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2. We turn to some material from an exam from http://www.math.umd.edu/ raw/ courses/436/436.html. (a) Let :[0,L]R2 be a unit speed closed plane curve. Show that, for any t1[0,L], there exists t2[0,L] such that (t1)=(t2). (Hint: Recall the expression (t)= (cos(t),sin(t)). (b) Consider the curve :RR3 given by (t)=(etcost,etsint,et). Compute the curvature and torsion , along with the Frenet frame {T,N,B}. (c) Let :(a,b)R3 is a unit speed curve with (t)>0 and (t)=0 for every t(a,b). Suppose that lies on a sphere of some radius R. Show that dtd(2)=. Hint: u2=R2 for some fixed vector u. Now differentiate! 2. We turn to some material from an exam from http://www.math.umd.edu/ raw/ courses/436/436.html. (a) Let :[0,L]R2 be a unit speed closed plane curve. Show that, for any t1[0,L], there exists t2[0,L] such that (t1)=(t2). (Hint: Recall the expression (t)= (cos(t),sin(t)). (b) Consider the curve :RR3 given by (t)=(etcost,etsint,et). Compute the curvature and torsion , along with the Frenet frame {T,N,B}. (c) Let :(a,b)R3 is a unit speed curve with (t)>0 and (t)=0 for every t(a,b). Suppose that lies on a sphere of some radius R. Show that dtd(2)=. Hint: u2=R2 for some fixed vector u. Now differentiate