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Polar Tangent Lines & Slope
AP Calculus BC · Unit 9 Polar Coordinates
Learning Objectives
Convert polar r=f(θ) to x,y parameters
Calculate dy/dx = (dy/dθ) / (dx/dθ)
Find horizontal and vertical tangents
Understand slope vs radial direction (dr/dθ)
Derivation
x(θ)
= r·cos(θ) = f(θ)cos(θ)
y(θ)
= r·sin(θ) = f(θ)sin(θ)
Slope dy/dx:
(f'(θ)sin(θ) + f(θ)cos(θ))
-------------------------
(f'(θ)cos(θ) − f(θ)sin(θ))
Tags
polar
tangent line
dy/dx
chain rule
POLAR CURVE r(θ)
Cardioid: r = 1 − sin(θ)
3-Petal Rose: r = cos(3θ)
Limaçon: r = 1 + 2cos(θ)
Lemniscate: r² = cos(2θ)
Angle θ (rad)
0.00π
Radius r
—
Radial Drv r'
—
Coords (x, y)
—
Slope dy/dx
—
Orange line
: Tangent Line (dy/dx)
Blue line
: Radial Position Vector
If dx/dθ = 0 (and dy/dθ ≠ 0), vertical tangent.
If dy/dθ = 0 (and dx/dθ ≠ 0), horizontal tangent.