exo22-23-24.mw

> restart;with(linalg);with(plots);

Warning, the protected names norm and trace have been redefined and unprotected

[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp, Wronskian, addcol, addrow, adj, adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix, charmat, charpoly, cholesky, col, cold...[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp, Wronskian, addcol, addrow, adj, adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix, charmat, charpoly, cholesky, col, cold...[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp, Wronskian, addcol, addrow, adj, adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix, charmat, charpoly, cholesky, col, cold...[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp, Wronskian, addcol, addrow, adj, adjoint, angle, augment, backsub, band, basis, bezout, blockmatrix, charmat, charpoly, cholesky, col, cold...

Warning, the name changecoords has been redefined

[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, cylinderplot, densityplot, display, disp...

>

> Exercice 22

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

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

y := proc (t) options operator, arrow; 2*sin(t)-sin(2*t) end proc

f := [x, y]

> En 2Pi/3 on a le DL

> N:=3:t_0:=2*Pi/3:

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

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

y(t) = (series(3/2*3^(1/2)-3/2*3^(1/2)*(t-2/3*Pi)^2-1/2*(t-2/3*Pi)^3+O((t-2/3*Pi)^4),t = 2/3*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)-2*sin(2*t), 2*cos(t)-2*cos(2*t)]

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

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

En 2pi/3 on a les dérivées successives

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

Diff(f(2/3*Pi), t) = [0, 0]

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

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

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=0.04,head_width=0.1,color=green):

> display(G,Ap,Aq);

[Plot]

>

> Exercice 23

> x:=t->t^2-2*t;y:=t->2*t+1/t^2;f:=[x,y];

x := proc (t) options operator, arrow; t^2-2*t end proc

y := proc (t) options operator, arrow; 2*t+1/t^2 end proc

f := [x, y]

En 1 on a le DL

> N:=3;t_0:=1;

N := 3

t_0 := 1

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

x(t) = (series(-1+(t-1)^2,t = 1))

y(t) = (series(3+3*(t-1)^2-4*(t-1)^3+O((t-1)^4),t = 1,4))

> 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*t-2, 2-2/t^3]

Diff(f(t), `$`(t, 2)) = [2, 6/t^4]

Diff(f(t), `$`(t, 3)) = [0, -24/t^5]

En 1 on a les dérivées successives

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

Diff(f(1), t) = [0, 0]

Diff(f(1), `$`(t, 2)) = [2, 6]

Diff(f(1), `$`(t, 3)) = [0, -24]

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=0.5..2],thickness=3,color=red):

> Ap:=arrow(f(t_0),fprime[p],length=0.8,width=0.02,head_width=0.05,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 24

> x:=t->1/t+ln(2+t);y:=t->t+1/t;f:=[x,y];

x := proc (t) options operator, arrow; 1/t+ln(2+t) end proc

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

f := [x, y]

>

>

Etrude des branches infinies en 0

> N:=3:t_01:=0:t_02:=-1:t_03:=4:

> Limit('y(t)/x(t)',t=0)=limit(y(t)/x(t),t=0);

Limit(y(t)/x(t), t = 0) = 1

> 'y(t)-x(t)' =series(y(t)-x(t),t=t_01,3);

y(t)-x(t) = (series(-ln(2)+1/2*t+1/8*t^2+O(t^3),t,3))

On en déduit une asymptote y=x+ln(2) pour t=0 et Gamma au dessus en 0+ et en dessous en 0-

>

>

>

Etude locale en -1

> 'x'(t) =series(x(t),t=t_02,N);'y'(t) =series(y(t),t=t_02,N);

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

y(t) = (series(-2-(1+t)^2+O((1+t)^3),1 = -t,3))

>

> On en déduit que p=2 au point de paramètre -1

> N:=4:

N := 4

> 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) = [-1/t^2+1/(2+t), 1-1/t^2]

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

Diff(f(t), `$`(t, 3)) = [-6/t^4+2/(2+t)^3, -6/t^4]

En -1 on a les dérivées successives

> for i from 1 to N do fprime2[i]:=eval(subs(t=t_02,V[i])); od:
for i from 1 to N do Diff('f'(t_02),t$i)=fprime2[i]; od;

Diff(f(-1), t) = [0, 0]

Diff(f(-1), `$`(t, 2)) = [-3, -2]

Diff(f(-1), `$`(t, 3)) = [-4, -6]

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

> p2:=2:q2:=3:

> "det(f1,f2)"=det([fprime2[p2],fprime[q2]]);

D'où pour le point de paramètre -1 on a p=2 et q=3  c'est un point de rebroussemnt de 1ere espèce

>

>

Etude locale en 4

> 'x'(t) =series(x(t),t=t_03,N);'y'(t) =series(y(t),t=t_03,N);

x(t) = (series((1/4+ln(6))+5/48*(t-4)+1/576*(t-4)^2+O((t-4)^3),t = 4,3))

y(t) = (series(17/4+15/16*(t-4)+1/64*(t-4)^2+O((t-4)^3),t = 4,3))

>

> On en déduit que p=1 au point de paramètre 4 Vérifions le

En 4 on a les dérivées successives

> for i from 1 to N do fprime3[i]:=eval(subs(t=t_03,V[i])); od:
for i from 1 to N do Diff('f'(t_03),t$i)=fprime3[i]; od;

Diff(f(4), t) = [5/48, 15/16]

Diff(f(4), `$`(t, 2)) = [1/288, 1/32]

Diff(f(4), `$`(t, 3)) = [(-49)/3456, (-3)/128]

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

> p3:=1:q3:=3:

> "det(f1,f2)"=det([fprime3[1],fprime3[2]]);"det(f1,f3)"=det([fprime3[p3],fprime3[q3]]);

D'ou au   point de paramètre 4 on a p=1 et q=3 c'est un point d'inflexion

>

> G:=plot([x(t),y(t),t=-2..6],thickness=3,color=red,view=[1.5..3,-3..6]):

> As:=plot(x-ln(2),x=-3..3,thickness=3,color=violet):

> Ap2:=arrow(f(t_02),fprime2[p2],length=0.8,width=0.07,head_width=0.2,color=blue):Aq2:=arrow(f(t_02),fprime2[q2],length=0.8,width=0.07,head_width=0.2,color=green):

> Ap3:=arrow(f(t_03),fprime3[p3],length=0.8,width=0.07,head_width=0.2,color=blue):Aq3:=arrow(f(t_03),fprime3[q3],length=0.8,width=0.07,head_width=0.2,color=green):

> display(G,As,Ap2,Aq2,Ap3,Aq3);

[Plot]