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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 PANTANO
 Calculus

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