Linear Approximation & Differentials

Estimate values and errors with the tangent line and differentials.

The idea

Linear approximation replaces a function near a chosen point by its tangent line there. Magnified closely enough, the graph of a differentiable $f$ is nearly indistinguishable from a straight line, and a line can be evaluated easily even where $f$ itself is awkward.

The tangent at a base point $a$ passes through $(a, f(a))$ with slope $f'(a)$, so in point-slope form it is $L(x) = f(a) + f'(a)(x - a).$ $L$ is the linearization of $f$ at $a$, and the approximation $f(x) \approx L(x)$ for $x$ near $a$ is the linear approximation: it starts from the known value $f(a)$ and corrects it by the rate of change times the displacement $x - a$. The approximation is exact at $a$ and worsens as $x - a$ grows, so we choose the base point to be a convenient value nearby — one where $f$ and $f'$ are easy to evaluate.

The same statement, phrased in changes rather than positions, is the differential. Write $dx$ for a small change in the input and $dy$ for the estimated change in the output: $dy = f'(x)\,dx.$ This form answers questions about small changes — how far a computed quantity moves when a measurement is off by a little.

Ways to work on it

• Walkthrough. Build the linearization and use it to estimate a square root.
• Practice. Estimate a function value with its tangent line.
• Hardest. Use differentials to estimate a propagated measurement error.

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