Study guide and 2 practice problems on:

Tangent planes to surfaces

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To find the tangent plane to a surface at $\bfx_0 = (x_0, y_0, z_0)$, compute a normal vector $\bfn$. The tangent plane is given by $$ \bfn \cdot \bfx = \bfn \cdot \bfx_0.$$
To find a normal vector to an implicitly defined surface, view that surface as a level set of some function $g(x,y,z)$.

A normal vector at $(x,y,z)$ is given by $\nabla g(x,y,z)$.
A normal vector to the explicitly defined surface $z = f(x,y)$ is $$\left \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, -1 \right \rangle$$

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