AP® Calculus AB · Unit 4 · Topic 4.6
Approximating Values Using Local Linearity and Linearization
Local linearity says that up close, a smooth curve looks like its tangent line. That tangent line, written as a function L(x), lets you approximate f near the point of tangency without ever touching the original messy function.
Zoom into any smooth curve far enough and the curvature disappears. What's left looks straight, because the curve's own slope barely changes over a tiny stretch. That straight-looking piece is just the tangent line at the point you zoomed into. Since the tangent line matches the function's value and slope exactly at that one point, and the two stay close together just nearby, the tangent line works as a stand-in for the function near there.
The stand-in has a name: the linearization, L(x) = f(a) + f'(a)(x - a). It's the same point-slope formula from algebra, just built from f(a) and the derivative f'(a) instead of two given points. To approximate f at some x near a, you don't compute f(x) directly — you plug x into L(x) instead, trading a hard calculation for a cheap linear one. The farther x drifts from a, the more the curve bends away from its tangent, and the worse the approximation gets.
This is the same tangent line you've been finding since derivatives were introduced, now repurposed as a prediction tool instead of just a slope. It sets up differentials, where dy = f'(a)dx measures the same approximation using a small change in x rather than a target value. It also previews error analysis: since L(x) only matches f at a, the gap between them grows with concavity, which is the seed of the error bound ideas that follow.
Key terms
- Local linearity
- The property that a smooth curve looks increasingly like a straight line as you zoom into a single point.
- Linearization
- The function L(x) = f(a) + f'(a)(x - a) giving the tangent line to f at x = a.
- Point of tangency
- The value x = a where the tangent line touches the curve and matches its value and slope exactly.
- Differential
- The quantity dy = f'(a)dx, a related tool measuring the approximate change in f from a small change dx.
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End of topic.