11 Calculus of Polar Curves - Handout - Tangent Lines Arc...

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Tangent LinesArc LengthAreaExercisesCalculus of Polar CurvesMathematics 54–Elementary Analysis 2Institute of MathematicsUniversity of the Philippines-Diliman1 / 21
Tangent LinesArc LengthAreaExercisesTangent Lines to Polar CurvesGoal: obtain slopes of tangent lines to polar curves of formr=f(θ)2 / 21
Tangent LinesArc LengthAreaExercisesTangent Lines to Polar CurvesParametrization of a Polar CurveA polar curver=f(θ) can be parametrized asx=rcosθ=f(θ)cosθy=rsinθ=f(θ)sinθRecall.slope of a parametric curvedydx=dydθdxdθdxdθ=f0(θ) cosθ-f(θ)sinθ,dydθ=f0(θ) sinθ+f(θ)cosθSlope of a Tangent Line to a Polar CurveGiven thatdy/dθanddx/dθare continuous anddx/dθ6=0, then the slopeof the polar curver=f(θ)isdydx=drdθsinθ+rcosθdrdθcosθ-rsinθ3 / 21
Tangent LinesArc LengthAreaExercisesTangent Lines to Polar CurvesExampleFind the (Cartesian) equation of the tangent line to the cardioidr=1+sinθat the point whereθ=π3.
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Tangent LinesArc LengthAreaExercisesTangent Lines to Polar CurvesExampleDetermine the points on the cardioidr=1-cosθwhere the tangent line ishorizontal, or vertical.Recall.For a parametric curveC:x=x(θ),y=y(θ),dydθ=0=⇒horizontal tangent linedxdθ=0=⇒vertical tangent lineprovided that they are not simultaneously zero.Note that for the above curve, we have the following parametrization:x=rcosθ=(1-cosθ)cosθ=cosθ-cos2θy=rsinθ=(1-cosθ)sinθ=sinθ-sinθcosθ5 / 21
Tangent LinesArc LengthAreaExercisesTangent Lines to Polar CurvesExampleDetermine the points on the curve

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Calculus, Arc Length, Polar coordinate system, Conic section, Tangent lines to circles

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