A tangent plane is the best first-order flat approximation to a smooth surface near a point.
The plane matches the surface’s height and both first partial derivatives at (a, b). The approximation is local; curvature makes the surface pull away as you move farther from that point.
A tangent plane need not stay on one side of a surface, especially at a saddle.
Drag to rotate, or use the view sliders. The surface and level curve are sampled; all point values and derivatives use exact formulas.