Review Formulas - Engineering Mathematics, Study notes of Engineering Mathematics

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T s ! o ll B6 GI lllo 2 Szioor$30t HE lrlr a-C++ +4r9o++ .c5 88 -a -ltE,E Fgg Y9Ev66 u g vE++8 88> I3ts $\r tl n v tl d l.l ro '0D aD Itlt< IT h, +t il) I ll x "3 +9s '--' !'5\ b, *=.' /.++;.6 ?",bhif A alc **lit:i6 El o+ rr i l $ e: rllTllrrlu rr i gltii I gl I a4. =o< P L/,(-) =I tl -? = l $ s N 3Ij{erpts6oo&tttttl i-o-sx x =. o ,6.1Ooto 6to iB gIE 9i .!t9s:4b: t\ \l E > llfll lo i* t]Jr ers6 _i'. llrt x5) I x z lol \- lt .< '-o J. i6 =ltEJ ? 1".x\,sx fx E 5\ ll J I x 2 s x + { sfJ-st ll 76:\ ZJ fSSxll rtgx; HT\r l-(, IDrsE + T f -pq- 3>g g x s\ l-o <l t l\ xls t-..r1 4Li 3 slo *rt $- fx 2 x N lt _o lc.l -o rl1 ,-a\ rt L r4w -*lg (a\w c,i'P ILrq M \,2 g v, Ix E u0gT rA$ E o gH B f s'li$ s a $Ii ?s €; g E vr E I ii ffi: I AI & 2 lD o ru ;HN} e ll)rl 7t! o LlJ 0r 3 @ UJ o...f ILzRt =g.E xls xls X lltlll =) s- -C -z = r 2 t$zIzx BEAIYS ANauvsts (conarurE Mu , 6lvEN Ad c Ag @ . fs=$e I flND'p @ Pmoo. -' OJ9 fbY rr*s=r.rt[ffP#ril @ T= A"fl @ C= o.gik'otb @ T=C 'Artg=o'esFab @ FIND'o' @ Mn =r(d-A= c$-q @ lulu =QMn vrva @ Asr= fr**bA @ Asr--As - As, , T,=C ' As,fg = o'9s t" ob @ Ftuo: a O Mn= Mo, +Mn2 = 't, (a- !) +r"(a- a') = As, f,, (d -s)* Asrfu (a - a) @ l'1rr = QMn @ o FIND:F , C"=o.8Sfo'ab , Cs'- As'fs' . 6=fo . ts''= GOo + fs' @ T--C*,+ Cs . A.$g --o'ss#@4b + A;(c*'$) FtuP' o @Lr/E: ot a=F' Mn= lvl^otMn. = cc(d-?) + cs(d -d) = o.ts fa ab (a - gt t Ar fs' (a -d') Mrr = t0 tvln =*o{-o'sfc, @ P--H bcl @*^ @ @ @ Cs Cc ll+I OtI c Tr +T2 EYACT Q,=o.tsli ab Q= As'fs' T =fufg 6pepEx C=o.tsfo'ab Tr= Asr{r, Tz= Asefi ,.1,qt471V',12 BTAMS Dcueux RerrlFr*t @ HND:ts * I,EBH@.THAT @MPnESg olt PllNHo EEt"tEur o As r= f,*,bd o \ffi.ni"r(eease) . T, = As, fg ri,,' t C, * o.$'l/ab @ Flgg3a, T, =C, @ SouvE, c ,.7 @ Mur =eT,(d-?) =o,qoAs,fg (d -?) @ Mrre = Mu - Mur o tAuz = eh(d-d') = o,qoAs2$g(d - d') O FIID , Asz * cHecK| . {s'=@6 c'=d' @ Tz=Cz . fis'{c' = Aszfg o FIND'As' As'=As, F,.?'$nl l i \-_ J..9LiNzAE 1z. T- BEAMS lr C)ilK,X: ^q.r=7 . fs = Aoo(J-c\ c DeSlGN (coMPurE [.,6tvEN Mu ,YSls ( cor'tpure Mu, 6NeN As C=T 0.86 fo' A" = Ar .fl FIND:AC @ ASSUTAE whot.F FIANGE @Mr,--oc(d-?) = 0,1(o.ts)$ibf t (d - i) lF M{r, )Mt, a(t @ M, = oc(a-) + = o.a@.$)|"'^\ (r-?) @ FIND: a @ C=T . O.gs $a\ = Asf, @ FtxtP,As tS lN cOtrlPCESglO}.t: b"r 0Orecx, ftoa--o.lsfto o a @ o @ @ @ bg(a)=Ae O FIND: or @ Mu = arrd- ?) = 0Asf.l U-?\ ,(tX U) + b,(a) = A" FIND: A Mu = Mu1 tMu2 = 0 "(d - ?) " o.('- i) P,'ol ) ffi ort A. tgr carne 4r fr"^(fi,ffi -!b o f' lF Mur 4 tlu a >t ' Mu 5 Mu, +Mu2 ' Muz= Mu -Mrr 6 C, +Cz=T . e.$6 fo'or bql + Z (o'cs)fi t U = A. fU O FIND: A3 @c*wx; pn^=fi . Pnir. af; o*l ' filin ? *i ose (=P'6n i As =ftln b,od cfu @cnec* lF fr >fe oK/ {s ( fq STEEL @es NcT vetD(q?gqt*r, {.<AI+7AB',. / !, dd N dJ (, i <a e{ {s Be ti" $$* i?r, o oJ ;z t., z lt q il F gI $l(6t --l6ol $il 6: EB .jt, x YIiI F .'; o e rt ij.;r.i !) t :,; , "_:r\.:.. ,.,.'a 2 &+ $ E ot u\, o.} u t- 2 u u,I oJ f,o {l} rR VA :la sf.e u0s€{d \\ r)a :- C} \1r) ..\ f e ii '1, a =t \i N(/l *diJ \l f'J^g t<J s \e-q*l \\ llll t zl ul &l 3 EEooJrr "-.\ i'- sc l:rl t>le _ts+s lt lt ct =,E ga. ouz Eicc rrr Bsl aat o ;ir-a..:':.-?..|.'.':....'..;-::."::s:...--...s.''-:a:..:. {, w ,o, € !/<f,, \, Ito3 (l*- I sl,\ S- t 9Ps3 ll c3 d2 E ^fiisF"S t st-g X-"-+ d-o ?l)- o 3; ll ++-S-oorL-i^-+ ! f YfiTt{s i ]d {o b cl$g Hf s El ii 5 .:i "P ii l- r-+\il\{ 9u.-' vo ul I v )1,- f- - (,,].) 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