Exercice 8On d\351fnit la param\351trisationrestart;x:=t->2*cos(t)+cos(2*t);y:=t->2*sin(t)-sin(2*t);

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Domaine d'EtudeR\351duisons l'intervalle d'\351tude \340 -Pi..Pi, par 2Pi p\351riodicit\351 puis \340 0..Pi car[x(-t),y(-t)]=[x(t),-y(t)];

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evalb(%);

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Donc la partie de courbe correspondant aux autres param\351tres se d\351duira par sym\351trie / \340 l'axe des abscissesCalculons le vecteur vitesseVt:=[D(x)(t),D(y)(t)];

NiM+SSNWdEc2IjckLCYtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjSSJ0R0YlISIjLUYpNiMsJEYuIiIjRi8sJi1JJGNvc0dGKkYtRjMtRjZGMUYv

V:=unapply(Vt,t);

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Cette courbe n'est donc r\351guli\350re, les point de param\350tre 0 est stationaireV(0);

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Tangente aux points A=M(0)=(3,0), B=M(Pi/2)=(-1,2) et C=M(Pi)=(-1,0)#les tangentes en B et C sont dirig\351es par les vecteurs

V(Pi/2);V(Pi);

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D'o\371 les \351quations des tangentesXTA:=x(Pi/2)+t*op(1,V(Pi/2));YTA:=y(Pi/2)+t*op(2,V(Pi/2));

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XTC:=x(Pi)+t*op(1,V(Pi));YTC:=y(Pi)+t*op(2,V(Pi));

NiM+SSRYVENHNiIhIiI=NiM+SSRZVENHNiIsJEkidEdGJSEiJQ==

EqTA:=[XTA,YTA,t=-infinity..infinity]:EqTC:=[XTC,YTC,t=-infinity..infinity]:Tangente au point stationnaire ACalculons la limite du taux d'accroissement en 0Limit((y(t)-y(0))/(x(t)-x(0)),t=0)=limit((y(t)-y(0))/(x(t)-x(0)),t=0);

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On en d\351duit donc une tangente horizontale au point de A param\350tre 0XTB:=x(0)+t;YTB:=y(0);

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EqTB:=[XTB,YTB,t=-infinity..infinity]:Etude des variations de x et yplot(x,0..Pi,-2..3);

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plot(y,0..Pi,-3..3);

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varx:=D(x)(t)/abs(D(x)(t)):vary:=D(y)(t)/abs(D(y)(t))/2:On trace en rouge le signe x' et en vert le signe de y'plot([varx,vary],t=0..Pi,discont=true,thickness=[3,3]);

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

Support de la courbetotal:=[x(t),y(t),t=-Pi..Pi]: partiel:=[x(t),y(t),t=0..Pi]:plot([total,partiel,EqTA,EqTB,EqTC],-3..3,-3..3,color=[blue,red,green,green,green],thickness=[1,2,2,2,2]);

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