SECTION 1: CURVATURE AND ARC LENGTH (Questions 1-3) Question 1 (8 points) Consider the ellipse: x24+y2=14x2+y2=1. Let P=(0,1)P=(0,1) and Q=(2,0)Q=(2,0) be two points on the ellipse . (a) (2 points) From the picture (imagine the ellipse shape), which one of the two points P and Q has a larger curvature? Briefly explain in words. Solution: The point Q has a larger curvature because it exhibits more bend compared to the point P. At Q, the ellipse is sharply curving around its major axis endpoint, while at P, the ellipse is relatively flatter at the top of its minor axis. (b) (6 points) Compute the curvature of the ellipse at the points P and Q. Solution: The ellipse can be parametrized as: r(t)=⟨x(t),y(t)⟩=⟨2cost,sint⟩r(t)=⟨x(t),y(t)⟩=⟨2cost,sint⟩ The curvature formula for a planar curve is: κ(t)=∣x′(t)y′′(t)−y′(t)x′′(t)∣(x′(t)2+y′(t)2)3/2κ(t)=(x′(t)2+y′(t)2)3/2∣x′(t)y′′(t)−y′(t)x′′(t)∣
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