m251F05ex1 - w Math 251 Name Test I Fall 2005 (Sept. 30,...

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Unformatted text preview: w Math 251 Name Test I Fall 2005 (Sept. 30, 2005) ID Section 508 (8—9:15 am) 509 (9:35—10:50 am) 1. 2. 3. 4. Total There are a total of 4 problems. No calculators are allowed. 1. Let n be a positive integer greater than 2, and (a) Show that f satisfies the Laplace equation 5‘2 f 82 f 62 f fOI' ($1,272,...71L'n)%(O,0,...,0). 2 (b) Evaluate (5%) Baa-363:2 ; (C) If n = 2 in (*), determine the value of f (1,3). (5%) 11¢ Exwfie'l WW“ ' ) 7 Wfi “7) $2.: (2”“>X2(“‘L+H.+Xy ‘ * "3"") _ 1 ’3 I z 3—12—- : (zen) XL (7974.“ Hgf) ( JEECXf-F'" "l"ij W4 n ex} ——-—/ «a» Name 2. Let a > 0 be a given constant. A quadric surface is given as ax2+y2—az=0. (a) Find the tangent plane to'the surface at the point (a,a, a + a2). (10%) (b) Use differentials to compute an approximate value for $/6(9.994)2 + 2505.91) (15%) with 4- decimal place accuracy. 1 7. q 1 J__ 7' (a) a: i-(ax+'l1)..x 1— of]. V5“: (mfg—‘7) = (209713“): (2‘92) ML“ 0’7):<a’a)x ' [iv 1% ez’w‘ww—W’W’W” ’5' («1%) = (am). (M, w): 24‘”) “’9’? (b) {Ax = 9-?‘14’l‘9;’0'””é A‘j ‘-‘- 15.9l—lé 1-0‘07 3 7‘ jL(x,j):iléxE{-L$m} : fionf—g—zfilé : \‘l 6073 +4.01 I ) M 9 (1er 9 WWA 1 -..L.—— ‘ I Z s- ”‘3'. 2 jg: 3 [oz Rwy 5%);1—3—(69‘ +1 l X «X (X,‘j)=U°)lé):-> 2%,; I 25' a; a ' [07’ 3i: ’(éth-Zs- ) 3 25‘ } I _L_. ., 0 _L,_.LL.Z;-. .0 3L(c1.?‘i4)zml)e7fv<!°,lé)+’f' .01 ’ZU") ( a” 0*”; ,0, (“0 7) r. I0_. 0_ 0024’”0' 0‘17; : {0”0‘609? : 19901. Name 3. Given two lines L1: 56:25, y=3+4s, z=—3+s, L2: $22—15, y=——1+t, z=5~3t, } S’tER’ '(a) Show that L1 and L2 are skew lines; (7%) (b) compute the distance between L1 and L2. (15%) (a) Fwd») M1; W the 91L A; w 4pm) M12.) (aft, 1) " (—55%), “hit, 101 Z] M A; 25:2,»? (x 2,): ‘ 45:4—11‘: 3+45=—|+t .—) 3;+4-$-—-l+‘t ~’; =5—3t9‘ 3f=8, t‘8/5' W55“? aphid/54m, 3=.—5+5=$—3t 5% 0%.; -3+<—i)=;_3<%) 7 =9 -‘:;‘=-5,« ‘mmaota: A4. «.4 41 W Pfizon 0K , . - ~ A : W Jaw? (2—): 59thMnM» trio-1%,; M A; fi/ifik f‘AfPW): ~ tux-2) +5 «1+: )~+(,(7;—3)= 0) ' Ea t ’) I ' —13;<+5x1+é3/:’l ~ _ TM out-m. m z. mm.) War at. W WWW t» Ham. d: —~————L-/"l"3‘c'l) : _ __._———’Z I 'Jg-Iafirk‘M‘. rd ¢3° Name 4. (a) Let —-7x2y = 'f . fist/3y) 254 + 3y271 (9:72;) 75 (070) Show that lim f (m,y) does not exist, (10%) (mm—40,0) (b) Let ' _7l.4y2 v f<$7y) = 9334 + 5y2 + 37 If (may) 7‘4 Prove that lim :3, = 3. (Iyy)—*(0,0) f( y) (You must check the definition for a limit rigorously by an (E, 6)—argument.) (13%) <‘7’L’ZW‘1 7 / {1L— l Eamme‘CJZ 1”“ “MET fiwwg:%fiw V ...
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m251F05ex1 - w Math 251 Name Test I Fall 2005 (Sept. 30,...

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