Linear Approximations

A tangent plane is a flat surface that touches a curved function at a specific point. In linear approximation, the tangent plane replaces the curved surface locally, allowing quick estimates of function values near a known point. The plane is defined by the function value and the partial derivatives in both the x and y directions at that point.

Partial derivatives measure how a function changes in the x and y directions at a specific point. The linear approximation formula L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) uses these rates of change to create a flat model. The partial derivatives fx and fy determine the slope of the tangent plane in each direction.

Linear approximation assumes the curved surface stays close to its tangent plane. Near the known point, this assumption holds well. However, as distance increases, the actual curved surface deviates more from the flat tangent plane. The approximation error grows because the function's curvature becomes significant at greater distances.

First, compute the partial derivatives fx and fy of your function. Evaluate these derivatives and the function value at your known point (a,b). Then substitute into L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b), replacing x and y with your target point's coordinates to get the approximation.

Linear approximation models real phenomena like pollution concentration across a city, where a monitoring station provides a known measurement. Rather than computing the entire curved surface, engineers estimate pollution levels at nearby locations using the tangent plane. This approach provides quick, practical estimates for environmental monitoring and other spatial modeling problems.

The function must be differentiable at the point of interest, meaning partial derivatives exist and the function changes smoothly. Differentiability ensures the tangent plane is well-defined and provides a reliable local model. Non-differentiable functions or those with sharp corners or discontinuities cannot be approximated using this method.

Accuracy depends on proximity to the known point. For points very close to the reference location, the approximation is typically accurate. For example, estimating f(2.1, 0.9) near (2, 1) yields 6.8 versus the exact value 6.84, a small error. As distance increases, error grows because the tangent plane diverges from the curved surface.