Math 317 Final Exam April 18th, 2015 Final Exam Math 317 April 18th, 2015 Last Name: First Name: Student # : Instructor s Name : Instructions: No memory aids allowed. No calculators allowed. No communication devices allowed. Use the space provided on the exam. If you use the back of a page, write see back on the front of the page. This exam is 180 minutes long. Rules governing examinations Each examination candidate must be prepared to produce, upon the request of the invigilator or examiner, his or her UBCcard for identification. Candidates are not permitted to ask questions of the examiners or invigilators, except in cases of supposed errors or ambiguities in examination questions, illegible or missing material, or the like. No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination. Should the examination run forty-five (45) minutes or less, no candidate shall be permitted to enter the examination room once the examination has begun. Candidates must conduct themselves honestly and in accordance with established rules for a given examination, which will be articulated by the examiner or invigilator prior to the examination commencing. Should dishonest behaviour be observed by the examiner(s) or invigilator(s), pleas of accident or forgetfulness shall not be received. Candidates suspected of any of the following, or any other similar practices, may be immediately dismissed from the examination by the examiner/invigilator, and may be subject to disciplinary action: (a) speaking or communicating with other candidates, unless otherwise authorized; (b) purposely exposing written papers to the view of other candidates or imaging devices; (c) purposely viewing the written papers of other candidates; (d) using or having visible at the place of writing any books, papers or other memory aid devices other than those authorized by the examiner(s); and, (e) using or operating electronic devices including but not limited to telephones, calculators, computers, or similar devices other than those authorized by the examiner(s) (electronic devices other than those authorized by the examiner(s) must be completely powered down if present at the place of writing). Candidates must not destroy or damage any examination material, must hand in all examination papers, and must not take any examination material from the examination room without permission of the examiner or invigilator. Notwithstanding the above, for any mode of examination that does not fall into the traditional, paper-based method, examination candidates shall adhere to any special rules for conduct as established and articulated by the examiner. Candidates must follow any additional examination rules or directions communicated by the examiner(s) or invigilator(s). Question Points Score 1 14 2 13 3 7 4 10 5 8 6 8 7 12 8 8 Total: 80
Math 317 Final Exam, Page 2 of 16 April 18th, 2015 ( ) r 1. (a) 3 points Compute and simplify div for r = x, y, z and r = x, y, z. r Express your answer in terms of r. (b) 2 points Compute curl (yz i + 2xz ) j + e xy k.
Math 317 Final Exam, Page 3 of 16 April 18th, 2015 (c) 3 points Find the length of the curve r (t) = 1, t2 2, t3 for 0 t 1. 3 (d) 3 points Find the principal unit normal vector to r (t) = cos(t) i + sin(t) j + t k at t = π/4. (e) 3 points Find the curvature of r (t) = cos(t) i + sin(t) j + t k at t = π/4.
Math 317 Final Exam, Page 4 of 16 April 18th, 2015 2. Let S be the surface obtained by revolving the curve z = e y, 0 y 1 around the y-axis where the orientation of S is where N points toward the y-axis. (a) 3 points Draw a picture of S and find a parameterization of S. (b) 5 points Compute the integral S ey ds.
Math 317 Final Exam, Page 5 of 16 April 18th, 2015 (c) 5 points Compute the flux integral S F d S where F = x, 0, z.
Math 317 Final Exam, Page 6 of 16 April 18th, 2015 3. Let C be the parameterized curve given by r (t) = cos t, sin t, t, 0 t π 2 and let F = e yz, xze yz + ze y, xye yz + e y. (a) 3 points Compute and simplify curl F. (b) 4 points Compute the work integral C F d r.
Math 317 Final Exam, Page 7 of 16 April 18th, 2015 4. (a) 4 points Use Green s Theorem to evaluate the line integral C y x 2 + y dx + x 2 x 2 + y dy, 2 where C is the arc of the parabola y = 1 4 x2 + 1 from ( 2, 2) to (2, 2). Hint: Green s theorem must be applied to a closed curve; note that the curve C is not closed. You dt may use the fact that 1 + t = arctan(t). 2
Math 317 Final Exam, Page 8 of 16 April 18th, 2015 (b) 4 points Use Green s Theorem to evaluate the line integral C y x 2 + y dx + x 2 x 2 + y dy, 2 where C is the arc of the parabola y = x 2 2 from ( 2, 2) to (2, 2). Hint: Consider carefully the point (0, 0) in your analysis of the situation.
Math 317 Final Exam, Page 9 of 16 April 18th, 2015 (c) 2 points Is the vector field F = y x 2 + y x i + 2 x 2 + y j 2 conservative? Provide a reason for your answer based on your answers to the previous parts of this question.
Math 317 Final Exam, Page 10 of 16 April 18th, 2015 5. 8 points Consider the curve C that is the intersection of the plane z = x + 4 and the cylinder x 2 + y 2 = 4, and suppose C is oriented so that it is traversed clockwise as seen from above. Let F(x, y, z) = x 3 + 2y, sin(y) + z, x + sin(z 2 ). Use Stokes Theorem to evaluate the line integral C F d r.
Math 317 Final Exam, Page 11 of 16 April 18th, 2015 6. Let E be the solid region between the plane z = 4 and the paraboloid z = x 2 + y 2. Let F = ( 1 3 x3 + e z2 ) i + ( 1 3 y3 + x tan z) j + 4z k. (a) 4 points Compute the flux of F outward through the boundary of E.
Math 317 Final Exam, Page 12 of 16 April 18th, 2015 (b) 4 points Let S be the part of the paraboloid z = x 2 + y 2 lying below the z = 4 plane oriented so that N has a positive k component. Compute the flux of F through S.
Math 317 Final Exam, Page 13 of 16 April 18th, 2015 7. A particle moves along a curve with position vector given by r (t) = t + 2, 1 t, t 2 /2 for < t <. (a) 1 point Find the velocity as a function of t. (b) 1 point Find the speed as a function of t. (c) 1 point Find the acceleration as a function of t. (d) 3 points Find the curvature as a function of t.
Math 317 Final Exam, Page 14 of 16 April 18th, 2015 (e) 3 points Recall that the decomposition of the acceleration into tangential and normal components is given by the formula ( ) 2 r (t) = d2 s T(t) dt ds + κ(t) N(t). 2 dt Use this formula and your answers to the previous parts of this question to find N(t), the principal unit normal vector as a function of t. (f) 2 points Find an equation for the osculating plane at the point corresponding to t = 0. (g) 1 point Find the centre of the osculating circle at the point corresponding to t = 0.
Math 317 Final Exam, Page 15 of 16 April 18th, 2015 8. 8 points Consider the following surfaces S 1 is the hemisphere given by the equation x 2 + y 2 + z 2 = 4 with z 0. S 2 is the cylinder given by the equation x 2 + y 2 = 1. S 3 is the cone given by the equation z = x 2 + y 2. Consider the following parameterizations: A. r (θ, φ) = 4 cos θ sin φ, 4 sin θ sin φ, 4 cos φ, 0 θ 2π, 0 φ π/6 B. r (θ, φ) = 4 cos θ sin φ, 4 sin θ sin φ, 4 cos φ, 0 θ 2π, 0 φ π/4 C. r (θ, φ) = 4 cos θ sin φ, 4 sin θ sin φ, 4 cos φ, 0 θ 2π, 0 φ π/3 D. r (θ, z) = 4 z 2 cos θ, 4 z 2 sin θ, z, 0 θ 2π, 1 z 2. E. r (θ, z) = 4 z 2 cos θ, 4 z 2 sin θ, z, 0 θ 2π, 2 z 2. F. r (θ, z) = 4 z 2 cos θ, 4 z 2 sin θ, z, 0 θ 2π, 3 z 2. G. r (θ, z) = z cos θ, z sin θ, z, 0 θ 2π, 0 z 1. H. r (θ, z) = z cos θ, z sin θ, z, 0 θ 2π, 0 z 2. I. r (θ, z) = z cos θ, z sin θ, z, 0 θ 2π, 0 z 3. J. r (x, y) = x, y, x 2 + y 2, x 2 + y 2 1 K. r (x, y) = x, y, x 2 + y 2, x 2 + y 2 2 L. r (x, y) = x, y, x 2 + y 2, x 2 + y 2 2 For each of the following, choose from above all of the valid parameterization of each of the given surfaces. Note that there may be one or more valid parameterization for each surface, and not necessarily all of the above parameterizations will be used. 1. The part of S 1 contained inside S 2 : 2. The part of S 1 contained inside S 3 : 3. The part of S 3 contained inside S 2 : 4. The part of S 3 contained inside S 1 :
Math 317 Final Exam, Page 16 of 16 April 18th, 2015 Extra page for work.