Mathematics 53.2, Fall 2008 Final Examination, 20 Decernber 114points total GSI:

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1 Mthemtis 53.2, Fll 2008 Finl Exmintion, 20 Deerner 114points totl NAME: I D: GSI: INSTRUCTIONS: The exmintion hs T.WO PAFUTS. of 90 rninutes eh. Prt I (54 pts) is MULTIPLE CHOICE n no justifition is neessry. Reor your ns\r'ers y irling the pproprite letters on the ANSWER SHEET (lst pge). Deth it from the exm pper, WRITE DOWN YOUR NAME, ID AND GSI on it n pss it on towrs the isle when so instrute. The nswer sheets will e oliete 90 minutes into the exrrr. In Prt II (60 pts), you must justify your nswers. All the u'ork for question must e on the respetive sheet. You my strt work on Prt II efore the 90 minutes for Prt I elpse (ut it is unq,ise to o so efore you finish Prt I). \u nee not turn in the lst sheet for roueh work.

2 Prt I: 18 questions in three groups. 3 points for orret nswers, 1 point penlty for wrong nswers. However, you will not reeive negtive totl on ny group. 1. A (ontinuously ifferentile) prmetri urve C is given y I * (r(t),0(t)), in polr oorintes, with 0 < t < 1. A formul for the r length of C is (.).fi \FTFT o@t r") Jo'ffit 2. If the prmetri urve t r* r(l), 1t 1, stisfies r(l).r'(t) () lr(t)l is onstnt () jr(t)l is inresing () "6Wt () } "[i r2 (t)rtt () lr'(t)l is inresing () None of the ove. > 0, it follows tht 3. The prmetri urve (r, y): (f (t),9(l)) must hve horizorrtl tngent t (/(t6),9(t6)) if ()//(*o) s () e'(ts; : s (") /'(to):0 n s'(to) + 0 () f'(to) * 0 n g'(to) :0 4. I f t h e t h r e e v e t o r s u, v, w i n R 3 s t i s f y : u x v f 0 n u x w f 0, u t ( u x v) '( u x w) : 9. then it follows tht () The plne spnne y {t,t} ()viw. ()uivnuiw. is orthogonl to tht spnne y {u,w}. () r, v n w lie in the sme plne. 5. If f (n,u):12+a3+z4,thenthetngeritplnetothelevelsurfe f(r,a,z):3 tthepoint (1,1,1) is given y the eqution ()2r*3y*42:0 () 2i + 3j + 4k:9 6. The funtion efine on R2 y f (r,y) : if (r,y) I (0,0) n /(0,0) : 1 () Is ifferentile everywhere ()2r*3y*42:9 () 2r + 3g :0. () Is ontinuous every'where. ut not ifferentile t (0,0) () Hs well-efine prtil erivtives ever)'where () Hs well-efine n ontinuous prtil erivtives everywhere 7. The funtion f (r,u) : 12 * 3ry + A2 + Un () Hs glol minimum t (0,0) () Hs lol minimum t (0.0), ut not glol minimurn () Hs lol mximum t (0,0) () Hs sle point t (0,0)

3 8. If.F'n G re ifferentile funtions of (r,a,"), G(P):0 t some point P n VG oes not vnish t P, then, sujet to the onstrint G : 0: () if F hs lol extremum t P, then VF : )VG t P, for some.\. () if -F hs lol extremum t P, then Y.F I YG t P. () we n e sure tht F hs lol extremum t P, if VF : )VG for some.\. () we n e sure tht F hs lol extremum t P, if VF I YG r P. 9. If f is ifferentile funtion of r,g, rvhih in turn re ifferentile funtions of u,'"-, tlren: (u) /" : f,.ru * f'au * f,'u, () /" : f,' r, * f.uu (") f": f,'u,-l fo'u, () /": f*.r,1fy.a, 10. The ngle etween the tngent plne to the surfe 12 + y2 * 222 : 4 t the point (1, 1, 1) n the zy-plne is () r l3 () rsin({2p) ( ) rosq/t/l ( ) rtn(ftp) 11. Let / e ontinuous 2-vrile funtion on the isk D of rius 1 entere t (0, 1). Whih of the following expresses IIo fli" G) [:l\rf 12sin0 f?,0) r,r"o () ff S2osq /(r os 0,r sin1) r,r,g sin /(r os, r sin 0) r"r,g () /;' f lm 12. Let RetheretngleinR2 efiney0 5 rl t/3,0<y ( 1. Inpolroorintes, tlte integrl II"f l of the ontinuous funtion f t uy e orretly written s: (") f{1, 1t'6lose f.r,r,o ) I;tt 1rt/ose f.r,r,o+ t:l:.f;,"t", f.r,rtl (") ffl^ f/osq f.rtr,o+ {:l:1t/sine fftu y'/'t"*e f.r,r.o+ t:l:1t/sin? f.r"r"o 13. Whih of the following pply to vetor fiel F : (r2 - yz)i - 2rAi + z2k? () Its ivergene vnishes () it is onservtive n its url vnishes () It is onservtive, ut its url oes not vnish () Its url vnishes, ut it is not onservtive 14. Green's theorem for plne region D enlose y simple, lose, positively oriente ifferentile urve C, n ifferentile vetor fiel F : Pi * Qj on D, sserts If o@, + Q)ry : f"f 'r () fln?, * Qs)rv : f"f 'ns k) ffoq" - Pr)ry : f"f. r () [["(Q, - Pr)rs : fe -n s In () n (), n is the outsie norml vetor n s the r length prmeter on C.

4 15. Let F e ontinuous 2-omponent mp, (.r,y) * (u,u) : F(r,il tking the region D in R2 to some other region f' (D) in R2. The following formul for hnge of oorintes from (u,u) to (r,y) in the oule integrl of the ontinuous funtion f on F(D), rr ll JJF(D) rf f (u,r)u,r: ll JJD () Applies to ny F, / s esrie () Applies whenever f is ijetive n ontinuously ifferentile () As in (), provie tht, in ition, / is positive () As in (), provie tht, in ition, LLru - uru, is positive 16. Let F e ontinuously ifferentile vetor fiel efine ner the smooth surfe,9 in R3, whih is prmetrise y (u, u) t- r(u,'--), (u,u) e D, n is oune y the pieewise smooth ounry urve C, oriente y the right-hn rule. Stokes' theorem sserts tht: (). //r(urlf) (r" x ru)uu :.r. $"F (U) /lr(urlf)llr" x r,lluu: $"F.r. k) [[ourlf. (", * 4) uu :0. ().(r, //"(urlf) x r,,,) uu:6f.ns, r-length prmeter on C. where n is the norml vetor to C n s the 77. Let S+,,9- e the upper n Iower unit henispheres in R3. The fluxes of the vetor fiel F - y2z2i* 12z2i + r2y2k ross 51 n S- re equl: () if S n,s- re oth oriente using the norml pointing outsie the unit sphere; () if S+ n S- re oth oriente using the upwr norml; () For the orienttion etermine y hoosing first n p seon in spheril oorintesl () for no hoie of orienttions. 18. For ontinuously ifferentile vetor fiet F in n open region D CPrz: ( ) urlf: 0 + F is onservtive () As in (), ut only uner the itionl ssumption tht ivf :0 () As in (), uner the itionl ssumption tht D is simply onnete () No sttement ove is orret. s well

5 Prt II Question 2 (15 pts) Using Lgrnge multipliers, fin the mximum n minimum vlues for the funtion f (z,y,z): *22, when sujeto the onstrint,n +yn I z4:3. Cu,t'ion: Min the possile ivisions y zero.

6 Question 3 (15 pts) Drw reile sketh of the polr urve C efine y r() :012r, for the rnge 0 < 0 16r. Write prmetri eqution for the tngent line to C t the point (1,0). Fin the re of the region in the first qurnt oune y the r-xis, the g-xis n the two rs of C swept out s 2n < 0 { 2n * n f2 n 4n I 0 < 4tr * nf2.

7 Question (15 pts) The helioi is the surfe in R3 prmetrise y (u,u) uost,i * usinuj * uk. u ) 0, u R. -. (Piture spirl rmp wining upwrs roun the z-xis.) Consier the portion S of the helioi giveny0(u{1,<usp. () () Express the surfe re of S s one-vrile integrl. (You nee not solve the integrl.) \\-ith o :0.0: n'. omprrte Ji!,/it+f l. () Bonus, 5 points: Evlute the integrl in Prt (). fonly if 1.ou hve solve () n ()]

8 Question 5 (15 pts) () Stte the ivergene theorem in R3, spelling out the ssumptions n explining the mening of the terms in the formul. () Compute the flux of the vetor fiel F: (r + z2 rtn'(y))i- ( + log(r2 * 1)sin z)j+ (z + 1)k ross the hemisphere,5 given v z : JT - t2 - A2, oriente upwrs. Hi,nt: ompute iv F n fin goo wy to use the ivergene theorem.

9 THIS PAGE IS FOR ROUGH WORK (not gre)

10 ANSWER SHEET FOR PAF T I NAME: I D: GSI: CI CI t o J. 4. tl. 7. B IJ. 1A r ttl D D C o 6 t) C o o 10