Find the difference between the Taylor polynomial of degree 4 about the point 0 for cos(x) evaluated at x = 1, and cos(1).

Question

Find the difference between the Taylor polynomial of degree 4 about the point 0 for cos(x) evaluated at x = 1, and cos(1).

	for some c between 0 and 1
	for some x between 0 and 1
	for some c between 0 and 1

Answer 1

P₄(x) = cos0 - sin0x/1! - cos0x²./2! + sin0x³/3! + cos0x⁴/4! = 1 - x²/2 + x⁴/24;

P₄(1) = 1 - 1/2 + 1/24 = 13/24;

P₄⁽¹⁾ - cos(1) = - sinC/5! for some C between 0 and 1.