Consider the curve given by $y ^ { 2 } = 2 + x y$. (a) Show that $\frac { d y } { d x } = \frac { y } { 2 y - x }$. (b) Find all points $( x , y )$ on the curve where the line tangent to the curve has slope $\frac { 1 } { 2 }$. (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 $y ^ { 2 } = 2 + x y$. At time $t = 5$, the value of $y$ is 3 and $\frac { d y } { d t } = 6$. Find the value of $\frac { d x } { d t }$ at time $t = 5$.
Consider the curve given by $y ^ { 2 } = 2 + x y$.
(a) Show that $\frac { d y } { d x } = \frac { y } { 2 y - x }$.
(b) Find all points $( x , y )$ on the curve where the line tangent to the curve has slope $\frac { 1 } { 2 }$.
(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 $y ^ { 2 } = 2 + x y$. At time $t = 5$, the value of $y$ is 3 and $\frac { d y } { d t } = 6$. Find the value of $\frac { d x } { d t }$ at time $t = 5$.