> restart;
> with(plots):
> circle := plot([cos(t), sin(t), t = 0..Pi/2]):
> display(circle, scaling = constrained);

> f := x -> sqrt(1-x^2):
> n := 10; x[0] := 0.0: y[0] := f(x[0]):
> for i from 1 to n-1 do
     x[i] := x[i-1] + 1.0/n:
     y[i] := f(x[i]):
     p[i] := [[0,0], [x[i], y[i]] ]:
  od:

 

> x[n] := 1.0: y[n] := 0.0:
> i := 'i':
  approx1 := plot([ [x[i], y[i]] $ i = 0..n ], style = point):
  approx2 := plot([ [x[n-1],y[n-1]],[x[n],y[n]] ], style = line): 
> display({circle,approx1, approx2});

> i := 'i';
  approx3 := plot([ p[i] $ i = 1..n-1 ], style = line, color = black):
> display({circle, approx3});



> n := 100; x[0] := 0.0; y[0] := f(x[0]);
> for i from 1 to n-1 do
     x[i] := x[i-1] + 1.0/n:
     y[i] := f(x[i]):
  od:
  x[n] := 1.0: y[n] := 0.0:

   

> i := 'i':
  approx1 := plot([ [x[i], y[i]] $ i = 0..n ], style = point):
  approx2 := plot([ [x[n-1],y[n-1]], [x[n],y[n]] ], style = line):
> display({circle,approx1}, scaling = constrained);

> k := 0: firstlen := sqrt( (x[k+1]-x[k])^2 + (y[k+1]-y[k])^2);
  k := n-1: lastlen := sqrt( (x[k+1]-x[k])^2 + (y[k+1]-y[k])^2);
  lastlen/firstlen;

> n := 1000; x[0] := 0.0; y[0] := f(x[0]);
> for i from 1 to n-1 do
    x[i] := x[i-1] + 1.0/n:
    y[i] := f(x[i]):
  od:
  x[n] := 1.0: y[n] := 0.0:

                        
> i := 'i':
  approx1 := plot([ [x[i], y[i]] $ i = 0..n ], style = point):
  approx2 := plot([ [x[n-1],y[n-1]], [x[n],y[n]] ], style = line):
> display({circle,approx1}, scaling = constrained);
> k := 0: firstlen := sqrt( (x[k+1]-x[k])^2 + (y[k+1]-y[k])^2);
  k := n-1: lastlen := sqrt( (x[k+1]-x[k])^2 + (y[k+1]-y[k])^2);
  lastlen/firstlen;