/* 1.1 */ f(x) := if x < 0 then %pi else 0; s(x,n) := %pi/2 + sum(((-1)^k/k-1/k)*sin(k*x), k, 1, n); sigma(x,n) := 1/n*sum(s(x,k), k, 0, n-1); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-1.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 1.2 */ f(x) := if x < 0 then -1 else if x = 0 then 0 else 1; s(x,n) := sum(2/%pi*(1/k-(-1)^k/k)*sin(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-1.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 1.3 */ f(x) := (%pi-x)/2; s(x,n) := %pi/2 + sum((-1)^k/k*sin(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-1.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 1.4 */ f(x) := abs(x); s(x,n) := %pi/2 + sum(2/%pi*((-1)^k/k^2-1/k^2)*cos(k*x), k, 1, n); plot2d( [f(t), s(t,5), sigma(t,5)], [t,-1.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 1.5 */ f(x) := x; s(x,n) := sum(-2*(-1)^k/k*sin(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-1.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 1.6 */ f(x) := x^2; s(x,n) := %pi^2/3 + sum(4*(-1)^k/k^2*cos(k*x), k, 1, n); plot2d( [f(t), s(t,5), sigma(t,5)], [t,-1.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 1.7 */ f(x) := exp(x); T : (exp(%pi)-exp(-%pi))/%pi; s(x,n) := T/2 + T * sum((-1)^k/(k^2+1)*cos(k*x) - (-1)^k*k/(k^2+1)*sin(k*x), k, 1, n); plot2d( [f(t), s(t,5), sigma(t,5)], [t,-3*%pi,3*%pi], [xlabel, "x"], [ylabel, "y"], [y, -2, 10] ); /* 2.1 */ f(x) := 1; plot2d( f(t), [t,-0.5*%pi,1.5*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 2.2 */ f(x) := x; s(x,n) := %pi/2 + sum(2/%pi*((-1)^k/k^2-1/k^2)*cos(k*x), k, 1, n); plot2d( [f(t), s(t,5), sigma(t,5)], [t,-2*%pi,2*%pi], [xlabel, "x"], [ylabel, "y"], [y, -1, 5] ); /* 2.3 */ f(x) := sin(x); s(x,n) := 2/%pi + sum(-2*((-1)^k+1) / (%pi*(k^2-1))*cos(k*x), k, 2, n); plot2d( [f(t), s(t,5), sigma(t,5)], [t,-2*%pi,2*%pi], [xlabel, "x"], [ylabel, "y"], [y, -1, 5] ); /* 3.1 */ f(x) := 1; s(x,n) := sum(2/%pi*(1/k-(-1)^k/k)*sin(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-2*%pi,2*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 3.2 */ f(x) := x^2; s(x,n) := sum(-2*(%pi^2*k^2*(-1)^k - 2*(-1)^k+2)/(%pi*k^3)*sin(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-2*%pi,2*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 3.3 */ /* 4.1 */ f(x) := x; s(x,n) := %pi + sum(-2/k*sin(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-2*%pi,2*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 4.2 */ f(x) := x^2; s(x,n) := 4/3*%pi^2 + sum(-4*%pi/k*sin(k*x) + 4/k^2*cos(k*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-2*%pi,2*%pi], [xlabel, "x"], [ylabel, "y"] ); /* 4.3 */ l : 3; f(x) := abs(x); s(x,n) := l/2 + sum(-4*l/(%pi^2*(2*k-1)^2)*cos((2*k-1)*%pi/l*x), k, 1, n); plot2d( [f(t), s(t,20), sigma(t,20)], [t,-2*l,2*l], [xlabel, "x"], [ylabel, "y"] ); /* 5.1 f(x) = x^2 abil [0,2pi] x=pi (4.2) */ /* 5.2 f(x) = x koosinusrea abil [0,pi] x=0 (2.2) */ /* 5.3 f(x) = sin(x) koosinusrea abil [0,pi] x=0 (2.3) vői 1/(2n-1)-1/(2n+1) vőttega */ /* 5.4 f(x) = x*(pi-x) siinusrea abil [0,pi] x=pi/2 */