uniform1web.html

MAPLE COMMANDS AND RESULTING ANIMATIONS ILLUSTRATING

CONVERGENCE OF SEQUENCES OF FUNCTIONS

> with(plots):

> animate( plot, [x^n, x=0..1, title = " f_n(x) = x^n "], n=1..50, frames=50, thickness=2 );

> animate( plot, [1/(n*x+1), x=0..1, title = " f_n(x) = 1/(n*x+1)"], n=1..50, frames=50, thickness=2 );

> animate( plot, [x^2/(x^2 + (1-n*x)^2), x=0..1, title = " f_n(x) =x^2/(x^2 + (1-n*x)^2)"], n=1..50, frames=50, thickness=2 );

> F := proc(n)
plots[display](
    plot(n^2*x, x=0..1/n, thickness=2),
    plot(2*n-n^2*x, x=1/n..2/n, thickness=2),
    plot(0, x=2/n..1, thickness=2) );
end:
animate(F,[n], n=1..50, frames =50);

> F := proc(n)
plots[display](
    plot(n*x, x=0..1/n, thickness=2),
    plot(2-n*x, x=1/n..2/n, thickness=2),
    plot(0, x=2/n..1, thickness=2) );
end:
animate(F,[n], n=1..50, frames =50);

> F := proc(n)
plots[display](
    plot(x, x=0..1/n, thickness=2),
    plot(2/n-x, x=1/n..2/n, thickness=2),
    plot(0, x=2/n..1, thickness=2) );
end:
animate(F,[n], n=1..50, frames =50);

> animate( plot, [n^2*x^n*(1-x), x=0..1, title = " f_n(x) = n^2*x^n*(1-x)"], n=1..50, frames=50, thickness=2);

> animate( plot, [n^2*x*(1-x^2)^n, x=0..1, title = " f_n(x) = n^2*x*(1-x^2)^n"], n=1..50, frames=50, thickness=2 );

> animate( plot, [sin(n*x)/sqrt(n), x=0..1, title = " f_n(x) = sin(n*x)/sqrt(n)"], n=1..50, frames=50, thickness=2 );

> animate( plot, [x^2/(x^2 + (1-n*x)^2), x=0..1,title = " f_n(x) = x^2/(x^2 + (1-n*x)^2)" ], n=1..50, frames=50, thickness=2);

> F := proc(n)
plots[display](
    plot(0, x=0..1/(n+1), thickness=2),
    plot((sin(x/Pi))^2, x=1/(n+1)..1/n, thickness=2),
    plot(0, x=1/n..1, thickness=2) );
end:
animate(F,[n], n=1..10, frames =10);

[Plot]