The equation xy-1=x+y^2 defines a function y=f(x) in a neighborhood of (-1,0). Determine f(-1), fx(-1), and fxx(-1) and the second -taylor polynomial t2 (x) based around x=-1

Answer 1

xy-1 = x+y^2
y + xy' = 1 + 2yy'

y' = (1-y)/(x-2y)

y" = [(-y')(x-2y) - (1-y)(1-2y')]/(x-2y)^2
= ((2-x)y' + y-1)/(x-2y)^2
= 2(-y^2+xy-x+1)/(x-2y)^3

That should get you started.