TD : Dérivabilité des fonctions

numériques d'une variable réelle

Exercice 1

> f:=x->(sqrt(1+x)-sqrt(1-x))/x;

f := proc (x) options operator, arrow; (sqrt(1+x)-s...

> plot(f,-1..1,0.9..1.5);

[Maple Plot]

> limit((f(x)-1)/x,x=0);

0

> D(f)(x);

(1/2*1/(sqrt(1+x))+1/2/(1-x)^(1/2))/x-(sqrt(1+x)-sq...

> limit(%,x=0);

0

Exercice 2

> diff(sqrt(1+sqrt(2+sqrt(x))),x);

1/8*1/(sqrt(1+sqrt(2+sqrt(x)))*sqrt(2+sqrt(x))*sqrt...

> diff(sqrt(1+sqrt(2+sqrt(x))),x);

1/8*1/(sqrt(1+sqrt(2+sqrt(x)))*sqrt(2+sqrt(x))*sqrt...

Il avait bon du premier coup !

Exercice 4

> diff(f1(x)*f2(x)*f3(x)*f4(x),x);

diff(f1(x),x)*f2(x)*f3(x)*f4(x)+f1(x)*diff(f2(x),x)...

> D(f1*f2*f3*f4);

D(f1)*f2*f3*f4+f1*D(f2)*f3*f4+f1*f2*D(f3)*f4+f1*f2*...

Exercice 7

> seq(diff(sin(x)^5,x$n),n=1..5);

5*sin(x)^4*cos(x), 20*sin(x)^3*cos(x)^2-5*sin(x)^5,...
5*sin(x)^4*cos(x), 20*sin(x)^3*cos(x)^2-5*sin(x)^5,...

> combine(sin(x)^5,trig);

1/16*sin(5*x)-5/16*sin(3*x)+5/8*sin(x)

> seq(diff(%,x$n),n=1..5);

5/16*cos(5*x)-15/16*cos(3*x)+5/8*cos(x), -25/16*sin...
5/16*cos(5*x)-15/16*cos(3*x)+5/8*cos(x), -25/16*sin...

> seq(diff(exp(x)*cos(x),x$n),n=1..10);

exp(x)*cos(x)-exp(x)*sin(x), -2*exp(x)*sin(x), -2*e...
exp(x)*cos(x)-exp(x)*sin(x), -2*exp(x)*sin(x), -2*e...

Exercice 8

> seq(diff(x^(n-1)*ln(x),x$n),n=1..12);

1/x, 1/x, 2*1/x, 6*1/x, 24*1/x, 120*1/x, 720*1/x, 5...

> seq((n-1)!,n=1..12);

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628...

Exercice 10

> limit((x-sin(x))/x^3,x=0);

1/6

> limit((ln(1+x)-x)/x^2,x=0);

-1/2

Exercice 19

> taylor(sin(Pi/3+h),h=0,4);

series(1/2*sqrt(3)+1/2*h+(-1/4*sqrt(3))*h^2-1/12*h^...

Exercice 21

> taylor(1/(1+t),t=0,9);

series(1-1*t+1*t^2-1*t^3+1*t^4-1*t^5+1*t^6-1*t^7+1*...

> taylor(ln(1+t),t=0,10);

series(1*t-1/2*t^2+1/3*t^3-1/4*t^4+1/5*t^5-1/6*t^6+...

> taylor(1/(1+t**2),t=0,9);

series(1-1*t^2+1*t^4-1*t^6+1*t^8+O(t^10),t,10)

> taylor(arctan(t),t=0,10);

series(1*t-1/3*t^3+1/5*t^5-1/7*t^7+1/9*t^9+O(t^10),...

> diff(arcsin(t),t);

1/(sqrt(1-t^2))

> taylor(%,t=0,7);

series(1+1/2*t^2+3/8*t^4+5/16*t^6+O(t^8),t,8)

> taylor(arcsin(t),t=0,8);

series(1*t+1/6*t^3+3/40*t^5+5/112*t^7+O(t^8),t,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);

0, 1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835, 0

> taylor(tan(x),x=0);

series(1*x+1/3*x^3+2/15*x^5+O(x^6),x,6)

> DL(18);

0, 1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835, 0, 13...

> taylor(tan(x),x=0,18);

series(1*x+1/3*x^3+2/15*x^5+17/315*x^7+62/2835*x^9+...