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Unformatted text preview: I have not combined the questions and answers into one ﬁle yet (12/23/04). Here are the
answers, for now. Most have references to where you can ﬁnd the question in your book.
Answers and Remarks: Most of these were taken directly from the exercises in the
textbook. The average grade was about 60. This does not include the grade of Mr.X. He
wrote me a note that he only took the ﬁnal to help lower the curve!
1) 14.7.1, ∂S 2) 14.5.7, 2π
0 F ·T =
3 2√
r3
0 2π
0 < 3y, −2x, xyz > · < −2 sin θ, 2 cos θ, 0 > dθ = −20π r dr dθ = √ 3 81π/2 3) 14.3.27, φ(x, y, z ) = xyz .
4) 13.2.25, √
4
y
√
0 −y x2 y dx dy = 512/21 5) 11.4.35, Parallel. Check the dot product of the plane’s normal vector, and the line’s
direction vector. Zero.
6) TFTTF
7) 11.6.9, The simplest method is κ = y /(1+ y 2 )3/2 = 1. To use the space curve formula,
set r(t) =< t, cos(t), 0 > and plug in.
8) See 13.4, Example 4.
9) 12.9.11, Set < y, x, 2 >= λ < 2x, 2y, 2z > and get y = x or y = −x etc. Max of 20 at
(4,4,2). Min of 20 at (4,4,2).
10) (proof) See the textbook.
Bonus) See the textbook (use polar coordinates), get 1 √ π/2. ...
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 Fall '06
 GRANTCHAROV

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