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Fall10Final1 - Math 1920 Final December 14 2010 2:00 PM to...

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Unformatted text preview: Math 1920, Final December 14; 2010, 2:00 PM to 4:30 PM You are NOT allowed calculators, the tezztt or any other book or notes. SHOW ALL W ORK! Writing clearly and legibly will improve your chances of receiving the maximum credit that your solution deserves. Please label the questions in your answer booklet clearly. There are 8 questions {3 on the End page) in total. Write your name and section number on each booklet you use. You may leave when you have finished but if you have not handed in your eram booklet and left the room by 4: 15pm, please remain in your seat so as not to disturb others who are still working. 1. (13 pts) Find the equation of the line tangent to the curve of intersection of the paraboloid z = 3:2 + y2 and the ellipsoid r2 + 2y2 + 22 = 7 at (1, 1, 2). 2. (20 pts) A vegetarian restaurant is'planning its daily menu. If it produces 1? pounds of hamburgers, y pounds of french fries, and 3 pounds of nuggets, then its profit in dollars will be P(m, y, z) = xgyz. The kitchen can convert any quantity of tofu to an equal weight of either hamburgers, french fries, or nuggets. If the 8 pounds of tofu available is all converted, i.e. :3 + y + z = 8, then what values of (at, y, 3) will maximize the profit P(:r,y, z)? 3. (20 pts) Find the volume and2 centre of mass (assume const2ant density 6) of the solid inside the sphere 11:2 + y2 + 22 — —a2 , outside the cylinder $2 + y2 — —b2 and above the plane 2— -— 0, where 0 < b < a are constants. 4. Consider the portion of the surface .3 = ’4 — 11:2 — y2 in the first octant. ' (a) (5 pts) Determine the parametric form of the surface in terms of the cylindrical polar coordinates r and 9 (b) (15 pts) Calculate the area of the surface using this pararneterization. 5. (12 pts) Write ‘True’ or ‘False’ in full. Give a brief reason for your answer. (a) If the domain of a vector field F iis3 and V X F is zero, then F is a conservative vector field. (b) If the divergence V - F is zero, then the outward flux of F over any oriented closed surface is zero. (c) If C is a closed curve given as a level curve of a function f (1:,y), then |%fL| is the outward unit normal vector. (NEXT PAGE) ...
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