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Unformatted text preview: 178 CHAPTER 2. CURVES Definition 2.4.1. A vector-valued function p ( t ) defined on an interval I is regular on I if it is C 1 and has nonvanishing velocity for every parameter value in I . (The velocity vector at an endpoint, if included in I , is the one-sided derivative, from the inside.) A vector-valued function p ( t ) on an interval I is a regular parametrization of the curve C if the image of I under p equals C (i.e., p maps I onto 15 C ). When p ( t ) is a regular parametrization of the graph of a function f ( x ), the tangent line agrees with the usual one, namely the graph of the linearization of f ( x ). This suggests that the tangent line is a geometric object, attached to a curve independent of the vector-valued function used to parametrize it. To study this question, we first study the different ways a graph can be parametrized. Lemma 2.4.2. If p ( t ) and q ( s ) defined on the interval I (resp. J ) are both regular parametrizations of a graph...
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- Fall '08