Concept explainerCalculus IFoundational8 min read

Linear approximation: use the tangent line as a local calculator

Near a known input, a smooth function behaves like its tangent line—and the derivative measures the approximation’s sensitivity.

L(x)=f(a)+f'(a)(x-a)

Concept explainer

What the idea means, why its conditions matter, and where it connects.

Reviewed July 11, 2026

ApproximationStart here

\boxed{f(x)\approx f(a)+f'(a)(x-a)}

Choose a nearby input a where both the function and derivative are easy to evaluate.

Why the tangent line works

Differentiability means the error between the function and its best linear model becomes small compared with the input change. The tangent line matches both the value and first-order rate at a.

The approximation is local. Moving far from a lets curvature accumulate and weakens the estimate.

Choose the center strategically

Pick a close value with simple arithmetic: a perfect square for square roots, a familiar angle for trig, or zero for exponentials and logarithms when allowed.

A closer center usually improves accuracy, but simplicity and distance should be balanced.

Differentials and error estimates

Writing dy = f′(a) dx expresses the predicted output change. It is useful for sensitivity and measurement error: a small input uncertainty is scaled by the local derivative.

Worked exampleEstimate a square root

Approximate √4.1 without a calculator.

f(x)=\sqrt{x},\qquad a=4

Center at the nearby perfect square.

f(4)=2,\qquad f'(4)=\frac14

Compute the known value and local slope.

L(4.1)=2+\frac14(0.1)

Apply the linear model.

Result

\boxed{\sqrt{4.1}\approx2.025}

Watch for

Common mistakes

Using x instead of x−a in the tangent formula.
Choosing a center that is easy but not close.
Treating a local approximation as exact.

Keep

Three takeaways

Match value and slope at a convenient nearby point.
The derivative predicts small output changes.
Curvature controls how quickly the approximation degrades.