Exercices d'initiation \303\240 Maple

a:=root(7+5*sqrt(2),3)-root(-7+5*sqrt(2),3); simplify(a); En fait 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 et 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... En effet : print('(1+sqrt(2))^3'=expand((1+sqrt(2))^3)),print('(-1+sqrt(2))^3'=expand((-1+sqrt(2))^3)); Compl\303\251ment : LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== est la solution r\303\251elle d'une \303\251quation du 3e degr\303\251... u:=root(7+5*sqrt(2),3);v:=-root(-7+5*sqrt(2),3); j:=exp((1/3)*(I*2)*Pi);jbar:=conjugate(j); z1:=u+v;z2:=j*u+jbar*v;z3:=jbar*u+j*v; P:=(z-z1)*(z-z2)*(z-z3);expand(P); coeff(P,z,1);simplify(%); Les nombres z1, z2, z3 sont les racines de l'\303\251quation du 3e degr\303\251 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(cf. formules de Cardan...). On v\303\251rifie a posteriori que 2 est racine. factor(z^3+3*z-14);solve(z^3+3*z-14); b:=(16*7^3-2*sqrt(2))/(4-10/3); c:=sin(2*Pi/3); evalf(sin(Pi/3),20); S1:=sum(k,k=1..50);S2:=sum(k^3,k=1..50); S2-S1^2;is(S2=S1^2); sum(exp(2*I*k*Pi/n),k=0..n-1); Attention au cas 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 ! n:=1;sum(exp(2*I*k*Pi/n),k=0..n-1); restart;Tn:=sum(k*binomial(n,k),k=1..n);simplify(%);

restart; solve(x^2+x+2); solve({2*x+3*y-z=1,4*x+y-z=0,x-z=2},{x,y,z}); eq:=x+2*y+z=1,2*a*x+3*y+2*z=1+a,a^2*x-y+z=a; solve({eq},{x,y,z}); Attention si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjVGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGOQ== (cas visible) ! eq5:=eval(eq,a=5);solve({eq5},{x,y,z}); Maple ne donne pas de r\303\251ponse, cela signifie qu'il n'y a aucune solution. Le syst\303\250me est incompatible ; en effet : eval(5*eq[1]-3*eq[2]+eq[3],a=5); Attention si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGOQ== (cas invisible) ! eq1:=eval(eq,a=1);solve({eq1},{x,y,z}); n a une relation entre les \303\251quations du syst\303\250me : eval(5*eq[1]-3*eq[2]+eq[3],a=1); ineq1:=1/(x+1)-x>2; ineq2:=(x^2-6*x+9)/(x-3)<-4; solve({ineq1,ineq2}); Remarque : il s'agit de la solution de 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 dont une valeur approch\303\251e est 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... ; en effet : solve(x^2+3*x+1); evalf(%);

restart; eq:=x^3-10*x+5=0; sols:=solve(eq,x); x1:=sols[1];x2:=sols[2];x3:=sols[3]; evalf(x1,20);evalf(x2,20);evalf(x3,20); Les parties imaginaires sont tr\303\250s petites, ce qui laisse supposer que les racines de l'\303\251quation sont toutes r\303\251elles, ce que confirme le calcul formel : simplify(Re(evalc(x1)));simplify(Re(evalc(x2)));simplify(Re(evalc(x3))); simplify(Im(evalc(x1)));simplify(Im(evalc(x2)));simplify(Im(evalc(x3))); Explication : evalc((-540+(60*I)*sqrt(399))*(-540-(60*I)*sqrt(399)));root(%,3); Donc 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 plot(x^3-10*x+5,x=-4..4);

restart; f:=x->x^2+exp(x); plot(f,-1..1); On voit graphiquement que l'\303\251quation admet deux solutions. solve(f(x)=1,x); Maple ne sait pas r\303\251soudre explicitement cette \303\251quation. R\303\251solvons-la num\303\251riquement : fsolve(f(x)=1,x); Maple ne trouve qu'une des deux solutions. Imposons un intervalle o\303\271 rechercher l'autre solution : fsolve(f(x)=1,x=-1..-0.5);

restart; f:=x->(x^3/3+x+1)*exp(x^2/2); plot([f(x),D(f)(0)*x+f(0)],x=-1.1..1.1,color=[red,blue]); fsolve(f(x),x); plot(f(t)*exp(-t^2),t=0..infinity); limit(int(f(t)*exp(-t^2),t=0..x),x=+infinity); Remarque : maple utilise l'int\303\251grale de Gauss sous-jacente 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... Int(exp(-t^2),t=-infinity..infinity)=int(exp(-t^2),t=-infinity..infinity);

restart; f:=x->ln(x+sqrt(x^2+1));g:=x->ln(x+sqrt(x^2-1)); plot([f(x),sinh(x),x],x=-2..2,color=[red,blue,brown]); Les deux courbes semblent sym\303\251triques par rapport \303\240 la droite d'\303\251quation 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; ce qui laisse penser que les fonctions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjc2hGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy9GM1Enbm9ybWFsRic= et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YvRjI= sont r\303\251ciproques l'une de l'autre... simplify(sinh(f(x)));simplify(f(sinh(x))) assuming x>=0; convert(%,exp);is(%=x) assuming x>=0; et 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(Maple a besoin de la condition LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOS8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnRjk= pour simplifier) donc LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0Y2USdub3JtYWxGJw== et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEjc2hGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnL0YzUSdub3JtYWxGJw== sont des fonctions r\303\251ciproques. De m\303\252me, plot([g(x),cosh(x)],x=0..3,color=[red,blue],scaling=constrained); et on v\303\251rifie que simplify(cosh(g(x)));simplify(g(cosh(x))) assuming x>=0;convert(%,exp);is(%=x) assuming x>=0; de sorte que LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== est la r\303\251ciproque de la fonction LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjY2hGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy9GM1Enbm9ybWFsRic= restreinte \303\240 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. plot(g(cosh(x))-x,x=0..2);

restart; f:=x->2*cos(2*x)/(1+cos(2*x));g:=x->tan(x)^2; plot([f(x),g(x)],x=-4..4,y=-4..4,colour=[red,blue]); Il semblerait que l'on ait 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. On v\303\251rifie : simplify(f(x)-(-g(x)+1)); h:=x->2*sin(2*x)/(1-sin(2*x)); plot([h(x),g(x)],x=-4..4,y=-4..4,colour=[red,blue]); Il semblerait que l'on ait 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 pour un certain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw==, on pourrait par exemple zoomer et comparer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGOQ== pour avoir une meilleure id\303\251e de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== : plot([h(x),g(x)-1],x=-2..1,y=-2..1,colour=[red,blue]); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== semble valoir \303\240 peu pr\303\250s LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEkMC44RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEjLi5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUYzNi1RIi5GJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZGSkZILyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGLw== On peut par exemple chercher \303\240 voir pour quelle valeur LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGOQ== s'annule entre 0 et 1. On obtient alors que 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, donc 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. On peut essayer de le v\303\251rifier en tra\303\247ant les courbes de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnRjk=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 et celle de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== sur un intervalle. plot([h(x),g(x+Pi/4)-1],x=-5..5); On peut aussi essayer de simplifier mais cela ne marche pas tr\303\250s bien... simplify(h(x)-g(x+Pi/4)+1); F:=x->2/(1+tanh(x)); plot([F(x),exp(2*x)],x=-3..3,y=-4..4,colour=[red,blue]); Il semblerait que 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. V\303\251rifions : simplify(F(x)-exp(-2*x)-1);convert(%,exp); JSFH Tra\303\247ons les 2 courbes : plot([F(x),exp(-2*x)+1],x=-3..3,colour=[red,blue]);

restart; F:=t->3*cos(3*t)+2*sin(3*t);G:=t->sqrt(13)*cos(3*t); plot([F,G],-5..5); On peut conjecturer qu'il existe un r\303\251el LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== tel que pour tout LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== dans \342\204\235, 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. 2. On admet qu'il existe deux r\303\251els LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw== et \317\225 tels que pour tout r\303\251el t, 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 En prenant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGOQ==, on obtient : 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. En prenant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJm1mcmFjR0YkNigtRiM2Ji1GLDYlUScmIzk2MDtGJy9GMEY9RjkvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJ0YvRjItRiM2KC1JI21uR0YkNiRRIjJGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GX28tRiw2JVEid0YnRi9GMkZYRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaW8vJSliZXZlbGxlZEdGPUZYRjk=, on obtient : LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJDRidGL0YyLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVi1GLDYlUSRjb3NGJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUScmIzk4MTtGJ0ZlbkY5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GYm8tSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVElJnBpO0YnRmVuRjkvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJ0Y5LUYjNiUtSSNtbkdGJDYkUSIyRidGOUZccEY5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZqcC8lKWJldmVsbGVkR0Y9RlxwRjlGOUY1LUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSEZhb0Zjb0ZPRlItRiw2JVEkc2luRidGZW5GOS1GZ242JC1GIzYlRltvRlxwRjlGOUZccEY5. Or 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, donc 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 De plus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RNCZHcmVhdGVyU2xhbnRFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGOQ==, donc LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2J1EiQ0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUSI9RidGLy9GNUY5RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1JJm1zcXJ0R0YkNiMtRiM2Ky1JJW1zdXBHRiQ2JS1GLDYnUSJBRidGL0YyRjRGNy1GIzYpLUkjbW5HRiQ2JlEiMkYnRi9GPkY3LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGLy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGPkY3LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GOzYvUSIrRidGL0Y+RjdGP0ZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZecC1GWTYlLUYsNidRIkJGJ0YvRjJGNEY3RmhuRmdvRl5vRi9GYW9GZG9GPkY3Rl5vRi9GYW9GZG9GPkY3. Par suite, comme 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, on a 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. De m\303\252me, comme 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, on a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2J1Ekc2luRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy1JKG1mZW5jZWRHRiQ2Ji1GIzYnLUYsNidRKSZ2YXJwaGk7RidGL0YyRjVGOC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRKFswLDAsMF1GJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNlEnbm9ybWFsRidGL0Y1RjgtSSNtb0dGJDYvUSI9RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zbby1GTjYtUSomdW1pbnVzMDtGJ0ZLRlFGU0ZVRldGWUZlbkZnbi9Gam5RLDAuMjIyMjIyMmVtRicvRl1vRmJvLUkmbWZyYWNHRiQ2KC1GIzYpLUYsNidRIkJGJ0YvL0YzRjEvRjZRLGJvbGQtaXRhbGljRidGOEZCRi9GXHBGSEZdcEY4LUYjNistRiw2I1EhRictRiM2KUZhcC1JJm1zcXJ0R0YkNiMtRiM2Ki1JJW1zdXBHRiQ2JS1GLDYnUSJBRidGL0ZccEZdcEY4LUYjNiktSSNtbkdGJDYmUSIyRidGL0Y1RjhGQkYvRlxwRkhGXXBGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRk42L1EiK0YnRi9GNUY4RlFGU0ZVRldGWUZlbkZnbkZhb0Zjby1GXHE2JUZpb0ZhcUZncUZCRi9GSEY1RjhGQkYvRkhGNUY4RmFwRkJGL0ZccEZIRl1wRjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRyLyUpYmV2ZWxsZWRHRjQtRk42LVEiLkYnRktGUUZTRlVGV0ZZRmVuRmduL0ZqblEmMC4wZW1GJy9GXW9GXXNGQkYvRkhGNUY4 Alors 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 3. Pour les fonctions d\303\251finies en 1, on a 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 et donc LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbXNxcnRHRiQ2Iy1GIzYnLUkjbW5HRiQ2JFEjMTNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGNEY3RjpGPUY0 . \317\225 doit v\303\251rifier 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, 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 et 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. Enfin, comme LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKSZ2YXJwaGk7RidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGMkYyLUkjbW9HRiQ2LVEiPkYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMEYnRjJGPUYy et 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, et 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, on a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEnJiM5ODE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZpc2ludjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1GNjYtUSJdRidGMi9GOlEldHJ1ZUYnRjsvRj5GUEY/RkFGQ0ZFL0ZIUSwwLjE2NjY2NjdlbUYnL0ZLUSwwLjExMTExMTFlbUYnLUY2Ni1RKiZ1bWludXMwO0YnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEsMC4yMjIyMjIyZW1GJy9GS0ZaLUkmbWZyYWNHRiQ2KC1GLDYlUScmIzk2MDtGJ0YvRjItRiM2Ji1JI21uR0YkNiRRIjJGJ0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvRjBGUC9GM1EnaXRhbGljRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl1wLyUpYmV2ZWxsZWRHRjEtRjY2LVEiLEYnRjJGOS9GPEZQRj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS1EsMC4zMzMzMzMzZW1GJy1GX282JFEiMEYnRjItRjY2LVEiW0YnRjJGT0Y7RlFGP0ZBRkNGRUZSL0ZLRlNGYm9GMg==. fsolve(cos(phi)=3/(sqrt(13)),phi=-(1/2)*Pi..0); On obtient donc LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn\342\211\210LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIwRidGLy1GLDYtUSIsRidGL0YyL0Y2USV0cnVlRidGN0Y5RjtGPUY/L0ZCUSYwLjBlbUYnL0ZFUSwwLjMzMzMzMzNlbUYnLUZHNiRRJDU4OEYnRi8vJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJ0Yv et donc LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0YvRjIvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy9GM1Enbm9ybWFsRidGQEY9RkA=\342\211\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 Pour la fonction LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0YzUSdub3JtYWxGJw==, on obtient 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 avec 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, 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 et 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 On cherche donc LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn entre 0 et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiRRIjJGJ0Y1LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkovJSliZXZlbGxlZEdGNEY+RjU= tel que 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= fsolve(cos(phi)=4/5,phi=0..(1/2)*Pi); On obtient 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

Attention \303\240 la position des crochets dans la commande plot pour obtenir une courbe param\303\251tr\303\251e ! LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= restart; plot([3*cos(t),3*sin(t),t=0..2*Pi]); plot([-1+3*t,2+t,t=-3..3]); plot([2+3*cos(t),1+2*sin(t),t=0..2*Pi]); On voit un cercle... For\303\247ons l'\303\251chelle : plot([2+3*cos(t),1+2*sin(t),t=0..2*Pi],scaling=constrained);

restart; x:=t->cos(t);y:=t->(1+cos(t))*sin(t); plot([x(t),y(t),t=-Pi..Pi]); Compl\303\251ment : trac\303\251 dynamique avec la commande "animate" with(plots):animate(plot,[[x(t),y(t),t=-Pi..a]],a=-Pi..Pi,frames=20); Il semblerait que la courbe admette l'axe LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiUtSSNtaUdGJDYlUSNPeEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnL0Y4USdub3JtYWxGJ0Y9RjpGPQ== comme axe de sym\303\251trie : 'x(-t)'=x(-t);'y(-t)'=y(-t); On a effectivement 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 et 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 restart; x:=t -> 2*sin(t)+cos(t); y:=t -> (sin(t))^3+2*(cos(t))^3; plot([x(t),y(t),t=-Pi..Pi]); Compl\303\251ment : trac\303\251 dynamique with(plots):animate(plot,[[x(t),y(t),t=-Pi..a]],a=-Pi..Pi,frames=20); Il semblerait que la courbe admette l'origine comme centre de sym\303\251trie : 'x(t+Pi)'=x(t+Pi); 'y(t+Pi)'=y(t+Pi); On a effectivement 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 et 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

restart; is(sqrt(x^2)=x); coulditbe(sqrt(x^2)=x); is(ln(exp(x))=x); is(sqrt(x^2)=x) assuming x>=0; is(ln(exp(x))=x) assuming x::real;

restart; z:=((1+I*sqrt(3))/(1-I))^20; w:=evalc(z);Re(w);Im(w);abs(w);argument(w);polar(w); with(LinearAlgebra): u:=<1,-1,3>;v:=<-5,-3,2>; u.v;DotProduct(u,v); VectorNorm(u,2); VectorNorm(v,2);

restart; f:=x->(2*x^4-3*x^2-10*x)/((x^2+1)*(x-2))*exp(abs(x)/(x+1)); limit(f(x),x=-infinity);limit(f(x),x=infinity); limit(f(x),x=-1,left);limit(f(x),x=-1,right);limit(f(x),x=2,left);limit(f(x),x=2,right); D1:=x->(2*x+2)*exp(1);limit(f(x)-D1(x),x=infinity); plot([f(x),D1(x)],x=-2..3,y=-10..25,color=[red,blue],discont=true);

restart; T:=x->(1-exp(-2*x))/(1+exp(-2*x)); T(x)+T(-x); simplify(%); Donc T est une fonction impaire. limit(T(x),x=infinity);limit(T(x),x=-infinity); D(T);(D(T))(2);evalf(%); Int(T(x),x)=int(T(x),x); plot(T(x),x=-5..5); g:=x->(1/2)*ln((1+x)/(1-x));plot([T(x),g(x),x],x=-5..5,color=[red,blue,black],thickness=[2,1,1]); Les deux courbes semblent sym\303\251triques par rapport \303\240 la droite d'\303\251quation 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; ce qui laisse penser que les fonctions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjdGhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy9GM1Enbm9ybWFsRic= et LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== sont r\303\251ciproques l'une de l'autre... simplify((T@g)(x));simplify((g@T)(x)); et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82JFEiMkYnRjIvJS5saW5ldGhpY2tuZXNzR0YxLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRjwvJSliZXZlbGxlZEdRJmZhbHNlRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYtSSNtaUdGJDYlUSNsbkYnLyUnaXRhbGljR0ZBRjItSShtZmVuY2VkR0YkNiQtRiM2KS1JJW1zdXBHRiQ2JS1GQzYtUS8mRXhwb25lbnRpYWxFO0YnRjJGRkZIRkpGTEZORlBGUkZUL0ZYUSwwLjExMTExMTFlbUYnLUYjNitGNUZCLUZaNiVRInhGJy9GaG5RJXRydWVGJy9GM1EnaXRhbGljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGXHAvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZfcEZicEZlcEZocEZqcEYyRjItRkM2LVEiPUYnRjJGRkZIRkpGTEZORlBGUi9GVVEsMC4yNzc3Nzc4ZW1GJy9GWEZkcUZoby1GQzYtUSIuRidGMkZGRkhGSkZMRk5GUEZSRlRGV0ZfcEZicEZlcEZocEZqcEYy

restart; f:=x->piecewise(-2<=x and x<-1,x+2,-1<=x and x<=0,-x,0<x and x<=1,1); plot(f(x),x=-2..1,discont=true);