Fonctions cyclométriques

Déterminer le domaine de définition, les éventuelles racines et la fonction dérivée de

n=0;
n++;f[x_,n]:=1/ArcTan[2x];
n++;f[x_,n]:=ArcCos[1-x];
n++;f[x_,n]:=ArcSin[x^3];
n++;f[x_,n]:=ArcSin[x]^3;
n++;f[x_,n]:=ArcCos[x^2-4];
n++;f[x_,n]:=ArcCos[1-2x-x^2];
n++;f[x_,n]:=ArcTan[(x-1)/(1+x)];
n++;f[x_,n]:=ArcCos[1-2x-x^2];
n++;f[x_,n]:=1/ArcSin[2x];
n++;f[x_,n]:=ArcCos[1-x^3];


Do[Print[i,") f(x) = ",f[x,i]],{i,n}];

Solutions

Do[Print[i,") Domf = ",Domaine[f[x,i],x]];
Racines[f[x,i],x,2];
 Print[" f'(x) = ",Derivee[f[x,i],x]],{i,n}]

Calcul de limites

Calculer les limites suivantes (éventuellement en utilisant la règle de l'Hospital)

n=0;
n++;Print[n,") ",Limite[HoldForm[ArcSin[2x-1]],x,1/2]];

n++;Print[n,") ",Limite[ArcCos[1-x],x,-1/2]];

n++;Print[n,") ",Limite[ArcTan[1/x],x,0]];

n++;Print[n,") ",Limite[ArcSin[x]/x,x,0]];

n++;Print[n,") ",Limite[3x/ArcTan[2x],x,0]];

n++;Print[n,") ",Limite[x (π/2-ArcTan[x]),x,+Infinity]];

n++;Print[n,") ",Limite[HoldForm[(ArcSin[x]-2x)/Sin[x]^3],x,0]];

Solutions

n=0;
n++;Print[n,") ",Presente[Limite[ArcSin[2x-1],x,1/2]]];

n++;Print[n,") ",Presente[Limite[ArcCos[1-x],x,-1/2]]];

n++;Print[n,") ",Presente[Limite[ArcTan[1/x],x,0]]];

n++;Print[n,") ",Presente[Limite[ArcSin[x]/x,x,0]]];

n++;Print[n,") ",Presente[Limite[3x/ArcTan[2x],x,0]]];

n++;Print[n,") ",Presente[Limite[x (π/2-ArcTan[x]),x,+Infinity]]];

n++;Print[n,") ",Presente[Limite[(ArcSin[x]-2x)/Sin[x]^3,x,0]]];