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Unformatted text preview: MATH 241 FINAL EXAM May 16, 2005 Instructions: Number the answer sheets from 1 to 9. Fill out all the information at the top of
each sheet. Answer problem n on page n, n = 1, - - - ,9. Do not answer one question on more than one sheet. If you need more space use the back of the correct sheet. Please write out and sign the
Honor Pledge on page 1 only. SHOW ALL WORK
The Use of Calculators Is Not Permitted On This Exam 1. (25 points) Let A z (2, 1, ~1), B 2 (5,0,1).
(a) Find parametric equations for the line L containing A and B. (b) Let F = yi + 2zj + 3xk. Find the work W done by the force F on an object moving from
A to B along L. 2. (20 points) A particle moves along a curve C with speed given by Hv(t)H = 5V t2 + 9 for all 13.
At time t : 4 the unit tangent vector to C is given by T(4) = %i + §j + gk.
(a) Find v(4), the velocity of the particle at time t = 4. (b) If the acceleration at time t = 4 is a(4) 2 4i + 4k, ﬁnd the normal component of the
acceleration at that time. 3. (15 points) Let f(:t, y) =2 6‘” cosy — sinus. Show that the plane tangent to the graph of f (at, y)
at (0, 2 , 0) is parallel to, but not the same as, the plane a: + y + z 2 0. 4. (20 points) The Ace Widget Company has determined that 56 units of labor and y units of
capital can produce f (:13, y) = 603:3/ 4311/ 4 heavy duty, left-handed Widgets. Also suppose that
each unit of labor costs $100 While each unit of capital costs $200. Assume that $40,000 is
available to spend on production. How many units of labor and how many units of capital
should be utilized in order to maximize production ? //s1n:r+y _ where R is the triangle with vertices (0 ,,0) (7r/2, 0) and (7r/2, 77/2). 5. (20 points) Compute 6. (20 points) Find the area A of the region bounded by the limacon 7" = 2 + cos 9. 7. (25 points) Use the transformation u 2 :c + y, 1) = :1: - y to ﬁnd f/R(:t — y)emz”y2 dA where R is the rectangular region bounded by the lines
x+y=0,m+y=1,x~y=1,av—y=4. 8. (25 points) Let A = (0,0,0), B = (2, 0,0), D : (0,2,1).
(a) Find the equation of the plane containing A, B and D.
F(;r, y, z) 2 —3y2i + 4zj + 696k Use Stokes’s theorem to calculate f0 F ' dr Where C’ is the triangle ABD oriented counter—
clockwise as Viewed from above. 9. (30 points)
(a) Compute 2
///D x/x2+y2+22 where D is the set of (my, z) for which 1 g m2 + y2 + 22 S 9. (b) Let
7 ’ x/rr2+y2+z2
Show that 2
V-F: V062 +1;2 +22 //EF-ndS where Z is the boundary of D and n is the outward unit normal. (c) Compute ...
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- Spring '08