Consider the curve given by $y^2 = 2 + xy$. (a) Show that $\dfrac{dy}{dx} = \dfrac{y}{2y - x}$. (b) Find all points $(x, y)$ on the curve where the line tangent to the curve has slope $\dfrac{1}{2}$. (c) Show that there are no points $(x, y)$ on the curve where the line tangent to the curve is horizontal. (d) Let $x$ and $y$ be functions of time $t$ that are related by the equation $y^2 = 2 + xy$. At time $t = 5$, the value of $y$ is 3 and $\dfrac{dy}{dt} = 6$. Find the value of $\dfrac{dx}{dt}$ at time $t = 5$.
Consider the curve given by $y^2 = 2 + xy$.
(a) Show that $\dfrac{dy}{dx} = \dfrac{y}{2y - x}$.
(b) Find all points $(x, y)$ on the curve where the line tangent to the curve has slope $\dfrac{1}{2}$.
(c) Show that there are no points $(x, y)$ on the curve where the line tangent to the curve is horizontal.
(d) Let $x$ and $y$ be functions of time $t$ that are related by the equation $y^2 = 2 + xy$. At time $t = 5$, the value of $y$ is 3 and $\dfrac{dy}{dt} = 6$. Find the value of $\dfrac{dx}{dt}$ at time $t = 5$.