Question

please answer both and circle the answers!! i always give thumbs up for correct answers!!

Find the Maclaurin series for f(x) using the definition of the Maclaurin series. f(x)= x cos(4x) a. 3 (-1)*4"x2+1 (2n)! -O b. į (-1)*43*2*** -0 n! C. 3 (-1)*4**** (2n)! d. (-1)*4*, *** (2n)! e. (-1)**:4" x 2" ** (2n)!

Answer 1

X 6! - (0)+(2-0) The Taylor serie's expansion of the function f (x)= cos x f (x) = f (0)+(x-0)/'(0)+ (2-0) - "(0) + f"(0) + 2! 3! 4! 5! f(x) = cosx= f(0) = 1 f '(x)=- sinx= f'(0)=0 F"(x) = -cos x="0)=-1 F"(x)=sin x= "(0)=0 **(x)= cos x=(O)= 1 ' (x)=-sinx=;"(0)= 0 ' (x) =-cosx=f(0)=-1 x COS X = 1- (-1)" 2! 4! 6! 8! (2n)! Replacing x by 4x (4x) (4x)* _(4x)"(4x) cos(4x) = 12 (-1)" (4x)** 2! 4! 6! 8! (2n)! *cos(4x) = xį (-1)= (43)* (2n)! 428x2x+1 *cos(4x) = E(-1)* *(4x) (-1)* (2n)! (2n)! xcos(4x)= 3(-1)" (2n)! + + + + -O 2x no O 2-0 2x+1 f(x) = 5e*** cos(4x) The Maclaurian series expansion of the function et about point c = 0 e* = 1+x+ + + + 2! 3! -+ 4! —+... **=1+(-x2)+ (-x+)?.(-x2).(-x+)(-x2). +... 2! 3! 4! 6 * = 1-X + + 2! 3! 5e- (4x)" (4x)" (4x)" (4x)* + + + + 2! 3! 2! 4! 6! 8! *cos(4x)=s[1-- so * cos(4x)=${1+2+375 +.]