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Local linearity

A curve is locally linear if it looks like a straight line when you zoom in on any point. A curve is locally linear exactly when it has a tangent line at each point. When you zoom in on a point, the curve becomes indistinguishable from the tangent line to the curve at that point.

tangent line zoom


The graph of a differentiable function is locally linear. The function f(x) = x1/3  has a locally linear graph but is not differentiable at x = 0.

A surface is locally planar if it looks like a plane when you zoom in on any point. A surface is locally planar exactly when it has a tangent plane at each point. The graph of a differentiable function of two variables is locally planar.


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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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