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Unformatted text preview: MATH 241 Final Examination Drs. D. Margetis, J. Rosenberg, and R. Wentworth Monday, December 13, 2010 Instructions. Answer each question on a separate answer sheet. Show all your work. A correct answer
without work to justify it may not receive full credit. Be sure your name, section number, and problem
number are on each answer sheet, and that you have copied and signed the honor pledge on the ﬁrst answer
sheet. The point value of each problem is indicated. The exam is worth a total of 200 points. In problems
with multiple parts, whether the parts are related or not, the parts are graded independently of one another.
Be sure to go on to subsequent parts even if there is some part you cannot do. Please leave answers such as
5\/§ or 37r in terms of radicals and 7r and do not convert to decimals. You are allowed use of one sheet of notes. Calculators are not permitted. 1. (20 points) Find the equation of the plane containing the points (3, —1, —2), (1, 1, 0), and (07 1,2). 2 2 2. (30 points, 10 per part) This problem deals with the ellipse a? + 99— = 1 in the 30—3; plane, viewed as
a curve C. (a) Give an explicit parameterization r(t) of C. (Trig functions are helpful.) (b) Find the formula for the curvature n of C at the point (x, y). Recall that the curvature can be
computed as n = ”v x a||/||v||3. (c) Find the maximum and minimum values of n, and ﬁnd the points (x, y) where the maximum and
minimum are obtained. Explain why your answers make intuitive sense in terms of the shape of
the graph. 1 1
3. (25 points) Evaluate the iterated integral: / / 6x2 dac dy.
0 y 4. (25 points) Find the surface area of the portion of the sphere $2 + y2 + 22 = 16 that is inside the
cylinder 3:2 + y2 = 1. 5. (25 points) Evaluate the surface integral ffz F - 11 d3, where
F(w,y,2) = myi-yj + (1 +Z)k and E is the boundary of the solid region in the ﬁrst octant bounded by the coordinate planes, the
plane 2 = 1 + a3, and the parabolic sheet a : 1 — y2, and n is the outward pointing normal. Continued on back of sheet. . (25 points, 10 for (a) and 15 for (b)) (a) Show that the surfaces described by the equations z = 4x2 + 3/2 — 2 and z = 9:2 + 4y — 6 have
the same tangent plane at the point (0,2,2). (b) Find the points on the surface 11:3 + y ~— 22 = 10 at which the plane tangent to this surface is
parallel to the plane 2730 + y —— 82 = 4. . (25 points) Consider the function f($,y)=yvl+at+x\/1+y wherew>~1andy>~1. Find all critical points (x,y) of f (x,y) with m > -—1 and y > —1, and characterize each one as a
relative maximum, relative minimum, or a saddle point. . (25 points, 10 for (a) and 15 for (b)) (a) Show that the vector ﬁeld F = (x2 + y2) i + 2533/ j + 32 k is conservative. (b) Compute the integral f0 F ' dr, where C is the portion of the curve 2 : 4 — $2, y = m from
(——2, —2,0) to (2,2,0). ...
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