Linearization of Multivariable Functions

Definition

Linearization $L(x, y)$ provides the best linear approximation of $f(x, y)$ near a point $(x_0, y_0)$ using the function’s value and first partial derivatives.

Why It Matters

Tangent lines become tangent planes in 3D. Linearizing multivariable functions is the essential “first-order” approximation for physics and economics, allowing us to predict the behavior of complex, multi-factor systems near a point of equilibrium.

Core Concepts

• Formula: $L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$

  • How to read: “The linearization L of x, y equals f at x zero, y zero plus the partial derivative of f with respect to x at x zero, y zero times the quantity x minus x zero, plus the partial derivative of f with respect to y at x zero, y zero times the quantity y minus y zero.”
  • Meaning / when to use: Tangent-plane approximation—matches $f$ and its first partials at $(x_0,y_0)$; reliable only for $(x,y)$ close to the base point.

• Geometric Link: The graph of $L(x, y)$ is the tangent plane to $z = f(x, y)$ at $(x_0, y_0)$.

• Accuracy: The approximation is reliable only when $(x, y)$ is very close to $(x_0, y_0)$.

Connected Concepts

• Partial Derivatives
• Tangent Planes
• Normal Lines
• Linearization
• Gradient Vector Properties