Linear Algebra
10.1810 min read

Local Linear Approximation

The local linear approximation (LLA) at x0x_0 replaces the function ff near x0x_0 with a line that matches the function's value and slope: f~(x)=f(x0)+f′(x0)(x−x0).\tilde{f}(x) = f(x_0) + f'(x_0)(x - x_0). This is the equation of the tangent line to the graph of ff at x0x_0.

The LLA is the best first-order approximation to ff near x0x_0: it has the same value (f~(x0)=f(x0)\tilde{f}(x_0) = f(x_0)) and the same slope (f~′(x0)=f′(x0)\tilde{f}'(x_0) = f'(x_0)). For xx close to x0x_0, f~(x)≈f(x)\tilde{f}(x) \approx f(x).

Why is this useful? Because linear functions are easy to analyze and optimize. Near any point, a smooth function looks like a line, and we can use linear algebra to approximately solve the optimization problem. This idea extends to multiple dimensions: the gradient and Jacobian are the multivariate generalizations of f′(x0)f'(x_0).

Formal View

Definition 10.13 — Local Linear Approximation
For a differentiable function ff at point x0x_0, the local linear approximation is
f~(x)=f(x0)+f′(x0)(x−x0).\tilde{f}(x) = f(x_0) + f'(x_0)(x - x_0).
This is the unique affine function agreeing with ff in value and slope at x0x_0.

The LLA is also called the first-order Taylor approximation or the tangent line approximation.

Interactive Visualization

Local Linear Approximation

Why This Matters

Local linear approximation is the foundation of all first-order optimization methods and the basis of calculus in multiple dimensions.

  • Newton's method: approximate ff near xkx_k by its LLA, then solve for the zero of the approximation.
  • Linearization of nonlinear systems: replace a nonlinear ODE with its linear approximation near an equilibrium.
  • Error propagation: Δf≈f′(x0)Δx\Delta f \approx f'(x_0) \Delta x — how input uncertainty propagates to output uncertainty.

Quiz

Question 1

The local linear approximation of f(x)=x2f(x) = x^2 at x0=3x_0 = 3 is:

Question 2

The local linear approximation has the same value AND the same derivative as ff at x0x_0.

Common Mistakes

  • Confusing the LLA formula with just f′(x0)⋅xf'(x_0) \cdot x — the correct formula has both f(x0)f(x_0) and f′(x0)(x−x0)f'(x_0)(x - x_0).
  • Using the LLA far from x0x_0 — the approximation is only good locally; the error grows as xx moves away from x0x_0.