Find an equation for the plane tangent to the cone
$r(r,\theta) = (r \cos \theta)i + (r \sin \theta)j + rk$, $r \ge 0$, $0 \le \theta \le 2\pi$
at the point $P_0(0, 2, 2)$ corresponding to $(r, \theta) = (2, \frac{\pi}{2})$. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together.
The tangent plane at a point $P_0(x_0, y_0, z_0)$ on a parametrized surface $r(r, \theta) = (r \cos \theta)i + (r \sin \theta)j + rk$ is the plane through $P_0$ normal to the vector $r_r \times r_\theta$, the cross product of the tangent vectors at $P_0$.
What is the equation of the tangent plane?
$\boxed{}$ = 0
(Type an expression using x, y, and z as the variables. Type an exact answer, using radicals as needed.)
What is a Cartesian equation for the surface?
$z = \boxed{}$
(Type an expression using x, y, and z as the variables. Type an exact answer, using radicals as needed.)
Choose the correct graph of the cone, point, and tangent plane below. The ViewPoint for both surfaces is (1.5, 0, 0.7). Scroll down for an additional view.
