Průběh funkce
Předpis funkce:
Pracujeme s reálnými funkcemi:
> with(RealDomain);
Warning, these protected names have been redefined and unprotected: Im, Re, `^`, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, eval, exp, expand, limit, ln, log, sec, sech, signum, simplify, sin, sinh, solve, sqrt, surd, tan, tanh
[Im, Re, ^, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, 

  arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, 

  eval, exp, expand, limit, ln, log, sec, sech, signum, simplify, sin, sinh, 

  solve, sqrt, surd, tan, 

  tanh]
> f := x^3/(x^4+1);
                                  3  
                                 x   
                          f := ------
                                4    
                               x  + 1
Definiční obor 
(nutno pro každou funkci zvlášť)
> solve(x^4+1=0,x);
Dále limity v bodech nespojitosti
> Limit(f, x=a, left)=limit(f, x=a, left);
                           /   3  \      3  
                           |  x   |     a   
                     lim   |------| = ------
                   x -> a- | 4    |    4    
                           \x  + 1/   a  + 1
> Limit(f, x=a, right)=limit(f, x=a, right);
                           /   3  \      3  
                           |  x   |     a   
                     lim   |------| = ------
                   x -> a+ | 4    |    4    
                           \x  + 1/   a  + 1
Je funkce sudá (příp. lichá)?
> eval(f,x=-x);
                                 3  
                                x   
                            - ------
                               4    
                              x  + 1
Je několik způsobů, jak ukázat, že je: LICHA:  f(x) = –f(–x) 
> evalb(eval(f,x=x)=-eval(f,x=-x));
                              true
> iszero(eval(f,x=x)+eval(f,x=-x));
                           iszero(0)
> type(f,oddfunc(x));
                              true
A SUDA... f(x) = f(–x) 
> evalb(eval(f,x=x)=eval(f,x=-x));
                             false
> type(f,evenfunc(x));
                             false
První derivace
> df := diff(f,x);
                              2         6   
                           3 x       4 x    
                    df := ------ - ---------
                           4               2
                          x  + 1   / 4    \ 
                                   \x  + 1/ 
> NuloveBody := solve(df = 0,x);
                                   (1/4)    (1/4)
              NuloveBody := 0, 0, 3     , -3     
> Rostoucí := solve(df>0,x);
> Klesající := solve(df<0,x);
                     /    /  (1/4)\         \           /             / (1/4)\\
Rostouc? := RealRange\Open\-3     /, Open(0)/, RealRange\Open(0), Open\3     //
                             /               /  (1/4)\\  
       Klesaj?c? := RealRange\-infinity, Open\-3     //, 

                  /    / (1/4)\          \
         RealRange\Open\3     /, infinity/
Druhá derivace
> d2f := diff(df,x);
> d2f := diff(f,x$2);
                                   5           9  
                     6 x       36 x        32 x   
             d2f := ------ - --------- + ---------
                     4               2           3
                    x  + 1   / 4    \    / 4    \ 
                             \x  + 1/    \x  + 1/ 
                                   5           9  
                     6 x       36 x        32 x   
             d2f := ------ - --------- + ---------
                     4               2           3
                    x  + 1   / 4    \    / 4    \ 
                             \x  + 1/    \x  + 1/ 
Inflexní body
> InflexníBod := solve(d2f = 0,x);
                                  (1/4)               (1/4)  
                     /      (1/2)\       /      (1/2)\       
  Inflexn?Bod := 0, -\6 + 33     /     , \6 + 33     /     , 

                  (1/4)  
     /      (1/2)\       
    -\6 - 33     /     , 

                 (1/4)
    /      (1/2)\     
    \6 - 33     /     
> with(Student[Calculus1]):
> InflectionPoints(f);
[              (1/4)                (1/4)                  (1/4)  
[ /      (1/2)\        /      (1/2)\          /      (1/2)\       
[-\6 + 33     /     , -\6 - 33     /     , 0, \6 - 33     /     , 

               (1/4)]
  /      (1/2)\     ]
  \6 + 33     /     ]
Extrémy
> eval(d2f,x=0);
> eval(d2f,x=NuloveBody[3]);
> eval(d2f,x=NuloveBody[4]);
                               0
                             3  (1/4)
                           - - 3     
                             4       
                            3  (1/4)
                            - 3     
                            4       
Konvexnost a konkávnost
> Konvexní := solve(d2f>0,x);
> Konkávní := solve(d2f<0,x);
                     /    /              (1/4)\      /              (1/4)\\  
                     |    | /      (1/2)\     |      | /      (1/2)\     ||  
Konvexn? := RealRange\Open\-\6 + 33     /     /, Open\-\6 - 33     /     //, 

           /             /             (1/4)\\  
           |             |/      (1/2)\     ||  
  RealRange\Open(0), Open\\6 - 33     /     //, 

           /    /             (1/4)\          \
           |    |/      (1/2)\     |          |
  RealRange\Open\\6 + 33     /     /, infinity/
                     /               /              (1/4)\\  
                     |               | /      (1/2)\     ||  
Konk?vn? := RealRange\-infinity, Open\-\6 + 33     /     //, 

           /    /              (1/4)\         \  
           |    | /      (1/2)\     |         |  
  RealRange\Open\-\6 - 33     /     /, Open(0)/, 

           /    /             (1/4)\      /             (1/4)\\
           |    |/      (1/2)\     |      |/      (1/2)\     ||
  RealRange\Open\\6 - 33     /     /, Open\\6 + 33     /     //
Asymptota se směrnicí
> k:=limit(f/x,x=infinity);
                             k := 0
> q:=limit(f-x*k,x=infinity);
                             q := 0
> Asymptota := k*x+q;
                         Asymptota := 0
A nakonec graf:
> plot([f,Asymptota],x=-10..10);