Engineering Calculus Notes 195

Engineering Calculus Notes 195 - −→ p − 1 = −→...

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2.4. REGULAR CURVES 183 This is illustrated by the Folium of Descartes , given by the equation x 3 + y 3 = 3 xy and sketched in Figure 2.25 . A regular parametrization (Exercise 5 ; see Figure 2.25: The Folium of Descartes: x 3 + y 3 = 3 xy also Exercise 12 ) is given by x = p 3(1+ t )(1 t ) 2(1+3 t 2 ) P (1 t ) y = p 3(1+ t )(1 t ) 2(1+3 t 2 ) P (1 + t ) − ∞ < t < . (2.24) The origin is a self-crossing of this curve:
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Unformatted text preview: −→ p ( − 1) = −→ p (1) . While Proposition 2.4.7 tells us that the restriction of our parametrization to a su±ciently small interval about t = − 1 ( resp . t = 1) gives the graph of a function, these two graphs are very di²erent: when t = − 1, dx dt = 3 2 dy dt = 0...
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