MTH 234 Exam 1 Oct 12, 2015Name:Section: Recitation Instructor:READ THE FOLLOWING INSTRUCTIONS.• Do not open your exam until told to do so.• No calculators, cell phones or any other electronic devices can be used on this exam.• Clear your desk of everything excepts pens, pencils and erasers.• If you need scratch paper, use the back of the previous page.• Without fully opening the exam, check that you have pages 1 through 9.• Fill in your name, etc. on this first page.• Show all your work. Write your answers clearly! Include enough steps for thegrader to be able to follow your work. Don’t skip limits or equal signs, etc. Includewords to clarify your reasoning.• Do first all of the problems you know how to do immediately. Do not spend toomuch time on any particular problem. Return to difficult problems later.• If you have any questions please raise your hand and a proctor will come to you.• There is no talking allowed during the exam.• You will be given exactly 90 minutes for this exam.I have read and understand the above instructions:. SIGNATUREPage 1 of 9MTH 234 Exam 1 Oct 12, 2015Multiple Choice. Circle the best answer. No work needed. No partial credit available.1. (8 points) Consider the surface given by equation below and answer the questions that follow.3z = x2+y29+ 2 (1)(a) Horizontal traces or cross-sections (i.e., planes parallel to z = k > 1) of the surfacedefined by (1) areA. CirclesB. ParabolasC. EllipsesD. HyperbolasE. None of the above(b) Vertical traces or cross-sections (i.e., planes parallel to either x = k or y = k) of the surfacedefined by (1) areA. CirclesB. ParabolasC. EllipsesD. HyperbolasE. None of the above2. (4 points) The contour plot (level curves) to the right could be from which function?xyA. f(x, y) =xsin yB. f(x, y) = x sin yC. f(x, y) = y sin xD. f(x, y) = x cos yE. f(x, y) =ysin xF. None of the aboveExtra Work Space.Page 2 of 9MTH 234 Exam 1 Oct 12, 2015Fill in the Blanks. No work needed. No partial credit available.3. (10 points) Find the velocity and position functions of a particle that satisfies the following conditions.a(t) = −sin t i + 12e−2tj − 8 kv(0) = i − 6 jr(0) = 3 j + k(a) v(t) =(b) r(t) =4. (8 points) Let u = 3 i + 2 j − k and v = i − 5 j + 2 k. Find each of the following.(a) 2u + 3v =(b) Let θ be the angle between u and v. Then cos θ =5. (6 points) The line segment from (3, 7, 5) to (−2, 4, 1) is given by the parametric equationsx(t) =y(t) =z(t) = , t ∈ [0, 1]Extra Work Space.Page 3 of 9MTH 234 Exam 1 Oct 12, 2015Standard Response Questions. Show all work to receive credit. Please BOX your final answer.6. (12 points) Find the length of the curve below.r(t) =8t3/23i − 4t j + t2k, 0 ≤ t ≤ 17. Let f(x, y) =1p2x2− 1 − y.(a) (8 points) Sketch the domain of f .xy(b) (4 points) Give the range of f. Express your answer using interval notation.Page 4 of 9MTH 234 Exam 1 Oct 12, 20158. (14 points) Find the following limits or show that they do not exist.(a) lim(x,y)→(0,0)x2sin2y2x2+ y2(b) lim(x, y)→(0,0)xy28x2+ 3y49. Let f(x, y) = x2ey− 6y√x.(a) (12 points) Find the linearization of f(x, y) at the point (4, 0).(b) (6 points) Use the linearization from part (a) to approximate f(3.9, 0.2).Page 5 of 9MTH 234 Exam 1 Oct 12, 201510. (12 points) Find the equation of the plane containing the points A, B, and C ifA = (3, 0, 1), B = (1, 2, 2), C = (6, −1, 3)11. (10 points) Let w = xy2+ yex−2where x = x(s, t) and y = y(s, t) are differentiable. Suppose also thatx(1, 3) = 2, xt(1, 3) = −4, xs(1, 3) = 5y(1, 3) = −3, yt(1, 3) = 6, ys(1, 3) = 0Find wt(1, 3) .Page 6 of 9MTH 234 Exam 1 Oct 12, 201512. (10 points) Let g(x, y) = 5x2y3−ln 2yx. Find gxyand gyy.gxy=gyy=13. (12 points) Find the parametric equations for the line tangent to the curver(t) = h2 − ln t, t2+ 3, 1 − 4t3i at the point (2, 4, −3).x(t) =y(t) =z(t) =Page 7 of 9MTH 234 Exam 1 Oct 12, 201514. (14 points) Let f(x, y) = 3xy2− 2y and answer the questions below.(a) Find the equation of the plane tangent to surface z = f (x, y) at the point P (2, 1, 4).(b) Find the distance from the point Q(−2, 4, 5) to the tangent plane from part (a).Page 8 of 9MTH 234 Exam 1 Oct 12, 2015Please have your MSU student ID ready so that is can be checked.When you are completely happy with your work please bring your exam to the front to be handed in.Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED .Congratulations you are now done with the exam!DO NOT WRITE BELOW THIS LINE.Page Points Score2 123 244 245 326 227 228 14Total: 150Page 9 of