Solution-to-12.1.66-Week3

Solution-to-12.1.66-Week3 - L is equal to the slope of the...

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Solution to Exercise 66 in Section 12.1. Solution: The cycloid has parametrization c ( t )=( t - sin t, 1 - cos t ) Let P =( t - sin t, 1 - cos t ). and let Q be the point at the top of the circle. We claim that Q =( t, 2). The y -coordinate is 2 because the unit circle has diameter 2, and the x -coordinate is t because t is equal to the distance the circle has rolled. Tangent line Cycloid y x P 1 t Q = (t,2) t Now let L be the line through P and Q . To show that L coincides with the tangent line at P , we need only verify that the slope of
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Unformatted text preview: L is equal to the slope of the tangent line (two lines through P of the same slope necessarily coincide). The slope of L is slope of L = 2-(1-cos t ) t-( t-sin t ) = 1 + cos t sin t The slope of the tangent line at P is dy dx = y ( t ) x ( t ) = (1-cos t ) ( t-sin t ) = sin t 1-cos t These two slope are equal: 1 + cos t sin t = sin t 1-cos t because 1-cos 2 t = sin 2 t . This completes the proof....
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This note was uploaded on 10/26/2011 for the course MATH 32A taught by Professor Gangliu during the Spring '08 term at UCLA.

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