import graph;
import geometry;
import math;

size(400,150,IgnoreAspect);

real f(real x) {return (x-4)^3-x+30;}
pair F(real x) {return (x,f(x));}

dotfactor=7;

int a=1, b=4;
int t=2;

xaxis("$x$");
yaxis("$y$");

for(int i=a; i < b; ++i)
  draw(F(i)--F(i+1));

path g=graph(f,a,b,operator ..);

draw(g,red,Arrow(position=time(g,2.5)));

int n=2;

perpendicular((t+1,f(t)),NW,blue);

draw(F(t)--(t+1,f(t)),blue);
draw((t+1,f(t))--F(t+1),heavygreen);

dot(F(t));
dot(F(t+1));


real sx=0.08;
real sy=1.5;
draw("$x(t+h)-x(t)$",F(t)-sy*I--(t+1,f(t)-sy),blue+dashed,Arrows,Bars,
     PenMargins);
draw("$y(t+h)-y(t)$",(t+1+sx,f(t))--F(t+1)+sx,heavygreen+dashed,Arrows,Bars,PenMargins);

labelx("$a$",a);
labelx("$b$",b);

label("$s(a)=0$",F(a),W,red);
label("$s(t)$",F(t),N,red);
label("$s(t+h)$",F(t+1),NW,red);
label("$s(b)=L$",F(b),ESE,red);
label("$P=(x(t),y(t))$",F(t),2.5W);
label("$Q=(x(t+h),y(t+h))$",F(t+1),NE);