Linear Approximation for Functions of Two Variables

The tangent plane approximation is the two-variable version of replacing a curve with its tangent line, and it is also known as a linear approximation multivariable. Use it to estimate f(x,y) near a known point (a,b) when computing the exact value is messy or impossible. The formula, from Stewart §14.4, is L(x,y) = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b). That plane matches the function's value and both partial derivatives at exactly one point. It is not the best fit over an interval; it is the unique plane that touches the surface at (a,b) with the same slope in both directions.

Newcomers most often get wrong that the plane works everywhere. It does not. The error grows with the square of the distance from (a,b), so the estimate degrades rapidly as you move away. Use it only for small displacements. The linearization of f(x,y) is the function L(x,y) itself; the tangent plane is its graph. They are the same object.

Step by Step Calculation

Take f(x,y) = x² + y² at the point (1,1). The partial derivatives are f_x = 2x and f_y = 2y, so f_x(1,1) = 2 and f_y(1,1) = 2. With f(1,1) = 2, the tangent plane is L(x,y) = 2 + 2(x - 1) + 2(y - 1) = 2x + 2y - 2.

To estimate f(1.1, 0.9), compute L(1.1, 0.9) = 2(1.1) + 2(0.9)-2 = 2.2 + 1.8-2 = 2.0. The true value is 1.1² + 0.9² = 1.21 + 0.81 = 2.02. The error is 0.02.

Error Growth With Distance

That is a 1% error. Move to (1.5, 1.5) and the approximation is L(1.5, 1.5) = 2(1.5) + 2(1.5)-2 = 3 + 3-2 = 4. The true value is 1.5² + 1.5² = 2.25 + 2.25 = 4.5. The error is 0.5, or 11%. The linearization error grows quadratically with distance from the anchor point.

The total differential dz = f_x(x,y) dx + f_y(x,y) dy is the infinitesimal version of the same idea. It tells you how a small change in x and y changes the function value, assuming the function is linear. The actual change Δz = f(x+dx, y+dy) - f(x,y) differs from dz by an error that involves second partial derivatives.

Use the total differential for error propagation. If you measure x and y with uncertainty dx and dy, the uncertainty in f is approximately dz. This is the same calculation as the tangent plane approximation, just written with differentials. The difference between dy and Δy in single-variable calculus carries over: dz is the change along the tangent plane, Δz is the real change on the surface.

For the tangent plane to exist at (a,b), the function must be differentiable there. That means both partial derivatives exist at (a,b) and the function is locally linear: the error between the function and its tangent plane shrinks faster than the distance from (a,b) as you approach it. If either partial derivative does not exist, or if the function has a corner or a cusp, the linearization does not exist either.

The differentiability condition is stricter than just having partial derivatives. A classic counterexample is f(x,y) = |x| + |y| at (0,0). Both partial derivatives exist (they are 1 and 1, in a sense), but the function has a corner along every direction. The tangent plane does not exist because the surface is not locally linear. Stewart §14.4 gives the formal condition: the function must be differentiable, meaning there exists a linear function L such that the error divided by the distance goes to zero as the distance goes to zero.

The pattern extends directly. For a function of three variables w = f(x,y,z), the linearization at (a,b,c) is L(x,y,z) = f(a,b,c) + f_x(a,b,c)(x - a) + f_y(a,b,c)(y - b) + f_z(a,b,c)(z - c). The total differential is dw = f_x dx + f_y dy + f_z dz.

Use this when estimating a function that depends on three measured quantities, each with its own uncertainty. The error propagation formula from the total differential becomes σ_f² ≈ (∂f/∂x)²σ_x² + (∂f/∂y)²σ_y² + (∂f/∂z)²σ_z² if the measurements are independent. The same caveat applies: the estimate is only good for small uncertainties, and the error grows quadratically with the distance from the linearization point.