Definitions — Question 3

Question 3 – Solution

Strategy. An implicit curve can contain multiple function branches. Select the branch using the initial value, then check both differentiability and the equation’s denominator.

Step 1: Select and verify. Solving for $y$ gives $y=\pm\sqrt{9-x^2}$. Since $y(0)=3$, choose $$\boxed{y(x)=\sqrt{9-x^2},\qquad -3<x<3.}$$ On this interval, $y'=-x/\sqrt{9-x^2}=-x/y$, and $y(0)=3$.

Step 2: Interval and endpoints. At $x=\pm 3$, the branch has $y=0$, so $-x/y$ is undefined and the derivative is not finite. For $|x|>3$, this branch is not real. Thus the largest open interval containing $0$ is exactly $\boxed{(-3,3)}$.

Step 3: Relation versus function. For each $-3<x<3$, the circle has two distinct $y$-values. The whole circle fails the single-valued requirement for a function $y(x)$. Even the undivided equation $yy'=-x$ cannot hold with a finite derivative at $(\pm 3,0)$: it would require $0=\mp 3$.

Step 4: Other initial value. For $y(0)=-3$, $$\boxed{y(x)=-\sqrt{9-x^2},\qquad -3<x<3.}$$ Its derivative is $x/\sqrt{9-x^2}=-x/y$, verifying the lower branch separately.