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Math 474 Exam #1 Due October 9, 2015 at 2:00 PM. Instructions: This take-home exam covers the material covered in class through September 30, 2015 (roughly corresponding to Chapter 1 from Shifrin). You may use: Maple/Mathematica/MATLAB, a calculator, Shifrin, your notes, notes posted on the website, your brain. You may not use: The internet, other books, other people’s notes, other people’s brains. 1. Find the curvature and torsion of the curve α(t) = (2 + cos t, sin t, 8 sin(t/2)). 2. The plane curve α(t) = ((2a cos t + b) cos t, (2a cos t + b) sin t) with t ∈ [0, 2π) is called a limaçon. Compute its rotation index. (If you’re getting lost in the symbols, try it with a = 1 and b = 1. Also, you are free to look up trig identities for this problem.) 3. Recall that we were able to parametrize the space of lines in the plane with coordinates (θ, p), where the line `(θ, p) satisfied the equation x cos θ + y sin θ = p. Now, consider an arclength parametrized curve α(s) = (α1 (s), α2 (s)) and for each s let `(s) be the tangent line to α at α(s). Then there exist functions θ(s) and p(s) so that `(s) = `(θ(s), p(s)). Give formulas for θ(s) and p(s) in terms of α(s) and α0 (s). 4. Suppose α(s) and β(s) are two arclength parametrized curves satisfying kα(s) − β(s)k = 1 for all s. Prove or give an explicit counterexample: α and β are either a double helix or are plane curves in parallel planes. 1