Asked by Mike
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
Answers
Answered by
oobleck
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.
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.
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