# lecture16(11.2,11.3) - Math 141 Lecture 16 Greg Maloney...

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Math 141Lecture 16Greg MaloneyUniversity of Massachusetts BostonOctober 28, 2010Math 141 (UMass Boston)Lecture 16October 28, 2010212 / 329
Outline1(11.2) Calculus with Parametric CurvesTangentsAreasArc Length2(11.3) Polar CoordinatesMath 141 (UMass Boston)Lecture 16October 28, 2010213 / 329
(11.2) Calculus with Parametric CurvesTangentsExample (Example 2, p. 667)Consider the cycloidx=r(θ-sinθ),y=r(1-cosθ).2Πr4Πr6Πrx2ry1At what points is the tangent horizontal?2At what points is the tangent vertical?Math 141 (UMass Boston)Lecture 16October 28, 2010214 / 329
(11.2) Calculus with Parametric CurvesTangentsExample (Example 2, p. 667)Consider the cycloidx=r(θ-sinθ),y=r(1-cosθ).2Πr4Πr6Πrx2ry1At what points is the tangent horizontal?
Math 141 (UMass Boston)Lecture 16October 28, 2010215 / 329
(11.2) Calculus with Parametric CurvesTangentsExample (Example 2, p. 667)Consider the cycloidx=r(θ-sinθ),y=r(1-cosθ).2Πr4Πr6Πrx2ry2At what points is the tangent vertical?Whenθ=2nπbothdy/dθanddx/dθare 0.To see if there is a vertical tangent, use L’Hospital’s Rule.limθ2nπ+dydx=limθ2nπ+sinθ1-cosθ=limθ2nπ+cosθsinθ10+Therefore limθ2nπ+(dy/dx) =.A similar argument shows limθ2nπ-(dy/dx) =-∞.Therefore there is a vertical tangent whenθ=2nπ.Math 141 (UMass Boston)Lecture 16October 28, 2010216 / 329

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Term
Fall
Professor
Maloney
Tags
Polar coordinate system, Conic section