TD : Dérivabilité des fonctions
numériques d'une variable réelle
Exercice 1
> f:=x->(sqrt(1+x)-sqrt(1-x))/x;
> plot(f,-1..1,0.9..1.5);
> limit((f(x)-1)/x,x=0);
> D(f)(x);
> limit(%,x=0);
Exercice 2
> diff(sqrt(1+sqrt(2+sqrt(x))),x);
> diff(sqrt(1+sqrt(2+sqrt(x))),x);
Il avait bon du premier coup !
Exercice 4
> diff(f1(x)*f2(x)*f3(x)*f4(x),x);
> D(f1*f2*f3*f4);
Exercice 7
> seq(diff(sin(x)^5,x$n),n=1..5);
> combine(sin(x)^5,trig);
> seq(diff(%,x$n),n=1..5);
> seq(diff(exp(x)*cos(x),x$n),n=1..10);
Exercice 8
> seq(diff(x^(n-1)*ln(x),x$n),n=1..12);
> seq((n-1)!,n=1..12);
Exercice 10
> limit((x-sin(x))/x^3,x=0);
> limit((ln(1+x)-x)/x^2,x=0);
Exercice 19
> taylor(sin(Pi/3+h),h=0,4);
Exercice 21
> taylor(1/(1+t),t=0,9);
> taylor(ln(1+t),t=0,10);
> taylor(1/(1+t**2),t=0,9);
> taylor(arctan(t),t=0,10);
> diff(arcsin(t),t);
> taylor(%,t=0,7);
> taylor(arcsin(t),t=0,8);
Exercice 22
>
DL:=proc(dernier)
local n,j,t;
t[0]:=0: t[1]:=1:
for n from 1 to dernier-1 do t[n+1]:=sum(t[j]*t[n-j],j=1..n)/(n+1) od;
RETURN(seq(t[n],n=0..dernier))
end:
> DL(10);
> taylor(tan(x),x=0);
> DL(18);
> taylor(tan(x),x=0,18);