Linear Approximation Calculator

Linear approximation replaces a curve with its tangent line near a chosen point. The idea is that a differentiable function looks almost straight when you zoom in close enough — the tangent line and the curve are nearly indistinguishable in a small neighborhood.

The linearization of f at x = a is: L(x) = f(a) + f’(a)(x - a)

To use it: pick a value a where f(a) and f’(a) are easy to compute, then estimate f at a nearby x using L(x) instead. The closer x is to a, the better the approximation.

Classic example: estimate sqrt(4.02). Pick a = 4 (nice perfect square). f(4) = 2, f’(x) = 1/(2sqrt(x)), f’(4) = 1/4. Linearization: L(x) = 2 + 0.25(x - 4). At x = 4.02: L(4.02) = 2 + 0.25(0.02) = 2.005. Actual value: sqrt(4.02) ≈ 2.00499. The error is tiny.

The approach breaks down when x is far from a, or when the function curves sharply near a (high second derivative). The error in linear approximation is roughly proportional to (x-a)^2 times half the second derivative — which is why choosing a close to x matters so much.

This is the same computation a calculator does internally when approximating transcendental functions. Engineers use linearization constantly to turn nonlinear systems into tractable linear ones — all of control theory and circuit analysis relies on small-signal linearization around operating points.

The percentage error shown here is the relative difference between L(x) and the true f(x), giving a practical sense of whether the approximation is “good enough” for your purposes.