(a) f(x)= e^x/2, f(0)= 1

F’(x)= ½ e^x/2, f ’(0)= ½

F’’ (x)= ¼ e^{x/2 ,} f ‘’ (0)= 1/4

F ‘’’ (x)= 1/8 e^x/2 , f ‘’’ (0)= 1/8 ….. and so on

Hence Maclaurin Series will be

(b) f(x)= sin3x, f(0) =0

F‘(x)= 3cos3x, f ’(0)=3

F ‘’(x)=-9sin3x , f ‘’(0)= 0

F ‘’’(x)=-27cos3x, f ‘’’(0)=-27

F⁴(x)= 81sin3x, f⁴(0)= 0

F⁵(x)= 243cos3x, f⁵(0)=243….. and so on

Hence Maclaurin series will be

(c) f(x)=x²/2 -1+cosx, f(0)= 0

F ‘(x)=x-sinx, f ‘(0)=0

F’’(x)=1-cosx, f’’(0)=0

F’’’(x)=sinx, f’’’(0)=0

F⁴(x)=cosx , f⁴(0)=1

F⁵(x)=-sinx f⁵(0)= 0

F⁶(x)= -cosx, f⁶(x)=-1…. And so on

Hence Maclaurin series will be

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