Tangent Planes & Linear Approximation

The Earth is round, yet a builder's level treats the ground as flat — and for a small backyard, that's perfectly accurate. Zoom in on any smooth surface and it looks flat. That flat sheet kissing the surface at a single point is the TANGENT PLANE. It's the 3D cousin of the tangent line, and it's the best flat approximation to a curved surface near that point.