1. Suppose the curves parameterized by r(t) = (1+2 sin t, e", 4t +2) and R(t) = (2t + 1, cos t - t,t +3 sin t+ 2) lie on a surface S. For the tangent plane to S at the point P(1, 1,2), (i) find a parametrization r(u, v), and (ii) find an equation (in x, y and z) for the plane. (Note r(0) = OP = R(0).) ,3t %3D %3D 2. Suppose a surface S is parameterized by r(u, v) = (2uv, u? - v², u? + v?), u + v? < 1. %3D a. Find an equation of the tangent plane to S at the point r (,4) = (4,- 1). b. Find the area of the surface in the given region. c. A sheet in the shape of the given surface has density proportional to the distance to the z- axis that. Find the mass of this sheet. 3. Let F(r, y, z) = (xy+ 3x, 4y- y + z, r2 + 10z + zy). Let S be the surface of the paraboloid z = 16 – r2 - y? for z > 0. Let D be the disk r? + y? = 16, z = 0. %3D %3D a. Evaluate f,F dS with the "upward" orientation. b. Evaluate [[F dS. with the "upward" orientation. You might want to think a bit before you start computing. (2u+3v, 2u-3v, uv), for (u, v) in region D in 4. Suppose a surface S is parameterized by r(u, v) = the uv-plane bounded by the curves v = u', v =8 and u = (0. For S oriented with the unit nor- mal vector pointing generally upward, evaluate JJs F dS, where F(r, y, z) = (r-y, z-y, (r+y)z). 5. Compute the line integral: fF dr, where F(r, y, z) = (r,-, y?- z2), and C consists of the three line segments that bound the portion of the plane z = 6- 3r with orientation of the curve compatible with the portion of the plane being oriented generally "upward". You may want to consider Stokes' Theorem here. y, in the first octant,