f:x^3+x-2;

pl(p1,p2):=[[parametric,(p1*(1-t)+p2*t)[1],(p1*(1-t)+p2*t)[2],[t,0,1]]]$

s:[3]$
plot:[[parametric,x,f,[x,0.24,4]]]$
for k from 1 thru 7 do (
    old:last(s),
    new:old-at(f,x=old)/at(diff(f,x),x=old),
    s:append(s,[new]),
    plot:append(plot,pl([old,0],[old,subst(x=old,f)])),
    plot:append(plot,pl([old,subst(x=old,f)],[new,0]))
)$
float(transpose(s));

plot2d(plot,[same_xy,false],[legend,false])$