Exercice 13restart;rho:=theta->(1+2*cos(theta))/(2*sin(theta)+1);

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Domaine d'EtudeOn r\351duit l'intervalle d'\351tude \340 -Pi..Pi par p\351riodicit\351rho(theta+2*Pi)=rho(theta);

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evalb(%);

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sPoints caract\351ristiquesIntersection avec l'axe des abscissesassume(t>=-Pi and t<=Pi);r:=[solve(rho(t)=0,t)]; avec le point O

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r:=[solve(rho(t)*sin(t)=0,t)];avec l'axe des abscisses

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r:=[solve(rho(t)*cos(t)=0,t)];avec l'axe des ordonn\351es

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t_0:=-2*Pi/3:M(t_0)=[rho(t_0)*cos(t_0),rho(t_0)*sin(t_0)],'rho'(t_0)=rho(t_0);

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t_0:=-Pi/2:M(t_0)=[rho(t_0)*cos(t_0),rho(t_0)*sin(t_0)],'rho'(t_0)=rho(t_0);

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t_0:=0:M(t_0)=[rho(t_0)*cos(t_0),rho(t_0)*sin(t_0)],'rho'(t_0)=rho(t_0);

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t_0:=Pi/2:M(t_0)=[rho(t_0)*cos(t_0),rho(t_0)*sin(t_0)],'rho'(t_0)=rho(t_0);

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t_0:=2*Pi/3:M(t_0)=[rho(t_0)*cos(t_0),rho(t_0)*sin(t_0)],'rho'(t_0)=rho(t_0);

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Calcul de la direction de la tangenteDrho:=D(rho);

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cV:=Drho(theta)/rho(theta);

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cV:=factor(simplify(cV));cotan V

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V:=unapply(arccot(cV),theta);On en d\351duit V modulo Pi

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D'o\371 quelques tangentes particuli\350rest_0:=-Pi/2:'V'(t_0)=V(t_0);

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t_0:=0:'V'(t_0)=V(t_0);

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t_0:=Pi/2:'V'(t_0)=V(t_0);

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les tangentes ont m\352me orientation relativement \340 la direction (OM)Etudions les asymptotes en -Pi/6 et -5Pi/6t_0:=-Pi/6:#En -Pi/6 \340 droite, on a bien une branche infinie car:Limit('rho'(theta),theta=t_0,right)=limit(rho(theta),theta=t_0,right);

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la direction asymptotique est -Pi/6. dans le rep\350re tourn\351 de cet angle on a la coordonn\351e YY:=theta->rho(theta)*sin(theta+Pi/6);

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Limit('Y'(theta),theta=t_0,right)=limit(Y(theta),theta=t_0,right);

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D'o\371 l'\351quation de l'asymptote dans ce nouveau rep\350re Y=1+sqrt(3)/3Position relative:delta:=simplify(combine(Y(tau)-1-sqrt(3)/3));#Expression de Y(tau)-1-sqrt(3)/3

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delta_u:=subs(tau=u-Pi/6,delta);#changement de variable

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Limit('delta_u'/u,u=0,right)=limit(delta_u/u,u=0,right);

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Comme delta_u est au voisinage de 0 (\340 droite) du signe de la limite delta_u/u, on trouve quela courbe est au dessus de l'asymptoteu:='u':#assume(u>-Pi/6 and u<-Pi/7);plot(delta_u,u=-0.1..0.1);

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t_0:=-Pi/6:En -Pi/6 \340 gauche, on a bien une branche infinie car:Limit('rho'(theta),theta=t_0,left)=limit(rho(theta),theta=t_0,left);

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la direction asymptotique est -Pi/6. dans le rep\350re tourn\351 de cet angle on a la coordonn\351e YY:=theta->rho(theta)*sin(theta+Pi/6);

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Limit('Y'(theta),theta=t_0,left)=limit(Y(theta),theta=t_0,left);

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D'o\371 l'\351quation de l'asymptote dans ce nouveau rep\350re Y=1+sqrt(3)/3Position relative:delta:=simplify(combine(Y(tau)-1-sqrt(3)/3));#Expression de Y(tau)-1-sqrt(3)/3

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delta_u:=subs(tau=u-Pi/6,delta);#changement de variable

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Limit('delta_u'/u,u=0,left)=limit(delta_u/u,u=0,left);

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comme delta_u est au voisinage de 0 (\340 droite) du signe contraire de la limite delta_u/u, on trouve quela courbe est en dessous de de l'asymptotet_0:=-5*Pi/6:En -5Pi/6 \340 droite, on a bien une branche infinie car:Limit('rho'(theta),theta=t_0,right)=limit(rho(theta),theta=t_0,right);

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la direction asymptotique est -5Pi/6. dans le rep\350re tourn\351 de cet angle on a la coordonn\351e YY:=theta->rho(theta)*sin(theta+5*Pi/6);

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Limit('Y'(theta),theta=t_0,right)=limit(Y(theta),theta=t_0,right);

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D'o\371 l'\351quation de l'asymptote dans ce nouveau rep\350re Y=1-sqrt(3)/3Position relative:delta:=simplify(combine(Y(tau)-1+sqrt(3)/3));#Expression de Y(tau)-1+sqrt(3)/3

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delta_u:=subs(tau=u-5*Pi/6,delta);#changement de variable

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Limit('delta_u'/u,u=0,right)=limit(delta_u/u,u=0,right);

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comme delta_u est au voisinage de 0 (\340 droite) du signe de la limite delta_u/u, on trouve quela courbe est en desous de l'asymptotet_0:=-5*Pi/6:#En -5Pi/6 \340 gauche, on a bien une branche infinie car:Limit('rho'(theta),theta=t_0,left)=limit(rho(theta),theta=t_0,left);

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#la direction asymptotique est -Pi/6. dans le rep\350re tourn\351 de cet angle on a la coordonn\351e YY:=theta->rho(theta)*sin(theta+5*Pi/6);

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Limit('Y'(theta),theta=t_0,left)=limit(Y(theta),theta=t_0,left);

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D'o\371 l'\351quation de l'asymptote dans ce nouveau rep\350re Y=1-sqrt(3)/3Position relative:delta:=simplify(combine(Y(tau)-1+sqrt(3)/3));#Expression de Y(tau)-1-sqrt(3)/3

NiM+SSZkZWx0YUc2IiwkKiYsLi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YlNiMsJkkkdGF1R0YlIiIiSSNQaUdGLCNGMSIiJCIiJyEiJEYxLUYqNiMsJkYwIiIjRjJGM0Y1LUkkc2luR0YrNiNGMCEjNyomRjQjRjFGOkY7RjEiIiUqJEY0RkBGOkYxLCZGO0Y6RjFGMSEiIiNGMUY1

delta_u:=subs(tau=u-5*Pi/6,delta);#changement de variable

NiM+SShkZWx0YV91RzYiLCQqJiwuLUkkY29zRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiU2IywmSSJ1R0YlIiIiSSNQaUdGLCMhIiIiIiMiIichIiRGMS1GKjYjLCZGMEY1RjIjISIlIiIkRjYtSSRzaW5HRis2IywmRjBGMUYyIyEiJkY2ISM3KiZGPSNGMUY1Rj5GMSIiJSokRj1GRkY1RjEsJkY+RjVGMUYxRjQjRjFGNg==

Limit('delta_u'/u,u=0,left)=limit(delta_u/u,u=0,left);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYlKiZJKGRlbHRhX3VHRikiIiJJInVHRikhIiIvRi4iIiFJJWxlZnRHRiksJiokIiIkI0YtIiIjI0YvIiInRjhGLQ==

comme delta_u est au voisinage de 0 (\340 gauche) du signe contraire de la limite delta_u/u, on trouve quela courbe est au dessus de de l'asymptoteVariations de rho et de Vplot(rho,-Pi..Pi,-10..10,discont=true);

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plot(V,-Pi..Pi,discont=true);

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Support de courbetotal:=[rho(t),t,t=-Pi..Pi]:asymp1:=[rho(t),t,t=-Pi/6..0]:asymp2:=[rho(t),t,t=-Pi/2..-Pi/6]:asymp3:=[rho(t),t,t=-5*Pi/6.. -5*Pi/6+0.4]:asymp4:=[rho(t),t,t=-5*Pi/6-0.2..-5*Pi/6]:plot([total,asymp1,asymp2,asymp3,asymp4],-2..5,-2..3,color=[red,blue,green,magenta,black],thickness=[2,2,2,2,2],coords=polar);

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En bleu la partie asymptotique correspondant \340 -Pi/6 \340 droite (courbe au dessus? voir sens du vecteur v(-Pi/6))En vert la partie asymptotique correspondant \340 -Pi/6 \340 gauche (courbe en dessous voir sens du vecteur v(-Pi/6))En magenta la partie asymptotique correspondant \340 -5Pi/6 \340 droite (courbe en dessous voir sens du vecteur v(-5Pi/6))En noir la partie asymptotique correspondant \340 -5Pi/6 \340 gauche (courbe au dessus voir sens du vecteur v(-5Pi/6))