2. (Sections 2.12,3.2,7.5,7.8) Find all points on the surface 2 = r' -ry + y' -2x + 4y at which the tangent plane is horizontal. [5] 3. (Section 7.9 and Chapter 8) Consider the 3-dimensional vector field F defined by E(x, y, z) = (3x yz, o'z, ry + 2). (a) Write down the Jacobian matrix JE(x, y, 2). 2) (b) Determine divE(I, y, 2). (c) Determine curlE(x, y, =). (d) Why does F have a potential function? Give reasons for your answer, referring to the relevant definitions and theorems in the study guide. (2) (e) Find a potential function of E. (6)4. (Chapter 9) Consider the R3 - R function defined by f(x, y) = e' In(1 + x). Find the third order Taylor polynomial about the point (0, 0). [10] 5. (Section 10.2) Consider the R2 - R function defined by f(x, y) = x3 +y' -3x - 3y. Find the critical points of f and their nature. [10] 6. (Section 10.3) Using the Method of Lagrange, find the point on the plane a + 2y + 2 = 1 that is closest to the origin. [6]