2.4. REGULAR CURVES191withdxdtnegationslash= 0 att=t0, and hence nearby (i.e.,onJ). Now considerthe two plane curves parametrized by−→py(t) = (x(t),y(t))−→pz(t) = (x(t),z(t)).These are the projections ofConto, respectively thexy-plane and thexz-plane. Mimic the argument for Proposition2to show that each ofthese is the graph of the second coordinate as a function of the first.Challenge problems:10.Arcs:In this exercise, we study some properties of arcs, as definedin Definition2.4.6.(a) Suppose−→p:I→R3is a continuous, one-to-one vector-valuedfunction on the closed intervalI= [a,b] with image the arcC,and suppose−→p(ti)→−→p(t0)∈C.Showthatti→t0inI. (Hint:Show that the sequencetimust have at least one accumulationpointt∗inI, and that foreverysuch accumulation pointt∗, wemust have−→p(t∗) =−→p(t0). Then use the fact that−→pisone-to-one to conclude that theonlyaccumulation point oftiist∗=t0. But a bounded sequence with exactly one accumulationpoint must converge to that point.)(b) Show that this property fails for the parametrization of the
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