The Taylor Polynomial

 

Example 9. (b) Find the Taylor polynomial of degree n = 5 for [Graphics:tp9.txtgr15.gif], expanded about [Graphics:tp9.txtgr16.gif].

First find the derivatives of f(x).

[Graphics:tp9.txtgr4.gif][Graphics:tp9.txtgr17.gif]

Second, evaluate the derivatives of f(x) at x = [Graphics:tp9.txtgr18.gif], and obtain a sequence of constants [Graphics:tp9.txtgr19.gif].

[Graphics:tp9.txtgr4.gif][Graphics:tp9.txtgr20.gif]

Third, substitute the constants [Graphics:tp9.txtgr21.gif] in Taylor's formula.

[Graphics:tp9.txtgr4.gif][Graphics:tp9.txtgr22.gif]

[Graphics:tp9.txtgr4.gif][Graphics:tp9.txtgr23.gif]

[Graphics:tp9.txtgr4.gif][Graphics:tp9.txtgr24.gif]

 

Want to see a higher degree Taylor polynomial ?

 

(c) John H. Mathews, 1998