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Unformatted text preview: 'T‘t 6221/3 1. (5) Find the angle between (1, —2, —5) and (3,2,1) 1 (20 points) at Cos" ( \,—2,—sf)(5’)2,')
Hf Fir/«(TQ— 2. (10) Find the angle between (1,1,1) X (1,2,3) and (0,1,1) ‘__,_.,— 2 (20 points)
1. (5) Draw level curves for f(a:,y) = 16:1:2 — y2 for k = —1,0, 1.
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(“ﬁx—33.1 Ohm) ‘ 2. (5) Let g(1:,y,z)= z — f(a:,y). Find V(g). 2 _/(p¥1+51' :: 3 (Ix’j' 2:) (“32% 23 / 4) 3. (5) Find the tangent plane to the surface g(:c,y, z) = 0 at the point “16" OH 93,0, %—(L)' {—31.0319 3 O 4. (5) At the point (1,0), in what direction is f(:z:,y) changing most rapidly? VS 1 < 31%] ’29 > 1' 0 : <32, 0) I“; dIW‘f/W Li/O)
«h—J 3 (20 points)
1. (5) If a(t) = (0,0,1) and v(0) = (1,0, 0),r(0) = (0,1,0), ﬁnd r(t) VL'l'): /c0, CU £+C1> 0 $100 5) V<+)’: (6/ 0745) {I \r U“) " \JCHm )CH 7/47,)l 2. (5) Use the r(t) you found above to determine T(t) T « 1 = L413
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3. (10) Find (1'1"an aN lg t = 1. Hint: draw a picture. r7 4 (20 points) Joe the bug follows a trail of strawberry jam tracked across
the kitchen ﬂoor by a four year old along the path :r(t) = \/ 1 + t, y(t) = 4+t2.
The density of jam at a point is given by the function J D(:1:, y). Determine
how Joe is feeling at t = 8 (speciﬁcally, determine change in jam density at
t = 8). Some data: JDz(1,4) = 5 JDy(1,4) = 9 . JD$(3,68) = 6 JDy'(3, 68) = 10
JD,(2, 3) = 7 JDj,(2, 3) = 11
JD(2, 3) = 8 JD(1,4) = 12 f é'élIHliT/O‘ZW
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ﬂél 5 (20 points) 1. (7) Find the distance from (1, 1, 1) to the plane x + 2y + 32 = 7 ray/I1 ll /“17 v: (40 )ll —?
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3. (5) Give the formula for the center of the osculating circle at a point p
(Hint: it should involve a subset of {1), li, T, N, B})
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 Fall '08
 Kim
 Math, Calculus

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