m251F05ex3key

m251F05ex3key - November 18, 2005 MATH 251 Fall 2005 Test...

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Unformatted text preview: November 18, 2005 MATH 251 Fall 2005 Test 111 gm“ Name 1. (15%) Section , 2. (20%) 508: (8:00-9:15) 3. (20%) 509: (9:35-10:50) 4. (30%) 5. (15%) Total (100%) There are a total of 5 problems. 1. If F (3:, y, z) 2 [—7e“9 Sin(""2)+sin(z2)]i+[7m sin(22)e‘y Sin("‘2)+4y] j +2(7ye_y S'i“(zz)+1):z:z cos(22)k, find (a) a function f such that V f = F. (12%) (b) /E' - where C’ is described by = (t, cos(7rt),sii1(7rt)), 0 S t S 1. (3%) s“ i" C ‘ 2v ”' ‘ " Ania L _ ° AM”: ) x L 60) ‘%§€‘ “I e i 3’) + M?) =~> WW; Wei 3’ +X 4w (2: W. (w; 39%.: 7x M®i>€fl M25) (.47 34> 3£CXJ‘1,}):~7X€T51®(X})L7—1Z (triaaém) ‘ ) $1 %:ziy%év%\%3% 1} X3~W(§) z; it‘d»??? ‘YX 63W£2§+ kaég); in; 0,1); 23%;]. W Wee (Mfg), W‘E WW I ‘ 5”" v A at ‘ \Pyfi W — ‘ L? (W Jr): 4% 9/ WW if“ *‘ X M‘fiwd . (a) 54m Ezvfiammmé jam) Pea; 9g? firmwmig Tin. 9% iiiaaw Inflacgméjh A ,. A v.9. , ‘ s“ , . W1 d»; ; (gfrdv :; ((W’Mmufi {‘éwzfliiévéi d, A ,. magafi (if. fl": 0 I! T (0):: (0 )mme)’ Mimwagx c a) :30) A ’> {fax/WWW. {tr-l; (U): (Ogmifiwg)p/MCW~1))= (5)429). flgw d : [WNW/(ma + 24—m- i-m¥«(®1”£*7{0)éi‘ f 0 Name 2. (a) Evaluate /F ' dr, where F(:v,y, z) = (3,12 — 1)i + xzj + x(y — 1)k, and C' is the twisted C cubic given by x—zt, yzta z=t3; 031532. (15%) (b) Is the above line integral independent of path? Explain Why or Why not. (5%) f’ . A V Sat”, (6") g; .4? : (big-de kkggéhg ‘3' 3601-043“ 96‘; a I [fl {T ,, {73: = S: {tau} 1Vth fagfilggfldf :3 Sanggfizgwdw .-.-.. £184“ ganf‘t» {I} E”: {25’ _. [0) g, $4-12,23Vm 1 U9) géwfi, Mamas! 5 ammfmwwfigg fig w. 9%} . {EMS/2,, gait/a j. nit Wit g3; nyga é ,a A a» '3 $53 ‘V X "’ a?» 2% ‘ P FE; 6%? E m-I >63 Name 3. Use change of Variables to evaluate the integral f/eg—Eg dA, D I ' Where R is the trapezoidal region with vertices (2,0), (3,0), (0,6) and (0,9). ' (15%) V Sketch the regions of transformation. ' ‘ (5%) Name 4. Given thecircular helix $(t) = 2cos(at) y(t) = 23in(at) 0 _<_ t g 27r/a;a,b > 0, z(t) = bt } a (i) Let 3(t) be the arclength variable of the circular helix: s=s(t)=/0tds. . Determine s(t) as a function of t. (7.5%) (ii) Determine the unit tangent vector on the curve Where 7" = (—2, 0, 9—:— . (7.5%) ' (iii) Evaluate the line integral / ysinz ds, Where b = a While a > 0 is arbitrary, by using C the 3 variable. ‘ (7.5%) g 03% (iv) Repeat part (iii) above, but by using the t variable. \ (7.5%). a} . ‘ t “V l A, are. ... “,s‘ “W V' .43» 69) 3:. at»): gala = gr WWW: ) “a (mvflmflgt’b mmém'z’t bi) - + “A . m. m a. ‘ Se“! (-mw {we} awmrwfi +~ 2:;— ‘M- “W 3* J‘M wig w). m’“ “A f {35,} ‘ %‘ 3w “Sf’ «43’- g: (4am; {a‘%); mafia gag; e)={0,m§t}) a) 'A WW» TCY): {0)2fl3§i}/S olfiadfvfb‘v Z , \\\ h » w I _ ‘ v 0"} S 3W3?“ «w- amgtg/mgarfm; .1 5% in.“ MM {hwéw Jr a ' 0 JW’ «#2331? I , “$3 was , A 2 a" V, (<4 1 ELM WW? “ W“ Fatwa-a 3’1 Le 212$ 4"“ ‘Qetféa S >d$ an» D‘Cfl‘k £3 1: 24??? m I u 24$ £924??? ‘ a » Wet-ant 114—??? 1/ ‘ - w 2. K \ W_ w mflwwgfifi" g :2: @Me(fl'€i >wvet2d$~ Mt 1: «km 5 («la- ceqmlfiar 7) . 9 ... «a. . ,. 211‘ ma ~ , “mw‘f‘fi m r, 3...... - "‘" 4WV a W R— . i Name 5. Use spherical coordinates to find the volume of the i ; solid that lies above the cone 2 = [(1122 + 1112)/3]1/2 and below the sphere in? + y2 + 22 (‘1de (12%) ' tar amen“) Sketch the region of integration. (3%) ' ’2; 2 JAM"; ” => fly: 4‘3" ...
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m251F05ex3key - November 18, 2005 MATH 251 Fall 2005 Test...

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