File:MVT.gif

import graph;

size(8cm, 0);

defaultpen(fontsize(14)+linewidth(0.8));

real dt = 0.04;
int n = 0;

void ship() {
	write(n);
	shipout(outprefix() + format("-%04d", n));
	++n;
}

real inf(real f(real), real a, real b, int n) {
	return min(map(f, uniform(a, b, n)));
}

real sup(real f(real), real a, real b, int n) {
	return max(map(f, uniform(a, b, n)));
}

real average(real f(real), real a, real b, int n) {
	return sum(map(f, uniform(a+(b-a)/(2*n), b-(b-a)/(2*n), n-1)))/n;
}

real f(real x) {
	return 2+4/(13-x)+sin(2sqrt(x^2+3)-3)/2-(x-4)^2/16;
}

real f(real x) {
	return 2+4/(13-x)+sin(2sqrt(x^2+3)-3)/2-(x-4)^2/16;
}

real xmax = 8;

xtick("$\xi$", xmax, invisible);
ytick("$f(\xi)$", sup(f, 0, xmax, 100), invisible);

for(real t = 0; t+dt/2 < 8; t += dt) {
	save();
	real x = (1-abs(t/4-1))*xmax;
	//real x = (0.9*cos(t/4*pi)+0.1*cos(t/4*pi*3)+1)/2*xmax;
	real y = f(x);
	fill((0, 0)--graph(f, 0, xmax)--(xmax, 0)--cycle, rgb(1.0, 0.5, 0.0)+opacity(0.5));
	fill(box((0, 0), (xmax, y)), rgb(0.0, 0.5, 1.0)+opacity(0.5));
	draw(Label("$f$", MidPoint, N), graph(f, 0, xmax));
	draw((x, 0)--(x, y), dashed);
	xaxis(0, xmax, above=true);
	yaxis(0, above=true);
	yaxis(YEquals(xmax), 0, above=true);
	yequals(y);
	xtick("$a$", 0);
	xtick("$b$", xmax);
	xtick("$\xi$", x);
	ytick("$f(\xi)$", y);
	dot((x, y));
	ship();
	restore();
}