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The University of British Columbia Final Examination - April 16, 2011 Mathematics 200 Closed book examination Time: 2.5 hours Last Name: , First: Signature Student Number Special Instructions:
- No books, notes or calculators are allowed. Rules governing examinations
- UBCcard for identification. Each candidate must be prepared to produce, upon request, a
- except in cases of supposed errors or ambiguities in examination Candidates are not permitted to ask questions of the invigilators, questions. • No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled startingtime, or to leave during the first half hour of the examination.
- est practices shall be immediately dismissed from the examination Candidates suspected of any of the following, or similar, dishon- and shall be liable to disciplinary action.(a) Having at the place of writing any books, papers or memoranda,ers/recorders/transmitters (including telephones), or other mem- calculators, computers, sound or image play- ory aid devices, other than those authorized by the examiners.(b) Speaking or communicating with other candidates. didates or imaging devices. The plea of accident or forgetfulness(c) Purposely exposing written papers to the view of other can- shall not be received. • Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take anyexamination material from the examination room without permis- sion of the invigilator. • Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.
Total 100
Page 1 of 18 pages
[10] x + 2 1.y +Let z = 3. A = (2, 3 , 4) and let L be the line given by the equations x + y = 1 and
(a) Write a vector equation for L. (b) Write an equation for the plane containing A and perpendicular to L. (c) Write an equation for the plane containing A and L.
[10] 2. According to van der Waal’s equation, a gas satisfies the equation (pV 2 + 16)(V − 1) = T V 2 , wherenow at pressure 1, volume 2 and temperature 5. Find the approximate change in its volume p, V and T denote pressure, volume and temperature respectively. Suppose the gas is if p is increased by .2 and T is increased by .3.
[14] 4. Let f (x, y, z) = (2x + y)e−(x^2 +y^2 +z^2 ), g(x, y, z) = xz + y^2 + yz + z^2. (a) Find the gradients of f and g at (0,1,-1). (b) A bird at (0,1,-1) flies at speed 6 in the direction in whichrapidly. As it passes through (0,1,-1), how quickly does g(x, y, z) appear (to the bird) to be f (x, y, z) increases most changing? (c) A bat at (0,1,-1) flies in the direction in whichbut z increases. Find a vector in this direction. f (x, y, z) and g(x, y, z) do not change,
[16] 5. Let h(x, y) = y(4 − x^2 − y^2 ). (a) Find and classify the critical points ofpoints. h(x, y) as local maxima, local minima or saddle
(b) Find the maximum and minimum values of h(x, y) on the disk x^2 + y^2 ≤ 1. (c) Find the maximum value of f (x, y, z) = xyz on the ellipsoid g(x, y, z) = x^2 + xy + y^2 + 3z^2 = 9. Specify all points at which this maximum value occurs.
[14] 6. Consider J =
0
∫ √ 4 −y 2 y
y xex (^2) +y (^2) dx dy.
(a) Sketch the region of integration. (b) Reverse the order of integration. (c) Evaluate J by using polar coordinates.
[14] 8. Let E be the ”ice cream cone” x^2 + y^2 + z^2 ≤ 1, x^2 + y^2 ≤ z^2 , z ≥ 0. Consider J =
E
√x (^2) + y (^2) + z (^2) dV.
(a) Write J as an iterated integral, with limits, in cylindrical coordinates. (b) Write J as an iterated integral, with limits, in spherical coordinates. (c) Evaluate J.
