Finding a point using partial differentiation

Question

A bug is traveling downward along the curve that is the intersection of $z=x^3-xy^2 +y$ with the plane $y=1$. At the point $(2,1,7)$ the bug went off the tangent line. Where did the bug hit the $yz$-plane?

Answer 1

Hint:

Find the equation of the tangent line to the curve lying on the plane $y=1$ by using the concept of partial derivative:

\begin{align} z-7&=\left[\frac{\partial z}{\partial x}_{|(x,y)=(2,1)}\right](x-2)\\[3pt] %z-7&=\left[3(2)^2-(1)^2\right](x-2)\\[3pt] %z-7&=11(x-2) \end{align} Now use the fact that $x=0$ when the bug hits the $yz$ plane.