Exercice 5On d\351fnit la param\351trisation,restart;x:=t->(1-t^2)/(1+t^2);y:=t->t*(1-t^2)/(1+t^2);

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Domaine d'EtudeR\351duisons l'intervalle d'\351tude \340 0..infinity, En effet[x(-t),y(-t)]=[x(t),-y(t)];

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evalb(%);

NiNJJXRydWVHSSpwcm90ZWN0ZWRHRiQ=

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

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Vt:=simplify(Vt);

NiM+SSNWdEc2IjckLCQqJkkidEdGJSIiIiwmRipGKiokRikiIiNGKiEiIyEiJSwkKiYsKCEiIkYqKiRGKSIiJUYqRixGNUYqRitGLkYz

V:=unapply(Vt,t);

NiM+SSJWRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJComOSQiIiIsJkYwRjAqJEYvIiIjRjAhIiMhIiUsJComLCghIiJGMCokRi8iIiVGMEYyRjtGMEYxRjRGOUYlRiVGJQ==

Cette courbe est donc r\351guli\350reTangente aux points M(0)=(1,0) et M(1)=(0,0)Ces tangentes sont dirig\351es par les vecteurs

V(0);V(1);

NiM3JCIiISIiIg==NiM3JCEiIkYk

D'o\371 les \351quations des tangentesXT0:=x(0)+t*op(1,V(0));YT0:=y(0)+t*op(2,V(0));

NiM+SSRYVDBHNiIiIiI=NiM+SSRZVDBHNiJJInRHRiU=

XT1:=x(1)+t*op(1,V(1));YT1:=y(1)+t*op(2,V(1));

NiM+SSRYVDFHNiIsJEkidEdGJSEiIg==NiM+SSRZVDFHNiIsJEkidEdGJSEiIg==

EqT0:=[XT0,YT0,t=-infinity..infinity]:EqT1:=[XT1,YT1,t=-infinity..infinity]:Branches Infinies \351ventuellesLimit('x'(t),t=+infinity)=limit(x(t),t=+infinity),Limit('y'(t),t=+infinity)=limit(y(t),t=+infinity) ;

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D'o\371 une asymptote verticale d'\351quation param\351triqueXA:=-1;YA:=t;

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EqA:=[XA,YA,t=-infinity..infinity]:Position Relative: pour cela on \351tudie le signe de x(tau)-(-1) pour tau grandassume(tau,real):is(x(tau)+1>0);

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On a donc l'arc est \340 droite de cette asymptote... Par sym\351trie / Ox on a la m\352me asymptote et la m\352me position relative pour t->infinityEtude des variations de x et yplot(x);

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plot(y);

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varx:=D(x)(t)/abs(D(x)(t));

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vary:=D(y)(t)/abs(D(y)(t))/2;

NiM+SSV2YXJ5RzYiLCQqJiwoKiYsJiIiIkYrKiRJInRHRiUiIiMhIiJGKywmRitGK0YsRitGL0YrKiZGLUYuRjBGLyEiIyooRi1GLkYqRitGMEYyRjJGKy1JJGFic0dJKnByb3RlY3RlZEdGNjYjRihGLyNGK0Yu

On trace en rouge le signe x' et en vert le signe de y'plot([varx,vary],t=-5..5,discont=true,thickness=[3,3]);

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

Support de la courbetotal:=[x(t),y(t),t=-infinity..infinity]: partiel:=[x(t),y(t),t=0..infinity]:plot([total,partiel,EqT0,EqT1,EqA],-1..1.1,-2..2,color=[blue,red,black,black,magenta],thickness=[1,2,1,1,2]);

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