(6) 1. Sketch and identify (a) x=tcost,y=tsint, z=5 zz+y2=z2 (6) 2. Write an equation in the indicated coordinate system. Also write a parametric equation and a vector equation for each of the following equation. (i) x2 + y2 = 25 in R3; cylindrical coordinates, spherical coordinates (ii) x2 + y2 = 100 in R2; polar coordinates (3) 3. x = t cost, y = taint is a parametric equation of a curve in R2. Write a polar equation of this curve. Answer: r = (3) 4. Find the length of the polar curve r = 6<, where="" 0="">< 9="">< 2r.="" (3)="" 5.="" find="" the="" length="" of="" the="" curve="" qt)="">< t,8sin="" t,="" 8cost="">, for 0 < t="">< 4r.="" answer:="" 4%,/r5tr="" (3)="" 6.="" find="" the="" curvature="" of="" rat)="">< t,="" t2,="" t3=""> at the point (1,1,1). Answer. (3) 7. Evaluate g'4 (cos 2ti + t23 + e-'k)dt. Answer: (1, 4, 1 - c-*/1) (5) 8. Find an equation of the tangent line to the curve f(t) = cos ti + c-' sin t3 + at (1,0,1). Also find the unit tangent vector f(t) at this point. Answer: F(t) 1— t, t, 1 — t > (6) 9. Find and sketch the domain of the function (x, y) = y2. (30) 10. Find the limit, if exists, or show that the limit does not exist.
+y. - 16y4 (a) lim Answer: (b) lim —. Answer: 32 04).4(1D) LT' (..0.(2.1) x - 2y + y3 x0 (c) (.. v).( lim0.0) 3x2 + y2' Answer: 0 (d) (..„..)illoomo x2 + 2y2 + 3y z • Answer: DNE (e) lim (x2 + y2) sin 1 ( ) Answer: 0