Answer to Question #1193 in Calculus for Mert Evren

Question

What is the first three non-zero terms of the Taylor series for the function f(x)=cos(x) about x=0?

Answer 1

The first terms of Taylor series for the function f(a) are
f(a) + f'(a) (x-a)/1! + f''(a)(x-a)² / 2! + ...
Let's find it for f(x) = cos(x) and a = 0:
cos(0) + (-sin(0))(x-0)/1! + (-cos(0))x² / 2! + sin(0) x³/3! + cos(0) x⁴ / 4! = 1 + 0 - x²/2 + 0 + x⁴/24
= 1 - x²/2 +x⁴/24