sion at the po is a measure of l (Optional) Torsion of a 4. Given the vector-valued function for curve C as r(t) = (3t2, 8et, 2t), answer the following. (a) Provide an arc length reparametrization of the curve measured from the point (0, 8,0) moving in the direction of increasing t. (b) Determine the curvature of the function r(t) at a general point (i.e. leave in terms of t). (c) Determine the curvature at the point (0, 8, 0). (d) Determine the normal vector at the point (0, 8, 0). la)g t 1 36 + 2 + 6 4 e 2 t + 9 ft ( 9 - 6 4 0 4 ) In ( 24 5 9 1 2 416ez( + 1 + 721 + 6 9 0 2 ) + (2* + 240 7 / 1 9 x 2 + 16 @ 2 x + + 37 ( 6 4 ez - 9 ) In ( 6 4 e2 + 24 1 - 2 4 ez 37 (b) ,' ( t) = 16t, Be#, 2) (" its = (6, 8 et, 0) " '( 6 ) x 1 " ( t j - ijk 6t Bet 2 = 1 ( 1 6 et ) 2 + 122 + ( 48 ( tet - ( 2 ) ) z 16 8 eto K - 1 16 ert + 14 4 + 482 (tet - t ) 7 (36 + 2 + 6le zt + 8/ 3/2 ( ( ) Klo) = 716 + 1 4 4 + ( 48 ) 2 12464 6 8 3/ 2 68312 ( d ) 1 " (0) = ( 6, Be" , 0) = >/lr " (0)ll = $62 +82 + 02 : 9100 = 10 N (O ) = 1 ( 0) . ( 6, 8, 0 ) 115" (0)11 15, 0 ) Answers: ( a ) clt)= (9-64 e# ) In (2459+2 +16 ext +1+ 72+ +64@2)+(27t+24@2)19x2+bez+1 +(64ez-9) In(64e+24)-24e 37 (6 ) K = 1 162 2 4 + 194 + 482 ( tet - 1 ) ? 12464 (36 (2 + 64 (2* + 9) 3/2 (C) 683/2 (2) ( 3 , 5, 0 )5. (Optional) Torsion of a curve is dened as T ||r'(t) >