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Unformatted text preview: Math 241 Final Examination Fall, 2007 Instructions: Answer each of the 8 numbered problems on a separate answer sheet.
Each answer sheet must have your name, your TA’s name, and the problem number (=page
number). Show all your work for each problem clearly on the answer sheet for that problem.
To receive credit you must Show enough written work to justify your answers. 1. (a) (10 points) Find the equation of the plane 73 that contains both the point (0, 1, 2)
and the line parametrized by the equations a: = 3 + t, y = 2 + 4t, and z = 1 — 5t. (b) (10 points) Find the projection of the vector a 2 4i — 2j + k onto the vector
b 2 2i ~ 2j + k. 2. The position of a moving particle for times t Z 0 is given by r(t)~~§t3/2i+t2j+4tk (a) (10 points) Find the distance travelled by the particle between times 1 g t g 3. (b) (10 points) Find the tangential component CLT of acceleration of the particle as a
function of t. 3. (25 points) In optics, the distance .r of an object, thedistance y of its image, and the
so—called focal length é satisfy the equation 1+1w1
a: gag where as, y, and E are all positive and the length ﬂ is ﬁxed. Use the method of Lagrange
multipliers to determine the minimum distance f (:r, y) = x + y between the object
and its image, subject to the constraint mentioned above. 4. (10 points EACH) Consider the function f(a:, y) : xiyg + 611*? \/§ 1 (a) Compute the directional derivative Duf(8, 8), where u : ~2— i — i j. (b) In which direction is f increasing most rapidly at the point (8, 8)?
(c) Find the equation of the plane 79 tangent to the graph of f at the point (8, 8, 9). EXAM CONTINUES ON THE OTHER SIDE 5. (25 points) Consider the double integral I=//RydA where R is the region in the any—plane bounded above by the circle 3:2 + y2 = 9 and
below by the line :r + y 2 3. Write I in terms of an iterated integral and evaluate it. 6. (25 points) Compute the line integral /F-dr
C where the vector ﬁeld F(JJ, y, z) = ——y i + a: j + z k and C is the helix parametrized by
r(t) : 3cos(t) i + 38in(t)j + 5tk for 0 S t S 27r. 7. (15 points each) Let D be the solid region bounded above by the sphere (1:2 +y2+z2 : 9
and below by the xy—plane. Let E be the boundary of D. (a) Evaluate z dV. (b) compllte the ﬁUX integral F - I1 d8, where F(at,y7 z) : Byi — 6:1:zj + 222 k
and the normal vector n is directed outward. [Hintz (7a) might be helpful] 8. (25 points) Let F(:1:7 y, z) = 4m2 i - 3x2} Compute the line integral / F - dr, where
C 2 — 3/2 and z : x2 + y2 — l, oriented C is the intersection of the paraboloids 2 : 7 — :c
counterclockwise when viewed from above. END OF EXAM — GOOD LUCK! ...
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