Part 5/8:
This polynomial effectively captures the behavior of the function by considering its derivatives at point ( a ), generating a series that can yield increasingly accurate approximations as ( n ) increases.
Example: Approximating the Cosine Function
To see Taylor polynomials in action, let’s approximate the function ( f(x) = \cos(x) ) at the point ( a = 0 ). We first evaluate the necessary function derivatives:
( f(0) = 1 )
( f'(0) = 0 )
( f''(0) = -1 )
( f'''(0) = 0 )
( f^{(4)}(0) = 1 )
From this, we observe a recurring pattern among the derivatives (1, 0, -1, 0). This allows us to construct the Taylor polynomial approximations, selectively including only even-order terms since the odd-order derivatives are zero: