Find the Taylor polynomial of degree n4 for x near the point

Solution

so taylor polynomial of degree n=4 is f(a) + (x-a) f\'(a) + (x-a)^2 f\'\'(a)/2! + (x-a)^3 f\'\'\'(a)/3! + + (x-a)^4 f\'\'\'\'(a)/4! so we need to evalute the function and the derivatives f(a) = cos(2*pi/2) = cos(pi)= -1 f\'(x) = - sin(2x) * 2 f\'(a)= - sin(2*pi/2) * 2 = -sin(pi) *2=0 f\'\'(x) = - cos(2x) * 4 f\'\'(a)= - cos(2*pi/2) * 4 = -cos(pi) * 4 =4 f\'\'\'(x) = sin(2x)*8 f\'\'\'(a) = sin(2*pi/2) * 8 = 8sin(pi)=0 f\'\'\'\'(x) = cos(2x)*16 f\'\'\'\'(a) = cos(2*pi/2)*16= cos(pi) 16 = -16 so taylor polynomial is -1 + (x-pi/2)^2 4/2! + (x-pi/2)^4 -16/4! -1 * 2 (x-pi/2)^2 - 2/3 (x-pi/2)^4