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Differential Geometry Problems Reading Assignment: 3.1 (through page 181), 3.2 Suggested Problems: 3.1: 1,3, 7, 15, 17 3.2: 1,5,11, 12 #1. Explain why the path parameterized by x (t ) (t 2 , t 3 ), 1 t 1 is not smooth. #2. Suppose that x is a C 2 path with non-zero velocity and x x 0 for all t. Show that x has constant speed if and only if its velocity and acceleration vectors are always perpendicular to one another. #3. Prove that for a C2 path x :[a, b] 3 , d x (t ) v (t ) x (t ) a (t ) . dt #4. Find the curvature of each path at the given point. a. x (t ) (t 2 , t 3 ), 1 t 1 1 1 p , 4 8 b. x (t ) (et cos 2t , et sin 2t ), t p e 2 ,0