Suppose T_4(x) = 7-3(x-2)+7(x-2)^2-6(x-2)^3+8(x-2)^4 is the 4th-degree Taylor polynomial centered at x=2 for some function f, what is the value of f^((3))(2)?

Apr 19, 2017

$- 36$

Explanation:

T_4(x)=(f^((0))(2))/(0!)(x-2)^0+(f^((1))(2))/(1!)(x-2)^1+(f^((2))(2))/(2!)(x-2)^2+(f^((3))(2))/(3!)(x-2)^3+(f^((4))(2))/(4!)(x-2)^4

By comparing the coefficient of the fourth term,

Rightarrow (f^((3))(2))/(3!)=-6

By multiplying both sides by 3!,

Rightarrow f^((3))(2)=-6cdot3! =-36

I hope that this was clear.