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math4250exam1_2012

# math4250exam1_2012 - Spring 2012 Math 4250/6250 Exam#1 This...

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Spring 2012 Math 4250/6250 Exam #1 This take-home exam covers the material on curves (class notes #1 through #6 and DoCarmo Chapter #1). Undergraduates must pick four questions and may substitute one or more graduate exam problems for undergraduate exam problems for bonus credit. Graduate students must pick one undergraduate problem and two graduate problems and may do all three graduate problems for bonus credit. You are permitted to use You are not permitted to use Maple (or Mathematica or MATLAB) The internet A calculator (or graphing calculator) DoCarmo Other books Your notes Other people’s notes Your brain Other people’s brains Class notes posted on the website Undergraduate Exam Problems 1. Find the (unsigned) curvature κ ( t ) of the parametrized plane curve given below in polar coor- dinates: - 2 2 4 6 - 4 - 2 2 4 α ( t ) = ( r ( t ) , θ ( t )) = (4 cos 3 t + 2 , t ) 2. Find the curvature and torsion of the curve given by α ( t ) = (1 + cos t, sin t, 2 sin t 2 ) . 3. We learned to write the set of lines in the plane using the coordinates ( θ, p ) where ( θ, p ) is the line which satisfies the equation: (cos θ ) x + (sin θ ) y = p, Suppose that α ( s ) = ( α 1 ( s ) , α 2 ( s )) is an arclength-parametrized plane curve, and that ( s )

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