# 11 Calculus of Polar Curves - Tangent Lines Arc Length Area...

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Tangent Lines Arc Length Area Exercises Calculus of Polar Curves Mathematics 54–Elementary Analysis 2 Institute of Mathematics University of the Philippines-Diliman 1 / 22
Tangent Lines Arc Length Area Exercises Tangent Lines to Polar Curves Goal : obtain slopes of tangent lines to polar curves of form r = f ( θ ) 2 / 22
Tangent Lines Arc Length Area Exercises Tangent Lines to Polar Curves Parametrization of a Polar Curve A polar curve r = f ( θ ) can be parametrized as x = r cos θ = f ( θ )cos θ y = r sin θ = f ( θ )sin θ Recall. slope of a parametric curve dy dx = dy d θ dx d θ dx d θ = f 0 ( θ ) cos θ - f ( θ )sin θ , dy d θ = f 0 ( θ ) sin θ + f ( θ )cos θ Slope of a Tangent Line to a Polar Curve Given that dy / d θ and dx / d θ are continuous and dx / d θ 6= 0, then the slope of the polar curve r = f ( θ )is dy dx = dr d θ sin θ + r cos θ dr d θ cos θ - r sin θ 3 / 22
Tangent Lines Arc Length Area Exercises Tangent Lines to Polar Curves Example Find the (Cartesian) equation of the tangent line to the cardioid r = 1 + sin θ at the point where θ = π 3 .
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Tangent Lines Arc Length Area Exercises Tangent Lines to Polar Curves Example Determine the points on the cardioid r = 1 - cos θ where the tangent line is horizontal, or vertical. Recall. For a parametric curve C : x = x ( θ ), y = y ( θ ), dy d θ = 0 =⇒ horizontal tangent line dx d θ = 0 =⇒ vertical tangent line provided that they are not simultaneously zero . Note that for the above curve, we have the following parametrization: x = r cos θ = (1 - cos θ )cos θ = cos θ - cos 2 θ y = r sin θ = (1 - cos θ )sin θ = sin θ - sin θ cos θ 5 / 22
Tangent Lines Arc Length Area Exercises Tangent Lines to Polar Curves