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Engineering Calculus Notes 203

# Engineering Calculus Notes 203 - 191 2.4 REGULAR CURVES...

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2.4. REGULAR CURVES 191 with dx dt negationslash = 0 at t = t 0 , and hence nearby ( i.e. , on J ). Now consider the two plane curves parametrized by −→ p y ( t ) = ( x ( t ) ,y ( t )) −→ p z ( t ) = ( x ( t ) ,z ( t )) . These are the projections of C onto, respectively the xy -plane and the xz -plane. Mimic the argument for Proposition 2 to show that each of these 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 defined in Definition 2.4.6 . (a) Suppose −→ p : I R 3 is a continuous, one-to-one vector-valued function on the closed interval I = [ a,b ] with image the arc C , and suppose −→ p ( t i ) −→ p ( t 0 ) ∈C . Show that t i t 0 in I . ( Hint: Show that the sequence t i must have at least one accumulation point t in I , and that for every such accumulation point t , we must have −→ p ( t ) = −→ p ( t 0 ). Then use the fact that −→ p is one-to-one to conclude that the only accumulation point of t i is t = t 0 . But a bounded sequence with exactly one accumulation point must converge to that point.) (b) Show that this property fails for the parametrization of the
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