# 5.2  Graphical Analysis
# Art Belmonte, Summer 1996
# These are the shaded gray examples which appear in Section 5.2.
# In-line plots have been removed to conserve disk space.
> restart; f:=exp(x) / (x^3 + x -1 + 0.2*exp(x));
> plot(f, x=-5..15, y=-10..10);

                                  exp(x)
                     f := ----------------------
                           3
                          x  + x - 1 + .2 exp(x)

> fsolve(denom(f)=0, x=0..1);

                             .5213890506

> Limit(f, x=-infinity); value(");
> Limit(f, x=infinity); value(");

                                         exp(x)
                     lim         ----------------------
               x -> (-infinity)   3
                                 x  + x - 1 + .2 exp(x)


                                  0


                                       exp(x)
                     lim       ----------------------
                x -> infinity   3
                               x  + x - 1 + .2 exp(x)


                                  5.

> Df:=diff(f, x); a:=fsolve(Df=0, x=2..4);

                                               2
                   exp(x)           exp(x) (3 x  + 1 + .2 exp(x))
     Df := ---------------------- - -----------------------------
            3                           3                     2
           x  + x - 1 + .2 exp(x)     (x  + x - 1 + .2 exp(x))


                           a := 2.893289196

> plot({f, Df}, x=0.5..15, y=-10..10);
> DDf:=diff(Df, x); b:=fsolve(DDf=0, x=6..10);

                                             2
               exp(x)             exp(x) (3 x  + 1 + .2 exp(x))
DDf := ---------------------- - 2 -----------------------------
        3                             3                     2
       x  + x - 1 + .2 exp(x)       (x  + x - 1 + .2 exp(x))

                    2                 2
         exp(x) (3 x  + 1 + .2 exp(x))    exp(x) (6 x + .2 exp(x))
     + 2 ------------------------------ - -------------------------
             3                     3        3                     2
           (x  + x - 1 + .2 exp(x))       (x  + x - 1 + .2 exp(x))


                           b := 8.131398912

> plot({f, DDf}, x=0.5..15, y=-10..10);
>