-.5))*uy)]withcolor bleu_m; draw 1/2[(f(-.5)*ux,f(f(-.5))*uy),(f(f(-.5))*ux,f(f(-.5))*uy)]--(f(f(-.5))*ux,f(f(-.5))*uy)withcolor bleu_m; %(f(f(-.5))*ux,f(f(-.5))*uy)--(f(f(-.5))*ux,f(f(f(-.5)))*uy) drawarrow (f(f(-.5))*ux,f(f(-.5))*uy)--2/3[(f(f(-.5))*ux,f(f(-.5))*uy),(f(f(-.5))*ux,f(f(f(-.5)))*uy)]withcolor bleu_m; draw 2/3[(f(f(-.5))*ux,f(f(-.5))*uy),(f(f(-.5))*ux,f(f(f(-.5)))*uy)]--(f(f(-.5))*ux,f(f(f(-.5)))*uy)withcolor bleu_m; draw((f(f(-.5))*ux,f(f(f(-.5)))*uy)--(1.5*ux,f(f(f(-.5)))*uy))dashed evenly withpen pencircle scaled 1.3bp withcolor bleu_m; draw ((-ux,0) shifted h)--((-ux,0) shifted -h); label.bot(btex $-1$ etex,(-ux,0) shifted -h); draw ((ux,0) shifted h)--((ux,0) shifted -h); label.bot(btex $1$ etex,(ux,0) shifted -h); draw ((2*ux,0) shifted h)--((2*ux,0) shifted -h); label.bot(btex $2$ etex,(2*ux,0) shifted -h); numeric n; n:=(1+ sqrt(5))/2; draw ((n*ux,0)--(n*ux,n*uy)) dashed evenly withcolor bleu_f; draw ((n*ux,0) shifted h)--((n*ux,0) shifted -h)withcolor bleu_f; label.bot(btex \large{$\f$} etex,(n*ux,0) shifted -h)withcolor bleu_f; label.ulft(btex $0$ etex,(0,0)); label.ulft(btex $1$ etex,(0,uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % E S C A R G O T % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=2cm; uy:=2cm; xmin :=-.5 ; xmax :=2.3; ymin := -.3; ymax :=2.5; pair d,h; d:=(.05*ux,0); h:=(0,.05*uy); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.rt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(62); axes; vardef f(expr x) = 2/(x+1) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,g(i)*uy) .. endfor (b*ux,g(b)*uy) enddef; path P,Q; P:=trace(f,-.2,2,.01) ; Q:=(-.2*ux,-.2*uy)--(2*ux,2*uy); draw P withpen pencircle scaled 1.3bp; draw Q withcolor bleu; label.rt(btex $\displaystyle y=f(x)$ etex,(2*ux,f(2)*uy)); label.rt(btex $y=x$ etex,(2*ux,2*uy))withcolor bleu; draw (.2*ux,0)--(.2*ux,f(.2)*uy)--(f(.2)*ux,f(.2)*uy)--(f(.2)*ux,f(f(.2))*uy)--(f(f(.2))*ux,f(f(.2))*uy)--(f(f(.2))*ux,f(f(f(.2)))*uy)--(f(f(f(.2)))*ux,f(f(f(.2)))*uy)withcolor bleu_m; %(.2*ux,0)--(.2*ux,f(.2)*uy) drawarrow (.2*ux,0)--1/2[(.2*ux,0),(.2*ux,f(.2)*uy)]; draw 1/2[(.2*ux,0),(.2*ux,f(.2)*uy)]--(.2*ux,f(.2)*uy)withcolor bleu_m; %(.2*ux,f(.2)*uy)--(f(.2)*ux,f(.2)*uy) drawarrow (.2*ux,f(.2)*uy)--1/2[(.2*ux,f(.2)*uy),(f(.2)*ux,f(.2)*uy)]withcolor bleu_m; draw 1/2[(.2*ux,f(.2)*uy),(f(.2)*ux,f(.2)*uy)]--(f(.2)*ux,f(.2)*uy)withcolor bleu_m; %(f(.2)*ux,f(.2)*uy)--(f(.2)*ux,f(f(.2))*uy) drawarrow (f(.2)*ux,f(.2)*uy)--1/2[(f(.2)*ux,f(.2)*uy),(f(.2)*ux,f(f(.2))*uy)]withcolor bleu_m; draw 1/2[(f(.2)*ux,f(.2)*uy),(f(.2)*ux,f(f(.2))*uy)]--(f(.2)*ux,f(f(.2))*uy)withcolor bleu_m; %(f(.2)*ux,f(f(.2))*uy)--(f(f(.2))*ux,f(f(.2))*uy) drawarrow (f(.2)*ux,f(f(.2))*uy)--1/2[(f(.2)*ux,f(f(.2))*uy),(f(f(.2))*ux,f(f(.2))*uy)]withcolor bleu_m; draw 1/2[(f(.2)*ux,f(f(.2))*uy),(f(f(.2))*ux,f(f(.2))*uy)]--(f(f(.2))*ux,f(f(.2))*uy)withcolor bleu_m; %(f(f(.2))*ux,f(f(.2))*uy)--2/3[(f(f(.2))*ux,f(f(.2))*uy),(f(f(.2))*ux,f(f(f(.2)))*uy)] drawarrow (f(f(.2))*ux,f(f(.2))*uy)--2/3[(f(f(.2))*ux,f(f(.2))*uy),(f(f(.2))*ux,f(f(f(.2)))*uy)]withcolor bleu_m; draw 2/3[(f(f(.2))*ux,f(f(.2))*uy),(f(f(.2))*ux,f(f(f(.2)))*uy)]--2/3[(f(f(.2))*ux,f(f(.2))*uy),(f(f(.2))*ux,f(f(f(.2)))*uy)]withcolor bleu_m; draw ((f(f(f(.2)))*ux,f(f(f(.2)))*uy)--(f(f(f(.2)))*ux,uy))dashed withdots withpen pencircle scaled 1.3bp withcolor bleu_m; draw ((.2*ux,0) shifted h)--((.2*ux,0) shifted -h); label.bot(btex $u_0$ etex,(.2*ux,0) shifted -h)withcolor bleu_m; draw ((f(.2)*ux,0) shifted h)--((f(.2)*ux,0) shifted -h); draw ((f(.2)*ux,0) --(f(.2)*ux,f(f(.2))*uy)) dashed evenly withcolor bleu_f ; label.bot(btex $u_1$ etex,(f(.2)*ux,0) shifted -h)withcolor bleu_f; draw ((f(f(.2))*ux,0) shifted h)--((f(f(.2))*ux,0) shifted -h)withcolor bleu_f; draw ((f(f(.2))*ux,0) --(f(f(.2))*ux,f(f(f(.2)))*uy)) dashed evenly withcolor bleu_f; label.bot(btex $u_2$ etex,(f(f(.2))*ux,0) shifted -h)withcolor bleu_f; draw ((f(f(f(.2)))*ux,0) shifted h)--((f(f(f(.2)))*ux,0) shifted -h)withcolor bleu_f; draw ((f(f(f(.2)))*ux,0) --(f(f(f(.2)))*ux,f(f(f(f(.2))))*uy))withcolor bleu_f dashed evenly; label.bot(btex $u_3$ etex,(f(f(f(.2)))*ux,0) shifted -h+d/2)withcolor bleu_f; draw((f(f(f(.2)))*ux,f(f(f(.2)))*uy)--(f(f(f(.2)))*ux,f(f(f(f(.2))))*uy))dashed evenly withpen pencircle scaled 1.3bp withcolor bleu_f; draw ((ux,0)--(ux,uy)) dashed evenly withcolor bleu_f; draw ((ux,0) shifted h)--((ux,0) shifted -h)withcolor bleu_f; label.bot(btex $\large{\ell}$ etex,(ux,0) shifted -h-d/2)withcolor bleu_f; label.ulft(btex $0$ etex,(0,0)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %% A R C T A N G E N T E TRIANGLE % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(63); u:=1cm; pair a,b,c,d,e; pair f,g,h,i,j,k; pair l,m,n,o,p,q; pair r,s,t,z; pair v,w,x,aa; a:=(0,0); b=(2u,0); c=(u,u);d=(3u,3u); e=(12u,0); x=(4u,4u); f=(5u,3u);g=(6u,2u);h=(7u,3u); i=(8u,2u);j=(9u,u);k=(10u,2u);l=(11u,u);m=(11u,-u);n=(10u,-2u);o=(9u,-u);p=(8u,0); q=(7u,-u);r=(6u,0);s=(5u,-u);t=(4u,0);z=(3u,-u);v=(2u,-2u);w=(u,-u);aa=(2u,2u); fill a--b--c--cycle withcolor 0.6white; fill c--b--d--cycle withcolor bleu; fill b--e--d--cycle withcolor bleu_ciel; pickup pencircle scaled 2pt; draw a--b--c--cycle; draw c--b--d--cycle; draw b--d--e--cycle; pickup pencircle scaled 0.8pt; %for i=1 upto 5 : draw (((a shifted (i*u,0)) )--((b shifted (i*u,0)) )--((c shifted (i*u,0)) )); endfor draw a--x; draw w--f;draw v--h;draw s--i;draw q--k;draw o--l;draw n--e; draw a--v;draw c--z;draw aa--s;draw d--q;draw x--n; draw h--m;draw k--e; draw halfcircle scaled 2u shifted(2*u,0); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %% % % A R C H I M E D E % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Un peu de couleurs color couleurtrace; couleurtrace := (0.3,0.4,1); %% Procédure d'affectation def affecte(suffix p)(expr coords,nom) = pair p.c; string p.n; p.c := coords; p.n := nom; enddef; %% Lecture des coordonnées vardef getcoordonnees suffix p = p.c enddef; %% Lecture du nom vardef getnom suffix p = p.n enddef; %% La « procédure » vardef placePolygone(text t) = pair g,c__[] ; string n__[]; path P; n_:=0; forsuffixes pp = t: c__[incr n_] = getcoordonnees pp; n__[n_] = getnom pp; endfor g := (0,0); for i=1 upto n_: g := g + c__[i]; endfor g := (1/n_)*g; draw c__[1]--for i=2 upto n_:c__[i]--endfor cycle withcolor couleurtrace withpen pencircle scaled 2bp; P:=c__[1]--for i=2 upto n_:c__[i]--endfor cycle; for i=1 upto n_: draw c__[i] withpen pencircle scaled 2bp withcolor couleurtrace; %label(n__[i],c__[i]+2*labeloffset*unitvector(c__[i]-g)); endfor; enddef; %Conversion de polygone.1 au format PNG beginfig(64); % Affectation des points pair A,B,C,D,E,F; A := (2cm,0); B := A rotated 60; C := A rotated 120; D := A rotated 180; E:= A rotated 240; F:= A rotated 300; affecte(a,A,"A"); affecte(b,B,"B"); affecte(c,C,"C"); affecte(d,D,"D"); affecte(e,E,"E"); affecte(f,F,"F"); % Construction du triangle ABC avec sommets nommés couleurtrace := bleu_f; placePolygone(a,b,c,d,e,f); fill P withcolor bleu_ciel; transform T; T := identity zscaled (1,0.5773502); pair AA,BB,CC,DD,EE,FF; AA := A transformed T; BB := B transformed T; CC := C transformed T; DD := D transformed T; EE := E transformed T; FF := F transformed T; affecte(aa,AA,"A' "); affecte(bb,BB,"B' "); affecte(cc,CC,"C' "); affecte(dd,DD,"D' "); affecte(ee,EE,"E' "); affecte(ff,FF,"F' "); % Construction du quadrilatère ABCD avec sommets nommés couleurtrace := bleu_m; placePolygone(aa,bb,cc,dd,ee,ff); draw fullcircle scaled 4cm; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù % % % % R E S E R V O i R E p % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ùùù beginfig(65); pair A[],B[],C[],D[]; path p,pp; A[0]:=(0,0); A[1]:=(10cm,0); A[2]:=(13cm,4cm); A[3]:=(3cm,4cm); for i=0 upto 3: B[i]:=A[i] shifted (0,2cm); endfor for i=0 upto 3: C[i]:=A[i] shifted (0,2.5cm); endfor for i=0 upto 3: D[i]:=A[i] shifted (0,3.5cm); endfor p:=C[0]--C[1]--C[2]--C[3]--cycle; pp:=B[0]--B[1]--B[2]--C[2]--C[1]--C[0]--cycle; fill p withcolor bleu_ciel; fill pp withcolor bleu_ciel; draw A[0]--A[1]--A[2]; draw B[0]--B[1]--B[2] withcolor bleu_m withpen pencircle scaled 2bp; draw C[0]--C[1]--C[2] withcolor bleu_m withpen pencircle scaled 2bp; draw D[0]--D[1]--D[2]--D[3]--cycle; draw A[2]--A[3]--A[0] dashed evenly; draw B[2]--B[3]--B[0] withcolor bleu_m dashed evenly withpen pencircle scaled 2bp; draw C[2]--C[3]--C[0] withcolor bleu_m dashed evenly withpen pencircle scaled 2bp; draw A[0]--D[0]; draw A[1]--D[1]; draw A[2]--D[2]; draw A[3]--D[3] dashed evenly; label.lft(btex $h_i$ etex scaled 1.5, B[0]); label.lft(btex $h_{i+1}$ etex scaled 1.5, C[0]); label.lft(btex $h_0=0$ etex scaled 1.5, A[0]); label.lft(btex $h_n=100$ etex scaled 1.5, D[0]); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % S U B D I V S I on P L U s F I N E % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(66); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ux:=2.3cm; uy:=0.8cm; xmin:=-.5; xmax:=7; ymin:=xmin; ymax:=6; coefficient:=1; % coefficient d'échelle % k:=-2.5; % intervient dans la fonction homothétique de x--> 1/x %%%%%%%%%%%%%%%%%%%%%%%%%% %Tracé des axes %%%%%%%%%%%%%%%%%%%%%%%%%% axes; pair a,b,c,d,e,f,bb,cc,dd,ee,xa,xc,xf,cu,cd,ct,gu,gd,gt,gq; a:=(ux,2.5*uy); b:=(2*ux,4*uy); c:=(3*ux,2*uy); d:=(3.4*ux,1.2*uy); e:=(4.5*ux,5*uy); f:=(5*ux,3.5*uy); bb:=b xscaled 0; cc:=c xscaled 0; dd:=d xscaled 0; ee:=e xscaled 0; xa:=a yscaled 0; xc:=c yscaled 0; xf:=f yscaled 0; cu:=(ux,2*uy); cd:=(ux,4*uy); ct:=(3*ux,4*uy); gu:=(ux,1.2*uy); gd:=(5*ux,1.2*uy); gt:=(5*ux,5*uy); gq:=(ux,5*uy); path p,pp,ppp; pp:=a..tension1.3..{dir 0}b{dir 0}..c..{dir 0}d{dir 0}..tension1.5..{dir 0}e{dir 0}..tension1.3..{dir300}f; p:=c--cu--cd--ct--cycle; ppp:=gu--gd--gt--gq--cycle; fill ppp withcolor bleu_ciel; fill p withcolor bleu_m; draw pp withpen pencircle scaled 1.5bp; dotlabel.top(btex $ $ etex,b); dotlabel.bot(btex $ $ etex,d); dotlabel.top(btex $ $ etex,e); dotlabel.urt(btex $ $ etex,c); label.lft(btex $M_i$ etex, ee); label.lft(btex $M'_i$ etex, bb); label.lft(btex $m'_i$ etex, cc); label.lft(btex $m_i$ etex, dd); label.bot(btex $a_i$ etex,xa); label.bot(btex $a'_i$ etex,xc); label.bot(btex $a_{i+1}$ etex,xf); draw ee--e dashed withdots withpen pencircle scaled 1.5bp; draw dd--d dashed withdots withpen pencircle scaled 1.5bp; draw f--xf dashed withdots withpen pencircle scaled 1.5bp; draw a--xa dashed withdots withpen pencircle scaled 1.5bp; draw bb--b dashed evenly withcolor bleu_m withpen pencircle scaled 1.3bp; draw cc--c--xc dashed evenly withcolor bleu_m withpen pencircle scaled 1.3bp; endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % I N T E G F O N C T I O N D e L A B O R N E S U P % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1cm; uy:=1cm; xmin := -.5 ; xmax := 9; ymin := -1.5 ; ymax := 3; pair d,h; d:=(.1*ux,0); h:=(0,.1*uy); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (xmin*ux,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,ymin*uy) -- (0,uy*ymax); % axe des y label.rt(btex $x$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(67); pair A,B; A:=(-0.5*ux,-uy); B:=(7*ux,3*uy); path C; C:=A{dir10}..B{dir10}; pair a,b; a:=A yscaled 0; b:=B yscaled 0; pair M,m; M:= point .8 of C; m:=M yscaled 0; path U; U:=(0,0)--(5*ux,0); pair X; X:= C intersectionpoint U; path V,K,W; V:=M--m; K:=m--X; W:=buildcycle(C,V,K); path Y,Z; Y:=A--a--X; Z:=buildcycle(C,reverse Y); fill W withcolor bleu; fill Z withcolor bleu; %%%%%%%%%%%%%%%%%%%% % tracés %%%%%%%%%%%%%%%%%%%% axes; pickup pencircle scaled 1.3bp; draw C withcolor bleu_f; label.rt(btex $y=f(x)$ etex,B shifted d)withcolor 0.3white; draw (A--a) withcolor 0.3white; draw (B--b) withcolor 0.3white; draw a--(a shifted h); label.top(btex $a$ etex,a shifted h); draw (b shifted h)--(b shifted -h); label.bot(btex $b$ etex,b shifted -h); draw (M--m) withcolor 0.3white; draw (m shifted h)--(m shifted -h); label.bot(btex $t$ etex,m shifted -h); label.rt(btex $\displaystyle\int_a^tf(u)du$ etex, (X+m)/2 shifted (-0.31ux,0.75*uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % C E N T R E D ' I N E R T I E % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(72); pair A,B; A:=(1*ux,0); B:=(7*ux,3*uy); path C; C:=A{dir10}..B{dir10}; pair a,b; a:=A yscaled 0; b:=B yscaled 0; pair M,m; M:= point .7 of C; m:=M yscaled 0; pair N,n; N:= point .78 of C; n:=N yscaled 0; path U; U:=N--n; path V; V:=M--(M shifted (6*ux,0)); pair X; X:= V intersectionpoint U; path W; W:=M--X--n--m--cycle; %path i,j; i:=(M shifted (-0.3*ux,0))--(m shifted (-0.3*ux,0)); j:=(N shifted (0.3*ux,0.3*uy))--(n shifted (0.3*ux,0)); path t,r; t:=C cutafter i cutbefore j; path r; r:=N--X--M; path Z; Z:=buildcycle(C,r); fill Z withcolor .2white; fill W withcolor bleu; path UU; UU:=(0,0)--(8*ux,0); pair XX; XX:= C intersectionpoint UU; path VV,K,WW; VV:=M--m; K:=m--XX; WW:=buildcycle(C,VV,K); path Y,ZZ; Y:=A--a--XX; ZZ:=buildcycle(C,reverse Y); path ZZZ,ZZZZ; ZZZ:=B--b--n--N; ZZZZ:=buildcycle(C,ZZZ); fill ZZZZ withcolor bleu_ciel; fill WW withcolor bleu_ciel; fill ZZ withcolor bleu_ciel; %%%%%%%%%%%%%%%%%%%% % tracés %%%%%%%%%%%%%%%%%%%% axes; pickup pencircle scaled 1.3bp; draw C withcolor bleu_f; label.rt(btex $y=f(x)$ etex,B shifted d)withcolor 0.3white; draw (A--a) withcolor 0.3white; draw (B--b) withcolor 0.3white; draw a--(a shifted h); label.top(btex $a$ etex,a shifted h); draw (b shifted h)--(b shifted -h); label.bot(btex $b$ etex,b shifted -h); draw (M--m) withcolor 0.3white; draw (m shifted h)--(m shifted -h); label.bot(btex $x$ etex,m shifted -2h); %label.rt(btex $\displaystyle\int_a^tf(u)du$ etex, (X+m)/2 shifted (-0.31ux,0.75*uy)); draw (N--n) ; draw (n shifted h)--(n shifted -h); label.bot(btex {$x\!+\!\d x$} etex,n shifted -h +(0.3*ux,0)); draw (X--(X xscaled 0)) withpen pencircle scaled 1bp; draw ((X xscaled 0)shifted d)--((X xscaled 0)shifted -d); label.lft(btex $f(x)$ etex,(X xscaled 0)shifted -d); dotlabel.top(btex $\mathbf{G_x}$ etex, ((M+N)/2) yscaled 0.5); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % %% S E R I E I N T E G R A L E % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO %OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO beginfig(69); ux:=0.5cm; uy:=ux; xmin:=-0.5; xmax:=8; ymin:=-0.5; ymax:=6; %%%%%%%%%%%%%%%%%%%%%%%%%% %Axe %%%%%%%%%%%%%%%%%%%%%%%%%% axes; %%%%%%%%%%%%%%%%%%%%%%% %Fonction, graphe %%%%%%%%%%%%%%%%%%%%%%% vardef f(expr x) =7*((0.8)**x) enddef; vardef fx(expr x) =x enddef; %vardef fy(expr x) =7*((0.8)**x) enddef; vardef trace (suffix g)(expr a,b,inc) = save i; numeric i; for i=a step inc until b: (i*ux,f(i)*uy) .. endfor (b*ux,f(b)*uy) enddef; %%%%%%%%%%%%%%%%%%%%%%%% %Rectangles %%%%%%%%%%%%%%%%%%%%%%% % une macro non parfaite pour obtenir le min de f sur un intervalle, % cela suppose qu'elle soit bien reguliere...... vardef minf(suffix fx,fy)(expr a,b) = save m,i; numeric m,i; m:=fy(a); for i=a step (b-a)/100 until b: if m>fy(i): m:=fy(i); fi; endfor; m enddef; %rectangle en dessous de la courbe colorié en bleu, utilise la macro precedente vardef trace_rectangles_min (suffix fx,fy)(expr a,b,inc) = save i; numeric i; for i=a step inc until b-inc: path p; numeric m; m:=minf(fx,fy,i,i+inc); p = (i*ux,0)--((i+inc)*ux,0)--((i+inc)*ux,m*uy)--(i*ux,m*uy)--cycle; %p := p scaled 1cm; fill p withcolor bleu_ciel; draw p; endfor; enddef; trace_rectangles_min(fx,f,1,xmax-1,0.75); draw trace(g,1,7.8,.1); label.rt(btex $y=f(x)$ etex,((xmax-.2)*ux,(f(xmax-.2))*uy)); %label.bot(btex $2$ etex, (1*ux,0)); %abel.bot(btex $3$ etex, (2*ux,0)); %label.bot(btex $4$ etex, (3*ux,0)); %label.bot(btex $\cdots$ etex, (4.5*ux,0)); %label.bot(btex $N\!-\!1$ etex, (6*ux,0)); %label.bot(btex $N$ etex, (7*ux,0)); %label.lft(btex $f(3)$ etex, (0,f(2)*uy)); %label.lft(btex $f(4)$ etex, (0,f(3)*uy)); %label.lft(btex $f(N-1)$ etex, (0,f(6)*uy)); %label.lft(btex $f(N)$ etex, (0,f(7)*uy)); %for i=2,3,6,7:draw (0,f(i)*uy)--(i*ux,f(i)*uy) dashed evenly;endfor endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % C U B E S R E C U R R E N C E % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% beginfig(70); numeric u; u:=0.5cm; pair a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,v,x,y,z; a:=(0,0); b:=a shifted (u,0); c:=b shifted (u,0);d:=c shifted (u,0); e:=d shifted (u,0); f:= e shifted (u,0); g:=(0,4*u); v:=g shifted (u,0); h:=v shifted (u,0); i:=h shifted (u,0); j:= i shifted (u,0); k:= j shifted (u,0); l:= a shifted (0,u); m:=l shifted (0,u); n:=m shifted (0,u); o:=f shifted (0,u); p:=o shifted (0,u); q:=p shifted (0,u); t:=f shifted (-u,u); s:=t shifted (-u,u);r:=s shifted (-u,u);x:=(u,3*u); y:=(2*u,2*u); z:=(3*u,u); path w; w:=a--g--v--x--r--y--s--z--t--e--cycle; fill w withcolor bleu_ciel; pickup pencircle scaled 1.3bp; draw l--t--e--a--g--v--b withcolor bleu_m; draw n--r--c withcolor bleu_m; draw m--s--d withcolor bleu_m; draw v--k--f--e dashed evenly; draw r--q dashed evenly; draw s--p dashed evenly; draw j--t dashed evenly; draw i--s dashed evenly; draw h--r dashed evenly; draw t--o dashed evenly; label.lft(btex $1$ etex, (g+n)/2); label.lft(btex $2$ etex, (m+n)/2); label.lft(btex $3$ etex, (m+l)/2); label.lft(btex $4$ etex, (l+a)/2); label.bot(btex $1$ etex, (a+b)/2); label.bot(btex $2$ etex, (c+b)/2); label.bot(btex $3$ etex, (c+d)/2); label.bot(btex $4$ etex, (d+e)/2); label.bot(btex $5$ etex, (e+f)/2); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % A D J A C E N T E S %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1cm; uy:=1cm; xmin := 0 ; xmax := 9; ymin := -.2; ymax :=3; pair d,h; d:=(.1*ux,0); h:=(0,.1*uy); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x % drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.rt(btex $\mathbb{R}$ etex,(xmax*ux,0) shifted d); % label de l'axe des x % label.urt(btex $y$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(71); axes; path t; % tirets t:=((0,0) shifted h)--((0,0) shifted -h); draw t shifted (ux,0); label.bot(btex $a_n$ etex,(ux,0) shifted -1.7h); draw t shifted (2.5*ux,0); label.bot(btex $a_{n+1}$ etex,(2.5*ux,0) shifted -1.7h); draw t shifted (3.75*ux,0); label.bot(btex $a$ etex,(3.75*ux,0) shifted -1.7h); % draw t shifted (4*ux,0); label.bot(btex $b$ etex,(3.75*ux,0) shifted 6h); draw t shifted (5*ux,0); label.bot(btex $b_{n+1}$ etex,(5*ux,0) shifted 6h); draw t shifted (6.5*ux,0); label.bot(btex $b_n$ etex,(6.5*ux,0) shifted 6h); label.top( btex $\cdots$ etex,(3.1*ux,0)shifted (0,-.6*uy)); label.top( btex $\cdots$ etex,(4.4*ux,0)shifted (0,.6*uy)); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % % % M E T H O D E D e S T R A P E Z E S %% % % % Trapèze %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Déclarations des constantes % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% numeric xmin, xmax, ymin, ymax, N; ux:=1cm; uy:=1cm; xmin := -.5 ; xmax := 8; ymin := -.5; ymax := 4; pair d,h; d:=(.1*ux,0); h:=(0,.1*uy); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Définitions des axes et labels associés %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vardef axes = drawarrow (ux*xmin,0) -- (ux*xmax,0) ; % axe des x drawarrow (0,uy*ymin) -- (0,uy*ymax); % axe des y label.rt(btex $t$ etex,(xmax*ux,0)); % label de l'axe des x label.urt(btex $x$ etex,(0,ymax*uy)); % label de l'axe des y enddef; beginfig(73); path t; % tirets t:=((0,0)shifted h)--((0,0)shifted -h); pair A,B,C,a,b; A:=(ux,1.5*uy); C:=(2.5*ux,2.5*uy); B:=(7*ux,uy); a:=A yscaled 0; b:=B yscaled 0; pair M,N,m,n; path P,Q,QQ,R,S; P:=A{dir-10}..C..{dir-10}B; M:=point .7 of P; N:=point 1.2 of P; m:=M yscaled 0; n:=N yscaled 0; S:=subpath(.7,1.2) of P; QQ:=M--N; Q:=N--n--m--M--cycle; R:=buildcycle(S,QQ); fill R withcolor bleu_f;%(0.829997,0.099994,0.119999); fill Q withcolor bleu;% (0.529405,0.807794,0.921598); axes; draw P; draw (A--a) dashed evenly; draw (B--b) dashed evenly; draw (M--m) dashed evenly; draw (N--n) dashed evenly; draw t shifted a; draw t shifted b; draw t shifted m; draw t shifted n; %label.bot(btex $a$ etex,a shifted -1.9h); label.bot(btex $b$ etex,b shifted -h); label.bot(btex $kT_e$ etex,m shifted -1.9h-d); label.bot(btex $(k+1)T_e$ etex,n shifted -h+4d); pair T; T:=M xscaled 0; draw (T--M) dashed evenly; pair TT; TT:=N xscaled 0; draw (TT--N) dashed evenly; path U,V; pair t; U:=N--n; V:=T--(T shifted (6*ux,0)); t:=U intersectionpoint V; %draw (M--t); draw (T shifted d)--(T shifted -d); label.lft(btex $x(kT_e)=x_k$ etex,T shifted -d); draw (TT shifted d)--(TT shifted -d); label.lft(btex $x((k+1)T_e)=x_{k+1}$ etex,TT shifted -d); %label.rt(btex $\mathcal{A}(x,x\!+\!dx)$ etex,(3.4*ux,1.5*uy)); drawarrow (3.4*ux,1.5*uy)--(2.3*ux,1.5*uy); endfig; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% P R O B a s patates beginfig(74); path a,b,aa,ab; a=fullcircle scaled 2cm; b=a shifted (0,1cm); aa=halfcircle scaled 2cm; ab=buildcycle(aa,b); picture pa,pb,px; pa=thelabel(btex $B\setminus A$ etex, (0,-.5cm)); pb=thelabel(btex $A\cap B$ etex, (0,0.5cm)); px=thelabel(btex $A\setminus B$ etex, (0,1.5cm)); fill a withcolor .8white; fill b withcolor bleu_m; fill ab withcolor bleu_ciel; unfill bbox pa; draw pa; unfill bbox pb; draw pb; unfill bbox px; draw px; endfig; end