Taylor Polynomials for f(x,y)

Calculus 3 · Axiom Academy

LESSON Taylor Polynomials for f(x,y) Approximating a surface near a point: the gradient builds the tangent plane, the Hessian bends it into a paraboloid, and higher orders sharpen the fit. 1. Linear Approximation with the Gradient The simplest approximation is linear, built from the function value and the first partial derivatives at a point (a, b) . Geometrically it is the tangent plane to the surface at that point. 2. Quadratic Approximation with the Hessian To do better, add the second-order terms. These are organized by the Hessian matrix of second partial derivatives, which measures how the surface curves away from its tangent plane. Worked example: f(x,y) = e^ x y near (0,0) The partial derivatives give the gradient f(0,0) = (1,\, 0) and the Hessian so the second-order polynomial is Test it at (0.4,\, 0.2) : the plane gives P_1 = 1.40 and the paraboloid gives P_2 = 1.46 , while the true value is e^ 0.4 0.2 1.4621 . The quadratic term shrinks the error from about 0.062 down to roughly 0.002 . Because it curves up along x and down along y, this surface is a saddle. 3. General n-th Order Taylor Polynomial The pattern continues. Including third-, fourth-, and higher-order partial derivatives yields ever more accurate approximations over a wider neighborhood of (a, b). The constant f(a, b) — a flat sheet at the point's height. The tangent plane — matches the gradient (linear). A paraboloid — matches the Hessian (quadratic).

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