CSB lektion 8, 29/3-2004Repetition kl. 1230-1305with(plots):with(CurveFitting):Rutchebanef := x -> sin(x);plot(f,0..Pi);L\346ngden af banen:g := x->sqrt(1+cos(x)^2);plot1:=plot(g,0..Pi): display(%);Beregning af stamfunktionint(g(x),x);Midtpunktsreglenplot2:=plot(g(Pi/2),x=0..Pi):display([%,plot1]);midt:=evalf((Pi-0)*g(Pi/2));Trapezreglenplot3:= plot((g(Pi)-g(0))/Pi+g(0),x=0..Pi):display([%,plot1]);trap:=evalf((Pi-0)/2*(g(0)+g(Pi)));Simpsons regelpunkter:=[[0,g(0)],[Pi/2,g(Pi/2)],[Pi,g(Pi)]]:l:=unapply(PolynomialInterpolation(punkter,x,form=Lagrange),x):plot4:=plot(l,0..Pi):display([plot1,plot4]);simp:=evalf((Pi-0)/6*(g(0)+4*g(Pi/2)+g(Pi)));Eksakt udregningexact:=evalf(int(g(x),x=0..Pi));Fejl evalf(abs(midt-exact));evalf(abs(trap-exact));evalf(abs(simp-exact));Forel\346sning kl 1505-1615MidspunktsreglenmyMidtpunktN:= proc(f,a,b,N)
local h, res, k;
h:= (b-a)/N;
evalf(h*sum(f(a+(k+1/2)*h),k=0..N-1));
end;myMidtpunktN(g,0,Pi,1);TrpezreglenmyTrapezN:= proc(f,a,b,N)
local h, res, k;
h:= (b-a)/N;
evalf(h/2*(f(a)+2*sum(f(a+k*h),k=1..N-1)+f(b)));
end;myTrapezN(g,0,Pi,1);Simpson regelmySimpsonN:= proc(f,a,b,N)
local h, res, k;
h:= (b-a)/N;
evalf(h/6*(f(a)+2*sum(f(a+k*h),k=1..N-1)+4*sum(f(a+(k+1/2)*h),k=0..N-1)+f(b)));
end;mySimpsonN(g,0,Pi,10);Eksempel 6 i Turnerf := x-> 1/(1+x^2);Vi vil udregneInt(f(x),x=0...1);Digits:=16;for n from 0 to 4 do
N:= 2^n;
midt:=evalf(myMidtpunktN(f,0,1,N)):
trapez:=evalf(myTrapezN(f,0,1,N)):
simpson:=evalf(mySimpsonN(f,0,1,N)):
fejlm:=evalf(midt-Pi/4):
fejlt:=evalf(trapez-Pi/4):
fejls:=evalf(simpson-Pi/4):
printf("%-1.9f %-1.9f %-1.9f %-1.9f %-1.9f %-1.16f\n",midt,trapez,simpson,fejlm,fejlt,fejls);
end:Antal iterationer for at bestemme integralet m pr\346cision 10^(-8)f4 := D(D(D(D(f))));plot(f4,0..1);M:=f4(0);evalf((M/(180*10^(-8)))^(1/4));