Mathematics

Mathematics

Problem

Consider the polar curve r=1+sin⁡θr=1+\sin\theta for 0<θ<π0<\theta<\pi. Determine the concavity of the curve at θ=π4\theta=\dfrac{\pi}{4} by evaluating d2ydx2\dfrac{d^{2}y}{dx^{2}}. Which of the following is correct? a) d2ydx2≈+10.2\dfrac{d^{2}y}{dx^{2}}\approx+10.2 and the curve is concave upward. b) d2ydx2≈−14.5\dfrac{d^{2}y}{dx^{2}}\approx-14.5 and the curve is concave downward. c) d2ydx2\dfrac{d^{2}y}{dx^{2}} is undefined because dx/dθ=0dx/d\theta=0 at that angle. d) d2ydx2≈+14.5\dfrac{d^{2}y}{dx^{2}}\approx+14.5 and the curve is concave upward.

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