# Math 54 Exam 2 Lecture 5 - Calculus of Polar Curves.pdf -...

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Tangent Lines Arc Length Area Exercises Calculus of Polar Curves Math 54 - Elementary Analysis II Institute of Mathematics University of the Philippines-Diliman 13 September 2017 Math 54 WFQ (Lec.: G.A.M. Velasco) 13 Sep. 2017 Calculus of Polar Curves
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 ) 0 π 2 Math 54 WFQ (Lec.: G.A.M. Velasco) 13 Sep. 2017 Calculus of Polar Curves
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 ( θ ) cos θ f ( θ ) sin θ , dy d θ = f ( 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 negationslash = 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 . Math 54 WFQ (Lec.: G.A.M. Velasco) 13 Sep. 2017 Calculus of Polar Curves
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 .
Math 54 WFQ (Lec.: G.A.M. Velasco) 13 Sep. 2017 Calculus of Polar Curves
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 . Solution (Cont’d). At θ = π 3 , dy dx = 1 , and r = 2 + 3 2 . Cartesian coordinates of the point of tangency parenleftbigg 2 + 3 2 , π 3 parenrightbigg are parenleftbigg 2 + 3 4 , 3 + 2 3 4 parenrightbigg . Equation of the tangent line: y 3 + 2 3 4 = parenleftbigg x 2 + 3 4 parenrightbigg 0 π 2 1 2 π 3 Math 54 WFQ (Lec.: G.A.M. Velasco) 13 Sep. 2017 Calculus of Polar Curves
Tangent Lines Arc Length Area Exercises Tangent Lines to Polar Curves