> Exercice 1
 

> evalf(Pi,1000);
 

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381...
(1)
 

> Exercice 2
 

> restart:
 

> f:=x->x^4-2*x^2+3;
 

proc (x) options operator, arrow; `+`(`*`(`^`(x, 4)), `-`(`*`(2, `*`(`^`(x, 2)))), 3) end proc (2)
 

> limit(f(x),x=+infinity);
 

infinity (3)
 

> derf:=D(f);
 

proc (x) options operator, arrow; `+`(`*`(4, `*`(`^`(x, 3))), `-`(`*`(4, `*`(x)))) end proc (4)
 

> solve(derf(x)=0,x);
 

0, 1, -1 (5)
 

> solve(derf(x));
 

0, 1, -1 (6)
 

> solve(derf(x)>0,x);
 

RealRange(Open(-1), Open(0)), RealRange(Open(1), infinity) (7)
 

> C'est l'intervalle ]-1,0[ U ]1,+infinity[
 

> plot(f(x),x=-5..5);
 

Plot_2d
 

> plot(f(x),x=-1.5..1.5);
 

Plot_2d
 

> Exercice 3
 

> x:=t->exp(-0.1*t)*((2/3)*sin(10*t)+(4/5)*cos(10*t));
 

proc (t) options operator, arrow; `*`(exp(`+`(`-`(`*`(.1, `*`(t))))), `*`(`+`(`*`(`/`(2, 3), `*`(sin(`+`(`*`(10, `*`(t)))))), `*`(`/`(4, 5), `*`(cos(`+`(`*`(10, `*`(t))))))))) end proc (8)
 

> v:=D(x);
 

proc (t) options operator, arrow; `+`(`-`(`*`(.1, `*`(exp(`+`(`-`(`*`(.1, `*`(t))))), `*`(`+`(`*`(`/`(2, 3), `*`(sin(`+`(`*`(10, `*`(t)))))), `*`(`/`(4, 5), `*`(cos(`+`(`*`(10, `*`(t))))))))))), `*`(e... (9)
 

> solve(v(t)=0);
 

0.6847386095e-1 (10)
 

> plot(v(t),t=0..2);
 

Plot_2d
 

> fsolve(v(t),t,t=0.2..0.6);
 

.3826331263 (11)
 

> fsolve(v(t),t,t=0.6..0.8);
 

.6967923917 (12)
 

> Exercice 4
 

>
 

>
 

>
 

>
 

> f:=x->piecewise(x<5,x^2+b*x+1,x=5,8,x>5,a*x+3);
 

proc (x) options operator, arrow; piecewise(`<`(x, 5), `+`(`*`(`^`(x, 2)), `*`(b, `*`(x)), 1), x = 5, 8, `<`(5, x), `+`(`*`(a, `*`(x)), 3)) end proc (13)
 

> solve({limit(f(x),x=5,left)=8,limit(f(x),x=5,right)=8},{a,b});
 

{a = 1, b = -`/`(18, 5)} (14)
 

> a:=1;b:=-18/5;
 

 

1
-`/`(18, 5) (15)
 

> plot(f(x));
 

Plot_2d
 

>
 

> Exercice 5
 

> restart;f:=x->3*x^4+a*x^3+b*x^2+c*x+d;
 

proc (x) options operator, arrow; `+`(`*`(3, `*`(`^`(x, 4))), `*`(a, `*`(`^`(x, 3))), `*`(b, `*`(`^`(x, 2))), `*`(c, `*`(x)), d) end proc (16)
 

>
 

> fp:=D(f);
 

proc (x) options operator, arrow; `+`(`*`(12, `*`(`^`(x, 3))), `*`(3, `*`(a, `*`(`^`(x, 2)))), `*`(2, `*`(b, `*`(x))), c) end proc (17)
 

> solve({-3=f(2),0=fp(0),fp(2)=0,f(0)=7},{a,b,c,d});
 

{a = -`/`(19, 2), b = `/`(9, 2), c = 0, d = 7} (18)
 

>
 

>
 

> c:= 0: d:=7: a:=-19/2: b:=9/2: plot(f(x),x=-1..2.5);
 

Plot_2d
 

> solve(fp(x),x);
 

0, 2, `/`(3, 8) (19)
 

> Exercice 7
 

> restart;
 

> f:=x->x^3-2*x^2+1;
 

proc (x) options operator, arrow; `+`(`*`(`^`(x, 3)), `-`(`*`(2, `*`(`^`(x, 2)))), 1) end proc (20)
 

> fp:=D(f);
 

proc (x) options operator, arrow; `+`(`*`(3, `*`(`^`(x, 2))), `-`(`*`(4, `*`(x)))) end proc (21)
 

> T:=x->fp(2)*(x-2)+f(2);
 

proc (x) options operator, arrow; `+`(`*`(fp(2), `*`(`+`(x, `-`(2)))), f(2)) end proc (22)
 

> S:=x->a*(x-2)+f(2);
 

proc (x) options operator, arrow; `+`(`*`(a, `*`(`+`(x, `-`(2)))), f(2)) end proc (23)
 

>
 

`+`(`*`(4, `*`(x)), `-`(7)) (24)
 

> C:=seq(plot([f(x),T(x),(-2+a*0.1)*(x-2)+1],x=0..3,color=[red,blue,green],thickness=[1,3,1]),a=-100..100):
 

> with(plots):
 

> display(C,insequence=true,view=[0..3,-5..5]);
 

Plot_2d
 

>