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Homework 2 Math 352, Fall 2011 Instructions: Solve at least two of the following three problems. Your solutions must be written in LATEX. Due Date: Monday, September 19 1. Let γ : [0, ∞) → R2 be the unit-speed curve with γ(0) = γ̇(0) = (1, 0) whose signed curvature is κs (s) = (2s)−1/2 . (a) Find a formula for γ̇(s). (Hint: Start by finding a turning angle for γ.) (b) Use your answer to part (a) to find a formula for γ(s). (c) Make a plot of γ(s) for 0 ≤ s ≤ 4π. Do you recognize this curve? 2. Let f : R2 → R be a smooth function, let C ∈ R, and let γ(t) = γ1 (t), γ2 (t) be a regular parametrization of the level curve f (x, y) = C. (a) Use the Chain Rule to prove that fx (γ(t)) γ̇1 (t) + fy (γ(t)) γ̇2 (t) = 0 for all t. (b) Take the derivative of the equation in part (a) with respect to t. (c) Let p = γ(0), and suppose that fx (p) = 0 and fy (p) 6= 0. Use parts (a) and (b) to find a formula for the curvature of γ at p in terms of fy (p) and fxx (p). (Hint: What is γ̇2 (0)?) (d) The curve x2 y + y 3 + x2 − 3y 2 = 0 is known as the trisectrix of Maclaurin: 3 2 1 0 -1 -3 -2 -1 0 1 2 3 Use your formula from part (c) to find the curvature of the trisectrix at the point (0, 3). 3. Find a smooth function f : (−π/2, π/2) → R such f (0) = f 0 (0) = 0 and the curvature of the graph y = f (x) is κ(x) = cos x.