Compute the curvature of the following arclength-parametrized curves:

Referencing Styles : APA 1. Compute the curvature of the following arclength-parametrized curves:Ë› . Calculate the unit tangent vector, principal normal, and curvature of the following curves: a. a circle of radius a: Ë›.t / D .a cost; a sin t / b. Ë›.t / D .t; cosh t / c. Ë›.t / D .cos3 t;sin3 t /, t 2 .0; =2/ Calculate the Frenet apparatus (T, , N, B, and ) of the following curves: Prove that the curvature of the plane curve y D f .x/ is given by  D jf 00j .1 C f 02/ 3=2 Use Proposition 2.2 and the second parametrization of the tractrix given in Example 2 of Section 1 to recompute the curvature. 6. By differentiating the equation B D T N, deerive the equationn B 0]7. Suppose Ë› is an arclength-parametrized space curve with the property that kË›.s/k  kË›.s0/k D R for all s sufficiently close to s0. Prove that .s0/  1=R. (Hint: Consider the function f .s/ D kË›.s/k 2 . What do you know about f 00.s0/?) 8. Let Ë› be a regular (arclength-parametrized) curve with nonzero curvature. The normal line to Ë› at Ë›.s/ is the line through Ë›.s/ with direction vector N.s/. Suppose all the normal lines to Ë› pass through a fixed point. What can you say about the curve? 9. a. Prove that if all the normal planes of a curve pass through a particular point, then the curve lies on a sphere. (Hint: Apply Lemma 2.1.) *b. Prove that if all the osculating planes of a curve pass through a particular point, then the curve is planar. 10. Prove that if  D 0 and  D 0 are nonzero constants, then the curve is a (right) circular helix.

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