m 226 notes

# m 226 notes - 9" mm “ Icoz*92)1<0)€‘<°35 =...

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Unformatted text preview: 9/ " mm “- Icoz*92)1,+<0)€‘<°35\ = 0 "C! S: Q -17 E r: /71 {C I, (squgod 9) ‘ '(9I ‘8) pm; (31 ‘9) ‘(77 ‘9) ‘(0 ‘0) squyod em Aq paumo; TII'BJBO[9[[’B.I‘QCI eqq go 'BGI'E‘ eqq pum (e) ‘1 6003 ‘9I Kremqea U0149UWXEI 391M 8 snlnoreo 953 [new syn/1.1470 5 Math 226: Calculus 3 First Examination February 13, 2009 (b) Reduce the equation to one of the stande forms, and sketch the surface, (8 points) \$2—y+z2—4a:—22+4=0 ' It}. 91% (x1-4X)—3 +(Zl—22.) +4— :0 [(x-2)2'_ 41—5 1— [(z— 01!] +4: 0 (X—QY—g +(z—IYT— I= 0 (X—Zf—r (z—))L =.'— (+31 . Lei: x’: xﬂz) 74/;2—1‘D 5’: 3+] XI; +_ Z/L -.= 7%, .3... g at X’2+ 2’1: 92' C (:7 HHO‘Y‘;CA ﬂ, loolqr ' i )— Confau_MtES) ﬂ = f y) ’ Math 226: Calculus 3 First Examination February 13, 2009 (c) Sketch the curve given by < tsin t, tcos t, 2t > (6 points) x; ‘t—Sl‘nt‘ L . 1 : i—ccot x451: t (Sm f—r Qt) )— Z: it t CON/“L22 but wove—«41% rad/{us , a. Sf‘mqﬁ (clockwise) but; 4/100 WW 2. Find the parametric equations for the line L2 through the point (0, 1, 2) that is parallel to the plane a: + y + z = 2 and perpendicular to the line L1: \$=1+t, y=1—t, z=2t (16points) L2 ‘to atva l? “die 371w X+ljﬁZv=2 (Pl) a,» [.2 ‘VJ __L_ D, )vl'lU. WW f5 (ﬂ': 4]) l) l> Alec? 1/5 given m Ll ——l— [—2 A 4‘ A .22.: L’UXY, __ Z \Il Ll ' I \ ~l 2- /\ . /\ 2 Z<2+l)—-—J(2—l)+ M-J—I) (squgod 01) I = 28 — fig + 3:9 en'eld aqq 02; [entered (eAmo sq: oq remogpuedlad AH'eooI auqd aqq s; syqq) auqd {mmou 9113 s; < 1,; ‘19 ‘9; >= (1)4 ammo eqq uo myod qeqm 21V '9 (Wk =7 ‘4-1:C ‘ 4% =>< =17 <zr‘{— “a? 3r+ <1 ‘I ‘07 2 ’- 77H.+ <2"! 40> =3 ‘aﬁ M?A|_Q v1\ ‘57 6008 ‘81 hemmed UOIQWHUWEI 2an s snInoIeo 955 MW Math 226: Calculus 3 First Examination February 13, 2009 4. The trajectories of two particles are given by the parametric equations r1(t) =< 2t2, 7t — 6, 2t2 > and r2(s) =< 4s — g, 2.52, 53 — 3 > (a) Determine Where (coordinates) and when (value of t) the paths intersect. (b) Find the angle of intersection. (c) Is there a collision? (16 points) (4) mt): 5(5) 2152: 45—%_ Q 7b—6 = 231 Q) Math 226: Calculus 3 First Examination February 13, 2009 5. A tetrahedron is deﬁned by the points (0,0,0), (1,0,0), (0,2,0) and (0,0,3). Obtain the equation of the sloping plane [the one deﬁned by (1,0,0), (0,2, 0) and (0, 0, 3)] and ﬁnd the angle that this plane makes with the tug-plane. (16 points) A(I,0,o) , BUD/2m), Hows) is: (‘l92>0>9 ﬂ:<_]90>?\ a: fixﬁf- ( : (é/3)2> m 10—1114. W {ﬁg—HQ Waking/0W We Wm t3 ﬁe X‘arlfvwi‘w t=<o,o,1> m ﬁve—5h lbw WW Kw “lb baiwzm m mimu 'we lel'ﬂ <é)3)l>°’<0)o)l> Clo/>8: —-——-—-———— |<c,3,2>| M» o, M , Ui____ ; ,2__ ’ ae+ol+4r -l 7 -1; 0; Ca 7 052+Qf¥9 )0 1 ﬁle. 0 = 0-2)??? + ("3)9" (ON/'- /_, o (newx (Oil 3(0 ‘G'C07 <(O)J"' Z 7 < O :(oz-Z) 3+ (Deﬁarﬂp (“adv (“71 ~<9 ’4] (X7: %’ Zr 0 f ’2. —— 1” Z/Z " <% Cﬁ—‘T> L, c J? ‘ v d 1" <‘_c,¢9>?:(o2+ (W X (OJ—La g 3 ‘ mm o)GTW(0/.\g WWW “‘14 W. ‘7) .5 <I‘I‘o> ; 23 ‘z _ _ —-————(——— -+ ~------- - r (U ') )<,_‘. 9; 4: ‘p‘o> ‘(ON [93+ 119) (1-9%)?) 2 _< < " ’ ’ 43,; q? zr> <}9‘49‘0> 92% (3’)“ (U (3‘a+}9) tr?— 39 69? t W[1(q__2+ #9)] <3r—a’ 61}? 69> : 91(922‘94‘ MEN—77 Jail ‘<;,3»29‘§j‘> (\$7.1: far/i- <}_9_ ‘49 99> = (HM— :92]: ‘ (ju'eId Buyqemoso eqq) (0)21. pm; (0); Sugurequoo eueId eqq JO uom'enba sq: (0) pm; (2 E“:qu (Immou mm 9 KHIQSSGDQII qou) (1)11 [131111011 ’8 (q) ‘(1)'L queﬁtmq {mm 3111 (1a) '4 . OZ) pug ‘< 1:9 39 ‘1 §/\ >= .L log 9 6003 ‘91 [frenqu UOIQQHIIU'BXQ’ 1333 8 snImIvo 5983 [139W ...
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