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RANGE FUNCTIONS} DOMAIN wt VALUE BRANCH) ae) Tepes] “Ye ost -1, 1) ~ Y= potte” “at R- (-197 a [-s, : x): Y= Bera R= (- 1) ’ ay =) t7 tam ‘x R en Ge ee R (9, T) mo t(-%) Atv eed ® Ain 36 = 3hne-4GhnFe -%)= - A. J ee y= - haw om = 0 Be = 4 corre - 3 Love —7 - Ras } Oe fom 36 = 3am 9 townie (OME 1- 3 hanie -1 @ soa -*(- x) = TH eosin 2 tom Se = aoc! 20 pt 2 (-a)= mt tata = tre" auch (-a) ee a a ae ® sin tase corns vee = = 2 tonite = dam? ( 20 Ke ee ae K/2 re) Om af ty. K i cour ta + dee a= Te @ 1- core = 2h &/2 ® kam t+ tan tys de tap ~ 1+ Kobe = 20047 8/y MATRICES — @RA= Klay] nan = CRO) ] mam @-A= (-1)A @ th A= [ij Jannn then Mae AT= [OjcJaxm QOoOAyvsa ii) AtB)'= A'+ 8" (i) (RAY = RA’ (iv) (A6)%= BA’ (@ A uw «a Symmebuc rrvolin. af A's A. A us a SRe0 Styrene Produ ug AN - A. DETERMINANTS © Qelermenont et oo tmodiiacl A = ie af | as Qo, querer. Any IAI = | On Vie Aaa Ar al _ A 1Ao5- Wi. As, Az bo C2 @ Delertrnama nd bf a maothix A = - by i As { a3 b3 Cin : inna Ang Ai —=s= |A] a b, Cy f= Os ba 1[°2 dx =~ cat m™4C — L+H? @ fsecra da = tanasc Is) J cpatcrn da = - cot a4€ je) [acr. damxda= AtcnrC fexdx = en +e @2) = (9) fa*dn 2 a _4¢ g [acc cotada = - eerie efx te dae geet © | Be = £ivi we C «Sart , = //i-# () i ~ eye H+ = mH Ix?-4 im 26) [Hadas Log 1% Wet ¢ GB | fame dx = hoa | cena) @ | [ten dia = Fog |s2e x-+tamx) B f cota da = hog | Merle 0) [essen dx = Big [Reese -0- cot x| @i) dx . 1 Nig /™8 fac 2 ee _ fin 4 = 2*- a? 20. 4 0 ge a oO 2) [4% 1 tag] 2**) 4c Qf AX. fog] a Sata go tlaavl”” El aea 4 ° [*c du _ t fant + d 23 eileen - £_ ee rc oe = 2 2 3 a7? +a? a a Jxiae fog [x + wa | +c | Fy(*)-f, (*) dx - f,(x)[Falada- iEe fi (x) fta(x) dx Jax — Se*[ fla)+ f'(a)Jdu= fer*f(x) date Sf ps1. = & Jaq? - 2° hog | «+ Sa?-a? | + a 3 ©) (EG an = KX farsa? + 0% hea m+ JxzxaF|+C o. 2 @ (lee 2 ie CD [IE da = K Jax? + ao hw? 2 4 ase< b, thin ta = [NGO = q. BETTER*MINGS DIFFERENTIAL EQUATIONS (Q) Order of 0 difpuuntial equation we thy order of thi Aucgheat voter dluinatie eCcusing, wn Abe kya tion, @ Negra of a dhifferentiok tquation ae the Aagheat Pauses of the ghaat word deuvaliioe un ut. @ Dutama Letiven parallee Lime K = 8, +d b' amd “Re sapb i | Bx (a) [BI © Distance from a foot (os Yas Ze) te the plane “Ax + By 4Cz4D=0 vr [ Sex tSae eed) A?+ B*+c + wna Vs Ao Ma. Wa osu the squeattiina oO} tie hints cthunr the voile Aorttiwun the Five Pinte ua ayaa Ai Aorg= |XL4ho+ ™m mt hima PROBABILITY Bovrditonal probability An wut E, 9 The see | of the daria -“ aa hoy cy re @ Os pP(ElF)S1 P(E IF)= 1 P(EIF) “p(EUE)IG) = PCEIG) + PCFIGY~ P ((ENF)IG) ® plene)= Ple) PEE ALE) 70 = (Ear) = PIF) PLEIF) » PIF) #0 @ 4 F and Front undlepsmelink » dram P(enr) = P(e) PCF) p(ElF)= P(E), P(F) FO P(FIE)= P(r), P(E) #0 © Gaye's Theorem, p(ec|A) = p(Ex) P( AlEx) > p(ej) PALE) jF1 
