MTH 234 Exam 2 November 20, 2017Name:Section: Recitation Instructor:INSTRUCTIONS• Fill in your name, etc. on this first page.• Without fully opening the exam, check that you have pages 1 through 12.• Show all your work on the standard response questions. Write your answers clearly!Include enough steps for the grader to be able to follow your work. Don’t ski p limit s orequal signs, etc. Include words to clarify your reasoning.• Do first all of the problems you know how t o do imme d iately. Do not spend too muchtime on any particular problem. Return to diffic ult problems later.• If you h ave any questions please raise your hand and a proctor will come to you.• You wi ll be given exactly 90 minutes for this exam.• Remove and u tilize the formula sheet provided to you at the end of this exam.ACADEMIC HONESTY• Do not open the exam booklet until you are instructed to do so.• Do not seek or obtain any kind of help f rom anyone to answer ques tions on this exam. Ifyou have questions, cons ult only the proctor(s).• Books, notes, calculators, phones, or any other electronic devices are not allowed on theexam. Students should store them in their backpacks.• No scratch paper is permitt e d . If you need more room use the back of a page.• Anyone who violates these instructi on s will have commit ted an act of academic dishonesty.Penalties for academi c dishonesty can be very severe. All cases of aca demic dishonesty willbe reported immediately to the Dean of Undergraduate Studies and add e d to the student’sacademic record.I have read and understand theabove instructions an d sta tementsregarding academic honesty:. SIGNATURE⋆ Page 1 of 12MTH 234 Exam 2 November 20, 2017Standard Response Questions. Show all work to receive credit. Please BOX your final answer.1. (5 points) Let f (x, y) = x ey−2+ y2ln x. Find ∇f at P (1, 2).2. (9 points) Let g(x, y) = 8x3−12xy + y2. Fi nd and classify each critical point of g as a local minimu m, a localmaximum, or a saddle point.Page 2 of 12MTH 234 Exam 2 November 20, 20173. Consider the integral below and ans wer the ques tions that follow.ˆ10ˆππxsin(y2/π) dy dx(a) (3 points) Sketch the region of integration.(b) (5 points) Evaluate the integral above by reversing the order of integration.4. (6 points) Evaluate the integral below.ˆ30ˆ√9−x2−√9−x21p5 + x2+ y2dy dxPage 3 of 12MTH 234 Exam 2 November 20, 20175. (6 points) Set up but Do Not Eva luate t he ite rated integral for computing the volume of a region D if Dis the right circular cy linder whose base is the disk r = 2 sin θ (in the xy-plane) and whose top lies on thesurface z = 8 − x2.xyzxyzz = 8 − x2r = 2 sin θ6. (8 points) Find th e area of the surface of the cap cut from the parabo loid z = 12 − x2− y2by the conez =px2+ y2.Page 4 of 12MTH 234 Exam 2 November 20, 20177. Let F = 2xy i + (x2− cos z) j + y sin z k and answer the questi o ns below.(a) (7 points) Find a function f so that ∇f = F.(b) (4 points) Let C be any path from (−2, 1, π/2) to (1, 3, π). Evaluate the integralˆC2xy dx + (x2− cos z) dy + y sin z dz(c) (3 points) Calculate c u rl F.Page 5 of 12MTH 234 Exam 2 November 20, 20178. (7 points) Let E be the portion of the bal l x2+ y2+ z2≤ 1 that lies in thefirst octant: x ≥ 0, y ≥ 0, z ≥ 0.Rewrite the triple integral below as an iterated integral in spherical coordinates and evaluate.˚E5y dV9. (7 points) Let F(x, y) =y2, x2. Use Green’s Theorem to calculate the work done by F on a particle movingcounter-clockwise around the triangle bounded by x = 0, y = 0, x + y = 1.Page 6 of 12MTH 234 Exam 2 November 20, 2017Multiple Choice. Circle the best answer. No work needed. No partial credit available.10. (4 points) Let P = P(1/2, 2) and u =1√2i −1√2j. If ∇f (P ) = 3 i + 2 j,then Du(P ) =A.112B.−32√2C.3√2i −√2 jD.1√2E. None of the above11. (4 points) Suppose that the line segm ent C is given by the parametrization: r(t) = (t + 1) i + 2t j, 0 ≤ t ≤ 3.Evaluate the integral below.ˆCx2, −y· drA. 57B. 39C. 21D. 3E. None of the above12. (4 points)ˆ20ˆ4−x20ˆx0sin 2z4 − zdy dz dx =A.sin 84B.−cos 82C.−cos 84D.−cos 42E. None of the abovePage 7 of 12MTH 234 Exam 2 November 20, 201713. (4 points) Pa rameterize the part of the plane x + z = 3 that li e s above the disk (x − 1)2+ y2≤ 1.A. r(s, t) = h s, t, 3 − si, with s ∈ [0, 2 cos t], and t ∈ [−1, 1].B. r(s, t) = hs, t, 3 −si with s ∈ [0, 2] and t ∈ [−1, 1].C. r(s, t) = h s cos t, s sin t, 3 − s cos ti with s ∈ [0, 2] and t ∈ [−π/2, π/2].D. r(s, t) = hs cos t, s sin t, 3 − s cos ti with s ∈ [0, 2 cos t] and t ∈ [−π/2, π/2].E. None of the above14. (4 points) Let F(x, y) = ∇arctanyx=−yx2+ y2i +xx2+ y2j.If C is the unit circl e param eterized by r(t) = cos t i + s in t j, 0 ≤ t ≤ 2π, thenˆCF · dr =A. 0B. πC. 2πD. UndefinedE. None of the above15. (4 points) Let f be a differentiable function of x and y with continuous sec on d order part ial derivatives. LetF = ∇f = M i + N j be a gradient field and let C be the (posi tively oriented) ellipse as s h own in the sketchbelow. Consider the foll owing statem e nts.(a) F is a conservat ive vector field.(b)ffiCM dx + N dy = 0(c) (curl F) · k > 0A. All three statements are true.B. Only (a) and (b) are true.C. Only (a) and (c) are true.D. Only (b) and (c) are true.E. None of the abovePage 8 of 12MTH 234 Exam 2 November 20, 201716. (4 points) Let F =x2y, −yz, z2. Which of t h e following is true?A. curl F = 2xy + z and div F = y − x2B. curl F =y, 0, −x2and div F = 2xy + zC. curl F = y − x2and div F = 2xy + zD. curl F = h2xy, z, 0i and div F =y, 0, −x2E. None of the above17. (4 points) Identify the surface give n by the vector equatio ns r(u, v) = hu, 5 sin 3v, 4 cos 3vi.A. planeB. elliptic paraboloidC. cylinderD. ellipsoidE. None of the above18. (4 points) Which of t h e following vector field plots could be F = xy2i − x j?A. B.C. D.E. None of the abovePage 9 of 12MTH 234 Exam 2 November 20, 2017Please have your MSU student …