Model 47132

Problema 1)

> restart;

1a)

> with(geometry):

> talls:=proc(c,r)
local p;
intersection(p,c,r):
if nops(p)=0 then
print("No es tallen");
elif nops(p)=1 then
print("Es tallen en un punt");
elif nops(p)=2 then
print("Es tallen en dos punts");
end if;
end proc;

>

1b)

> circle(c,[point(A,1,3),2],[x,y]);

> detail(c);

> line(l,2*x-y-1=0,[x,y]);

> detail(l);

> talls(c,l);

1c)

> intersection(p,c,l);

> detail(p);

> draw([c(color=blue),l,p[1](color=black),p[2](color=black)]);

Problema 2)

> restart;

> plot(ln(x)-1/4*x,x=0..20);

> sol1:=evalf(fsolve(ln(x)=1/4*x,x=0..4),15);

> sol2:=evalf(fsolve(ln(x)=1/4*x,x=8..10),15);

> fsolve(ln(x)=1/4*x,x=10..infinity);

Model 39051

Problema 1)

> restart;

1a)

> with(LinearAlgebra):

> solucions:=proc(M,k)
local sol;
sol:=LinearSolve(M-k*IdentityMatrix(RowDimension(M)),ZeroMatrix(RowDimension(M),1)):
if IsMatrixShape(sol, zero)=true then
[];
else
sol;
end if;
end proc;

1b)

> M:=<<1|-2|1>,<2|-1|-1>,<-1|2|1>>;

> solucions(M,1);

> solucions(M,2);

Problema 2)

> restart;

> a:=n->(n^n*sqrt(n))/((n)!*exp(n));

> a(1);

> a(2);

> evalf(a(15));

> l:=limit(a(n),n=infinity);

> with(plots):

Warning, the name changecoords has been redefined

> dib1:=plot(l,x=0..16,colour=green):

> dib2:=plot([seq([n,a(n)],n=1..15)],style=point,colour=blue):

> display([dib1,dib2]);

Model 86183

Problema 1)

> restart;

1a)

> suc:=proc(n)
local i,a,b;
if n<1 then
print("n ha de ser mÈs gran o igual que 1");
elif n=1 then
a:=10:
else
a:=10:
for i from 2 to n do
b:=(a^2+7)/(2*a):
a:=b:
end do;
end if;
evalf(a,20);
end proc;

> suc(1);

> suc(2);

1b)

> for i from 1 while (evalf(abs(suc(i+1)-suc(i)))>10^(-7)) do

> i;
end do;

>

> evalf(abs(suc(7+1)-suc(7)),15);

> evalf(suc(8));

> evalf(sqrt(7));

Problema 2)

> restart;

> with(LinearAlgebra):

> M:=<<1|2-a|-2+3*a|2-a+2*a^2>,<2|1+a|-1-2*a|4+a-a^2>,<3|4-a|-4+3*a|6-a+2*a^2>>;

> RowOperation(M,[2,1],-2,inplace);

> RowOperation(M,[3,1],-3,inplace);

> RowOperation(M,[3,2],-2/3,inplace);

> M0:=subs(a=0,M);

> LinearSolve(M0[1..3,1..3],M0[1..3,4]);

Per a=0 es sistema compatible indeterminat

> M1:=subs(a=1,M);

> LinearSolve(M1[1..3,1..3],M1[1..3,4]);

Error, (in LinearAlgebra:-LA_Main:-LinearSolve) inconsistent system

Per a=1 es un sistema incompatible

> LinearSolve(M[1..3,1..3],M[1..3,4]);

Per a<>0 i a<>1 Ès compatible determinat.

Model 30970

Problema 1)

> plot([(ln(x))^4,x],x=0..10,y=0..20);

> sol1:=evalf(fsolve(ln(x)^4=x,x=0..1),15);

> sol2:=evalf(fsolve(ln(x)^4=x,x=3..5),15);

> sol3:=evalf(fsolve(ln(x)^4=x,x=5..10000),15);

> fsolve(ln(x)^4=x,x=5504..infinity);

> plot([(ln(x))^4,x],x=5503..5504);

Problema 2)

> restart;

> with(geometry):

> configuracio:=proc(P,r,argument)
local l;
if argument<>1 and argument<>-1 then
print("El tercer parametre ha de valer 1 o -1");
elif argument=1 then
PerpendicularLine(l,P,r):
draw([l(color=blue),P,r(color=green)]);
elif argument=-1 then
ParallelLine(l,P,r):
draw([l(color=blue),P,r(color=green)]);
end if;
end proc;

> point(P,3,1);

> detail(P);

> line(r,x-y+2=0,[x,y]);

> detail(r);

> configuracio(P,r,1);

Model 94264

Problema 1)

1a)

> restart;

> with(geometry):

> talls:=proc(C1,C2)
local In;
if center(C1)=center(C2) and radius(C1)=radius(C2) then
printf("Les dues circumferencies son la mateixa");
else
intersection(In,C1,C2):
printf("El numero de punts de tall es:");
nops(In);
end if;
end proc;

>

1b)

> point(A,1,3);

> circle(C1,[A,2]);

> circle(C2,x^2+y^2-2*x+2*y-2=0,[x,y]);

> detail(C1); detail(C2);

assume that the names of the horizontal and vertical axes are _x and _y, respectively

> talls(C1,C2);

El n™mero de punts de tall Ès:

1c)

> intersection(In,C1,C2);

> draw([C1(color=blue),C2(color=green),In(color=black)]);

Problema 2)

> restart;

> x:=n->(1/sqrt(n))*(product(2*i,i=1..n)/product(2*i-1,i=1..n));

> limit(x(n),n=infinity);