The polar curve is traced once for .

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

The polar curve r=3sin⁡(2θ)r = 3\sin(2\theta) is traced once for 0≤θ≤π0 \le \theta \le \pi. (a) Show that the curve passes through the origin and list every value of θ\theta in [0,π][0,\pi] for which this occurs. (b) Determine all angles in the first quadrant (0<θ<π2)(0<\theta<\tfrac{\pi}{2}) at which the tangent line to the curve is horizontal. Provide exact values. (c) Determine all angles in the first quadrant at which the tangent line to the curve is vertical. Provide exact values. (d) Choose one of the angles found in part (b). Compute d2ydx2\displaystyle \frac{d^{2}y}{dx^{2}} at that angle, and use the result to decide whether the curve is concave up or concave down at the corresponding point. Justify your conclusion thoroughly.

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