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Unformatted text preview: MATH 241/241H, Final Examination Drs. M. C. Laskowski, J. Rosenberg, and T. Fisher
Thursday, December 15, 2005 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
5J2 or 37r in terms of radicals and 7r and do not convert to decimals. You are allowed use of a calculator and one sheet of notes. 1. (20 points, divided as indicated) Let P = (0, 0, 1), Q = (27 O, 0), and R = (0, 3, O). (a) (8) Give the equation of the plane containing the points P, Q, and R.
(b) (6) Find the area of the triangle PQR. (c) (6) Find (in any form) the equation of the line through P that is perpendicular to the plane
containing P, Q, and R. 2. (25 points, divided as indicated) A curve C is deﬁned by the parameterization r(t)=1n<t+\/1+t2) i+ 1+t2j. (a) (10) Show that the parameterization has speed 1 for all values of t.
(b) (15) Find formulas for the unit tangent vector T(t), the curvature n(t), and the unit normal
vector N(t) of C as functions of t. (Hint: Part (a) makes this much easier than usual.) 3. (20 points) Find the minimum value, and the p1ace(s) where it is attained, for the function f (at, y) =
x + 2y on the portion of the curve 3323; = 16 in the ﬁrst quadrant. 4. (20 points, 10 points per part) Let f (x, y, z) = x2y+xyz+ezy and consider the point p = (1, —2, 0). (a) Find the directional derivative of f at p in the direction of v = i + Zj — 2k. (b) In what direction is f increasing most rapidly at p? 5. (25 points) Compute f f R a: CM, where R is the region in the 113-}; plane bounded by the parabola a: = 3/2
and the line at + y = 2. (Hint: First determine where the two curves intersect.)
(over) 6. (30 poipts, 15 per part) Let D be the solid region above the cone 2 = x/zv2 + 3/2 and inside the sphere
$2 + y + 2'2 z 4. (a) Compute fffD de. (b) Let E be the boundary of D. Compute // F - 11 d3, where F = mzi + 3yz j + $23; k and the
2 unit normal vector 11 is oriented outward. You will get credit for this part of the problem if you
correctly express the answer in terms of the answer to part (a). 7. (30 points, 10 per part) Using MATLAB, one gets a sketch of the vector ﬁeld F =: —y2 sin(7rx)i +
(a: + y)j in the rectangle O S a: S 2, O S y S 2 (see ﬁgure). 0.8
0.6 0.4 /
i 02 a I ..... 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (a) Is (curl F) - k positive or negative at the point (0.4, 0.6)? Explain your reasoning.
(b) Is div F positive or negative at the point (0.8, 1.4)? Explain your reasoning. (c) Let C be the boundary of the rectangle formed by the lines as = 1.0, x = 1.8, y = 0.2, y = 1.8
(dotted line in the ﬁgure above) and assume that C is oriented counterclockwise when viewed
from above. Is [0 F - dr positive or negative? Explain your reasoning. 8. (30 points) Compute / F - dr, where C is a portion of the parabola y =: x2 in the any-plane starting
at (~1, 1,0) and ending at (2,4,0), and F = (2233/ — z sin(:L‘ + y)) i + (x2 — zsin(a: + y))j + cos(:c + y) k. ...
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