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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  < for sufficiently small > 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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 Fall '08
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 Calculus, Derivative

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