Sample Questions for Final Exam, MATH 2A03/2X03, Spring 2014 Ruipeng Shen June 13, 2014 The duration of test is 180 minutes. Total marks = 50. Thus you have an average of 7 minutes to spend on a 2-mark problem. Attention The questions marked with “?” are expected to be solved via Green’s theorem or Stokes’ theorem. These materials will be covered by Tuesday’s class. You may choose to solve them after the last lecture. 1. Consider the function f (x, y, z) = x2 + y 2 + z 2 + exy . [1] (a) Find the gradient ∇f of the function f . [2] (b) Find the Hessian Matrix of the function f . [1] (c) Calculate div (∇f ). [1] (d) Evaluate the limit lim f (x, y, z). (x,y,z)→(0,0,1) RR 2. Consider the double integral D xdA. Here D is the region bounded by the curves x2 + y 2 = 1, x2 + y 2 = 4, y = x and y = 0 in the first quadrant. [1] (a) Sketch the region D in a coordinate system. [3] (b) Evaluate the double integral ZZ (x + y)dA. D 1 3. Let D be the region bounded by the lines x − y = 1, x − y = 5 and the coordinate axes, as shown in figure 1. [3] (a) Evaluate the double integral ZZ x+y sin dA. x−y D [1] (b)? Determine the positive orientation of ∂D and mark it in figure 1. R [2] (c)? Evaluate the line integral ∂D xdy. y -1 x x-y=1 D x-y=5 -5 Figure 1: The region D 4. Consider the solid W bounded by the surfaces z = x2 + y 2 and z = 8 − x2 − y 2 [2] (a) Find the volume of the solid W . [3] (b) Find the triple integral ZZZ (x2 + y 2 )2 dV. W [2] (c) Find the flux of the vector field F = ey i + zk out of the solid W . 2 5. Let S be a surface as shown in figure 2. The blue and red shapes show the positive and negative sides of the surface, respectively. [1] (a)? Mark the positive orientation of the boundary ∂S of the surface S in the figure. [2] (b) Suppose that r(u, v) = (u sin v, v, u cos v), (u, v) ∈ [−1, 1]×[0, π] is a parametrization of the surface S, Calculate the tangent vectors Tu = ∂r/∂u, Tv = ∂r/∂v and the normal vector N(u, v) = Tu × Tv . [1] (c) Determine whether the parametrization r is orientation-preserving or orientationreversing. RR √ [2] (d) Calculate the surface integral S x2 + z 2 dS. z y x Figure 2: The Surface S 6. Let S be the ellipsoid x2 /9 + y 2 /25 + z 2 /25 = 1. [2] (a) Give a parametrization of S. [3] (b) Calculate the surface integral ZZ (x2 i + zk) · dS. S [2] (c) Find an equation of the tangent plane to S at the point (0, 3, 4). 3 7. Consider the function f (x, y) = x2 + xy + y 2 defined in the disk D = {(x, y)|x2 + y 2 ≤ 1}. [3] (a) Find the absolute extreme values of the function f . [2] (b) Find an upper bound of the double integral ZZ (x2 + xy + y 2 )1/3 dA. D 8. Let c(t) = (et cos t, et sin t) with t ∈ [0, 2π] be a spiral . [2] (a) Calculate the length of the spiral. [2] (c) Calculate the following path integral Z p cos( x2 + y 2 )ds. c 4 9. [1 mark each] Determine whether the following statements are true or false. Simply give your answer in one word. No argument, reasoning or proof is required. (a) The tangent planes of the cone x2 + y 2 = z 2 always contain the origin. (b) The line integral (c) The integral RR S R c [(e x + y + z 2 )i + xj + (2xz + z 2 )k] · ds is independent of path in R3 . x2 dS over any surface S is positive. (d) The parametrization r(u, v) = (u + v, u + v, u + sin v), (u, v) ∈ [−1, 1] × [−1, 1] is a smooth parametrization. (e) The function f (x, y) = |xy| is differentiable at the origin. (f)? The curl of a gradient vector field is still a gradient vector field. 5