Tangent Planes & Linear Approximation

The tangent plane is the linearization that approximates a surface.

The idea

The tangent plane to a surface $z = f(x, y)$ at a point is the plane that stands in for the surface nearby, as the tangent line at $x = a$ stands in for the curve $y = f(x)$.

A plane through the point $(a, b, f(a, b))$ is pinned down by two slopes, one for each horizontal direction, and $f$ supplies them: $f_{x}(a, b)$ and $f_{y}(a, b)$. Matching the height and both slopes gives the tangent plane $z = f(a, b) + f_{x}(a, b)\,(x - a) + f_{y}(a, b)\,(y - b).$

Read as a function of $(x, y)$, the right-hand side is the linearization $L(x, y)$ of $f$ at $(a, b)$: the constant term sets the height at the base point, and each slope multiplies the displacement in its own direction. $L$ is the only function of that form agreeing with $f$ in value and in both partial derivatives at $(a, b)$, and the approximation $f(x, y) \approx L(x, y)$ holds for $(x, y)$ near $(a, b)$.

The approximation is local. The plane is flat and the surface is not, so the error grows with the distance from the base point.

Ways to work on it

• Walkthrough. Partials as slopes, the tangent plane, and the linearization.
• Practice. Estimate a nearby value from a given linearization.
• Hardest. Linearize a nonlinear function and approximate a hard value.

Not sure where to start? Take the ten-question placement test.