restart;x:=t->t^3/(3*(abs(t)-2));y:=t->t^2/2-3*abs(t);

NiM+SSJ4RzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiJCwmLUkkYWJzR0kqcHJvdGVjdGVkR0YyNiNGLUYuISInIiIiISIiRiVGJUYlNiM+SSJ5RzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIjIyIiIkYvLUkkYWJzR0kqcHJvdGVjdGVkR0Y0NiNGLiEiJEYlRiVGJQ==

R\351duction du domaine d'\351tude[x(-t),y(-t)]=[-x(t),y(t)];

NiMvNyQsJComSSJ0RzYiIiIkLCYtSSRhYnNHSSpwcm90ZWN0ZWRHRi02I0YnRikhIiciIiIhIiJGMSwmKiRGJyIiIyNGMEY0RishIiRGJA==

evalb(%);

NiNJJXRydWVHSSpwcm90ZWN0ZWRHRiQ=

On en d\351duit la r\351duction du domaine d'\351tude aux param\350tres positifs, le reste du support s'obtient par sym\351trie / \340 (Oy)Vecteur vitesseassume(t>=0);"x'(t)"=factor(D(x)(t));"y'(t)"=factor(D(y)(t));

NiMvUSZ4Jyh0KTYiLCQqKEkjdHxpckdGJSIiIywmRigiIiIhIiRGK0YrLCZGKEYrISIjRitGLiNGKSIiJA==NiMvUSZ5Jyh0KTYiLCZJI3R8aXJHRiUiIiIhIiRGKA==

Au point stationnaire M(3) on a une tangente de pente 1/6

ta:=factor((y(t)-y(3))/(x(t)-x(3))):('y'(t)-'y'(3))/('x'(t)-'x'(3))=ta;Limit(('y'(t)-'y'(3))/('x'(t)-'x'(3)),t=3)=limit(ta,t=3);

NiMvKiYsJi1JInlHNiI2I0kjdHxpckdGKCIiIi1GJzYjIiIkISIiRissJi1JInhHRihGKUYrLUYyRi1GL0YvLCQqJiwmRipGKyEiI0YrRissJkYqRisiIidGK0YvI0YuIiIjNiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkKiYsJi1JInlHRik2I0kjdHxpckdGKSIiIi1GLjYjIiIkISIiRjEsJi1JInhHRilGL0YxLUY4RjNGNUY1L0YwRjQjRjEiIic=

Au point M(0) on a une tangente verticaleta:=factor((y(t)-y(0))/(x(t)-x(0))):('y'(t)-'y'(0))/('x'(t)-'x'(0))=ta;Limit(abs(('y'(t)-'y'(0))/('x'(t)-'x'(0))),t=0)=limit(abs(ta),t=0);

NiMvKiYsJi1JInlHNiI2I0kjdHxpckdGKCIiIi1GJzYjIiIhISIiRissJi1JInhHRihGKUYrLUYyRi1GL0YvLCQqKEYqISIjLCZGKkYrISInRitGKywmRipGK0Y2RitGKyMiIiQiIiM=NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkLUkkYWJzR0YnNiMqJiwmLUkieUdGKTYjSSN0fGlyR0YpIiIiLUYxNiMiIiEhIiJGNCwmLUkieEdGKUYyRjQtRjtGNkY4RjgvRjNGN0kpaW5maW5pdHlHRic=

Quelques valeurs caract\351ristiquest_0:=-3: M(t_0)=(x(t_0),y(t_0));

NiMvLUkiTUc2IjYjISIkNiQhIiojRioiIiM=

t_0:=0: M(t_0)=(x(t_0),y(t_0));

NiMvLUkiTUc2IjYjIiIhNiRGKEYo

t_0:=3: M(t_0)=(x(t_0),y(t_0));

NiMvLUkiTUc2IjYjIiIkNiQiIiojISIqIiIj

Tableau de variation

H:=t->abs(t)/t:U:=t->H(t)/2:xp:=H@D(x):yp:=U@D(y):

plot([xp,yp],thickness=[3,3],color=[green,red]);

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

En vert le signe de x'(t)En rouge le signe de y'(t)Branches infiniesEtude en 2Limit('x'(t),t=2,right)=limit(x(t),t=2,right);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYlLUkieEdGKTYjSSN0fGlyR0YpL0YuIiIjSSZyaWdodEdGKUkpaW5maW5pdHlHRic=

Limit('x'(t),t=2,left)=limit(x(t),t=2,left);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYlLUkieEdGKTYjSSN0fGlyR0YpL0YuIiIjSSVsZWZ0R0YpLCRJKWluZmluaXR5R0YnISIi

Limit('y'(t),t=2)=limit(y(t),t=2);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkLUkieUdGKTYjSSN0fGlyR0YpL0YuIiIjISIl

D'o\371 une asymptote horizontale d'\351quation y=-4La position relative \351tant donn\351 par le signe de la diff\351rence dedelta:=factor(y(t)+4);

NiM+SSZkZWx0YUc2IiwkKiYsJkkjdHxpckdGJSIiIiEiI0YqRiosJkYpRiohIiVGKkYqI0YqIiIj

pos:=H(delta):plot(pos,t=1..3,thickness=2);

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

D'o\371 la courbe est au dessus pour t<2 et en dessous pour t>2Etude en +inifinityLimit('x'(t),t=infinity)=limit(x(t),t=infinity);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkLUkieEdGKTYjSSN0fGlyR0YpL0YuSSlpbmZpbml0eUdGJ0Yw

Limit('y'(t),t=infinity)=limit(y(t),t=infinity);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkLUkieUdGKTYjSSN0fGlyR0YpL0YuSSlpbmZpbml0eUdGJ0Yw

D'o\371 une branche infinie lorsque t->+infini'y'(t)/'x'(t)=factor(y(t)/x(t));

NiMvKiYtSSJ5RzYiNiNJI3R8aXJHRiciIiItSSJ4R0YnRighIiIsJCooRikhIiMsJkYpRiohIidGKkYqLCZGKUYqRjBGKkYqIyIiJCIiIw==

Limit('y'(t)/'x'(t),t=infinity)=limit(y(t)/x(t),t=infinity);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkKiYtSSJ5R0YpNiNJI3R8aXJHRikiIiItSSJ4R0YpRi4hIiIvRi9JKWluZmluaXR5R0YnIyIiJCIiIw==

D'o\371 une direction asymptotique d'\351quation y=3/2'y'(t)-3*'x'(t)/2 =factor(y(t)-3*x(t)/2);

NiMvLCYtSSJ5RzYiNiNJI3R8aXJHRiciIiItSSJ4R0YnRigjISIkIiIjLCQqKEYpRiosJkYpRi9GLkYqRiosJkYpRiohIiNGKiEiIkY0

Limit('y'(t)-3*'x'(t)/2,t=infinity)=limit(y(t)-3*x(t)/2,t=infinity);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkLCYtSSJ5R0YpNiNJI3R8aXJHRikiIiItSSJ4R0YpRi4jISIkIiIjL0YvSSlpbmZpbml0eUdGJywkRjchIiI=

Il s'agit donc d'une branche parabolique de direction y=3/2Support de la courbe

t:='t':with(plots):

Warning, the name changecoords has been redefined

NiM3Zm5JKGFuaW1hdGVHNiJJKmFuaW1hdGUzZEdGJUktYW5pbWF0ZWN1cnZlR0YlSSZhcnJvd0dGJUktY2hhbmdlY29vcmRzR0koX3N5c2xpYkdGJUksY29tcGxleHBsb3RHRiVJLmNvbXBsZXhwbG90M2RHRiVJKmNvbmZvcm1hbEdGJUksY29uZm9ybWFsM2RHRiVJLGNvbnRvdXJwbG90R0YlSS5jb250b3VycGxvdDNkR0YlSSpjb29yZHBsb3RHRiVJLGNvb3JkcGxvdDNkR0YlSS1jeWxpbmRlcnBsb3RHRiVJLGRlbnNpdHlwbG90R0YlSShkaXNwbGF5R0YlSSpkaXNwbGF5M2RHRiVJKmZpZWxkcGxvdEdGJUksZmllbGRwbG90M2RHRiVJKWdyYWRwbG90R0YlSStncmFkcGxvdDNkR0YlSSxncmFwaHBsb3QzZEdGJUktaW1wbGljaXRwbG90R0YlSS9pbXBsaWNpdHBsb3QzZEdGJUkoaW5lcXVhbEdGJUksaW50ZXJhY3RpdmVHRiVJMmludGVyYWN0aXZlcGFyYW1zR0YlSS1saXN0Y29udHBsb3RHRiVJL2xpc3Rjb250cGxvdDNkR0YlSTBsaXN0ZGVuc2l0eXBsb3RHRiVJKWxpc3RwbG90R0YlSStsaXN0cGxvdDNkR0YlSStsb2dsb2dwbG90R0YlSShsb2dwbG90R0YlSSttYXRyaXhwbG90R0YlSSltdWx0aXBsZUdGJUkob2RlcGxvdEdGJUkncGFyZXRvR0YlSSxwbG90Y29tcGFyZUdGJUkqcG9pbnRwbG90R0YlSSxwb2ludHBsb3QzZEdGJUkqcG9sYXJwbG90R0YlSSxwb2x5Z29ucGxvdEdGJUkucG9seWdvbnBsb3QzZEdGJUk0cG9seWhlZHJhX3N1cHBvcnRlZEdGJUkucG9seWhlZHJhcGxvdEdGJUkncmVwbG90R0YlSSpyb290bG9jdXNHRiVJLHNlbWlsb2dwbG90R0YlSStzZXRvcHRpb25zR0YlSS1zZXRvcHRpb25zM2RHRiVJK3NwYWNlY3VydmVHRiVJMXNwYXJzZW1hdHJpeHBsb3RHRiVJK3NwaGVyZXBsb3RHRiVJKXN1cmZkYXRhR0YlSSl0ZXh0cGxvdEdGJUkrdGV4dHBsb3QzZEdGJUkpdHViZXBsb3RHRiU=

support_plus:=plot([x(t),y(t),t=0..10],-30..30,-5..5, color=blue,linestyle=DASHDOT,thickness=2):support_moins:=plot([x(t),y(t),t=-10..0],-30..30,-5..5, color=red,linestyle=SOLID,thickness=2):asymptote_plus:=plot(3*t/2,t=-30..30,color=green,linestyle=DASH):asymptote_moins:=plot(-3*t/2,t=-30..30,color=green,linestyle=SOLID):asymptote_horizontale:=plot(-4,t=-30..30,color=magenta,linestyle=SOLID,thickness=3):tangente_plus:=plot(t/6-6,t=-30..30,color=black,linestyle=DASH):tangente_moins:=plot(-t/6-6,t=-30..30,color=black,linestyle=SOLID):

display(support_plus, support_moins,asymptote_plus,asymptote_moins,asymptote_horizontale,tangente_plus,tangente_moins);

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En pointill\351s le support et les droites caract\351ristiques associ\351s aux param\350tres positifsEn trait plein le support et les droites caract\351ristiques associ\351es aux param\350tres n\351gatifsEn rouge le support des param\350tres n\351gatifsEn bleu le support des param\350tres positifsEn vert les directions asymtotiques pour t->infiniEn noir les tangentes en M(3) et M(-3)En magenta l'asymptote horizontale pour t->2 et t->-2