Question

How to find the fourth order Taylor polynomial of the function cos(x^2)

Answer 1

gonna need a few derivatives to play with

Answer 2

let

cos(x^2) = f(x) = ax^4 + bx^3 + cx^2 + dx + e

[cos(x^2)]' = f'= 4ax^3 + 3bx^2 + 2cx + d

[cos(x^2)]'' = f''= 4.3ax^2 + 3.2bx + 2c

[cos(x^2)]''' = f'''= 4.3.2ax + 3.2b

[cos(x^2)]'''' = f''''= 4.3.2a

we can therefore solve for a, then b, then c, then .... all the way to e and our 4th degree polynomial function is defined