Linear Approximation

A rigorous, example-driven guide to linear approximation, including hypotheses, method choice, verification, and practice.

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A derivative is the limiting slope of nearby secant lines. A smooth curve has a highlighted point, a tangent line there, and a nearby secant line. A small delta-x bracket shows how the secant approaches the tangent as the second point moves closer.

The central idea

If $f$ is differentiable at $a$, its linearization is $L(x)=f(a)+f'(a)(x-a)$. Differentiability means $f(x)-L(x)$ is small compared with $x-a$ as $x\to a$. The approximation $f(x)\approx L(x)$ is most reliable close to the base point $a$.

Definitions, hypotheses, and notation

Linearization is centered: the same formula can be accurate for several nearby inputs, but the error generally grows with distance from $a$. Taylor's theorem refines this by showing a typical second-order error of roughly $f''(\xi)(x-a)^2/2$. Thus choosing a convenient base point is balanced against keeping $|x-a|$ small.

The tangent approximation also supports sensitivity analysis. If an input is uncertain by $h$, the output changes by approximately $f'(a)h$; a large derivative signals amplification. However, a small derivative does not guarantee a small error over a wide interval, because curvature may change the slope. The approximation is local by definition.

A dependable method and decision rule

Choose a base point where the function and derivative are easy to evaluate.
Compute $f(a)$ and $f'(a)$.
Write the complete tangent-line formula $L(x)=f(a)+f'(a)(x-a)$.
Substitute the nearby target input only after constructing $L$.
Use concavity or an error estimate to assess direction and size of error.

Fully worked example

Common mistakes and why they fail

Choose the anchor strategically

The linearization $L(x)=f(a)+f'(a)(x-a)$ works best when $a$ is close to the target and both $f(a)$ and $f'(a)$ are simple. It matches the function value and slope at $a$, but curvature makes the error grow with distance. If $f''$ keeps one sign, concavity predicts the error direction: a concave-up graph lies above its tangent and a concave-down graph below it. Choose a nearby perfect square, familiar angle, or other convenient anchor, and write the signed change $x-a$ explicitly. Report only precision supported by the step size and curvature; extra decimal places do not improve an approximation.