Exercice 1On d\351fnit la param\351trisation, puis on construit le support partiel et sir la totalit\351 des param\350tresrestart;x:=t->(cos(t))^3;y:=t->(sin(t))^3;

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total:=[x(t),y(t),t=0..2*Pi]: partiel:=[x(t),y(t),t=0..Pi/4]:plot([total,partiel],color=[blue,red],thickness=[1,2]);

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RmpfbQ==

Calculons le vecteur vitesseV:=unapply([D(x)(t),D(y)(t)],t);

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V\351rifions si les points A,B,C de param\350tre 0, Pi/4 et Pi/2 sont bien r\351guliers

V(0);V(Pi/4);V(Pi/2);

NiM3JCIiIUYkNiM3JCwkKiQiIiMjIiIiRiYjISIkIiIlLCRGJSMiIiRGKw==NiM3JCIiIUYk

Seul le point B est r\351gulier d'o\371 la tangente en ce point dirig\351e par V(Pi/4) (pente=-2), dont voici alors une \351quation param\351tr\351eXTB:=x(Pi/4)+t*op(1,V(Pi/4));YTB:=y(Pi/4)+t*op(2,V(Pi/4));

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V\351rifions 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);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkKiYtSSRzaW5HRiY2I0kidEdGKSIiJCwmKiQtSSRjb3NHRiZGLkYwIiIiISIiRjVGNi9GLyIiIUY4

Et donc au point A de param\351tre 0, on a une tangente horizontaleXTA:=x(0)+t;YTA:=y(0);

NiM+SSRYVEFHNiIsJiIiIkYnSSJ0R0YlRic=NiM+SSRZVEFHNiIiIiE=

V\351rifions la limite du taux d'accroissement en Pi/2Limit((y(t)-y(Pi/2))/(x(t)-x(Pi/2)),t=Pi/2)=limit((y(t)-y(Pi/2))/(x(t)-x(Pi/2)),t=Pi/2);

NiMvLUkmTGltaXRHNiRJKnByb3RlY3RlZEdGJ0koX3N5c2xpYkc2IjYkKiYsJiokLUkkc2luR0YmNiNJInRHRikiIiQiIiIhIiJGM0YzLUkkY29zR0YmRjAhIiQvRjEsJEkjUGlHRicjRjMiIiNJKnVuZGVmaW5lZEdGJw==

Essayons alors celle de la valeur absolue du taux d'accroissementLimit(abs((y(t)-y(Pi/2))/(x(t)-x(Pi/2))),t=Pi/2)=limit(abs((y(t)-y(Pi/2))/(x(t)-x(Pi/2))),t=Pi/2);

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On en d\351duit donc une tangente verticale au point de C param\350tre Pi/2XTC:=x(Pi/2);YTC:=y(Pi/2)+t;

NiM+SSRYVENHNiIiIiE=NiM+SSRZVENHNiIsJiIiIkYnSSJ0R0YlRic=

Tracons ces tangentesEqTA:=[XTA,YTA,t=-1..0]:EqTB:=[XTB,YTB,t=-0.5..0.5]:EqTC:=[XTC,YTC,t=-3..0]:plot([total,EqTA,EqTB,EqTC],color=[blue,red,magenta,green], thickness=[1,2,2,2],axes=none);#

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MSQiM0dtKmZtS1trNCRGMTckJCIzeWU5Ij5tK2Z3JEYxJCIzJnAsYSpccjswTEYxNyQkIjMkKTNfMDkjcDJhJEYxJCIzKm9FNXlmKUhJTkYxNyQkIjNdKkh0PCI0I3BJJEYxJCIzQ3dANCtwOWtQRjE3JCQiMzBHdz54X04uSkYxJCIzb1p5bU1Ecm5SRjE3JCQiMyl5NSQqXCIpKVwkKUdGMSQiMyV5T3NvKipvdj0lRjE3JCQiM1NKYmN3Yk9jRUYxJCIzS1cqKkhOQXE5V0YxNyQkIjMqUSZlLjorO01DRjEkIjMlPWlIb3oycGolRjE3JCQiM3kiemUnM1o7PkFGMSQiMyVSbzFLNS4+JltGMTckJCIzJzRNQm0yXC8pPkYxJCIzL05AQ04oPTE0JkYxNyQkIjNTT0ckUllfZnciRjEkIjNoUkUkek06XkkmRjE3JCQiM0ZjL11ZOiRwYCJGMSQiM1k+XU9saThNYkYxNyQkIjMqR14waGIvJUg4RjEkIjNSaipmZERqO3UmRjE3JCQiMzhKRGFTRl8tNkYxJCIzKVslSEtyXWFvZkYxNyQkIjN5XjU2TWpXISopKUY6JCIzKSpwVlh5Si0jPSdGMTckJCIzaSZ6IlEwXSkqZW1GOiQiM3MnSEY4SnBeUydGMTckJCIzbzNEYiNvQnRaJUY6JCIzVUQtaFZhTEJtRjE3JCQiMzVFJilSamtLJD4jRjokIjM3QmNfbF50Xm9GMTckJCEzYiI9MmE6PVZWJyEjQSQiMyFmLCJvVjdycnFGMTckJCEzcSd6JHlOdStjQUY6JCIzJVsmUVdiJm9tSChGMTckJCEzJypvS0BfSCZwWyVGOiQiMz0ubzMySnc+dkYxNyQkITN0bFB4JT1ncGAnRjokIjNJX0dNSVF3Q3hGMTckJCEzUUtjWzspKlwnKSkpRjokIjNTUVNeJHo8KGZ6RjE3JCQhM08rYUAyY3opNCJGMSQiMzV3MzM+TScpcCIpRjE3JCQhM1RzXGd1WSZHSyJGMSQiM3FbL1onW0FSUilGMTckJCEzJSoqZipRInoydGAiRjEkIjNBd11ELmNQMycpRjE3JEZhaWxGX2lsLUZbX2w2JkZdX2xGYF9sRl5fbEZgX2xGaGhsLUYmNiU3UzckRi0kISIjRiw3JEYtJCEzJioqKioqXFAmM1kkPkZecDckRi0kITMhKioqXGl2PHJ4PUZecDckRi0kITMxKytEO2FzOD1GXnA3JEYtJCEzOSsrdjhcSlw8Rl5wNyRGLSQhMz0rXTdWMEAmbyJGXnA3JEYtJCEzJykqKlxpJmV4ZGkiRl5wNyRGLSQhMz0rXWkrI1FVYyJGXnA3JEYtJCEzKioqKlxpITMlZis6Rl5wNyRGLSQhMyEqKipcN29TOlA5Rl5wNyRGLSQhMzErK108IykqPVAiRl5wNyRGLSQhMykpKioqXChHM1U5OEZecDckRi0kITMxKytdLVxyXDdGXnA3JEYtJCEzISoqKioqXChHVlo9IkZecDckRi0kITMxKytdKDRKQDciRl5wNyRGLSQhMyEqKipcaUlLRmw1Rl5wNyRGLSQhMycpKioqKipcRlBtKCoqRjE3JEYtJCEzKCkqKioqKioqNCcqUVMqRjE3JEYtJCEzPysrRGM+bVAoKUYxNyRGLSQhMyczKysrJj0kejkpRjE3JEYtJCEzTioqKlxpWC80XShGMTckRi0kITNDKioqXChvOHklKW9GMTckRi0kITMzKioqKlxpOiM+QydGMTckRi0kITMvKioqXDdldjpsJkYxNyRGLSQhM0crK10obzJbLCZGMTckRi0kITNDKytEMVtRYFZGMTckRi0kITNXKyt2VlVoeFBGMTckRi0kITNPKioqKlxpaXdiSkYxNyRGLSQhMzAsKytda0w4REYxNyRGLSQhMykzKytEQFdbKT1GMTckRi0kITM5KytEY2N1dzdGMTckRi0kITNULysrdillYiwnRjo3JEYtJCIzZCMqeSoqKioqXEA4JkZlbzckRi0kIjNGMCsrXTcsSGxGOjckRi0kIjNqKSoqXFA0dylSN0YxNyRGLSQiM3MsKyt2WmYiKT1GMTckRi0kIjMneioqXFAvLWFbI0YxNyRGLSQiM1IrK3Y9WWI7SkYxNyRGLSQiM3MpKioqKlxpQE90JEYxNyRGLSQiM2cpKipcUGZMJ3pWRjE3JEYtJCIzKD0rKyshKj49KyZGMTckRi0kIjNFKytdaV80UWNGMTckRi0kIjN1LCt2Vj41cGlGMTckRi0kIjNVLCsrXTokKltvRjE3JEYtJCIzJjMrK0RyIls4dkYxNyRGLSQiMysrKysrTCd5NSlGMTckRi0kIjNyKyt2ViEpZlQoKUYxNyRGLSQiM20rK0RjSTtbJCpGMTckRi1GKi1GW19sNiZGXV9sRl5fbEZgX2xGXl9sRmhobC0lKkFYRVNTVFlMRUc2IyUlTk9ORUctJStBWEVTTEFCRUxTRzYkUSE2IkZfYm4tJSVWSUVXRzYkOyQhMm9XNlZHJikqUjUhIzskIjJzM2Q5cioqKlI1RmdibjskISQxI0ZnaG0kIiQxIkZnaG0=