Quiz (Tangent Vectors, Arc Length, and Curvature)

Show all work and justify each step.

(1) Find the unit tangent vector for which at the point where

(2) Given find so that



(3) Given find so that



(4) Find the length of the given curve described by the vector function



over the interval

(5) Determine so that the length of the given curve described by the vector function



has length over the interval

(6) Express the vector function in terms of the arc length parameter measured from the point where in the direction of increasing

(7) Determine so that the vector function written in terms of the arc length parameter measured form the point where in the direction of increasing is

(8) Given the vector function and find the unit tangent vector and the curvature at

(9) Find the point(s) where the ellipse has maximum curvature.

(10) If is a smooth curve in described by the parametric equations and then the curvature is given by



Use this formula to find the curvature of the curve described by the parametric equations   and

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