restart: with(plots):
alias(u=Heaviside):
plot(u(x+2), x=-5..5);
plot(u(-x), x=-5..5);
plot(u(x-4), x=-5..5);
plot(u(2*x), x=-5..5);
plot(u(2*x+1), x=-5..5);
plot(u(3-x), x=-5..5);
plot(2*x+1, x=-5..5);
plot((2*x+1)*u(x+2), x=-5..5);
plot((2*x+1)*u(2-x), x=-5..5);
plot((2*(x-2)+1)*u(x-2), x=-5..5);
plot(sin(Pi*x), x=0..5);
plot(sin(Pi*x)*u(x-1.5), x=0..5);
plot(sin(Pi*x)*u(1.5-x), x=0..5);
plot(sin(Pi*(x-1.5))*u(x-1.5), x=0..5);
plot(exp(-2*x), x=0..2);
plot(exp(-2*x)*u(x-1), x=0..2);
plot(exp(-2*x)*u(1-x), x=0..2);
plot(exp(-2*(x-1))*u(x-1), x=0..2);
plot(u(x+2)-u(x-2));
plot(u(x+2)-u(x)+2*u(x)-2*u(x-4));
plot((2*x+1)*(u(x)-u(x-3)));
plot(x^2*(u(x+1)-u(x-1)),x=-1..1);
f:=(x)->(1/(2*a))*(u(x+a)-u(x-a));
limit(f(x),a=0);
g:=(x)->(1/a)*((1+(x/a))*(u(x+a)-u(x))+(1-(x/a))*(u(x)-u(x-a)));
limit(g(x),a=0);
alias(d=Dirac):
Integrate(exp(-x^2)*d(x-2), x=-infinity..infinity) = integrate(exp(-x^2)*d(x-2), x=-infinity..infinity);
Integrate((5*x-3)*d(x+1),x=0..7) = integrate((5*x-3)*d(x+1),x=0..7);
Integrate(cos(tan(exp(x)))*d(x-1),x=-infinity..infinity) = integrate(cos(tan(exp(x)))*d(x-1),x=-infinity..infinity);
Integrate(d(x+3)+d(x-3),x=-infinity..infinity) = integrate(d(x+3)+d(x-3),x=-infinity..infinity);
;Integrate(exp(-x^2)*d(x),x=0..infinity) = integrate(exp(-x^2)*d(x),x=0..infinity);