Unformatted text preview: 182 CHAPTER 2. CURVES −→ p ( t ) would fail to be regular. But −→ p ( t ) is assumed regular, so its first component function x ( t ) has a nowhere zero derivative on the interval I and hence is strictly monotone. Thus each xvalue occurs at most once: points associated to different xvalues are distinct, so the vectorvalued function −→ p ( t ) is onetoone, as required. The case when C is the graph of y as a function of x is analogous. 2. Now suppose only that −→ p ( t ) is regular. Given t ∈ I , at least one of x ′ ( t ) and y ′ ( t ) must be nonzero. Assume x ′ ( t ) negationslash = 0; without loss of generality, assume it is positive. Since −→ p ( t ) is C 1 , so is x ( t ), and in particular x ′ ( t ) is continuous. Thus, if it is positive at t , it is also positive for all t with  t − t  < ε for sufficiently small ε > 0. But then the xcoordinate is strictly increasing over this subinterval, and in particular the restriction of −→ p ( t ) to this subinterval can cross any...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Derivative

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