Section 10 2 Parametric Curves some formulas Tangent line at a point on a parametric curve Let x f t and y g t where t is a parameter The slope of the tangent line to a parametric curve at the point where t is a given value is given by dy dx dy dt dx dt dx dt cid 54 0 Horizontal tangent dx dt Vertical tangent dy dt 0 and 0 and dy dt dx dt cid 54 0 cid 54 0 To determine concavity we calculate the second derivative Arc Length If a curve C is described by y f x where a x b and f cid 48 is continuous then the length L of a curve C is given by If a curve C is described by x g y where c y d and g cid 48 is continuous then the length L of a curve C is given by If a curve C is described by the parametric equations x f t and y g t t and C is traversed exactly once as t increases from to then the length of C is d2y dx2 d dx cid 19 cid 18 dy dx cid 19 d dt cid 18 dy dx dx dt cid 115 cid 90 b L 1 cid 19 2 cid 18 dy dx dx cid 115 cid 90 d L 1 cid 19 2 cid 18 dx dy dy a c L cid 115 cid 18 dx dt cid 90 cid 19 2 cid 19 2 cid 18 dy dt dt 1 Section 10 3 and 10 4 Polar Coordinates some formulas If a point has Cartesian coordinates x y and polar coordinates r then x r cos y r sin r2 x2 y2 tan y x Tangents to Polar Curves Let a polar curve r f where is as a parameter and its parametric equations x r cos f cos y r sin f sin The slope of the tangent line to a polar curve r f at the point where is a given value is dy dx dy d dx d dr d dr d sin r cos cos r sin Areas of a Polar Region Let the polar equation r f the area of the polar and enclosed region where a b is or Arc Length The length of a curve with polar equation r f a b is A r2 d cid 90 b 1 2 a 1 2 cid 90 b a A f 2 d L r2 cid 115 cid 90 b a cid 19 2 cid 18 dr d d 2
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