AP Calculus AB and BC glossary

Local linearity

Local linearity means that a differentiable function, zoomed in near a point, looks almost exactly like its tangent line there. That is why the tangent line closely estimates function values near the point of tangency, and it is the basis of linear approximation.

This is what differentiable looks like up close. Zooming in on a smooth point flattens the curve toward one straight line, its tangent, and the closer you zoom the better the match. That fact is what makes the linearization $L(x) = f(a) + f'(a)(x - a)$ a usable estimate.

How quickly the estimate degrades away from $a$ depends on concavity. Where $|f''|$ is large the curve bends away from its tangent fast, so the range over which a linear estimate stays accurate is short.

The mistake

Treating local linearity as global. The tangent line matches the curve only near the point of tangency, and the estimate worsens as you move away. It also fails at corners and cusps, where zooming in never settles on a single line, and at vertical tangents, where the graph approaches a vertical line and so has no finite-slope tangent to use.

Appears in: Unit 4: Contextual Applications