Engineering Calculus Notes 155

Engineering Calculus Notes 155 - 143 2.2. PARAMETRIZED...

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2.2. PARAMETRIZED CURVES 143 The circle x 2 + y 2 = 1 consists of two graphs: if we solve for y as a function of x , we obtain y = ± r 1 x 2 , 1 x 1 . The graph of the positive root is the upper semicircle, and this can be parametrized by x ( t ) = t y ( t ) = r 1 t 2 or −→ p ( t ) = ( t, r 1 t 2 ) , t [ 1 , 1] . Note, however, that in this parametrization, the upper semicircle is traversed clockwise ; to get a counterclockwise motion, we replace t with its negative: −→ q ( t ) = ( t, r 1 t 2 ) , t [ 1 , 1] . The lower semicircle, traversed counterclockwise, is the graph of the negative root: −→ p ( t ) = ( t, r 1 t 2 , t [ 1 , 1] . Displacing Curves The parametrizations so far concern ellipses and hyperbolas in standard positions—in particular, they have all been centered at the origin. We saw
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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.

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