MATH 251 Spring 2014 Final Exam Simon Fraser University 23 April 2014, 7:00-10:00pm Instructor: Weiran Sun Last Name: First Name: SFU ID: @sfu.ca Signature: Instructions Question Maximum 1. Please do not open this booklet until invited to do so. 1 5 2. Write your last name, first name(s) and SFU ID in the box above in block letters, and sign your name in the space provided. 2 5 3 20 4 25 5 20 6 25 7 10 8 10 9 20 10 20 11 20 Total 180 3. This exam contains 9 questions on 9 pages (after this title page). Once the exam begins please check to make sure your exam is complete. 4. The total time available is 180 minutes, and there are 180 points, so allow about a minute per point; for example, you should aim to spend about 10 minutes on a 10-point question. Attempt all problems! 5. Use the reverse side of the previous page if you need more room for your answer, and clearly indicate where the solution continues. 6. Show all your work, and explain your answers clearly, unless it is explicitly stated that no explanation is needed. 7. This is a closed book exam; no books, papers or electronic devices should be within the reach of a student during the examination. No calculators are allowed; cellphones off, please. 8. This exam is administered under the SFU Code of Academic Honesty. During the examination, communicating with, or deliberately exposing written papers to the view of, other examinees is forbidden. 9. Good luck! Score Math 251 : Final Exam 2 1. [5pts] Show that if ~u + ~v and ~u ~v are orthogonal, then the vectors ~u and ~v must have the same length. 2. [5pts] Find the limit, if it exists, or show carefully that the limit does not exist: 3xy(2 cos x) . (x,y)(0,0) x2 + 2y 2 lim Math 251 : Final Exam 3 3. Suppose the surface S is given by 2z + x + 12 = z 3 + (1 + 2x2 + p 4 + y 2 )2 . (a) [10pts] Find the equation of the tangent plane at the point P (1, 0, 2). (b) [10pt] Suppose the point Q(1.01, 0.02, z) is on S. Estimate the value of z. Note that your answer should be close to 2 but not exactly equal to 2. Math 251 : Final Exam 4 4. (a) [5pts] Use a scalar projection to show that the distance D from the point P (x0 , y0 , z0 ) to the plane ax + by + cz = d is given by D= |ax0 + by0 + cz0 d| . a2 + b 2 + c 2 (b) [10pts] Find all the values of a such that the following two lines intersect each other: L1 : (1 t, 2t, t + 2) , L2 : (3 + at, t + 1, 2t a) . Find the corresponding intersection point(s). (c) [10pts] Let a = 1 in L2 in (b). Find the distance between L1 and L2 . Math 251 : Final Exam 5 5. (a) [10pts] If C is the line segment connecting the point (x1 , y1 ) to the point (x2 , y2 ), show that Z x dy y dx = x1 y2 x2 y1 . C (b) [10pts] Use (a) to find the area of the pentagon with vertices (0, 0), (2, 1), (1, 3), (0, 2), and (1, 1). Math 251 : Final Exam 6 6. Consider the vector field given by ~ F(t) = (2xy 1)~i + (x2 + z)~j + y~k ~ is conservative by finding a scalar function f such that F ~ = f . (a) [10pts] Show that F Show your work. (b) [3pts] Suppose ~r(t) is the intersection of two surfaces: y + z = 2 and x2 + y 2 = 1. Find a parametrization of ~r(t). (c) [12pts]Suppose an object moves from P (1, 0, 2) to Q(1, 0, 2) along ~r(t) in (b) under ~ given in (a). Use two methods to find the work done by F. ~ the action of F Math 251 : Final Exam 7 7. [10pts] Suppose f is twice differentiable. We say f (x, y) is harmonic if it satisfies the equation fxx + fyy = 0 . Show that if f is harmonic, then z(x, y) defined by z(x, y) = f (x2 y 2 , 2xy) is also harmonic. 8. [10pts] Find the surface of the part of the sphere x2 + y 2 + z 2 = 4z that lies inside the paraboloid z = x2 + y 2 . Note that the integral will simply to a form that can be evaluated without too much effort if your calculation is correct. Math 251 : Final Exam 8 9. The computation for both parts will not be heavy if you solve the integrals in proper ways. (a) [12pts] Evaluate the integral ZZ D sin(x + y) dA , x+y where D R2 is bounded by x + y = 1, x + y = 2, x-axis, and y-axis. (b) [8pts] Evaluate the integral Z 0 8 Z 2 4 ex dx dy . y 1/3 You need to draw the integration domain. Math 251 : Final Exam 9 10. (a) [12pts] Evaluate the integral ZZZ z dx dy dz , V where V is the region between the spheres x2 + y 2 + z 2 = 2z and x2 + y 2 + z 2 = z. You need to sketch the integration region. Using appropriate coordinates can help. (b) [8pts] Find the volume of the ellipsoid ax2 + by 2 + cz 2 1 , for some a, b, c > 0. The computation will be quick if you apply proper change of variables. Math 251 : Final Exam 10 11. (a) [10pts] Use the method of Lagrange multipliers to show that the plane S that passes through (1, 1, 1) and cuts off the smallest volume in the first octant is given by x + y + z = 3. (b) [10pts] Let D be the solid region bounded by the coordinate planes and plane S in (a) in the first quadrant. Note that D includes its boundaries, that is, the coordinate planes and plane S. Find the largest distance of the points in D to the point (1, 1, 2)