Linear Approximation

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.

Conceptual meaning

A smooth graph looks like its tangent line under sufficient magnification. The linearization keeps the function's value and first derivative at $a$ while discarding curvature and higher-order effects.

A dependable method and decision rule

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

Fully worked example

Graphical or geometric meaning

The tangent line and curve meet at $x=a$ with the same slope. For a concave-down function the tangent lies above the graph nearby, so linearization overestimates; for concave-up behavior it underestimates.

Common mistakes and why they fail

Verification and reasonableness checks

• Verify $L(a)=f(a)$ and $L'(a)=f'(a)$.
• Compare against a rough calculator value or algebraic back-check.
• Use concavity to predict over- or underestimation.

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.

Practice

1. Linearize $e^x$ at $0$.
2. Use it to estimate $e^{0.03}$.
3. Linearize $1/x$ at $x=2$.

Connections and next steps

• Differentials
• Derivative
• Taylor Polynomial