to a surface at a specific point. This plane provides the best linear approximation of the function near that point, similar to how a tangent line approximates a curve in 2D. ResearchGate Multivariable Calculus - Khan Academy

, compute the dot product of the gradient and the unit vector:

For ( f(x, y) = 4 - x^2 - y^2 ) (a downward paraboloid), ( \nabla f = \langle -2x, -2y \rangle ). At ( (1, 1) ), ( \nabla f = \langle -2, -2 \rangle ), pointing directly downhill.