Problem: Express the implicit curve given by
Solution: Is this a good question or a trick question? We start by checking that the given point really satisfies the given equality: Check. So at least this makes sense, the point lies on the given curve.
We could try to use theory (Implicit function theorem, partial derivatives) to prove that finding a function is possible, but it would not help us actually solving the problem. They want us to find a concrete function, so unless this is a trick question, such a thing is possible. Therefore we simply try it and we will see whether it works. We will try to express y from the given equation.
We start by taking the square root on both sides of the equality and won't forget to put both possible signs.
Which sign is the right one? We know that the function we obtain must work
for the point
On what neighborhood of
We check that for x from U, if we substitute this
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