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bed Untk- 3 (Partial DiYerediation)) — chagter-l —_— Topica: Lina, coniuatk, , deviveiivu , maxima b minima af funchiived of several variablesg. a va piped $C xediag Wight) Temper afte = + ( bottitact , lovig tus. time) Dep : Suppase Dik a pet of n-Tuple 4 peal wes (Cy) ae ™) . # waa value funciion t ™ " n le that apr qua o meal number ure BCH 9% p17 7H) to eae demente ™ > ob ia db, & called Aaveain . . LULA’ , —— Mae A a es pe tf v4 oe Coble mangerk 4 S alu Rave 64: Fan cum 47 oe [o, >) 2 —, , we dy - d gatixe vy plane Gd = tw ee Bd v) (0°92) (0, #) we ae oq) Recall that for fun clio of Ve variable , ¥ =f) domalv- tur by Unow q interval . mm terval x tar tee oon ned , half ope [hott clerad , Inle Know im tenor porte ava ena ant of mterrt » Nove wee well oli tune yr two ding ar 'C- ay plane . Trterior pot * f point (G=2) bd wa ae R te aw mten ae prt of R if tex ve the aq ~plane gush a Ai kk ceuberca at (a,b) tnat UK ent vet ow R. Boundary point: al povuct (@nb) m a veg On Rip calleal boundary point “+ Quy ak cutercd at (2,4) MA ouithiadl ab R as well 2d conten f poivdl “that paint that Ua mw R, . ee! © The get of all tntenor poiede ta called 0. tow . The ge all jwteror pet - inten or O Ff " bouved ary Tat U called boundary of a vplow . (or R wa lanw i& gad be 9) Regio : A wg st consist eudioty oF sts pnteder pork . Opwr Livut of 409) : Let £0x%, y Le A “ale re efired ant reg tow R ate a a es CX Yo) , _ of kiloly af (%Yo). le f dat % 4 “stay ee UB c L as Cary) L eh volue of deny) appa cpp te (mayo): Notetin : hin Cx) = L (Gy) “acd The Lint & ini te puro Ungpe- Def: The Lut , Fonte at C0) Yo) c& pord +p Dp euch avd egal 0 if for giver & 70 thew euctt § 70 aul * silat | £C%y) - LI < & when ver p <[e-el <§ oe ava 0 < ly-gol <8 never Ce the olusk vodtuf § Cutered ak wey. bee r Remark : Lim Cay) = ivr [ (x, N\ Lim (bi (ray) (eye) Foy 9 Ho yet ° Pe bis provided a\\ Bit enith . Exawples : (t) Lin ed fut = tina | Lin e inn) = bin ur. t (uy) (ove) wo Ud? Ww 200% @. Dine my Y= eR = Jinn x_ loner) = Qin — (et) uy)> Cty) —z- 1 73 \ | peo aw) =<-1L. Joos path tet . in pate ah pe nor evttnce pA lle ower Be Hat there axe flit ” pout (%1ye) wm plans . Patt} “to ap prvach a 719) exe tt thelr tt mutt Le However , | Lim +40 Ul N Quy) (oro) W Gre thot mean} whith er pate Yor follow te tint ghed be fame . Thus if tk rA velut ef Limite fur chan ; PAL cou fw o oli fpere du ut patif ten tut un giver t t ol pes not eu pt . a +o ty enwureda trot Lina of. tat . pw We name it an fer P Remark : Thus tut % balp fol only rf thu Lindt clear vot ey pt pare of Bist pas tu AeDek provide avy jn pomalion pate He Points mG. Keep m wind wale chansiwe parts © proud a pork throu le be glen point (xex#) wlucl. Tle variable % © Geverally df (dt r4o) > (© ,©) Us a, Yr ax ye% > We aw tee ote uti Mm Sein Ca’) (mide pre poth fam ava grrx po opproach. (0,0) . ‘ .. Then Lm Wye Bn Uw 2 f _— (%4) 4007?) Joey } yo Ju] Pe es ee tay) || yone | | veloc of porit & disl'net fe destinck pot, Sine pes not expe « the byt L. (dy (>) Conor le Ue patie ge mm te oppor (0,0) then Lin Xu [xv - Lm a (e-me ') > 2z 0 ae tke = Lim WA ( I-m*) = p) y Cor) bem” mit & od. fw oll velut m th value & Lint INES pyrite Ah ou flout genily ¥ curs yrs au tbe - 2 qn dim CRY) 2 bw x( 2% ax™ +b) me ; oS, oho Le (oy Flor) uty” um wet ax tbe = him Ww qu fhe = 2 =D nI0 lo apart We how cantidere! fee tad curs ( Gemn, Y= act), al Lette pow We ns vii volun Dinit pe 0. a""d we tut aloe not porvide any tan ‘mn hore | Hane Limit moy ex'st 5. watt on w ured -to ensure Lh, ) _- Nv Limit exicke, volucl & 0 iy syllabus) Lek im ttape L avs| Py C% a) 7 Jin ( fCmy) + qty) ) = Lam a Bim dtm — a) = LM im (ECD BPE EM y, um (R stay) = RL Bim ACroy = M ren (2G4 D9 (% sho) sum fee = BOM atx) " C. We rm ava be in teen on Yn _ a povided gin (HO) = 2 Tm is tel nunker: Feerige: Show tet ToT , Cay) #02) 6) , Cy) = no ) in cont nueus ot onthe point: except the orlatn Paper bina (1) Sum $oegt JOH (3) Yoduck F tag) alg) (3) Compo how + let re Fly) _ hew gre we gl Few) | alo pancliort age H { 235 4% catiuvers, all cavctimuos - iti P axtiol Acnvaiveg Dervaivr m me yanade Cordimary evivaion) tne sor fem dion tm one yam ole ys fle) ondinay clenivaon = dy = y= £'(*) Wok Nate Au at gives vole, f chavs © ce 4 pot avo ay | Bw Flaw) = AOD poo ded Lutte Aw Y20 — KY } en ote - U=Uso Recall Gemetal ally a \ poder plepe op tanpot +4 Care Ye fla) at to 2x) he cape of tie inolapenerst variable poy Ze lg), b, vartable wt wart +d we Caw Cooper al toli'c arent we hop Bowe ‘le , appears the concept of parti at devivell siclion dak fn tee FUN (a) = wr 24) puprsedl partic! olpxivelit yo wt XL: eo ig pode op change of fly) wer’ bag Kupiva ¢ conttauk - wy 22 we (14 ) papel patie! olivate ww Wt Y p) o Tt J rote chavs a fly) wr vty by Keeping oe conateed - uw a Fini quod okt # chawre i fling) ot ow (Xoo) ; pot Cx igo) fie My cchin, of 1 Qyid: Sivuilatly) we Cae btive > few) 7 thot gues note edchaye af Flay) ot pant (% nyo) 7 tle cbiveclign of 4 aie, defo: oie | - AFl4) I= 1° 24 Cto,4o) dy roloo! Lit = Lim (%., +h) _ C%6 4 ) yov' ho 4 ye ae ‘ Prvenst | Qeomett only sf ot (erfo) gud shpe yf tarpat t Cure Ze $59) at pot: ( % Yo ; P(X ry0) ) ww pe plane We Xe. Notebout : tor Ze F (x14) [D Partial duxivelue torn * az , at ) te on on & fa olen vobiie vliie vrt ¥ , af, ty xy a bof. 2 [tr 4a)= ) - tos C at a4 1’) ty Po Pay Crtere we axe Raph x comnstet’) Exerche. (1 bel lu 4) = xd. tr Fae aud Fay . Gre fpr bax Ft pte (Hiwt : gn b= dnb Ln. O Eq The plane r=| ode Ze ate he OO parrabata, ~~ Fiad slope ef tryed ty the porrehote at (1,25) . co fle chevivele ca to a= San : Slice quenp lout wm w=! ; — ie taken wrt ¥ oe = Ay oe ob eee =H, The a Diethion 4 ak U4) by Find 22 if yZ- Anz = x44 dibineg z nh Ow of tn Indleendewt vanriablee % avd Oh | nction called implicit function, Thi type of fonction callaa Impies fut Lurcbiin Seth Sine f2- Qnz= mY Di ppandioty bot. des wrt x veeeploy x cotoet 4 22 — 132 = 1 ~xo ow 2D an >) = (g-d)e | ° & ( e)s| 5 Be 2 hy Ow 2 Da ye- | E a ans ‘ a ‘ Kerct ge Bind = ce *y + 22n dyes D olifive Z Ab a uch ow tis Mnol eudle ‘4 aed qe ¢ ped it variable J. | Qi\atron, lacloeer covith nut le avd partial erivalh ues me] inv dun dean or WL vania ble ; if fun taon 18 AG coctin uove the om valu oboe not est: 5 bt penction d poverel varialde , partial desivatlws mad euch exe if “fun dion te AK collet. G (arg) = ° ge 2) + ? 1 y= 0 Cleazty 3 i out ¢ Com Timwors wt tor) . Bat 4 (oe) = fim ffoteye) = Slo”) - de \ctle . bh dy br) = dies J (09 otk) = fle). dim I-! < O y k30 —_- eno ts nist Oo). e et FO = ca Tye Find to tyapen tuys dey » Fy - Sty : ae oO (x emy tye) Ou = O%.w aL _ Fd ‘ de = fopyt yo = wAyT A® 55 = (~ BY x yer) 2 a Dept ZS = —% Biny ae ay o4 = — x gmyt © . . Jan * = ea = 2 (erst 4c") = Or 4¢ =Ye ~ 2 \- 2[-xAM eer) = KRY *O ty oY [Z } Oy ( J ) _ vay = 3 - 3 wo + =< —-fslny T © fy 2 (B)- 50% yh) = AMY Gx: \leniba stu = dye sr 4 tug) = Lon (Ae +3y . <_ WwW . Chain Rule Recall “trol tor a fundien of ome vanablk y= $00, kd m wih, Pe chore py al Ou if WOR Axe interes CAT UL x= g(t) thon we carn dex ae a h and calelati, vrby fe choy’ sg aint dye dy dx ota | ott eo At « te ie ; 4) Fo 4, a gin = wot At = At wf r x indlipew d ot “Tred Ning Here wi Cor (made {= arn and X ‘ ; t eudet vanablef Chain Rule for fonchion f a bonne di att vtarriables) be a dippouaiable fendi ard Fen chions + thn wdewt a\,/2 a: © at £ _— dmdloper tab Tree, Degraw [ Gaveneut te vemennleY, agra ( wry farela >) 
