Engineering Calculus Notes 159

Engineering Calculus Notes 159 - that the actual points p (...

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2.2. PARAMETRIZED CURVES 147 Figure 2.15: The curve y 3 = x 2 Using the relation between polar and rectangular coordinates, this can be parametrized as −→ p ( θ ) = ( f ( θ )cos θ,f ( θ )sin θ ) . We consider a few examples. The polar equation r = sin θ describes a curve which starts at the origin when θ = 0; as θ increases, so does r until it reaches a maximum at t = π 2 (when −→ p ( π 2 ) = (0 , 1)) and then decreases, with r = 0 again at θ = π ( −→ p ( π ) = ( 1 , 0)). For π < θ < 2 π , r is negative, and by examining the geometry of this, we see
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Unformatted text preview: that the actual points p ( ) trace out the same curve as was already traced out for 0 &lt; &lt; . The curve is shown in Figure 2.16 . In this case, we can Figure 2.16: The curve r = sin recover an equation in rectangular coordinates for our curve: multiplying both sides of r = sin by r , we obtain r 2 = r sin...
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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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