Theorem ( Taylor polynomial ).  Assume that the function    and its derivatives    are all continuous on  .  If  both    and    lie in the interval  ,  and    then  

    ,  
    
is the n-th degree Taylor polynomial expansion of    about  .  The Taylor polynomial of degree n  is

       
and  
    .

The integral form of the remainder is  

    ,

and Lagrange's formula for the remainder is

    

where depends on and lies somewhere between  .  

Exploration.

(c) John H. Mathews 2004