2. (5 points) Find the parametric equation of the tangent line to the curve FM = (1/F 3, ln(t2 + 3), t) at (2,1n4,1) 3. (4 points) Find the velocity and acceleration at t = 2 of the object with the position function F(t) =< 2="" sin="" t,="" 4t2="" +="" t,="" 5et2="">. Answer: < 2="" cos="" 2,="" 17,="" 20e4="">, < —2="" sin="" 2,="" 8,="" 90e4=""> 4. (6 points) Given an acceleration 5(t) =< 1,="" t,="" 4t="">, initial velocity < 20,0,0=""> and initial position < 0,="" 0,="" 0="">, find the velocity and position vectors for t > 0. Answer: (t + 20, g, 2t2), (q+ 20t, it3) 5. (4 points) Find the arc length: (t) .< et,="" e-t,l/lt="">, for 0 < x="">< 1n3.="" answer:="" 6.="" (4="" points)="" find="" the="" length="" of="" the="" complete="" cardioid="" r="4" +="" 4="" sin="" 0.="" answer:="" 32="" 7.="" (5="" points)="" find="" the="" unit="" tangent="" vector="" t="" and="" the="" curvature="" k="" for="" f(t)="">< 4+t2,="" t,="" 0="">. Answer: 2 (4t2+1)3/2 8. (20 points) Find the limit, if exists, or show that the limit does not exist. (a) lim X4 — x2y6 — 4x. Answer: — (x,y)—,(1,-1) x5 + xy4 + 3 x3+ X2y + x2+xy+x+y 3 (b) (x,y)-,( lim . Answer: 1,-1) x2y + xy2 + 3x + 3y x2y2 (c) (x,y1)i—,(o,o) x2 + 2y3' Answer: 0 sin (x2 + y2) (d) lira Answer: 0 (x,y)-(0.o) ?s2 + y2 X4y (e) (x,yli)—,m(0,0) 5y2 Answer: does not exist (f) fim(0,0,0) 3x2 x2y + /i 4y2 — 3 x . Answer: Does not exist (x,y,z)—> _z_ 4