I am looking for the intuition behind the following problem, so as to not only understand this, but similar problems like it. The problem is as follows:
What's the equation of the line tangent to the intersection of the surface $z = \arctan(xy)$ with the plane $ x = 2$ at the point $\left(2, \frac{1}{2}, \frac{\pi}{4}\right)$?
Thank you!
You have two surfaces here: the surface $z=\arctan(xy)$ and the surface $x=2$. And you have the point $P=\left(2,\frac12,\frac\pi4\right)$, which belongs to both of them. And you are after the tangent line to their intersection at $P$. Using partial derivatives, you can compute the tangent plane at $P$ for each of them (which is an overkill in the case of the plane $x=2$, of course; the tangent plane will be $x=2$ too). And then the tangent line will be the intersection of those two planes.