Exercice 10restart;rho:=theta->1+2*cos(theta)-4*cos(theta)^2;

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Domaine d'EtudeOn r\351duit l'intervalle d'\351tude \340 -Pi..Pi par p\351riodicit\351#Puis \340 0..Pi carrho(-theta)=rho(theta);

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evalb(%);

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#Le support complet se d\351duit alors par sym\351trie par rapport \340 l'axe des abscissesPoints caract\351ristiquesIntersection avec l'axe des abscissesassume(t>=0 and t<=Pi);r:=[solve(rho(t)=0,t)];

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r:=[solve(rho(t)/sin(t)=0,t)];

NiM+SSJyRzYiNyYtSSdhcmN0YW5HNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjLCQqJiwmIiM1IiIiKiQiIiYjRjEiIiMhIiNGNCwmRjJGNEY0RjEhIiJGNCwkRidGOCwmLUYoNiMsJComLCZGMEYxRjJGNUY0LCZGNEYxRjIjRjhGNUY4RjRGMUkjUGlHRipGMSwmRjtGOEZCRjg=

r:=simplify(r);

NiM+SSJyRzYiNyYtSSdhcmN0YW5HNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjKiYsJiIjNSIiIiokIiImI0YwIiIjISIjRjMsJkYxRjBGMEYwISIiLCRGJ0Y3LCYtRig2IyomLCZGL0YwRjFGNEYzLCZGN0YwRjFGMEY3RjdJI1BpR0YqRjAsJkY6RjBGP0Y3

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

NiQvLUkiTUc2IjYjIiIhNyQhIiJGKC8tSSRyaG9HRiZGJ0Yq

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

NiQvLUkiTUc2IjYjLCRJI1BpR0kqcHJvdGVjdGVkR0YqIyIiIiIiJDckI0YsIiIjLCQqJEYtRi9GLy8tSSRyaG9HRiZGJ0Ys

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);

NiQvLUkiTUc2IjYjLCRJI1BpR0kqcHJvdGVjdGVkR0YqIyIiIyIiJDckIyIiIkYsLCQqJEYtRi8jISIiRiwvLUkkcmhvR0YmRidGNA==

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

NiQvLUkiTUc2IjYjSSNQaUdJKnByb3RlY3RlZEdGKTckIiImIiIhLy1JJHJob0dGJkYnISIm

Calcul de la direction de la tangenteD\351riv\351e de la param\351trisationDrho:=D(rho);

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

NiM+SSNjVkc2IiomLCYtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjSSZ0aGV0YUdGJSEiIyomLUkkY29zR0YqRi0iIiJGKEYzIiIpRjMsKEYzRjNGMSIiIyokRjFGNiEiJSEiIg==

cotan VcV:=simplify(cV);

NiM+SSNjVkc2IiwkKigtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjSSZ0aGV0YUdGJSIiIiwmISIiRi8tSSRjb3NHRipGLSIiJUYvLChGMUYvRjIhIiMqJEYyIiIjRjRGMUY2

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

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D'o\371 quelques tangentes particuli\350rest_0:=0:'V'(t_0)=V(t_0);

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t_0:=Pi/3:'V'(t_0)=V(t_0);

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

NiMvLUkiVkc2IjYjSSNQaUdJKnByb3RlY3RlZEdGKSwkRigjIiIiIiIj

Variations de rho et de Vplot(rho,0..Pi);

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plot(V,0..Pi);

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Support de Courbetotal:=[rho(t),t,t=-Pi..Pi]:partiel:=[rho(t),t,t=0..Pi]:plot([total,partiel],-1..5,-2..2,color=[red,blue],thickness=[3,3],coords=polar);

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