May 2025 Section II

Show that $\frac{dy}{dx}=\frac{-x}{2\left(3y^2-2y-1\right)}$.

There is a point $P$ on the curve $G$ near $(2,-1)$ with $x$-coordinate 1.6. Use the line tangent to the curve at $(2,-1)$ to approximate the $y$-coordinate of point $P$. Show the work that leads to your answer.

For $x>0$ and $y>0$, there is a point $S$ on the curve $G$ at which the line tangent to the curve at that point is vertical. Find the $y$-coordinate of point $S$. Show the work that leads to your answer.

A particle moves along the curve $H$ defined by the equation $2xy+\ln y=8$. At the instant when the particle is at the point $(4,1)$, $\frac{dx}{dt}=3$. Find $\frac{dy}{dt}$ at that instant. Show the work that leads to your answer.