Study Guide for Math 206 Final

Study Guide for Math 206 Final - Mid-term Exam air; 3 1....

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Unformatted text preview: Mid-term Exam air; 3 1. {49154 For theide dmulntim retina-u rltl -I:Eu5=r.li.l13-C:l. |I.| {flpul Find. the u.|1.i1tnmgcnt water T. v - i' - 1-3.1.1331». E] I! -. DI'n'Eb—flmuifib-S ; ms“:- 3r). I]. {#11an a — 1" — yd. :- — E11. and. y — 1* — r'. find the inn-swing using 11-: their. rule. {bl {flpul Find. the pl'iJ'LGipillfl'lil: 11mm] 1.th N. |I.| Ell-'5'; . Sol-ell ion: Hun s...- gm 5'. :. u..- 1.».- .rr' . . . ?- 1—3-5133. _e.msr_. . I. _ flint” _ h_ :yl_ 1:3": _ I.._ - -. Elena: E — Dlin'Sr - 3 E Jr {M 32.15:. fl: 1. . .. . . . . . ; . 1 {2915' L“ If: 5.} _ Va;- _ y? 59 I” - .1:__!I_ — ijI‘EI - bl —-15I'. - l._3l 'E — r‘_. |I.| {lpel Find. End nh‘bchfl-edmmined'f. Sol-alien: fisdminoéj in {::.l‘: :J- l" a ll]. [E I lefl (ha 5 |I.| njflplzll Fh'ndfluetmgmu plmenjtheluflleez —k' —!.I'i'wh-In-'_-— l. 9—]. r_-4._— $-13. f.:1.1_ -4. _r.:1.1:-2 Elna-I:- Lindy-J. "Ll-1.- -3.Hlmmhw:l.].3:. Tm tune-I" Flu-Iii :—3- i::—I.: —2:y—]: :- h—Ey—fl {bl {lpel Find. the rugged}. -:b| cepul 15mm. ma nerltedinnegrn] I”: 11-9 .192. 1 .. Exa m # 2 Holulian: . 9-5 . :s-gi :-g-=': .: :. , , . . . _ , ._ . , . ;. J11. .. ___,. '..r!'::... “a. ' “$1. .,-‘ "9'? - '1 1. [Sp-t3]. Fmd an equetJ-z-n er the plane that passes through 171—1. 1. 11. fi.)n_1.—1. 1.1. and 1911.1. —1.|. '3b' Solution: we need to find a vector perpendieuiar to the pine that threugh the heme three points. Let a = m and b = Then we have a. = ::2.—2.|:|II and b = {ill—2:; Then shimmy 11 = a \ h is Ferpendieuinr to the plane. Using these, we hare II Lmrrufli : - D: . . _ 1' : l J k “Jenn” 11: a \h = 2 —2 U = 11— (—133 +41; = 11+ 13+“; = :j4.1.1j;. :- n'mn-m'u :; - D: 2 U —2 . D 1- Jun“ 0T;- ' a Let 1'.) be the persitien meter of P. Then we have L'n = ::—1.1.1::. Hence an equa tien efthe piane is Hum rh- Jim'c dam-main. M 4|:1'+1:I+4(g-—1)+4[t—1)=IJ I‘lfgrlllmiiy, 41-111“: TT_t'9-flid§Lml.IJidE-y;|. x+y+¢= 1. Bolulian- r nfls-‘ui- 21"": 2:. [Zr-ts} Find the distance from the point [3.8.5.] to the ]:-lene : — 231 — 2: : 1. :n‘ . .. um :I _ n- ' n“ . . I, “Guam-I H _ D: 5011114011: James“ _M D =|a1'L+hi_n+c:L+d|=|1-3+(—2]-S+(—2)-5+(—1)|=|—2c1|= S- . .-....—-.a ' +5” + c” 1*“- + 1—21“ + (—212 3 I'm —'L - lim .1"— - _L. :-rI-:HI 1-‘—y“ Inl—wnci‘e' 2 I I _ Hm. mun-1 ham“. 3. rep-ts.- TWC- planes are gn’en by Fl'j :1- + g.- : 1 and en : r + : : 1. d. Hp‘lnl. I'Lewrinbln : '1: Mud Whirl? 1.1 l function d:- Ind 11 11" 1M squad-m 2'— f—zg—%yz —.'1. m rind egg»: Solution: First. we need to find a point on L. J3]: setting; = CI. we hares..- = 1 and c, = 1. .50 the point [0.1, 1] fies en the fine L. Nave. we need to find a reeth v paraiiei to L. This is given by the cross pru‘l'uct of the nernmr1 [e] Find the symmetric equations for the line of interseeLi-Jn L of these two planes. Selulien: rectors: '1 j k 1; =11: \112 = 1 1 CI = -;1_—1_—1;-. 1 U 1 {bl rind £43.13! Therefore. the symmetric aqua tiens of L can be written as semi“: %.= -9 __ 4:9. -11 Hy—;_——1r_-—T:¥—-EI E. r-_'-' n- =—:y Mid-term Exam 4 1. {5915] For theiunetinn :12: 5|:- — £2 — 9': answer the finlltre'ingqueetiom. [I.] flip-tn] Find. the gndient ocfflr: 91- et fl: L]. Solutiom Solution" final If. - Zrltm' lfv - fly. it 'I halal-El H-I: IIl'.||.:iI:-- Elna— Iii-:1: l} - 2. Ir. - LJ _ 4!. IIrII _ 4.9,: _ ILL-I m"-":- 1: "mil- EIu'ahr. -t-.a.'§.-e,nat¢='"=—y-onay‘—=-e. Sehh'i-efiu‘gg-r: tuna-5n: Eb] {iptfl Find. the directinn in which the funfllon LnereeI-e: meet npitllf L1 [1: lj-. [I - 1' —: - DC: — l:-|:— ill-:1: — l|||_1:I — l:- He-Iurioul Elaine-id lI-:l.:]lj.l'1;lrl1-I_ - Thul i not-nun In I-IIIII III-Emoh' :- l1: L:—|.. fin hlai'xnl "LII-l ll: [:1 d-:—J:—1I. H'Iclhuhbl iii::nndprfilidlifirlthr1niflZ:=EI:-' I'l 912'- L l u . - _I_._I_ - _.— . Ia". — If!“ TE .Iru ' '2‘: I... - -4 fw - LEE-l D:::y:- -‘|'. .5... —-:f.||' - HILL-'9' — 13. in] {flptfl Find. flie- directinn in whinl: the funfl'mm dent-me: m1 rapidly at it: L]. Fun DIDIIJ‘ - —lB -'-":'.|'1:['|I'-rlfle7=lll paint an [HILL-1:15:- nm‘r" 1| - 12 :- n, nu:- - —l irllncll IIII'III'ImIn Solul‘lflal ItrIe'i-Irlinnir 5;}qu .53 —L:—l:- - 125. LLB-ne- .._—]= —]| - L: -.I n, ‘l'|—l:—lj- - —l am...“ an] IIII'III'IuIIII ] —I\.'I - — —-.——- 1"2' 1'! fl--]]lfldfl-l-l .. . . ,2... v.2: v.2 - E. -,2IItI_-Cnlo.lll.1e the .flere'bedinxeynl em? I19”. Selurieal 5|. {2915]an the “ELEM pLene tn1he51n'Inee: — is!“ — 9' when r- l: 5| — 1. l' I "Fig.5.? it“, - .' 13¢, .' I" .ing - 2 y |_- 2, 'e 'n- n- :- 3 I r' + H In" I]. {2 p1:]Es-Lin1.ete Low :n'nteh. 1L:- nJue of Ice: 9:2] — Quin! — £92 Iel'lll ehenge H the yoint F1}. 55::- J. - 1: if. -2g. mow: Ill 'lmll: homPhflJ: LEI:- :.1:rei5h.t 1mm IFS-22:2 —2]. f.|:l:l.:--l. Ill-ZLJI-i Elm. I - [Indy -] In 1-1; -3. Hum- Irvl hn-:L:]:3:-. mining-It Flt-mil I 'f'. [I] {bonus flp-tl] Enlu1e the Irlere'bed. inteng If“ :in[£— we]: where R— II: r39: 3-: II: r.-"_“:. :—3- L::—I.:-—2|5I—]| ‘ _ “52E _3 Solomon. Laying _ I... - linronly— “may, II». 1... II' III Iiu|x—I,-|J.J.- III III :Iinrmly—oolzliny:-eu - III III Iiuzac-EdA—II' III uni-Edd. a. HII IIII fimII'IIII l t rims-l - 2‘ - I'- 429 - L: 5"“ mum-the {dinning qnm'iems. I'I I'In‘hflfllhu- 3...?” - [I] gap-II] Shm- 11m IIIJI moulme m gin-n by [n to: -:L: l]: uni [-l: -1]- .' .' mm?“ - . .;b] .32ng tnII'IfiIr ItII .II'II'IIILI-III'IIII emu EIIIIcI'IIII. ' '" I'" eh] {bonus fits:- Ilt'elnete 1h: itemed integral {fa-:41- — 33in: where D i: the region hnundaetl by thepenbole: 9—b9enn9—l —1'"‘. Helulionl _ I I I .-1 I F1—u' I I H: -33.; - I I -jL-— Buy-1'.- ' ' E- ' I. -I '.— II: I p1- I: “I __:I I fluid. - II .:L-— 3H,,- - 4.15-— 35- - -:i.:— 3:? '1'- 1..|-. 'v-Il: I-.l -.:I._-—3I 9 " -II.I-—3II:I—:’:.-e.I—e.:‘ 4—39 . “h! I I"'I III I: {dz—3:431“.- Iu—tI-‘—3—:.I':-II'.=- :_-'—I-‘—:_-—I-' 4—2-1. E I. -I ' "'1 Exam fi' 1 2. [Epts] Two vectors a = {1. I5. t5? and b = 1: III} are given. [9.1 [31:13] Fin-1th»; scalar projection [swim component] of 13 onto a. Solution: _ _ -“1.I.-'2.I.r’§‘.r.--'o.1.u‘- _o’i _ I t11=+tfi1=+tfi1= Solution: PHIij = Compib- I1 = n ...
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