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

• Notes
• ffcamiguing
• 21

This preview shows page 1 - 7 out of 21 pages.

The preview shows page 5 - 7 out of 21 pages.
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.
4 / 21
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

Course Hero member to access this document

Course Hero member to access this document

End of preview. Want to read all 21 pages?

Course Hero member to access this document

Term
Summer
Professor
NoProfessor
Tags
Polar coordinate system, Conic section, Tangent lines to circles
• • • 