Calc III · §14.4 tangent planes

Tangent plane to z = 2x² + y² at (1, 1, 3)

The tangent plane is the flat sheet that just kisses the surface at P, built from the two partial‑derivative slopes. Its equation is z = f(a,b) + fx(a,b)(x−a) + fy(a,b)(y−b). Drag (a,b) to slide P along the bowl and watch the plane tilt to stay tangent; drag the graph to rotate.

Tangent plane = point height + (x‑slope)·(Δx) + (y‑slope)·(Δy). 🟡