Another service from Omega

Using the maple commands: plot, diff, D


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> f := x -> (x+2) / (3+(x^2+1)^3);

                                          x + 2
                            f := x -> -------------
                                            2     3
                                      3 + (x  + 1)

> D(f);

                                                    2     2
                              1           (x + 2) (x  + 1)  x
                   x -> ------------- - 6 -------------------
                              2     3              2     3 2
                        3 + (x  + 1)        (3 + (x  + 1) )

> # Notice that D(f) returns the derivative of f as a function.
> # Think of D(f) as the usual f' (i.e. f-prime the derivative of f)

> plot(f);
> # The first time let Maple adjust the x and y ranges.


                                       +A                                     
                                   0.5 * A                                    
                                       * A                                    
                                      A* A                                    
                                      A+  A                                   
                                   0.4A+  A                                   
                                      A+  A                                   
                                      A+  A                                   
                                     A +  A                                   
                                   0.3 +  AA                                  
                                     A +   A                                  
                                     A +   A                                  
                                     A +   A                                  
                                   0.2 +   A                                  
                                    A  +   AA                                 
                                    A  +    A                                 
                                    A  +    A                                 
                                   0.1 +    AA                                
                                   A   +     A                                
                                  AA   +     AA                               
                                 AA    +      AAA                             
  ********************************-+---+---+--+-***************************** 
 -10               -5                0 0                  5                10 
> f(0);

                                      1/2

> # Do you see that the global max of f is at an x a bit bigger than 0 ?

> plot(f(x), x=-5..5, y=-0.1..0.1);



                                A  0.1 +          A                           
                               AA      +          A                           
                               A       +           A                          
                               A       +           A                          
                               A       +            A                         
                              A   0.05 +            A                         
                              A        +             A                        
                             A         +              A                       
                             A         +              AA                      
                           AA          +                AAA                   
  -+--+--+--+--+--+--+--****--+--+--+--+--+--+--+--+--+--+-****************** 
  AAAAAAAAAAAAAAAAAAAAAA               +                                      
        -4             -2            0 0              2              4        
                                       +                                      
                                       +                                      
                                       +                                      
                                 -0.05 +                                      
                                       +                                      
                                       +                                      
                                       +                                      
                                       +                                      
                                  -0.1 +                                      
                                                                              
> d := simplify(D(f)(x));

                             6      4      2       5       3
                    - 4 + 5 x  + 9 x  + 3 x  + 12 x  + 24 x  + 12 x
             d := - -----------------------------------------------
                                      6      4      2 2
                                (4 + x  + 3 x  + 3 x )

> fsolve(numer(d));

                           -2.485165927, .2700964050

> # Let's grab this list of two numbers for later use.
> s := ";

                         s := -2.485165927, .2700964050

> # A quick check...
> D(f)(s[2]);

                                         -9
                                    .1*10

> # VERY close to 0 as it should. Why isn't exactly zero?
> # Now the inflection points...
> # We use now the command "diff" instead of "D".... See the difference?

> diff(d,x);

      5       3             4       2
  30 x  + 36 x  + 6 x + 60 x  + 72 x  + 12
- ----------------------------------------
                 6      4      2 2
           (4 + x  + 3 x  + 3 x )

                   6      4      2       5       3             5       3
         (- 4 + 5 x  + 9 x  + 3 x  + 12 x  + 24 x  + 12 x) (6 x  + 12 x  + 6 x)
     + 2 ----------------------------------------------------------------------
                                       6      4      2 3
                                 (4 + x  + 3 x  + 3 x )

> # Use 'simplify' to combine the two fractions into a single one.

> simplify(");


            5      11       9       7       3              4       10       8
   6 (- 19 x  + 5 x   + 17 x  + 18 x  - 37 x  - 12 x + 14 x  + 14 x   + 54 x

              6       2        /       6      4      2 3
        + 76 x  - 30 x  - 8)  /  (4 + x  + 3 x  + 3 x )
                             /

> fsolve(numer("));

                    -2.964270928, -.5472819999, .8702050673

> # So these are the x locations of the three inflection points.
> # Had we used 'solve' instead of 'fsolve' we would have obtained the
> # exact values but usually at the expense of complicated expresions.

> # We can now adjust the x and y ranges to see what's going on
> # to the left of 0.

> plot(f(x), x=-4..0, y=-0.01..0.01);


                                               A                          0.01
                                               A                            + 
                                              A                             + 
                                              A                             + 
                                             A                              + 
                                            AA                           0.005
                                            A                               + 
                                           AA                               + 
                                          AA                                + 
                                         A                                  + 
  +---+--+---+---+--+---+---+---+--+---**--+--+---+---+---+--+---+---+--+---+ 
  AAAAAAAAAAAAAAAAA                 AAA                                     |0
 -4                -3AAAAAAAAAAAAAAA  -2                 -1                 0 
                                                                            + 
                                                                            + 
                                                                            + 
                                                                        -0.005
                                                                            + 
                                                                            + 
                                                                            + 
                                                                            + 
                                                                         -0.01
                                                                              
> # Do you see the inflection pt. around -3 ?
> # Also, do you see the global min of f.. at around s[1] = -2.5 ?

> 


Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Sep 4 18:31:06 EDT 1996