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Engineering Calculus Notes 190

Engineering Calculus Notes 190 - 178 CHAPTER 2 CURVES...

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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 C = gr ( f ) , then there is a unique function t ( s ) defined on J and mapping onto I such that −→ q ( s ) = −→ p ( t ( s )) for all s J. The function t ( s ) is C 1 with nonzero derivative everywhere in
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