math200_december2002_2

math200_december2002_2 - .Be sure that this examination has...

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Unformatted text preview: .Be sure that this examination has 16 pages including this cover The University of British Columbia Final Examinations - December, 2002 Mathematics 200' THE UNlVERSlTY OF BRlTlSH COLUMBIA' St clCotulgieselling Services . u en velopment & Se ' All 5 rwces ( “W‘s . 1200-1874 East Mall 7 .-~ Time: 2.5 hours - _ - p r __Vancower. Bic. V6T 121; Student Number _____—— Instructor’s Name I Section Number Special Instructions: Students are allowed to bring into the exam an 8%x11” sheet of paper filled on both sides No calculators are allowed. ' Rules governing examinations: 1. Each candidate should be prepared to produce his/her library/AMS card upon request. 2. Read and observe the following rules: No candidate shall be permitted to enter the examination room after _ _ the expiration of one half hour, or to leave during the first half hour of the examination. », ' Candidates are not permitted to ask questions of the invigilators, fexcept in casesof supposed errors or ambiguities in examination qUestions. '_ . CAUTION - Candidates guilty of any of the following or similarpractices - shall be immediately dismissed from the examination and shall be liable to disciplinary action. . (a) Making use of any books, papers or memoranda other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (C) Purposer exposing written papers to the View of other candidates. The plea of accident or forgetfulness shall not be received. 3 Smoking is not permitted during examinations. Page 2 of 16 1. The position of a particle at time t (measured in seconds .9) is given by 7ft . . 7rt . r(t) = tcos(3)1 + tsm(-2-)_] + tk (a) Show that the path of the particle lies on the cone 22 = + ' (b) Find the velocity vector and the speed at time t. (c) Suppose that at time t = ls the particle flies off the path on a line L in the direction tangent to the path. Find the equation of the line L. - (d) How long does it take for the particle to hit the plane a: = —1 after it started'moving along the straight line L? . . [11 marks] Page 3 of 16 2. (a) Let f be an arbitrary differenfible function defined on the entire real line. Show that the function 111 defined on the entire plane as _ . Many) = e‘yf(=v - 1/) satisfies the partial differential equation: wi—§21+-QE'-O 8:1: 81; _ The equations a: = u3 — 3uv2, and y = 3232; —- v3 and z = u2 —— v2 definez as a function ofx and . (b) , 1/ Determine 53—: at the point (11., v) = (2,1) which corresponds to the point (at, y) = (2, 11). [15 marks] Page 5 of 16 3, You are Standing at a lone palm tree in the middle of the Exponential Desert. The height of the sand dunes around you is given in meters by I _ h(2:, y) = 100€-(m2+2y2) - ' where :v represents the number of meters eaSt of thepalm tree (west if a: is negative), and y represents the number of meters north of the palm tree (south if y is negative). , . (3.) Suppose you walk 3 meters east and 2 meters north. At your new location, (3,2), in what direction is the sand dune sloping most steeply downward? (b) Ifyou walk north from the location described in part (a), what is the instantaneous rate of change .of height . of the sand dune? (c) If you are standing at-(3,2) in what direction should you walk to ensure that you remain at the same height? (d) Find the equation of the curve through (3,2) that should you move along in order that you are always ' pointing in a steepest descent direction at each point of this curve. [12 marks] , Page 7 of 16 4. Find all the critical points of the function - _ flay) =$4+y4—.4:z:y defined in thg m—y plane. Classify each critical point as a local minimum, maxnn' um, 01' saddle point. Explain your reasoning. [12 maiks] Page 8 of l6_ 5. (a) By the points of tangency determine the values of c for Which x + y + z = c is a tangent plane to the surface 4:.'z:2+4yz+z2 = 96. _ _ ' (1)) Use the method of Lagrange Multipliers todetermine the absolute maximqu and minimum values of the function f(z? y, z) = :L’ + .y + 2 along the surface 9(2, y, 2) = 4:1:2 + 4112+ 22 = 96. (c) Why do you get the same ansWers in (a) and (b)? j [12 marks] Page 9 of 16 Question 5 (continued) Page 10 of 16 '. Exialuate the following ifitegral: [8 marks] Page 11 of 16 7. Let D be the region in the zy-plane which is inside the circle 2'2 + (y — 1)2 = 1 but outside the circle 2:2 + y2 = 2. Determine the mass of this region if the density is given by:’ 2 May): 2 . :1: +y [15' marks} Page 13 of 16 8. Evaluate ff z W, whefe E is the region boimded by the planes y = O, z = 0, '2: + y = 2 and the cylinder E ~ , y2+22=1infimfimummm..* , ‘ x I I HSUEflE] ...
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math200_december2002_2 - .Be sure that this examination has...

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