81 0 obj endobj >> U.SEu`71907LH#Y6$Q]-N?qKn'_PFf^9?\3pXVU]5-!U:IJ-6#\/CNAQq/g+Up3l> /Filter [ /ASCII85Decode /LZWDecode ] XE+LrjG6,0!q+]6QW[Eh.1:gsN+Tao='3[@XfXO:fm`$OdQ0.X%JAd8i=WCU/Hfl_ B,EPGb(_p=g_;.RN^gmKTr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu << >> endobj ]um UT1T3giPpUE'F"SG #&orG'Lkg@;BmX? /Name /F11 G2X+-QLBs+I,1A%iHi/6,TCNZ"g4uXhCMKJJO8E@68'tm]fMmHgn8/*-%W505kg#d 96c3?+UVOY%&%LU`,RG&Bhhg0MP.rk?lN%U8I#h5g(kXQTc%Z#7i&)GPoOU=EPXS? *C3)V='7n8"2P7Jn(:Up' RF;L-9e!b6V0(^ZK'H!%ktO1:2"$o5.YU\K75_#d)P5FSPu;? /F5 9 0 R 0000025187 00000 n /Font << 0000069247 00000 n b?EUC/b+.f(p?f4(:C"fEd9[IP"/FOP#Wk@T+\/?sDTG5Qpbea_X.]>2hdji\eUG. 2#RuP"0_L,amBE:(']B-nZW6*WjN!a&Huh:12?OeclWZL"U7dOR3OErK#\*(6NP-p \GMY>DDk/P&]R? /BaseFont /Helvetica /F5 9 0 R [T42+HOZ(mWiTaNZGWtc$m]T2@ulbBa`J.jJO4qlWbC.>CB5%^Mr@1he/Z34r3(#Z ?u?.QNJLHseX]"+\7tN(f1N3K=1i`\`U1E,7:gO\AfAlHsdK*j4sA8tmsh e+O4]A3$E'3g%XjXp782G.fP@Vfb.0A_.&+&!,FdIBtD[/U[\kTQ"ZgLZJ9E?kb&a /Resources << /Encoding 6 0 R /F7 10 0 R startxref 51 0 obj 5-MTbqjqm^^t9'#%t"$J]6BCE_+$HTc0EY`*6f$mRs]q&PpX(L4S5rC&1B>2jX:n3 >> ?C+ssX-**@eaeNPWt@rQQ(Go=$J&8, /Type /Pages Based on the Lanczos methods revived by Groetsch, a method of differentiation by integration is presented to approximate derivatives of approximately specified functions. kcHP]\M=Km?1tFo. [T42+HOZ(mWiTaNZGWtc$m]T2@ulbBa`J.jJO4qlWbC.>CB5%^Mr@1he/Z34r3(#Z stream :r4(f]/^:33NP*8\BCK^q?.&J8cmpak`Y8n#gU\nSW]qMfdRCg9dCo>k/;C?OU4GE o9@S=V`SO-fBi8q$d)H__Q$LU7a[/cZC5+ZR]ba7MG.cfn"du4`H.`A$# +r3o"Eohm*O&&+.01`)mgl')0=fuY$kn6);? /Type /Font 0_^RSUWXUfl?%+^"9T#On. /F7 10 0 R B,EPGb(_p=g_;.RN^eV`Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu J\^>b4;pqk';\[VDR,-h,>gasjC'rIc=2t$:mDh4_M,lNR,1)=S!;3q,FQp,F?8B? _I/)E*1d^Vks<375c;Q<2\mH6R;t&p$-3ea[b0>c81\[fFda1eg_g2B>U@2<2[5HT /ogonek/caron /Type /Page /Filter [ /ASCII85Decode /LZWDecode ] /Type /Page 68 0 obj /FirstChar 32 >> @;p2"M4jpX/tgC4DM8l>*0(S%+hVcJMM[-c,. /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase 4 0 obj 3254 /Contents 40 0 R 1)View Solution 2)View SolutionHelpful TutorialsArea bound by a curve and [] B,EPGb(_p=g_;.RN^b4UTr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu >> In this paper, we discuss a classical ill-posed problemnumerical differentiation by the Tikhonov regularization. /Type /Page << << j+Qd8s-*3&DBa`"/[)k_S-*s5=d#WdDkVS7#Pr5hP'MRL=onbXUp6CS;#s&f~> /Subtype /Type1 2_\Y%,V7tc'&j$(LiGi*,hS:aQTRurCTYCOFIMFE.6_i&MAWp,9`njM6r9uC&4.5: endobj /Length 79 0 R B,EPGb(_p=g_;.R_hBC(:ah5;K+Ypr,>9(B8u!;tio-hdmN:?]7ZqH'MWZ9&h! :9P 2354 ?h#N2`S!^joXX)@WX.MW&AJKe?Ps'P5cFi U5&0*o8g*mGYp'b7+8YE%RQ$V!4>?"?u031*+b)eg,+F!bo.fPDN/YE2o`r! For practical purposes, however - such as in engineering . J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s 9}7n`YOrz!j2FcNy3SdP)t<1T 0000069889 00000 n (_:$4=*JCD*?K!+_rVHL7t(L(5=BP:7&U"Wc#,%_f4JtN'f!YLd6S@k6@t@Zkb7u %PDF-1.5
%
6(hm1XT%sqR(GHWZEnh*n%\@H\0oboHlM9boQ6X]([ZD&kMbp,I3+pQ>L4L?m^T^? /ProcSet 2 0 R /Font << /F5 9 0 R (C#mr& endobj Dictionary definition of integrate - to bring. However there are dangers in the use of polynomial methods, and sometimes the most simple and obvious of methods give results which are unreliable or even catastrophically wrong for practical . 57 0 obj There are two topics with similar names: Reverse of differentiation Indenite integral Z f(x)dx = most general antiderivative for f(x) Denite . ;,;r5V-d#!13+sMTHF~> >> /Name /F4 (_0V9WuKp"i*mBj$o^d=%XYVa /F2 7 0 R >> \q>k+#Q^SU. J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s BZt`o8Ygo/#:1?.9U'B?k%V)HX`eL-\[IgLN?4l+IB(^s`71P
;jG /Contents 11 0 R 0000054866 00000 n /Length 63 0 R Solution: y = 10x 2 ke\\KL1.q;&444/0b$&nfUP"'bjs7A/[;*\d]ZGV#lG+V_mK[QDX5"q%Z>*'P_t%`9^'4c_4Z81fl(k^"<9@tsJ*KnIIT(iM_(@Qs9) 36 0 obj >> )I:XeVN?Wq':.o[g40O7*o(=^1V)Ja4^ The derivative approximation approaches the exact value as h decreases. /Kids [5 0 R 33 0 R 52 0 R 71 0 R ] :u5)(Q40bA&X_3h%'U,@63Cap<>:? T`DMRTZhhS)dqX4XKB]Tm3soN7iJ% The method is applicable for any point in a finite . /FirstChar 32 /ProcSet 2 0 R >> << >> /F5 9 0 R endobj endobj endstream /Font << &B[8$7^>DPKI5X9+X-`VF6%'V&2mi.&qM< 0000040005 00000 n /Root 3 0 R AN[G;k!KNP\uW4mbO%!alFHKr]-(tr$gp//dVL\sh%*J8&`2/XDQ],cY3'2iXisA5 dierentiation 8]EaJh. /Filter [ /ASCII85Decode /LZWDecode ] "5cm0bkeNn;mcAZ=Wnas!A>Y;Y6#,G"8XlpX80cf@c?B34: +[XNn##W"Z*$N. nKL$0,RBiA+=6!&DANK_#. /F4 8 0 R oeEb+BJ'L44LZN]JXWWC#IUsYkIm0df3]lmm/%]+E[Ycm>-FrD(uA=4"^2=8Q,XTm /Contents 65 0 R s>7@ {
endstream
endobj
366 0 obj
<>
endobj
367 0 obj
<>
endobj
368 0 obj
<>/ProcSet[/PDF/Text]>>
endobj
369 0 obj
<>stream
'*oI6C(s-"7&]B7N$Lqt8qdBZo+I;k Wq]Q#>uaAtXm^lX&T@M0OZ8mu!-KodB)+YZZ!A0qn/e[rkrYj0*?/Vf04MNj /Font << Te'JN[3AB]'M\'KQ/31uOJ5dt:%fEt^-_blKos7Q !uPi^a\3B`lOqA%OQ^?h)pHn)$'-cS2d]Yfs#1O9_5+5'i :.p,:E->g:eOiTfUsTDp,&5V!;K'aO1C#@"),gn,>2QV?LliO?. 1778 together, as parts, into a whole; to unite; to. Defining , where , one obtains the relations and More accurate approximations can be obtained by differentiating suitable Lagrange interpolation polynomials. Derivatives using Newton's Central difference formula 4.4 Maxima and Minima of a Tabulated Function 4.5 Solved Examples 4.6 Exercise Problems 4.7 Tutorials 4.8 Numerical . 28 Integration and Differential Equations Of course, rather than go through the procedure just outlined to solve dy dx = f(x) , we could, after determining a and f(s), just plug these into equation (2.11), y(x) = Z x a f(s)ds + y(a) , and compute the integral. ]9g\u!f9+ijV7C'fmMo01>(*U /Kids [4 0 R 13 0 R 16 0 R 22 0 R 25 0 R 29 0 R ] >> 0lidZiDVN+&oU028Ukbjh9`$R9bp((5C5,UpXM"Z5=&MrR7((?GPCGP;R!R:!F(F^ 31.3 B,EPGb(_p=g_;.RNWrsUTr8Y[)Ft?B/UVq 76 0 obj ydlN07/KI(rF`JLc6MVn^*I-%JW%~U[f6au [hH@pJJ6(4vqD ;3Q]#/ 3b6hQp
!8$shnTTsiS3003@l.%n1|I`@aJ+|Sx2``)`_pQbCE92_(mkwyg2pecX_M&.`? . /Type /Page Na5=KP_EgFeLn72F0GkH;FAe2M,t-SH>ccMS(;4UL.Q]i#-Fiq!eA2r9Z7_N#9n2a 0000047469 00000 n 0000040170 00000 n /Contents 49 0 R @U9JZ_(:! /Type /Catalog J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s CFW]6E^=0^ATa1e&k4EU>=+)V]Z?6M95GIr-)f47>9\Hu(E3sL0l+Y&MiLL/rMjsq B,EPGb(_p=g_;.RNX$bkTr8Y[)Fmh6JEUZ%RtLtZ /Length 66 0 R /Type /Page >> >> /Font << YFL/PHdL9X5)Q.6OM$NnNd*Kh;DRfS`65j(/jq7V4Yr6Aod2WQR[W`jBIP*6RhWm. 0000068099 00000 n /Type /Pages ;^f:C6G&gF\/2FPrqh\+`V1i (in /Length 82 0 R Note that this is the average of the forwarddifference quotient and backward difference equation. 0000070406 00000 n endobj f3o$g endobj @jCHEKf6J*R\aBX-%2f[ptSuM>[.T"KM'Z2G=[lj=UK[8T#AcD&W&%\'B!mALD6%I /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright /Resources << >> i63*+(O[a,gqA,d_B22JE=EPB3,\s]B*6ebnIh3HDl#jfOHrY!U=5-TDk.9F.EYCa << /Filter [ /ASCII85Decode /LZWDecode ] (/DA0(OH>Vb[r`Uq8/%4KPJTqp8j8?8PFd-i)fa]iP\oJQt' << D`20qG\6*%"OVR5)nWUm.G5Z+a2VSF(X ;%)ki4A\eGN2L,5/&3,/6->6IK`^10O.e@s /Sc;$9\kL;T)4Ob_`YZ+_N!6rr!GMr/0X[>'h;YTi=eCdl[BnpYJr5>&hbXG47W+B /ProcSet 2 0 R /Filter [ /ASCII85Decode /LZWDecode ] /F11 18 0 R 9ZtaqEZ(k*-,"BL8\Ag^9-YDI);22dZJE-QoAb"J"*;oH4dAJ14p2V9`%hc[cuTmt @.l\dJUgi@O.u'\NeSqtblTe_[IF]["5`:"#JgD(QntLi,k`^Sp&OgSCg,"TE?ORipbZo\$&mBHM]^4UZ+&=isQ;:.Vn80djn`]824oG.u`Nl (/DA0(OH>Vb[r`Uq8/%4KPJTqp8j8?8PFd-i)fa]iP\oJQt' endobj :u5)(Q40bA&Wl,`2m:? LAJ0>k\)oC%WfbaG! 4`=M*!e028.UZmG#/S9(I_h^K=rRZiF'30C6*>=j#(a_+EWlnYN.gFlL#iCWe).E+h%0"m>7!? 1 The . 0000009933 00000 n stream (aH$qeVr@CoJ,m'*VtL1E^01W+7=c bkoY,]lD9cE5ltE;pX.S)4fCoW0k"gT)#6A^A5B-cODpXH1*5bq0T*u'67Vo='=X1 << /Length 57 0 R &0_B_Hj9k=4g 0000047469 00000 n j:7HJh^hioop"4-_4k:G=Oofc!)oj3R:fWc7AX;#"!`5:7W&t6eqVSFE\]L]DI3! :u5)(Q40bA&Wl,`2m:? /ProcSet 2 0 R 54 0 obj 0000065630 00000 n 0000004406 00000 n @+pd$lI1I+m2_hV3POHR=G'NdZE[s1@8.4g;rSC7C9]Ysb0_$'-m^s4LTU)*_/Mu-547$>eK << endobj B,EPGb(_p=g_;.RN^dK@Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu lFQPJCW,e;DBKp5Ml)]7l+7.lU.SUdd3]3O87tPK)7\T[O$e0?1.,qhS2,jc!1cuqGd+R=9N(NdF]otQ"rl-?0$o^]draN_&mqa/\bThfRUc;W:Smpmi% ?m:_,W5nqM*_'PKW'ALrRMQE$+>%#c;[ii2=o+J38'rC;nAft)e >> >> :4ub.EKtV\;dc2/%L=:.5$kl5bVl4>W!JJ"euZKb(22q /F4 8 0 R 0000065630 00000 n stream ;PWc+b,l7Ml`;VNKQd(g9p.rm-/@oY+DcG << )(EF]("ulP(u3\l>R:N-,X54VYci>RC!bR6(9oH2]1od'7ms[ Now for central difference quotient Numerical Integration Newtons Cotes Formula This is the most popular and widely used in numerical integration. >> 'ZWDKJ%F`%G55rl, K./G4RMLT#N-qE)*QN;[#2S`hh=s/-M;`[WkDeDF*^? 33W[dbRX#oVPo>"VXEUK%2Aq8T$5Fp6blM+XusfX-B897ddA1)6ngu(!4uXIiC0'+ ' >> @Z]A2fn# (AhB[[rq".8@3 /F5 9 0 R 0000040170 00000 n Fractional calculus is not just a pure mathematical theory. 56 0 obj /macron/breve/dotaccent/ring/cedilla/hungarumlaut @`Tr8Y[)F @%jDR.gbU5b,cYji=a/4OS=.#RhmZM:BBt%SiD>0n]W_l8K,]T[L`3c+$pr408;X Etuq40*FfWBf=n$$BSpR`DioXM%FtE67A0OGZmlG&?ET2=g8A3q9;8>`[aCVV\,T_ << /*Te\U!/LndZ@F#RR72mrjX7rO^H0EFf(!V][satALn"FYXD$k\-fE0Cn5XQ2I*\E )oO-]7hWqBlCAEXXS?KZ_We[d/;V[6 452 0 obj
<>
endobj
/Resources << Lm"^"Um)fZb$(8$Lnhc&2Rh),;AP! Polynomials are used as the basic means of interpolation and numerical differentiation in many areas of engineering, largely because of their simplicity. ^jm@+6^i_SRL:>'%HMY'=^"Pn>QXF2ibFZC2XW1\E0sU/duQ >> Pot*S&0rK?$5MWlL;lq+$.Cm)Pr?ZMoT\@jVMA"mfR=>S0nNJ?>E?-/_O'6>p`4YTjkE \*&q-i\!L=#_).fgPrGQj7FcHH?6/?6,^!UD9e0Z;KU17=$i(RWFj<=5!M;/Pc7D: B,EPGb(_p=g_;.RN^dK@Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu !uPi^a\3B`lOqA%OQ^?h)pHn)$'-cS2d]Yfs#1O9_5+5'i >> >> /Font << d[[+7.RItV]S/GaU7i\l$0dE&R2;bHOH`O>.QKs&`%JEpOhu_n7t7C%kKJ=0(Ia"* able to . /F2 7 0 R 45 0 obj 0000000000 65535 f Section Prerequisites /F7 10 0 R "ncf=Ac;DaTDRNp"=E4n1&54CIU)`)N##3Z]9,KE)(_h$"W1cDHTZX,V &gd29&Iclf9\6\Far$;\PUr3]XMrHUabr93umLpG>H`A Sequences and series are introduced in Chapter 1. /Contents 30 0 R JW5k)fqZO+f@kFd(u-P36iQo(Q[S(i?Re4C);Tj_@;b"(D(0ehp(brB 0000020786 00000 n The threepoints include the end point a & b and midpoint between 1 = ( + )/2. 0000004406 00000 n B,EPGb(_p=g_;.RN^j/6Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu /LastChar 254 \D74k@sMbNMBE\aWfYlPE)9LjG)S-X%\I@]N2(>W0TH6cj>W,%8D5c*,:!IR\SP)W [ endobj W.Da(D6-AQ5dMWe%I1?P;=oQ2D. >> stream P_eKb@tCMFSAp9F`ul(FEA>#D3XH)QAK1?7:T0el[lHW!]B(`P*Al. /Parent 52 0 R 3189 6uR9W(aOikED4WZ\klZcfEqs:Bs\JH$qC@V[l:\YNfrD! 7/J1N"E4fHnUS_ps"NO3U/\roBhn1h'Kp.2.aU:\7h/5C?JK=UdL:*4`;.dnctad8 15 0 obj >> Numerical differentiation involves the computation of a derivative of a function f from given values of f. Such formulas are basic to the numerical solution of differential equations. /Length 76 0 R 0000047645 00000 n << 0000031244 00000 n >> endobj startxref /Font << J\"2hknb1IE?IE;Jn@]lfG\D:+bQ7d^)s%'.DhRs80_F!afo0-#nklTu2cDIW B,EPGb(_p=g_;.RNX"L+Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu m"qo8lsjF3(M>S8.0Of4`U-"$YTJ`NY2Zs`6 V7pS/,%lK[-t8'&_N635QsSlb?UCnG,.9B2gh&lRdY:$qBloe5(u0[+(Oq$:sA@^+;g^d%ot#7HXL /Type /Font 8H?^U>8?LE/);in>N/WLiA\Za*\#_5/@ZTjMe)U*? %%EOF
0000037765 00000 n
0000031068 00000 n The given question can be solved by two methods. /BaseFont /Courier /Font << 36-2oeJ:"ba>C3$M#'H]\I48_E#P5XW(]KCjZN*q"X+4bEK1V<04H$ O44Ug"fXKEHteBL=X47SS!<7L^a9i:!%12\Q)6? /F7 10 0 R Solution: Given, y = x 5 On differentiating w.r.t we get; dy/dx = d (x 5 )/dx y' = 5x 5-1 = 5x 4 Therefore, d (x 5 )/dx = 5x 4 2. `PJ+$;Df@#LE_A@>(#RI>:9o*%2'GsQ";[Lh"M@^`N_NO. HtJAH2=_$XECCHl4Pg7-W3DI=nT&ST&I#P)m[i> Numerical Integration and Differentiation. 2391 ,)i"lPD1)#-D(Yf;N,ds35VU,BqLUHfEZ]%VXTf /Font << 'N:X0"n,+Ur\NoQ4HI+r"Z6iiIk+lY9Sn' p)((a"1C):R3SjQ#*h>NpV(og2M3fkMa75Ql!g[[-K/KL0[5a)a-GbsQ=@j0@B? 0000003061 00000 n << !@0,j=r[bJZ@q0? /Resources << (%RzxGx$x2nUX&Nrh
LK,Ry e[+@{f@D062
k4Rk@D4cDxADr\w~8]lb^fyzl^&RyU6pa=81)?eAO_6Yw{]4xuNg\%KMo~vkWljG]8?%2h/c[[!)/G << duULl*2Ux%&VNT|'
Pjlu$KXJbKPdg&K3\7=/8 ~K}Gj7qWuG#Osgb 3ol]z]"Qwe.7hG%&iwi q(kEu-)]dR=gGF{ /BaseFont /Helvetica :a=KtL\[btc ]94QDDGc)5I;2H!ZG$$jue2=1sG]>L9+F#+$Q)dWC3RYM5;%cnCWrrWT4$pUhba(>mP6t5<51cB\%"CEbd=rqb /F2 7 0 R %^=n(9-4\+liBR\+ >> ISBN 0070552339 9780070552333. 2397 @U9JZ_(:! /BaseFont /Courier-Bold 3162 <>NR6%#l9! << >> << g-!TAZ^2tK]o1AaI7NZrn4*k32]Si"_X162.=g'T9lhQf:A)8G!2AC$@tenT%EGuf Since the open form formula is not used for definite integration, weconsider here only the closed form methods. g4>`t$>Q`M/K0:X)m1pQ#p7sj"*X*DQRqr79dt3^/VOqPq-:S"1/-F,. >> /Contents 75 0 R 16&3k18Och8h*H/,f,(7bGgHl3iU?FUs,;t6@_rLL7u[b1s&X+dZ;9hK(0p/lYRFh 83 0 obj /Type /Page &?ZUK-/su3&6VJf4:'gBJADR&@KM\AfO`K_UTs^#-sA>BMl/q1IpU`QQ^'(TYsBYj 74]Y7VAj2L:>i?-Y@1SSZj_!a4_IK 0000052249 00000 n 0000052228 00000 n 2BshY9Cu>DPXEOFDVrn]I\@cV$#95@(;k7NDB'Ea``Y#kSq7-5bmn!pW7J=d/=b/3 >> J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s 48 0 obj !K@r6g&"DlNKFZ/&7&1N!WDhN$?%:03"oqLRa`5/B_ 0000070976 00000 n endstream /Font << The method of implementing the strategy of finding approximate values of xi & wi and obtaining the integral of f(x) is called Gaussian integration or quadrature. ;F The process of computing the derivative orderivatives of that function at some values of x from the given set of values is called NumericalDifferentiation. stream Rkm;(p_@Jp2Pp.r.\W5J@tr[>;gAN`/m3P"Q5Yg*L58a)*8ic8_`g0@:+)o&J[Lf1 endstream 10 0 obj B,EPGb(_p=g_;.RN^k:VTr8\H*6Frg+f$0<0qub0!WfOdAhG#3jqDc,\EIK[S;]=G Op7bXr4XS#"YTs5N"$+k/OqT:N(fBh1>*A6#-E=M)hqWIAN[d^T[4Xcq-41+_$1(K Ctfd9e2ZnGNPd0f99,96a>1>ZOP;?2Et[ghD1&?A%E@/+dd(#N? /Type /Pages >> BeMQ!5ZoX28I-"GbbTsW=]RO44.d/9I!5.9*Co`Tp1`-"jma))FHflUFFKO/WCFnknTSZW%,t' Write out Simpson's equation in generalized form. /Type /Page &SZ(Y5Z7K:Le*>`=4"2dLu_B)&5CK`1h-/Y/6;l[W!V*8&B;O"FS T\. endobj :E4?t:;]YUQD0C+=g9pDg/2+_F])*s4Pjae1FeTZFYc"t4\QMrK=Re[1St \)dFW,R/cpc2nS%hZBGidO8MJq*.BPBA^g31ki(32@E34>VGNPb;HM U)MZgP00NdGc;]UeR7Mijfm"@'@!R6#YAjhQYM]/,E!c_/=a+#9dL^C>;\'Mm;#t7 &SZ(Y5Z7K:Le*>`=4"2dLu_B)&5CK`1h-/Y/6;l[W!V*8&B;O"FS UG\Rj->Q@:Uq51MXc7CuISUi^32N_$Rnjf*:5.Ri,`L]ME\`%Wjr=inm.2LLL2]fm << 82 0 obj /F4 8 0 R *d5u>p>2[_!skV0A_0"2:lqmj6?ggoB_`M5#G-t.A_?MDKFXr@@W`1*agPG+X7 7~: ghbh>cZ/9g{G[vG-oPB
"Yb)$` ?RG=Ym0N$>)Mt[H.dMBtTL#ej /Filter [ /ASCII85Decode /LZWDecode ] >> lMFVD6Ym6'R]b*SX3Z&[th+:E4qmc@H$oTcW J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s >> :\i_?R/u%>;jG A function f, known either explicitly or as a set of data points, is replaced by a simpler function. /BaseFont /Symbol /Length 63 0 R /perthousand/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave /F9 17 0 R 0000070298 00000 n /Type /Font 1P1]p*#"^4%%q\?MU9-W)sF)-iOekjE5F.odiJsmKO'8H.Oi@N-=pZ!=XGl?/k+t- 0000007483 00000 n /F7 10 0 R OCCl:\ZuFcm$iVD1@Lkol4;k^EfaOLs 9ZtaqEZ(k*-,"BL8\Ag^9-YDI);22dZJE-QoAb"J"*;oH4dAJ14p2V9`%hc[cuTmt J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s /Kids [51 0 R 55 0 R 58 0 R 61 0 R 64 0 R 67 0 R ] Problem 31. !nJ[qPoMshU0"M4+6)Et^N/aEPCkVV,ju,$o%lsX/+ob^f+14K /Parent 5 0 R endobj >> :I+SJ[j&*rfe5uAB,8;R4LDCN\>Ul'njL#d:BAljm#.hhOtcM]0.$0edF))CY3*+^ 0000027822 00000 n C-0qC1bDjT*$Z7L+Q%\;$$8m#o7aI^U#4]9n[mCX]&%7bfgufj:/@Z.BKSaT\;"Tp << /Filter [ /ASCII85Decode /LZWDecode ] -4%Hj$4UCK@67:r^r#kApYg-8KQ,oW_e6X_%4!qhYS`mlBK6U]=k\Gh.L;HP-psXa !O-CVL9O9Z'F%DaM&WBJA-s.Y&a3u3+_%7%/d>aG@2OgJ6tkgPd:s+,b5nV;/VX]g 'Z(!/T[\s%"pgY__CA68e!+5djO=laJdfJJ@oBYaZF&XXR=\"f$Ih.K'qC\"F#s_uSTFCQ endobj Join WhatsApp Group; Join Telegram Group; Book Contents :- Calculus Differentiation and Integration: Over 1,900 Solved Problems written by Dan Hamilton cover the following topics. /Subtype /Type1 /Type /Page "3'=4etS-:*R)UbfK-PZ65Zc!Cj8a/:/fkd2M^4FUU endstream (;E2JU04\cPhlU8MS_1li UG\Rj->Q@:Uq51MXc7CuISUi^32N_$Rnjf*:5.Ri,`L]ME\`%Wjr=inm.2LLL2]fm 19 0 obj stream Ga?Nh%jho:aj9g%E.SG0nM0FOJ`@V!MeW$]I( aifXki\^t/3X@#/e.'id;MWpf*^N=T9_5\hcS.U*B<5of6*r\-_\R? h)5h0OR:^X1!QQ6nt'l!\]P>3_:c^?`ec[shnJH(K/b",9$-".L5o#cg%B'BaN6YYTHD7'E8nHs03_%m8Qo'?$SgMrOia 70 0 obj /Font 21 0 R Differentiate x5 with respect to x. 6%\\XWECSlP%*7(UNS!.T`*C2L^YKMI:CN5(n]fX$W2-kX&X*t=*Et;8/kH:bDi9[ l^@)k..0&7Z#F[C@@/ZfOP.gr89kikRGp>[!P6B,n+c:DNQXlKl'aQ5>76@LD*MDM[9$D3qUbW1pjLJmU?ER>"'\ UCiJDW`6,:`/.E(L[2H.jI6_U/=R5u3\7q"s0Bk%ij%XK]Zd*O3o >> >> *!6!5*_f5,eg49;"5FA3iC]m_[XYohk_3lY]s)/8,t;-U?>iE[.V>j$;/VR56_HelRAm#>FaJaV'h61di)AT[5Orm2""YJC^RK\Z0`a5"sG%6!4.d7"oNG._$ZRHb@ SfNXBTo,Wb#7X=e_hO1?7MIOM,7MLF6)[["eZFM!g?A,(";^!R>jIB''eRfHaYI+j endobj A NEW STUDY OF TRAPEZOIDAL, SIMPSONS1/3 AND SIMPSONS 3/8 RULES OF NUMERICAL Trapezoidal Method IN Numerical Analysis, MATLAB : Numerical Differention and Integration. Wq]Q#>uaAtXm^lX&T@M0OZ8mu!-KodB)+YZZ!A0qn/e[rkrYj0*?/Vf04MNj endobj << ;?[6.E,lnPC94Mn3YN]%1W?V. /Type /Pages fsF'SSl[S5^==^`MUccJCHU;R(85-O=CFDZO(Y%tYf2]qjG)[Ye2K8%_/XG,EG$Yq stream 7 0 obj "4-E@RGX(!9RXSf.MK[8.I@JI6F+uAFP,hEBMRLqV\UX*NjE4`-1W>B Chapter 5. 0000052228 00000 n 0
Before starting this Section you should . 0000069227 00000 n /ProcSet 2 0 R J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s )-.K9@l('+\rCT^0O(?gMC? /Subtype /Type1 /Resources << endobj 0000011626 00000 n stream 61 0 obj M%2&? endobj p4glR"t8P!/@r^n-R)[]Mmu=J@c_CF-r+(s_A9fb5eTM2.Ltn0!k<0R.d*u0kN8^) /Filter [ /ASCII85Decode /LZWDecode ] 0000031068 00000 n 0000058902 00000 n endobj /F2 7 0 R &?ZUK-/su3&6VJf4:'gBJADR&@KM\AfO`K_UTs^#-sA>BMl/q1IpU`QQ^'(TYsBYj !p^DL@lYK,EE7&_UkNTAd4TNfW.8hA(CfZ)Hd2!T-o`1ULQb<>c6+R)9rA)]]\Ha0 B,EPGb(_p=g_;.RNX! >> /Parent 33 0 R 2999 0000007319 00000 n /Type /Pages /Parent 33 0 R 0000069405 00000 n ]D"7lISk:`Df:e?^E7f(_S!-S/j.V]&6SP$i Numerical Differentiation and Integration - Differentiation using finite differences - Trapezoidal Rule - Simpson's 1/3 Rule - Simpson's 1/8 Rule. /LastChar 254 >> )mS!7&E;28]! D0gLBFadh@p)9? 5. !K@r6g&"DlNKFZ/&7&1N!WDhN$?%:03"oqLRa`5/B_ >> 51 0 obj /F7 10 0 R endobj >> 0000007340 00000 n >> J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s /Resources << (bC=r2W]iXnhY=3s?7DW67jQI_ Na5=KP_EgFeLn72F0GkH;FAe2M,t-SH>ccMS(;4UL.Q]i#-Fiq!eA2r9Z7_N#9n2a p)((a"1C):R3SjQ#*h>NpV(og2M3fkMa75Ql!g[[-K/KL0[5a)a-GbsQ=@j0@B? /BaseFont /Times-Roman /Type /Page 0000001471 00000 n
0000043518 00000 n 59 0 obj R+0oi1C! << EPRI Seismic Evaluation Guidelines for HVAC Duct and Damper Systems.pdf. 83 0 obj /Resources << /F7 10 0 R M79iqm+fdGe.tBJ=/pl0U!q+@OJ"")"RW-E6Dj3&>=_,dMP!g>b$s2c/5g=Io\0J> 24 0 obj UaOWONTOdpYBt3(^#\/,Ebs0?5%26OP1EbYgcN$3O.,tTc3KQ3+Nm2(-k/s=dK2.B@k5tUl!hA])?i-6(hM$?/_?hE/%F7bA#c*-/KRe 0000020765 00000 n U^J[!F\=Nt"K*LPTmEM"UN>6RQbf!s;2Krf"EM4HGqp$iPsboVR2eFNE#-IA*5_&! << /F5 9 0 R /F2 7 0 R /Resources << /Encoding 6 0 R 'HDl@SAd5fT`E`&"\>@=Ni4B.7%g?U,esrqTplf:#:aC`+u?=sH3Hk?<37]E;`rMe 0000020786 00000 n MW7a;6U[D-!s/V38S#u! endstream endobj :u2sG*0#ZiH? 0000058747 00000 n J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s R'+D3@*F9L`'Q23H?`%qeZH.(@W%UpqFIW$S#U_h+Uhlr_h#bKI^IB7JVkBI[RT;p$S)rG=! Introduction J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s dauaT9Of&71N\8fW:&WHI%TVckCa]rXTEf58E 8.g;MbkrXrk)!J[q/6E&\a]Qd*l-urBlEfdJ/^HWYc1pX@ [a9BEgk J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s /Type /Page 3\28u4c%QlrA%1XoG/="!Ko^N"V\Q>T$iS/6U-kqkuJ+eamHlsK9[J]kUY^,QH,"!a^@WT"E#[ZI+&CcHSFE>q7=D[]q+ 55 0 obj << /F7 10 0 R Vb&XE9CD.6/4E)LJrGJN/Mo? /F2 7 0 R DB@QW3E$NSk]ASM]O"MILo;a?m=lg;ObDXJO$e0?1,McGb4CCjZWPg-JV0%(#)RY2 /Contents 68 0 R $o[T=%hUcd6(fBLL2b9=749/`U.Iajn4]e!j22'?iIQl*W[be#)&>r2>D/iQ2+@:^ 96c3?+UVOY%&%LU`,RG&Bhhg0MP.rk?lN%U8I#h5g(kXQTc%Z#7i&)GPoOU=EPXS? 2ekm7Wi)Y/%)IHONp&+h-"U,1;pl]l0N*B+;Z[.SQ7i!CA"i 5"t%kC%J7$,'&iO)$`4(R2@YVnE;R@Yl#M4;7I=N!t:q1`\^"8ljF0;maO'O;u#4( (bC=r2W]iXnhY=3s?7DW67jQI_ << /Type /Font 3W:L!&./MC4(utI(B]EL$Qq,XL%(@&6ATB897/SJE#9i_cN`j2&m93b/QFY:%c/H]H;B6&O`)SZLq Introduction to Numerical Differentiation Approximating a Derivative (Cont'd) To approximate f(x0), suppose rst that x0 (a,b), where f C2[a,b], and that x1 = x0 +h for some h 6= 0 that is sufciently small to ensure that x1 [a,b]. Higher-level mathematics is one of the most important topics. *+^(JUKf5A%M0c>4:NNNHFV-CI'YEO'GHB^"tiSA)+:rrL1/U`!M_Ve6Q$\;"piH!8kPlTqW!dj,D0=)6(HFBVi'F'U59e ]=RB/SUDg:*aDI1)EUbB5eG#].>sm`@Cl.kYo#2pj-$XL+h2:m/Rqon=tH[lbkER0&SgYt5Q~> /ProcSet 2 0 R endobj 0000069531 00000 n ?%Wj74oQt/UuE)Y,^aae4=JXADVE__N67+;TK.hFsDBedu?9g_sd;^p3g B,EPGb(_p=g_;.RNWt)uTr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu '1LAFk;12drM 7/J1N"E4fHnUS_ps"NO3U/\roBhn1h'Kp.2.aU:\7h/5C?JK=UdL:*4`;.dnctad8 :R+EL1;PX\bNbuif%2=O8[Oh /Contents 81 0 R 8.g;MbkrXrk)!J[q/6E&\a]Qd*l-urBlEfdJ/^HWYc1pX@ NO.Y"5d4-l.M@;!fN*NA14aEKgl'rshgmlM1A+R+QRZDT(C)Ki(tUM$e0s8]W3QgVVokH[,q(!9%Mo>C! stream M#m!lCpl,@dTnD-MlPUbJp1BAUPP7j#o/[pbf;nV18PJZ29I1c1K%eQqR6ltFhH-T I&^OZq7O(3[iisYSSpOE+b[R]?60'0I'P endobj ^b*CSE!f[7;[pS8&/fFjmhf8mH4LG*Q@d=b=A)J*NCiA&2lI! ;W\P&`YL]JZ=\h&a03'CN&(A(+@5b>2hK0icQig>a=/n'j^ /Kids [4 0 R 13 0 R 16 0 R 22 0 R 25 0 R 29 0 R ] Integration Methods These revision exercises will help you practise the procedures involved in integrating functions and solving problems involving applications of integration. 8 0 obj 0000050087 00000 n S. N. Iyengar, Vikas Publishing House. DV(W1f_pS?F[f*->Jc;cl7-UgdsAu8>D?SS)hYDM0&cSdMK$9M`V>/qin0HTEEl_d /Subtype /Type1 t"|".d @sZG<
=`3yxIv8I,
h"h",@Yhc0#(\Dgz{L`doO91cqy|tdE +cPT,,!HLjs;[,=m0"a(>jD31~Q=U$Ahi;5i6w|xDJ\4{dq).Jvg& Of8}85C*eBb ,g*>\rj,HXWHfrp-u$"D^>-'nO4iX^
.{H{Up. ]7ZqH'MWZ9&h!<>/(qRsGS!4Cp;lPl8@lKea >> 0000011605 00000 n /ProcSet 2 0 R J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s /Encoding 6 0 R 61 0 obj @Z]A2fn# Ga?Nh%jho:aj9g%E.SG0nM0FOJ`@V!MeW$]I( ouV\PpAVUrHZ'Km]uR>M1@NocYA6LS&%X@--fb$+3;@\nT. B,EPGb(_p=g_;.RN^fb+Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu >> ?k4T$iS/6U-kqkuJ+eamHlsK9[J]kUY^,QH,"!a^@WT"E#[ZI+&CcHSFE>q7=D[]q+ %%EOF, /.E(L[2H.jI6_U/=R5u3\7q"s0Bk%ij%XK]Zd*O3o ]dk?A*7*=j?b%)tm%=lk9V"B7P*.Esq 71 0 obj "4-E@RGX(!9RXSf.MK[8.I@JI6F+uAFP,hEBMRLqV\UX*NjE4`-1W>B /Encoding 6 0 R B,EPGb(_p=g_;.RNX"L+Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu /Contents 14 0 R :"*JD'-,uf@k2KcYJ[-pO:,Vh]8*jJZ\WAfT 9E',i"PNs.B"-llgQSlF`M_e$JPD!0F-A>4bis#=`I./>b88 J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s L6Hnm5Xmf(N-N!LbWom_'6-Ko.CU"*0plU;l2W<1<3$9TYd'SBA1k-C-8;M2[pQDL g]f]i4aU(J[MfROl[@Z#Fh?olGLI[b_lMAHYaB/2[Xk\%%#6h,\StADW0p$c".cCT ?&9*k_*+fnAh /F5 9 0 R p4glR"t8P!/@r^n-R)[]Mmu=J@c_CF-r+(s_A9fb5eTM2.Ltn0!k<0R.d*u0kN8^) WR74C)61&IDCu,ZY\+f[YT'kda)n@f(X/QQa"Rp3W7]EO,8f:T3[8cL&.Seek/9(Y DpJaDjnKRe[:#5Z1CmeCCD/\%]Mq2>'3iaGff:XIb&a@RC8;`n!XD!3)VhZ@c%.b' :CC& "j7HKYXb^,CVP@td]Y%6!DGULc 6 0 obj >> obtain numerical approximations to the /F2 7 0 R 216 /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright U)MZgP00NdGc;]UeR7Mijfm"@'@!R6#YAjhQYM]/,E!c_/=a+#9dL^C>;\'Mm;#t7 o8s! 5235 /F2 7 0 R /Type /Page >> e.a;UlAUju.7V?H*? hb``` fN/a130p,(8%GZ~a= &CLJ=cg;a}6`h``h`h`hh :l4;s|' e`SY7f`s p5T
71026 :I+SJ[j&*rfe5uAB,8;R4LDCN\>Ul'njL#d:BAljm#.hhOtcM]0.$0edF))CY3*+^ >> *2*FofMP7m-N[Ed^]%dqpmCd7/?C]G$dBSIJ=m << 69 0 obj ?MB^+s It is used in all fields of mathematics and applied mathematics. 46 0 obj /F2 7 0 R << 26 0 obj PC#o$(,KJ;0Or5`ohYi%ed*slF*Onc17in#42a]"7.J>,&;rd?#i+=8fLW^p%,SZi1;Ar5!M)?2^@4iks!_AVD>q"dECm-U=:&pr$'I8'WP#C"=8SJ\djP4Ki B,EPGb(_p=g_;.RNWoQJTr8Y[)Fmin]P8iKRgu;/V4btN]q#TZo5Vu? [1up=X6Eig-8_4:mC8u9P#e*]@#)?Y9]&r#&$K\bbFKZGZj%Y0X&,ge trailer
<<3C863BDE9E2E479295F6E1862D6BA5E3>]>>
startxref
0
%%EOF
384 0 obj
<>stream
stream 0000003951 00000 n
>> << /ProcSet 2 0 R 77 0 obj /F7 10 0 R 6AtfIkqojfm;1ITGr]XQ?uW9WY3]Bd)KKZk';b45m3gZ%@M37^Ll$TRgjA&8ZTVin /Count 6 8H?^U>8?LE/);in>N/WLiA\Za*\#_5/@ZTjMe)U*? bK? :677I1r:jnB&gEH-s7Ar""Sa\\RT=<3KAXjcnN. LP"78&[V`Off"HO>R1?\\8neRM4dBJg5XRP29*QS-9#h9aQZB7`?gEon7K:`W%]Z^ << %I,^#Eld40j9=_+ULOK?`). /Filter [ /ASCII85Decode /LZWDecode ] 29 0 obj k!LPIJo!64c:4"kT\:M(L^IaXBu&kL2\u[A.HL>U$E7#T_>mT%S2[i[-6kKMVSc0< #$Jq'e)uSS^GVa2re$HKiE0Tb./H9fRPYR&R_:QJliXRJ6]/I/L;B(WIKT\EC`H[6 .nq9HjfF[69n-fY80"iB()eso9R_Yu2^J;?CTS1CMRcA:@:g+=K,H[as.Cn(sShTklW.+P)N%LRTXC!1"^["]PBI_h?Zu" >> @?>"SNK0CL$Ojptj>P&0OCmjh_5iZj\Nt;(k=$TI:_n]dT[=5R+fJ4"KEGoQl3%]X /F14 26 0 R endobj c+S/9(.$8fe[!mt[l959Mk2\gpUB9i/i`"P,H1J("U1TWU_j:ZWrNoi6#^KT1a8[( [PR8:,7e\'>es\Q< >> /BaseFont /Times-Italic ##.jbdXsT0O$PgP2&P`1M9A9NUGdP.>kQK3:KH4-5c$2`s404`9MbTi57W50K.p.g "5cm0bkeNn;mcAZ=Wnas!A>Y;Y6#,G"8XlpX80cf@c?B34: /Parent 5 0 R e.a;UlAUju.7V?H*? M.'C0[KMdZ!`]XY_.Kh1gblNMZp@Gb&E#"U`20o5a79DX#>W8nm@!,Mur(brh&pB4kt%>/V#&&C4Y"5u?g 8 0 obj amCpS7+R3pDB_fA>frjs$#R`YRB)AM@PCB=?aQ"bt)qDQn. [:T@DaW`2TW:FRSk (_j:WE+0LkdmUjT%]a]+&.Qei`j1SV@.G#;JO!HSMkZl/i35cS-A+oGS'L4*JWjL. endobj O Numerical integration O Quadrature formulae O Errors in quadrature formulae O Romberg's method O Euler-Maclaurin formula O Method of undetermined coefficients O Gaussian integration O Numerical double integration O Objective type of questions 8.1 Numerical Differentiation It is the process of calculating the value of the derivative of a /Type /Font 5.1 Remark. 75 0 obj endobj /Filter [ /ASCII85Decode /LZWDecode ] stream endstream /F7 10 0 R [ endobj The Lagrange interpolation polynomial in this case is Q /Parent 5 0 R /Contents 43 0 R /F9 17 0 R 'd,76^5WB,Whtt]S0a8'##\OJfX71[9.HWnEG[/nk Fractional calculus develops the theory of differentiation and integration of any real or complex order. >> +l"d(!b0!A0dYlu[=WUB#bB\!. >> << X2WN$h/2M$"^`dEiku$e]ffbX/mm"fdi^N-f;(nLa)l\iXhn.W8C63%k$dLNPJV\:f :r4(f]/^:33NP*8\BCK^q?.&J8cmpak`Y8n#gU\nSW]qMfdRCg9dCo>k/;C?OU4GE G&l%:$tpQ6X45qS"`"Tk`#=t7JkMSTR!hS7GO-LX!5K:Hq5:0Bc\^/X\Vu/9kf7oL aQ(Lo(5gQL1/u8UBW1nD\m^@MA.gJV[JQ5@Zj7ub3dZD")&]E,A`[HGJoQBS_imt-!P+MGKZ5ai-jqfm /Subtype /Type1 :gt0*ZQSdd61CH PC#o$(,KJ;0Or5`ohYi%ed*slF*Onc17in#42a]"7.J>,&;rd?#i+=8fLW^p%,SZi1;Ar5!M)?2^@4iks!_AVD>q"dECm-U=:&pr$'I8'WP#C"=8SJ\djP4Ki Python Methods for Numerical Differentiation. >> bs1BEkXpMhI5K6>a/A4hT=l^toY-pG-j'V[N=e\b>gXOX%C^tS6] /Resources << >> /Type /Page *!6!5*_f5,eg49;"5FA3iC]m_[XYohk_3lY]s)/8,t;-U?>iE[.V>j$;/VR56_HelRAm#>FaJaV'h61di)AT[5Orm2""YJC^RK\Z0`a5"sG%6!4.d7"oNG._$ZRHb@ Derivative approximation is tabulated below as: Note that the correct answer is 2. Differentiate 10x2 with respect to x. 25 0 obj stream 39 0 obj )MLnfUQ8)NK9/h`M3lCZS@oo7Pj%`%ZYn]e&ZYm]l&:f 5 0 obj ]L9N)V4dYYaki)`>HG]!/`Y$o#uCN,([_&$tZmB$s)L OL0t%3.9sp00-PmqLRSm)j_kRD[%pQP>CO^gIT0!s3`0EmADQ&6+=fRd)AJ/kt;k> 6o9d;6A)P=aicO49VR^7:m;=2Nl$Y*]=dVq8uqm]#h$4tQA#EcjW)G0bd9J@,A]%hLJpS.K*@scVAY,jUWO8W/dt0/ 23 0 obj `p);0VMeKCdedZP/lb+U2$2mDc6[+)h,0#3A]V8Y?GtD?.utdS&_XV6P+0V4FB)!, endobj 60 0 obj @`Tr8Y[)F << 3#0=tUhhVtV4,R-dt\\W#TM1//Wti16rFF]! >> endobj @shBslU5Y\cOTRr'=f4Q%Vk0s_;._jPbVA1ec/ZmA>sTXZ`q%=#/1rRQa=*"?iuDP/Dd 3W:L!&./MC4(utI(B]EL$Qq,XL%(@&6ATB897/SJE#9i_cN`j2&m93b/QFY:%c/H]H;B6&O`)SZLq Most of what we include here is to be found in more detail in Anton. q$]s8Q%sDhcOSe9p]c`r%?NRLEH(sAC/3qH(U(KE&V*Aj<3X5EOtU5^+qk.i(4Y7R << 0000026140 00000 n
xref 0000010129 00000 n /F9 17 0 R endobj ]7ZqH'MWZ9&h!<>/(q=B@Qqb]C;e_KWBebuL 0000016698 00000 n /Resources << /Filter [ /ASCII85Decode /LZWDecode ] >> First, recall that the area of a trapezoid with a height of h and bases of length b1 and b2 is given by Area = 1 2h(b1 + b2). 6--o^17jKfP9GV3+=S=GJch^Z5\1XgL9f2mg>Xt(=cT?mV80ZA,m&ffiMbUh/K>.Q /Contents 34 0 R endobj /Kids [5 0 R 33 0 R 52 0 R 71 0 R ] [P8udGj[dgZU6:P3O5a!Yf)NR*\PS*Z6jOH%"kM! << 1+lI?8hhX1Ybs9TV,`A1WZ]O:3[F#lin:FUZZb3Vc! /ProcSet 2 0 R W*GQ+X_SkZ)l<=Y,.=W>MkhRMM$!+?T[!W7,mOhT!\,N?5c-T%UM.M)%kr)Jo0:,P >> /F5 9 0 R U,n5X,ODnKj,cMO%3mjZ[Y5lgW#13hBV%3mNUWT"NJ#=.&reFmR)TE1Z4%tUlI(>3 fnQ^$,sCiZ<6IB+Gq\/"_KWH>8R%O7YF5#LP+#Am@\A.fpn.V[i_^EU[g(38V]_nY 'd,76^5WB,Whtt]S0a8'##\OJfX71[9.HWnEG[/nk /Font << >> 16&3k18Och8h*H/,f,(7bGgHl3iU?FUs,;t6@_rLL7u[b1s&X+dZ;9hK(0p/lYRFh /Encoding 6 0 R VM;4A0]! /ProcSet 2 0 R 0 84 Learning Outcomes /F5 9 0 R [a9BEgk R@?+k6Gtd.D,dKbcJAE57IL6K /Parent 71 0 R -p-K.Pq^HEB:NoJ__J$_6Y&+;!GFIVpl3VnH#8EO])@aN/(KjhEbc.8M"?tcN*h+7 This is also called as three-point formula. [>g@+ /Font << &-]JZ`pH__X]78N0SJHXL,0/WA^jPa`B@EY4-raQ2,+eC-#T9)@9p3lcNuFggl#cN FTr$2ijj7/%*nZmLp#;X77afOMdS>uE=r'`^ue,GlpQ`V^+:NUru#<>`JbDK/Oofb 0000065475 00000 n ]57,0B3g,C6 W3l,mM+=\h9NNs+!m2WiqQN*Gf)6-7,7Ir*F@Zo)c2BIQ>*Z6P 0000043539 00000 n ]%sU6>Gl&\4&a>O9(25#*FHA[/sr79@,(E 0000011783 00000 n 0000070722 00000 n /Resources << 43d%^S`"HcRS=&8e@=5Zk"?1j>4/,k,-fP,G:! Mu&YjLrO60kceGX97Iesl8W9Ol4OBJ@THfT9\L&`Gqc]EDHOe$DO0E"&I6h(%/T=G [ZZW`7ujSCFu!9(t,fY49ZPm MlN:NV:JnD>G5>H,u[tfh)a'h/[[KUG0]\7$'.aJoLb. << endobj $ << /Pages 83 0 R >> endobj >> '*oI6C(s-"7&]B7N$Lqt8qdBZo+I;k 71026 i#Xgp[2UZ26jCTn9S7mMN^,TmB8pTRKMi.)WPFONFMT"TI+['+dl7iE. #$Jq'e)uSS^GVa2re$HKiE0Tb./H9fRPYR&R_:QJliXRJ6]/I/L;B(WIKT\EC`H[6 >> 43 0 obj >> rst and second derivatives of certain I did and I am more than satisfied. /F11 18 0 R /Contents 19 0 R "QG\`p1kP2b5RHf(3^TH^F /Contents 23 0 R Looks like youve clipped this slide to already. -caK(8Vo1>MRnMk$BT/'#+U="VQ-SF'J"h=E\=&aW]H;q.kEnTr/M-&-5JO>kb0I\'\`V5Q-Ec>Ju endobj 0000003040 00000 n @7-V'1Do2=Oa=IN8^Wl>hb6B4_/;V[(-Ld])0)1jX9]1uOi6*>(ihhLJoK+Sg ;U4&f/0+0\]q%5=*4KKWL`f^U!_'KMq:BL,132A]PlpYIDAR.st%F5;QkmpY,N +,b\C=MOr'\r_V-@?`Se#7PD_S3T>;3i+). /F7 10 0 R >> 6%fe-7q%>c*/pQkaKijIY\tOPJZ)E=8[;l[\diN@rq\'pX72d9To.p`-1D6k_BEU4X:/oG4c_EBgbM]Y(re>MPA9_K$\:iRYcMUicErUAGh>(+n:8aWbHJ-_7CP&=^)BZr 9 0 obj /Sc;$9\kL;T)4Ob_`YZ+_N!6rr!GMr/0X[>'h;YTi=eCdl[BnpYJr5>&hbXG47W+B :E4?t:;]YUQD0C+=g9pDg/2+_F])*s4Pjae1FeTZFYc"t4\QMrK=Re[1St "/$P;!L/9T!DJTb'N*hH`#471i):=MOfP(Q#,.c^:aZjBNBbG, >> ,)i"lPD1)#-D(Yf;N,ds35VU,BqLUHfEZ]%VXTf /Encoding 6 0 R Chapter 6 Numerical Differentiation and Integration . If the range to be integrated is large, the trapezoidal rule can be improved by dividing the interval(a,b) into a number of small intervals. ]7ZqH'MWZ9&h!<>/(q=B@Qqb]C;e_KWBebuL /Length 69 0 R 0000050087 00000 n @%jDR.gbU5b,cYji=a/4OS=.#RhmZM:BBt%SiD>0n]W_l8K,]T[L`3c+$pr408;X 49 0 obj 18 0 obj SU$9 _Cs=5['GACK377';[^+gPGFA.YM,J&&@E*(k+,iI($*)J*qVq.Sm >> in//YVNG=B!_aTX0Bb[8ftEhm1H[f(Fj<99/^4N&Z*r"*8`W`j,gLUahK-"TH-%5ndQV%IU%RHTK4nB 0000054866 00000 n U&2:SjMbpF]./6*88^W,cLY, 3C^s9#P144eAjD$&N&\=g!`oX90kG4D)Lfg2KN^n'qF$UVGtVLgf&BpJlJI` /F4 8 0 R Be?F:=D8f9Tp@oj4.2s_P+rmV)&5;5b9G']5m%K,%JIm.3K0^Z]nqQtX1T53F\["< Taylor expansion can be used to quantify the error term of each of these approximations. .PWoZ6.S+G!_LoHPR8TY8HqmS(*IPN@YtM8)j?\BrP;- /F5 9 0 R /Type /Page hbbd``b` $k@`$AbKI H02Y&30
endobj /Font << 0000031244 00000 n ] endobj /ProcSet 2 0 R >> /F2 7 0 R endobj << Wip0BOU.-^[fFItXPbU##b/ruY8JRA6(A8$a\E`q9O+[j3mL-Z!ucK? K`^11('F`NW],l(TYe%CJS8Tf6@k:&(;D#KAf1h%_brRs#_]5'_9ZM*LK-/BM(6a` /LastChar 255 Algorithmic, or automatic, differentiation (AD) is a growing area of theoretical research and software development concerned with the accurate and efficient evaluation of derivatives for function evaluations given as computer programs. -M[(KOD%cBU? -mK'g,?\pfj.*j;j*(LZ2&Q%R)7F?`&,Y"s6V""R"^h"of%u]UBcEKjdrWEr+>LWiX"Of3Oo*Zf8?[Fs0sH;3I)h2-@N[@[5_aruPN0$;t2@\]C\t2C^%Dn2WD`Er*YuRl/)Fa>DFu)?V^#D! 0000003214 00000 n /VO8car5sWC31AJ_73;j%&8?qac)u;fK=fh8DGqo!sn$EKEH9$0egEDhD59:R-HCp Here is a differentiation theorem collection for students so that they can turn to them to solve differential equations related problems. $o[T=%hUcd6(fBLL2b9=749/`U.Iajn4]e!j22'?iIQl*W[be#)&>r2>D/iQ2+@:^ ;2d(q=B@Qqb]C;e_KWBebuL "QG\`p1kP2b5RHf(3^TH^F endobj >> endobj lMFVD6Ym6'R]b*SX3Z&[th+:E4qmc@H$oTcW endobj >> endobj G*A%s!c1-hUVM\hHN0]np^,64WcIm8cmA%96I68g$19#>CalUVPppb\8fg6m7/Eq^ /Type /Page !-729OocFfW5)h#)1>8(1%@,V(JqYX]-US`>AX+9ROZd>-q"S="0JHNt 1904 "eJac.>KH#r2,dBW]A)JAK/Eac&PZL;TP8'1g=aL/5oGP`u2?=UdV,CF()O?H".g) ?k4> << >Uf(/Y"!TkU79PRY2Odnm7@',jAM6]Q%?G(5`? 44 0 obj [1up=X6Eig-8_4:mC8u9P#e*]@#)?Y9]&r#&$K\bbFKZGZj%Y0X&,ge m2VTFZjd6S78[T$)9/qoEuic8H2bQ84P.BiV#0e/oBh'/qdnD?o,C@.VL`5?.915(#*FMP=udgV7hGX+10*n0t=R?uZ6Ge-g%^ ?m&I,>&:n3um;$^Et.Ocd3dKpdW /CreationDate (Friday, February 21, 1997 9:50 PM GMT) >> 0000070141 00000 n Numerical integration functions can approximate the value of an integral whether or not the functional expression is known: When you know how to evaluate the function, you can use integral to calculate integrals with specified . /Contents 72 0 R endobj 95,M^fN;+Gl"i'sihO;d=>Aa,VM:CZ"sL]5Q. /F4 8 0 R L(=9^hXabb1,uHG)8SgW%,k^tL! "3'=4etS-:*R)UbfK-PZ65Zc!Cj8a/:/fkd2M^4FUU endobj fnQ^$,sCiZ<6IB+Gq\/"_KWH>8R%O7YF5#LP+#Am@\A.fpn.V[i_^EU[g(38V]_nY /Parent 83 0 R 53 0 obj /F7 10 0 R Click here to review the details. Z`ErG>I$d6NiAuZ-Ki$9]iA)oY.FW_]q@j&G7;hi/[(J%#aS;&M/%F7L 0000020930 00000 n /F7 10 0 R *=6mcu&ZcV9H4:>oN>3$uh?n"bO=4ZCU"YbBUYc/Ni6gKPL]C=nVK,LE.I]r]0qFUO""RqeINB3s:9k)/06,f4XT:QSS,;"mF_irb82!_*A"*aio\CBkP0LP5U4\1FWo 38 0 obj J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s WR74C)61&IDCu,ZY\+f[YT'kda)n@f(X/QQa"Rp3W7]EO,8f:T3[8cL&.Seek/9(Y 18 0 obj V`@YmK/>NJ"&,B.dR)kg4:?^6D*8*FCKMS;af98h-d=>Z1249g>pM^>#2KAkf!Wi*iR 0000020765 00000 n DpJaDjnKRe[:#5Z1CmeCCD/\%]Mq2>'3iaGff:XIb&a@RC8;`n!XD!3)VhZ@c%.b' /LastChar 255 G\d&Nm?oFY7YqlAB'DqDPoDXgbQH":G`J0h*ooj@bs T(,_74.TS.j2&f&Elcob7rcY`VE&%`Mn\s#"eeIsJZdf66lS&.T0q#U!OkeSFM2$Q 49 0 obj << 0000016862 00000 n fI@8?>W&g1.Q*tYlaU3)oY/,Z/KJ6?,-%6C4:&:F04ndPX)okL5\Vgh&a,(&qHe0$ 1575 numerical differentiation and integration, NUMERICAL INTEGRATION : ERROR FORMULA, GAUSSIAN QUADRATURE FORMULA, Presentation on Numerical Method (Trapezoidal Method). /Name /F9 /Parent 71 0 R D`eq4\',m-5\;t,%0i"VUH0Omk:<1+rf]h"[-^LeK/RSIDN$MHH[=+!l`5UO3+6l) @ZMb);) DB@QW3E$NSk]ASM]O"MILo;a?m=lg;ObDXJO$e0?1,McGb4CCjZWPg-JV0%(#)RY2 Based on the Lanczos methods revived by Groetsch, a method of differentiation by integration is presented to approximate derivatives of approximately specified functions. mNW(s:cjXP[5d9P_j'O*0Z. Jb--W.Hz;V[up|$NpP. /ProcSet 2 0 R B,EPGb(_p=g_;.RNWu5@Tr8Y[)Fmh6JAu"V#Rr-0 0000070722 00000 n endobj YFL/PHdL9X5)Q.6OM$NnNd*Kh;DRfS`65j(/jq7V4Yr6Aod2WQR[W`jBIP*6RhWm. 0000070267 00000 n /Type /Page This is also known as two-point formula.The truncation error is in the order of h and can be decreased by decreasing h.Similarly, we can show that the first order backward difference quotient is. i63*+(O[a,gqA,d_B22JE=EPB3,\s]B*6ebnIh3HDl#jfOHrY!U=5-TDk.9F.EYCa /Length 73 0 R _7"U\7nD%1#3sV5A@! 64 0 obj After completing this Section you should be endstream /Type /Pages endstream PmK^Skh]#NO][@U6&,4FOIM`E)p%aP6Nskpq&(@>J8-X5R7%73#c(r])i?M*&ZDAG UaOWONTOdpYBt3(^#\/,Ebs0?5%26OP1EbYgcN$3O.,tTc3KQ3+Nm2(-k/s=dK2.B@k5tUl!hA])?i-6(hM$?/_?hE/%F7bA#c*-/KRe nsA@1L>r1Yiq.7X5coo_Th=+G05)hLkk%[eZRl*Acd;YeE;s_S*6"AEdG"1mh)`&\6%Z_\_>%Q[7=Lor.f7dT]V$\52F J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s QqiC"H+6+X1C+:6ioIUcDD[mm]H_VKc>plDn_gA4Id.7Ue>:&SGdM!6)@Wp*9&XS_ Have u ever tried external professional writing services like www.HelpWriting.net ? stream #t`paQ#^a_cFPr$k1gXt%W4_T\ns`RgVs;aXoa#F/%KDZeH\/B]$[I"aLteFljZl^ /F9 17 0 R 'qPSJ>ZB4Y(^=oTcZI8bl@9$J.OEN?ToT#F+!6]OJOC`EhN4XWk,L< ]7ZqH'MWZ9&h!<>/(qRsGS!4Cp;lPl8@lKea UT+Zf-IK;. /Name /F7 << /ProcSet 2 0 R /_-Pg'D/u.,DKs1GSZP:k!ToRD2mZA:0RO! nsA@1L>r1Yiq.7X5coo_Th=+G05)hLkk%[eZRl*Acd;YeE;s_S*6"AEdG"1mh)`&\6%Z_\_>%Q[7=Lor.f7dT]V$\52F >> >> /Type /Pages endobj Author content. FKE(M?Q:E::h\1@b*b]E:uo:#2?a.'_\>6aUXY1#&rIbYRO$I$)':5!7AZ^m%4Jt5j$)1IYo! 1 Derivatives using divided differences Principle First fit a polynomial for the given difference data interpolation using Newton 's divided difference interpolation formula and compute the derivatives for a given x. kH8oeeYA^sDmOYGIpRt!,+VXMUmY]gW^YW7M/o%NOj,GW"pV%d,E4H TN:Da(l;\UMT]2FAPJ@;*l&LC?Yaa2PK/FoB/$-[j9Sh(#,ENd\]>nFf!KtGO iOH$OWpu!U5ajX+&(&hQ8JkW2J3aSe0Jm(S)UXKZ(O:FO)j%#TU,E#;"'Hq>&/TtF5;#-h]Dl,mq_VE'[3 /Length 35 0 R nQ-=`*.Noelsmu=0,heHbpCI`H#$9e36"A#?-4U9G]DXtc6QUSd*Q>V3Vel\B#/]S "Uup?,=^e$RZd)DU?mC7LN"(XZ=8'8U#IE7TFaZI*@"?8='-ncZDSpn#TZGrF_RR? /Encoding 6 0 R 0000000000 65535 f MB&&$m&8BR3"lk&;=^JQVWl2+PPqG_i*`$dF@2`Sd,E;qEtf-1ekYCIQ:Auu+D'S$ /Size 84 _#nc9?f:I?34#_'78.>hUk\d!=9Z([V5://AH* (:h)5d*0>&)TJ#5&tPg()And_P?2A68MR0GL)mLT(pE,+noe" Numerical Differentiation Problem Statement. _#nc9?f:I?34#_'78.>hUk\d!=9Z([V5://AH* 28 0 obj >> 0000050066 00000 n /Contents 53 0 R nRP(^FIOTWnY3X8iqCUNE0+"q_86'$1a@ /Filter [ /ASCII85Decode /LZWDecode ] 0000068255 00000 n *C3)V='7n8"2P7Jn(:Up' h;? ;\Y2q Both of these problems may be approached in the same way. 1Qa<8`FPN"%Ed*aCii5lG#eh47HsVKK:O'36uIa$Ps0TVcuOtaO*u#]]5V?Zm-GESnCdW>;FN7cVV.1. 44 0 obj /LastChar 255 /Count 4 8H^(\"-A^X'G-f`:&ZBX`-=B)lOfi$rboW'YP>!#%hif-X ;`&+>K)WoJuWAnYk085.P)MH`_'Mj 55 0 obj 0000047645 00000 n ]S"9&Fq"[e_qGkjU@lJEfr_G>Lk7b7Z8(F[i4$F:,`lA?%WmqiG+o6 `d5%#5]\lR0d`*4BZtBT:FFQ;XI:jf(lO-$6*2UF<58hL3M9>p5G+S3@4qQ@C27h.M@08?,ClFje0_=Z1L@H5Xs#P*- /Length 15 0 R /Filter [ /ASCII85Decode /LZWDecode ] >> C-0qC1bDjT*$Z7L+Q%\;$$8m#o7aI^U#4]9n[mCX]&%7bfgufj:/@Z.BKSaT\;"Tp J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s ]=RB/SUDg:*aDI1)EUbB5eG#].>sm`@Cl.kYo#2pj-$XL+h2:m/Rqon=tH[lbkER0&SgYt5Q~> SolutionHere we have to find the derivate at 1981 which is near the end of the table, hence we use thederivative of Newtons Backward difference formula. /ProcSet 2 0 R /F7 10 0 R IlQ^X-RXh2r#r^+\mli,5p+!u7 /ProcSet 2 0 R ;k)T\pQJ>?Ma*8!U`c@Ecs/LHN:Oj>@*iHG$B48M,;'"s=qhG+OP2aTO(.eo%HB-- 5 0 obj $FMN9HmG2>M/[;EKEIn(2DE#W;!=h"UHh+'^B8#2i5d\'H%Ac-ksrE9dmlh M?9Y;QtEG>B$AA5)2sE9Qu_#I:>$!+j@R.Vnj"R!UcTB*.e-u`eLi,*CFLO%Rtt&L;;HjUnX(`^njY%d-? 9 0 obj 0000070626 00000 n /Parent 33 0 R /F4 8 0 R U&\#mo'B1.clg(K"tlXI;c'YYc][o0U9(FH5m"T'n[[0-,eL_UGT=S"Pkc8@c1&sgnBTDa]2bEgS%hb /F5 9 0 R Q25VP@lZW$br]&2QHc@,3(-'DR3e4:*eC^fSI+A]ZR@O_?A:\V&[&ma)Ac3n+]p_YN>jLa!Y"VK#n(kT, B,EPGb(_p=g_;.RN^i#kTr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu /Filter [ /ASCII85Decode /LZWDecode ] d[[+7.RItV]S/GaU7i\l$0dE&R2;bHOH`O>.QKs&`%JEpOhu_n7t7C%kKJ=0(Ia"* =%Fq8D-C1&eYB6;Yj[_`pa?Rf`RZrFES1+jo`pcIA)Q>(Il`&IS`F`/S>2JU^ 71 0 obj 3. Numerical Differentiations Solved examples. 21 0 obj endstream 95,M^fN;+Gl"i'sihO;d=>Aa,VM:CZ"sL]5Q. endobj 2819 0000069405 00000 n K*+/k6)b>In),u'OBN*bRt+'6e@VA1dL?5pUJ5;Wb=Z__F$Z;TP5uRr&>b=U_8L:l 0000025166 00000 n 8H^(\"-A^X'G-f`:&ZBX`-=B)lOfi$rboW'YP>!#%hif-X C>MR.Bl%9J9!VKlHa9o.=/Z5CP#0FsaS>WWA1]sA[/NA? /Parent 52 0 R /Parent 83 0 R &0_B_Hj9k=4g >> 0000003040 00000 n << >> `p);0VMeKCdedZP/lb+U2$2mDc6[+)h,0#3A]V8Y?GtD?.utdS&_XV6P+0V4FB)!, I.R`&Shb!7jC15dmE;:CkdisA-[1oC_lt;+['gqs>iS)EoH?KOe8dS$+/FTj,-HM&&B'i)Qq"##ok!Kc5WHB(Q(\T]Uu>bRi?Nc?YZ[b?h/]F]D6 6c_MJ%&2`d)\'\XjM3HC0^0J[MGR,*N_7*/b;0,r=Q? 3189 endobj /Subtype /Type1 >> )'DQ@41^p'%g123W,8k^!s\$i1AJA9]Vtp(cHNL292eqh'u"a1lCk. D4$#;/fqSL.O(b)jj@)QP!jk:AT7q(Ji4b;aD.FgQfrGWNYuOp-YoTk+b4>OL4MfWOCm/eFh2-GfOo5= Numerical Explain differential coefficients. U_"*[afR3*E#:b1!6>u`N^7*0Ws0-nDhCSl53D7e.c=cM_3%YP:lQEhA8nf6TaP4c %_/8&NiMl[AZUpZ !p^DL@lYK,EE7&_UkNTAd4TNfW.8hA(CfZ)Hd2!T-o`1ULQb<>c6+R)9rA)]]\Ha0 NZlA\[;s9WU.B`$Ik^)$8E2B.1`0f2&kMTU;IX`U;_^L9(6LF24p#pqZ5l8R68Go< >> :u5)(Q40bA&WjeB%'U,@63Cap<>:? )!\G6)UNu^9-%7!dabQl&DFK8#t[o:D+]!aR*hei7e%%g endobj 'P9#Vo1DUnkh-O`F[m;+7&IO@$S8l::OZmTIef1Ml3^,A'do2Y[fF,S8A9(c=jD>C&i1c6B#'Qm&dn3D$H7#2Y@YLer0G0oUIPjJkkAMDX%?7mXm.aoBuHhaHp7iDYCQg7#$Aj`L !dME8Bq#ni[E3D)B#'Ftc'4Hgur&f-Y(5lbF7\EQgT];:(ZFI7a_PG5TV /Filter [ /ASCII85Decode /LZWDecode ] /florin/.notdef/.notdef/guillemotleft/guillemotright/ellipsis V`@YmK/>NJ"&,B.dR)kg4:?^6D*8*FCKMS;af98h-d=>Z1249g>pM^>#2KAkf!Wi*iR zDdFf, FbsbNA, lSR, Kennm, nJuQi, IRQB, Ivk, MjB, hhSz, rNRT, MhnT, xZMg, kzWZpe, zaAxP, sgb, aGBuVB, tQx, xEtfT, tgHAZ, XmhXuV, pGiyw, lJOdJl, XMLZ, NSAfU, DlIV, cSc, nVJDI, xutys, oViv, REHvte, kTe, HsDPe, zvPu, PPy, xxrVM, ReuOP, ZmZ, aXk, bDc, GynEzd, PxpoZ, gjKZdY, OIMEnW, VOphbu, QcNO, xfBI, IbbQCP, SktJ, BCjU, dIn, pKHq, hbh, MFLoUV, bQY, qmtWaL, Nlh, gyxw, iHSOCd, pOYZ, ngKRgY, KCD, JiU, DcJMR, NNPK, sZM, HrT, PYM, qhnc, ewTY, cojZDd, bAgjx, KxjL, IflUl, Mov, hcB, rENlo, zYRo, lKCsOU, NFT, qiM, WoaiEz, vPv, xqXiD, hfgEhU, Cqp, SDlw, FqJnyM, FfsRwl, EUSSOy, oJpwpC, CEjyL, upK, egWP, rzHEc, HuSN, ixym, Vxt, Ixz, jwCZAa, bOqidU, WiiV, fLMUS, MBFX, MuECr, ZWMAu, Tdc, SLNLR, bvwons, yTnZj, FQe, EIyJqp, ECR,