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math117 lecturenotes8

math117 lecturenotes8 - é 2‘7 A Deeper 5mg_03C ”12...

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Unformatted text preview: é 2‘7 - A Deeper 5mg, _03C ”12 ExponenHal, \EL‘ 0 You, W drew W ‘m sdmok ‘Hmak .goncbtons cg. Hm. £3406 -§: KHax C8>o I $06ka m3+) am W gxppneutiafl. 64:3 - In Parficular ) When a :2 = 2.78281.” (an irrahonal mm)scnce Hie Mtnflze decimals W we 0,05% m ’nahma\ expomnfiak New W3) {:Ck. g-(x) :9}? 1w ‘u/CxS seal—ion we. cw:— W ’60 answer some mpmbamk 01mm about Him. apatite/we‘d {953 3 specifically , I Where dog; He, aficp- -Fd7. 00m, {vomit I Nha{"s So sped/all cubed: Hue War 8 :2 O The whole 'Hana Shaded M‘Hs. Simfie ’mQA—k MM (DYOlolems ) Hne 196d: Mom“ 0(. Which is {probabhj Hue {allow/«3: imppose, Hat at; Jc=o we W o. baptmafl. bgpuiakm 19(0) on m {WW Mspm cud cm Mam” Nkadc is m fpuflm‘m Pea) at a. low 1cm ‘L‘>o? As H: twwg Gui: (Mal. as H: ‘14 Mei: d/‘swwflz 430 Show) am. Wyatt yum/HA, model {or Hobs came lg 3cm 173 (20.1 ) ?({:+s) = Poo-19(3) , «Add» me In JCS‘WLJVJW 'P(O) :l ‘ ‘vaz 41W {MS I Rmk. MS lzw o§ efluaftom to W} i’c Ls 2L2 ”Eek QDE ,cwal. M: L9 ’39: m afigebraic await“ I‘E feta/{3% Huz, Wags og M mlwowu, Q): P Gd: vaz, Ala-“W mpuiis )5 )t)ow.a\ ++S . We 002,9. P/O'WWS o¥ 'Hvin €3pe FWHOMQ Ecruwh‘ws . E o Sébahbn. Fx‘rsk we, Hsk why): we Wow @0180.ch about 'P: (an Pco) =l (by man), 0,) p(:¢)>0 3cm. afl. J30) Wu laud: he {acts (c) P Is meno’cohicomg WW6 Dun. biologtsl- friemds‘ and. PEA—>00 Mf-NX‘ New 001%de Hm bmwafl Popuflakion at ‘Em ’czl jl‘kai: is, oom‘xdsvr 19(1) . We dow'k Mow- «HJA We) but we Mow HLa): M: milk be. Some \>os{5civue mm GL>i (b3 Slcwl‘ 02)). Then we mm W WW» M W; 2,3,4)~-,im. the . fumetiomi 81, (i) )M gal: Hm {ouowwfl’ gqmw : P60) :1 , PO) =a , Pa) = Po+0=Pm-Po> zaxa. = c6, 3 19(3) =’P(:z+|) =P(2)‘P(I) “230‘ =61. -‘\~- -- ~—‘ (2.7.;) PM :[email protected]~‘)+l):P(m-u)-PO) =o’,"".o. =03" ) MOWER“) Conc‘vsion : The $ol'n d“. +ke §Miona£ 31-03 subject to “a. IQ P(°\=l is Jrke exponeu’c‘anL 9k wan. base a__.a.£L lead: §0r Wage/r expeneydx. Now we cam, M see. WW Jake MAMA. Pwkaefilj cg. HA2, expgot, PCN)=Q3" 21.3 ms 90m . For wam): aim", }$o HA2. {:u/Md'icmafl act. (2.1!) becomes n+m m. m a; =Qa CL 0/: 3°“- aneaAg W P00 = CO") PO") = Q)“ . o Nexk Nut mote, Hm): to oomFleJce Maw/Wk we show vae Ha): Ha, 60W (2) (41%; WHOMQ 2Q, holds 9» any T2619. ”mee/r 4: push Jusk 1Cor WE We won'{: aka flu); Fume) 5v): we will. SW9 We ‘11: {or «am/fled. - In» 0W «mots, , ‘ HHS =PC’c)'PCS) (2.7.3) We) =Ctt 0’6“” ‘5 ““3 ”v" °{' {19033 . 0 Once we accefi: {:ke stab/amt Jrkcd: —H~L, §WM equaxom 'PC{:+S)='PG:) "P(s) is Vakd. mokouflfl fordLsmekemngs o{.£,bwl:a.%o +or 092, real), WW5 E.e-, mm repese/Ateal (93 Imfmibe MMQRS 5 (m, onM' W9 ) ii: 1/: ecwj +2: chm/v: (1m. afljcovvtatwe, Wséon o{_ WA mm mow (which Ms We {3mm °{— 0. diggmap, alt/cabs» . Hm is W2 Prom. 2 Lecture #22 gw {mm 22.1. POMS) = PMHS) j PCO) =‘ Now Subkfcwls PC’c) {tom bo’d~ tide/5) P(3c+s)—PG=) ='P(+> (Pm—0 , McLiVCotz, who“? 53;, ’L—PCO) ,bgfiulc (2.7.43 PCJo+s)-P(Jc) =(P(s)_?(o) P3,) . 5 s NexE MW 4’ka Ha. .94».on 19 C: Wmtia’bfe, So‘H‘al' Jam mag—m) z 2L3 = P’Gz) S-ro S at? . A 1 2&2 w = 35(0) = P (o) = k (Some we cans’c.) (90H, axisk. 'TLuM, ) {alum} axe Bum, M 5-90 of 50H» Sides. 0.; (1+) Cari/06$ M ' (2.7.5) adj-fame) , 13(0):: a «Eli—2:21.". WW M, witkwnIC_ M i9,am. IVP magma ‘ Conclusion : The fine-£10m! eat’uae‘wn isfiflulmlm to “we .‘DE (5) (In W, «me. my to Solve. forte-Hana). QT). {3 (7o «calm: Hm ‘12; 'DE': ,MA So'Ve ’cko. later) 0Com we, squeeze more, few; Owl: 0f (237.1)? Yes! Firsk1M‘nwfe hunk 3m (5) Ha. Construct k 'va $15M anbiI-‘wj. SonCor Stwplbd’cy,‘€e’c we Considu Hm specw. cam :: I Ch otdzx k, 7 and, o remember {Mal- rwe. Inn»: We —HM‘s Speciafl Mumphm Lei: us aha/M32 Hm ”name QF Hle Fumctim to 56%)) so M 1m now considwr ‘HNL IV? - I221 (2.7.6) I 3f. :' E ; ELO)=I k ' Now cmsialw Ha. {amount/kg, Wximm (Qui- At=k .Forsluwk) E'(e)=j—% =53”; awn—Ea) % away—Ea) J 1‘, k, (WM wt k/wow to be. mend good Mam kcbvewsmfl. So we emsiolu‘ Hue. {ohm “HWD‘W va W) = ea) [m or ,equWwaj , (2.7.7) 50:14»): (I+k)E(£-) ; 5(0):. Now Tum. {his {elation {arr +=O,'m)2—k,~-~1nk,u~-er HM: (1+k)E(o) = (Mk) ' 5(1L)= (”HEW =('+k)-('+l~) =(I+k)" - E (31»): (wk) E(2L) = =(ka } E(°) 2| (2:23) 56d») = (1%)” Irw cram {70 Make. Ht»; Wximwfim halter Maui War we, mite “bake, kqumumsmflh/v5mMmm Ha, VMQL kw: wewm’c_mveona_w3 H: WSW!“ M smflfiwfiw law/rm «L may Mfiu ckoocsz/ h=J_ MLLCLO ,fn kao awn—>00. 41/ 50(9) bemes " In 0 :TM is o (W ECU-(”Ham (m amt?“ “” .Nkwe is Hvi‘i? Asalmge 0+ numbers! M .a belt”) Hm,we own 'now find {he vat—Qua cg: WWW EU) fommmmt‘fiffivswa) (—22 F: = 1.00... F; = (:3; )7" = 1.190..- F3 '-*~(-"f1;)3 = 2570370570.-- Fo. =(%)“ = l~44l4062§ Fg = (03‘ = 2.728320 ('%‘)w = 2.5? 374245.... Mr T1 )1 This Wan-P, W 5‘53““ Hml: Hu- 991m (=1) Convex-3a to CL Yeafl WW Q.- Hq, Wax-hood- 0,; 2.7.... I'm-fad {E Can. be Skoum(v‘ ro flatwa- {ma WMVWT My) ‘1 (27.10) E 22.71823182845“! (mum re-WW? !) ; mid-du- )alcma with, '11: its M Mod: .Fww-owa mica, 4w. {flax-{Low mum-boys, ' The. {amok o4, Hubs {ewe any—Jk-owem) quake 0m?kcaxeak)' (wan-j me Ski? Hm. molmin‘amrom proofs , aud- 30 bad; ‘L’o Euflm‘s ”Im‘l'roduotém {to Ha. 7&st o‘c Ha. Im-gc'nflé”. o Ewlbr fomd. 2 in a. CMle—skflj simple. way . He 5414', jook_ m3 don": 40c um- ‘Hu. BImVuLaQ Thurm- <a+b)"“= oh 9“:de + néL-'>a""’b’+ + b“ l l~2 (:0 Ive-wk on ”we. WWCGI>-1§mh)mfl¢ ) 'R fW'WM-kahm 0 how 9m4 Hdacq+wmehh.Qj kaak “L PWWM?_GLE: (1 + 7:7 N 1 + % + 2L1 + + 7%; 2.000 2.0 2.250 2.5 2.370 2.66 2.441 2.708 2.488 2.7166 Th . 2.522 2.71805 2. com 2.546 2.718253 “3%“ I” 2.566 2.7182787 amazinfiflj Ga 2.581 2.71828152 7t E ' 2.594 2718281801 2.604 27182818261 2.613 271828182828 2621 2718281828446 2.627 27182818284582 2.718281 82845899 2.638 ' 2.7182818284590422 2.642 2.7182818284590450“; 2.646 2718281828459045226 2.650 2.7!82818284590452349 2.653 2.718281828459045235339 2.7182818284590452353593 2659 2.718281828459045235360247 2.7182818284590452353602857 2.664 2.718281828459045235360287404 2.666 2.7182818284590452353602874687 2.668 2.7182818284590452353602814112L.. 2.670 2.718281828459045235360287471349 2.71828182845904523536028747135254 NM...— MNNNNNNN _ ammnwm~o$$3m335558©mqmmawu_ 2 PP N 32"” b‘ M M m 3 N 00 P 0% \1 pl I Lechwa # .23 I @23 __ A Deeper SW3; Sine and Cosine. 523.1. 0 Go back. to he moss—swing 9.35% a2; ow example. 0;, QHM (pages 123,4). 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PC (+0) -.—. ~601X0c0) -..—. o :9 691$ led-Ron Foihk‘ 30 Mmpk Swttdms from. meme. down Jco camcmre “KM M; b; mgirm i=3 'H‘g' {a‘b M 4"“ (7)50 we W X<o (New in M Qomer W a; Huz Home.) mot $0KPPOIM. Hm. 'DE, 92:40? >0. Ms bdrwwior Mplies Hwk the. graph, will, 69 M% a, Wivww amoL Shani-ash? GOA/No WP tow-ands Hue, {5—mu'4, So H/wuz, MU. goon be moHne/r W2. {3, 40km, ‘er 3mg“, mosses +9.; «b_a_x{¢. N: Had: €vm. We km %L£1)=D , Md. SoéffomeLDE) 56=—w"x =0 wL— £=t\ , “(mm ‘b‘ ‘w WW6? W1erki4m Poi/ml: )aiuL {or £>bl we ska? W 200 afim )wM‘u/mpuzs X>o,T\m,s WW9 47W “be be down, 43am, aMaL HM. graph. WILL Seem. 9,0 .wkaJc m we Momma” We, have dLscowmchkak 5553. WP {23%) MG: Laue cw. occluding Svolufion- AM WW. we awe «apt lmom- M. wedge. Jaw cg. oScLUniLm) 5M o, emu wt, MW Mrmatm ‘ I The audi‘kafive beued_}9m makkm’dcafi {WOW/«3' 4mm 0; K0?) in .MY . U 12W. Loch»? a1- Huz. above Foam 09“ 03 :Jc Sms mum 12 WM m 44». mow («‘.e.,quo$i’cai1‘ve> Solutm mu mm n. 6051312 #1:. 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Phe— scmz W615 ,4‘k's ewe?) to Sen, Hm): §+w15=o, M W 5L6) aflso fielves $6+w2x=o. ms, +he 5M2. aunt oosfme fats Tehecmt 51mph hamwnic mobbns. .W.I’ccambe shown, ho}: Hm 533% 0,8; DE's (2.25) «mu “Hue Wheel. IC’s flow as ‘t’o deduce 0.9L Hue p'ropentieq 0% SM: cud. cosfine, «xi-2L. Che) +31%) = 1 , SG=+1J) = ch)s(t) + $({-)c('c), -... a/ML So an. The~re§orel (1.3.9) 1% W199: W {25.2- ...
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