رو V レス Question 4 dx2 0=30° 11 tan O tan 30° COS =2. Find the radius of curvature [15] (a) Given a = 1 + sin 0, y = sin 0-cos 20, d²y dx2 show that and the coordinates of the centre of curvature for the point on this curve where 0 = 30°. (b) Find the range of values of x for which the following series is absolutely convergent X x2 2:3 24 x5 2 x 5 + 3 x 52 4 x 53 + 5 x 54 + 6 x 55 Question 5 (a) If V = In(x² + y²). Prove that -743²)-2x( (x²+y2)2 a²v av + = 0. მე2 Jy2 [10] [15] 2. Jy 27 Jx (b) Find the sum of the series 242-202 2(x²+2)-2y(24) 2x2 x²-2y2 (x²+y2)² 8 1+2" n=1 3n-1 Jaz 4 буг [10] - 2y²-2x² 2x²=2yz 2 +-

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Which of the following statements are true (circle all that apply). (a) An arc-length parameterization gives a paramaterization for which the curve travels with constant speed 1. (b) The curvature at a point on a curve doesn't depend on the choice of parameterization. (c) The unit tangent vector at a point on a curve doesn't depend on the choice of parameterizaion.

A highway has an exit ramp that begins at the origin of a coordinate system and follows the curve y = (1/ 32)x5/2 to the point (4, 1) (see figure). Then it follows a circular path whose curvature is that given by the curve at (4, 1). What is the radius of the circular arc? Explain why the curve and the circular arc should have the same curvature at (4, 1).