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Local linear approximation

Concept

A smooth curve can often be approximated by its tangent line near a chosen point.

Explanation or usage

The tangent line at a point matches the curve’s value and slope there. For a differentiable curve, it can approximate the curve near that point; how accurate it is depends on the neighborhood and the curve’s behavior. A good local approximation need not describe a large interval well.

Common misunderstandings

A tangent line that works nearby must be correct everywhere.

In How Not to Be Wrong

General context reviewed · book attribution and sources pending

This concept can help distinguish nearby straight-line approximations from globally curved relationships.

Linked books

Practice this item

A tangent line closely matches a smooth curve near one point. What does that establish? For f(x) = x² near x = 2, the local slope is 4. What linear estimate describes the change in f for an increase of 0.01 in x?