math4250homework1

math4250homework1 - Math 4250/6250 Homework #1 This...

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Math 4250/6250 Homework #1 This homework assignment covers DoCarmo 1.1 - 1.5. It accompanies Lectures 1 and 2 in the course notes. Please pick 5 of the following 9 problems. Remember that undergraduate students should average one challenge problem per assignment, while graduate students should average two challenge problems per assignment. 1. R EGULAR P ROBLEMS 1. (Do Carmo, 1-3, #2) A circular disk of radius 1 in the xy plane rolls along the x axis without slipping. The curve described by a point on the rim of the disk is called a cycloid . (1) Find a parametrization α ( t ) of the cycloid. (2) Compute the arclength of the portion of the cycloid corresponding to one complete rotation of the disk. 2. (Do Carmo, 1-3, #4) The curve α ( t ) = ± sin t, cos t + log tan t 2 ² . is called the tractrix. Show that (1) α is a differentiable parametrized curve, regular except at t = π/ 2 . (2) The length of the portion of the tangent line to the tractrix between α ( t ) and the y -axis is always equal to
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math4250homework1 - Math 4250/6250 Homework #1 This...

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