restart;with(plots):secu:=1000:secu une valeur de s\351curit\351 pour \351viter les boucles trop longues

Warning, the name changecoords has been redefined

Exercice 1Question a)Dichotomie:=proc(f,a,b,epsilon) local aa,bb,compteur,milieu; aa:=a;bb:=b; avant d'amorcer la boucle assurons-nous que nous sommes dans un cas particulier if is(f(a)*f(b)>0) then return([infinity,infinity]); si f(a) et f(b) sont de m\352me signe on renvoie une solution erronn\351e elif is(f(a)=0) then return([0,a]); si f(a)=0 alors a est solution elif is(f(b)=0) then return([0,b]); si f(b)=0 alors b est solution fi; compteur:=0; while is(abs(bb-aa)>epsilon) and is(compteur<secu) do compteur:=compteur+1;milieu:=(aa+bb)/2; if is(f(milieu)=0) then aa:=bb au cas o\371 milieu soit une solution on sort de la boucle en posant [an,b_n] de diametre nul elif is(f(milieu)*f(bb)<0) then aa:=milieu; else bb:=milieu; fi; od; [compteur,milieu]; end: Question b)Premier exemplef:=x->exp(-x/2)-1;a:=-4;b:=4;epsilon:=10^(-8);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSSRleHBHRiU2IywkOSQjISIiIiIjIiIiRjNGNUYlRiVGJQ==NiM+SSJhRzYiISIlNiM+SSJiRzYiIiIlNiM+SShlcHNpbG9uRzYiIyIiIiIqKysrKyI=

sol:=Dichotomie(f,a,b,epsilon);solution propos\351 par la proc\351dure Dichotomie

NiM+SSRzb2xHNiI3JCIjSSMhK0I9dXQ1IipjYVZvIw==

'f'(sol[2])=evalf(f(sol[2])); valeur de f en ce point

NiMvLUkiZkc2IjYjIyErQj11dDUiKmNhVm8jJCIrJTNjISpRJyEiKg==

Deuxi\350me exemplef:=x->x^3-4*x+1;a:=-2.5;b:=-2;epsilon:=10^(-8);

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sol:=Dichotomie(f,a,b,epsilon);solution propos\351 par la proc\351dure Dichotomie

NiM+SSRzb2xHNiI3JCIjRSQhK1d2IVw2IyEiKg==

'f'(sol[2])=evalf(f(sol[2])); valeur de f en ce point

NiMvLUkiZkc2IjYjJCErV3YhXDYjISIqJCEjQ0Yq

Troisi\350me exemplef:=x->x^3-4*x+1;a:=0;b:=0.5;epsilon:=10^(-8);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIkIiIiRi4hIiVGMEYwRiVGJUYlNiM+SSJhRzYiIiIhNiM+SSJiRzYiJCIiJiEiIg==NiM+SShlcHNpbG9uRzYiIyIiIiIqKysrKyI=

sol:=Dichotomie(f,a,b,epsilon);solution propos\351 par la proc\351dure Dichotomie

NiM+SSRzb2xHNiI3JCIjRSQiKyoqbyxURCEjNQ==

'f'(sol[2])=evalf(f(sol[2])); valeur de f en ce point

NiMvLUkiZkc2IjYjJCIrKipvLFREISM1JCEiJyEiKg==

Quatri\350me exemplef:=x->x^3-4*x+1;a:=1.5;b:=2;epsilon:=10^(-8);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIkIiIiRi4hIiVGMEYwRiVGJUYlNiM+SSJhRzYiJCIjOiEiIg==NiM+SSJiRzYiIiIjNiM+SShlcHNpbG9uRzYiIyIiIiIqKysrKyI=

sol:=Dichotomie(f,a,b,epsilon);solution propos\351 par la proc\351dure Dichotomie

NiM+SSRzb2xHNiI3JCIjRSQiK19lITMnPSEiKg==

'f'(sol[2])=evalf(f(sol[2])); valeur de f en ce point

NiMvLUkiZkc2IjYjJCIrX2UhMyc9ISIqJCEiKEYq

Illustration Graphique

SDichotomie:=proc(f,a,b,epsilon) local aa,bb,compteur,milieu,suite_m; aa:=a;bb:=b; if is(f(a)*f(b)>0) then return([infinity,infinity]); elif is(f(a)=0) then return([0,a]); elif is(f(b)=0) then return([0,b]); fi; compteur:=1;suite_m:=NULL; milieu:=(aa+bb)/2;suite_m:=suite_m,milieu; while is(abs(bb-aa)>epsilon) and compteur<secu do if is(f(milieu)*f(bb)<0) then aa:=milieu; else bb:=milieu; fi; compteur:=compteur+1;milieu:=(aa+bb)/2;suite_m:=suite_m,milieu; od; [compteur,suite_m]; end:Graphique:=proc(f,a,b,N,s) local i,segments,courbe,points; segments:=NULL;points:=NULL; for i from 1 to N do segments:=segments,plot([[s[i],0],[s[i],f(s[i])]],color=blue,thickness=2); points:=points,plot([[s[i],0],[s[i],0]],color=black,thickness=2,style=point); od; courbe:=plot(f,a..b,color=red,thickness=3); display(courbe,points,segments); end:

f:=x->x^3-4*x+1;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIkIiIiRi4hIiVGMEYwRiVGJUYl

sol:=SDichotomie(f,a,b,epsilon):N:=sol[1]:s:=sol[2..N+1]:Graphique(f,a,b,N,s);

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

sur l'axe des abbscissdes apparaissent les points milieu xn de la suite de segments [an,bn] construits par dichotomie la courbe est cell de fpoints:=seq([i,s[i]],i=1..N):plot([points],color=black,thickness=2,style=point);

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Les points trac\351s sont ceux de coordonn\351es (n,xn)