exo25-26.mw

> Exercice 25

> x:=t->cos(t)^2+ln(sin(t));y:=t->sin(t)*cos(t);f:=[x,y];

x := proc (t) options operator, arrow; cos(t)^2+ln(sin(t)) end proc

y := proc (t) options operator, arrow; sin(t)*cos(t) end proc

f := [x, y]

>

En Pi/4 on a le DL

> N:=4:t_0:=Pi/4:

> 'x'(t) =series(x(t),t=Pi/4,4);'y'(t) =series(y(t),t=Pi/4,4);

x(t) = (series((1/2+ln(1/2*2^(1/2)))-(t-1/4*Pi)^2+4/3*(t-1/4*Pi)^3+O((t-1/4*Pi)^4),t = 1/4*Pi,4))

y(t) = (series(1/2-(t-1/4*Pi)^2+O((t-1/4*Pi)^4),t = 1/4*Pi,4))

On en déduit que p=2, Vérifions le

>

> V:=NULL:for i from 1 to N do V:=V,map2(diff,[x(t),y(t)],t$i); od:
for i from 1 to N do Diff('f'(t),t$i)=V[i]; od;

Diff(f(t), t) = [-2*sin(t)*cos(t)+cos(t)/sin(t), cos(t)^2-sin(t)^2]

Diff(f(t), `$`(t, 2)) = [-2*cos(t)^2+2*sin(t)^2-1-cos(t)^2/sin(t)^2, -4*sin(t)*cos(t)]

Diff(f(t), `$`(t, 3)) = [8*sin(t)*cos(t)+2*cos(t)/sin(t)+2*cos(t)^3/sin(t)^3, -4*cos(t)^2+4*sin(t)^2]

Diff(f(t), `$`(t, 4)) = [8*cos(t)^2-8*sin(t)^2-2-8*cos(t)^2/sin(t)^2-6*cos(t)^4/sin(t)^4, 16*sin(t)*cos(t)]

> En Pi/4 on a les derivées successives

> for i from 1 to N do fprime[i]:=eval(subs(t=t_0,V[i])); od:
for i from 1 to N do Diff('f'(t_0),t$i)=fprime[i]; od;

Diff(f(1/4*Pi), t) = [0, 0]

Diff(f(1/4*Pi), `$`(t, 2)) = [-2, -2]

Diff(f(1/4*Pi), `$`(t, 3)) = [8, 0]

Diff(f(1/4*Pi), `$`(t, 4)) = [-16, 8]

Verifions la non colinéraité de f''et f'''

> p:=2:q:=3:

> "det(f2,f3)"=det([fprime[p],fprime[q]]);

D'où p=2 et q=3 c'est un point de rebroussemnt de 1ere espèce

>

> G:=plot([x(t),y(t),t=-4..4],thickness=3,color=red):

> Ap:=arrow(f(t_0),fprime[p],length=0.8,width=0.04,head_width=0.1,color=blue):Aq:=arrow(f(t_0),fprime[q],length=0.8,width=0.02,head_width=0.05,color=green):

> display(G,Ap,Aq);

[Plot]

>

> Exercice 26

> x:=t->exp(t)/(t+1);y:=t->t*exp(t)/(t+1);f:=[x,y];

x := proc (t) options operator, arrow; exp(t)/(t+1) end proc

y := proc (t) options operator, arrow; t*exp(t)/(t+1) end proc

f := [x, y]

>

Etude des branches infines en +infini et en -1

> Limit('x(t)',t=infinity)=limit(x(t),t=infinity);;Limit('y(t)',t=infinity)=limit(y(t),t=infinity);

Limit(x(t), t = infinity) = infinity

Limit(y(t), t = infinity) = infinity

> Limit('y(t)/x(t)',t=infinity)=limit(y(t)/x(t),t=infinity);

Limit(y(t)/x(t), t = infinity) = infinity

> On a donc une branche parabolique dans la direction Oy pour t->+infini

> Limit('x(t)^2+y(t)^2',t=-1)=limit(x(t)^2+y(t)^2,t=-1);

Limit(x(t)^2+y(t)^2, t = -1) = infinity

> Limit('y(t)/x(t)',t=-1)=limit(y(t)/x(t),t=-1);

Limit(y(t)/x(t), t = -1) = -1

> 'y(t)+x(t)'=series(y(t)+x(t),t=-1,3);

y(t)+x(t) = (series(exp(-1)+exp(-1)*(t+1)+O((t+1)^2),t = -1,2))

On a donc une  asymptote y=-x+1/e pour t->-1, la courbe est au dessus en 1+ et en dessous en 1-

>

>

Etude locale en 1

> N:=3:t_0:=1:

> V:=NULL:for i from 1 to N do V:=V,map2(diff,[x(t),y(t)],t$i); od:
for i from 1 to N do Diff('f'(t),t$i)=V[i]; od;

Diff(f(t), t) = [exp(t)/(t+1)-exp(t)/(t+1)^2, exp(t)/(t+1)+t*exp(t)/(t+1)-t*exp(t)/(t+1)^2]

Diff(f(t), `$`(t, 2)) = [exp(t)/(t+1)-2*exp(t)/(t+1)^2+2*exp(t)/(t+1)^3, 2*exp(t)/(t+1)-2*exp(t)/(t+1)^2+t*exp(t)/(t+1)-2*t*exp(t)/(t+1)^2+2*t*exp(t)/(t+1)^3]

Diff(f(t), `$`(t, 3)) = [exp(t)/(t+1)-3*exp(t)/(t+1)^2+6*exp(t)/(t+1)^3-6*exp(t)/(t+1)^4, 3*exp(t)/(t+1)-6*exp(t)/(t+1)^2+6*exp(t)/(t+1)^3+t*exp(t)/(t+1)-3*t*exp(t)/(t+1)^2+6*t*exp(t)/(t+1)^3-6*t*exp(...

>

> En 1 on a les derivées successives

> for i from 1 to N do fprime[i]:=eval(subs(t=t_0,V[i])); od:
for i from 1 to N do Diff('f'(t_0),t$i)=fprime[i]; od;

Diff(f(1), t) = [1/4*exp(1), 3/4*exp(1)]

Diff(f(1), `$`(t, 2)) = [1/4*exp(1), 3/4*exp(1)]

Diff(f(1), `$`(t, 3)) = [1/8*exp(1), 7/8*exp(1)]

Ceci est aussi un résultats des DL en 1 des fonctions x et y (d'après Taylor-Young)

> 'x'(t) =series(x(t),t=t_0,N+1);'y'(t) =series(y(t),t=t_0,N+1);

x(t) = (series(1/2*exp(1)+1/4*exp(1)*(t-1)+1/8*exp(1)*(t-1)^2+1/48*exp(1)*(t-1)^3+O((t-1)^4),t = 1,4))

y(t) = (series(1/2*exp(1)+3/4*exp(1)*(t-1)+3/8*exp(1)*(t-1)^2+7/48*exp(1)*(t-1)^3+O((t-1)^4),t = 1,4))

>

Verifions la  colinéraité f' et f'' puis la non colinéraité f' et f'''

> p:=1:q:=3:

> "det(f1,f2)"=det([fprime[1],fprime[2]]);"det(f1,f3)"=det([fprime[p],fprime[q]]);

D'où au point de paramètre 1 on a p=1 et q=3  c'est un point d'inflexion

> Limit('x(t)',t=-infinity)=limit(x(t),t=-infinity);;Limit('y(t)',t=-infinity)=limit(y(t),t=-infinity);

Limit(x(t), t = -infinity) = 0

Limit(y(t), t = -infinity) = 0

>

>

Etude locale en -infini

On prolonge la courbe en rajoutant pour t->-infini le point O

> Limit('dy/dx',t=-infinity)=limit((V[1])[2]/(V[1])[1],t=-infinity);

Limit(dy/dx, t = -infinity) = -infinity

En O on  a donc une tangente verticale

> G:=plot([x(t),y(t),t=-infinity..10],thickness=3,color=red,view=[-3..3,-3..6]):

> As:=plot(-x+exp(-1),x=-3..3,thickness=3,color=violet):

> Ap:=arrow(f(t_0),fprime[p],length=0.8,width=0.07,head_width=0.2,color=blue):Aq:=arrow(f(t_0),fprime[q],length=0.8,width=0.07,head_width=0.2,color=green):

> Tg:=plot([0,t,t=0..10],thickness=3,color=cyan):

> display(G,As,Ap,Aq,Tg);

[Plot]

>