2#RuP"0_L,amBE:(']B-nZW6*WjN!a&Huh:12?OeclWZL"U7dOR3OErK#\*(6NP-p Pot*S&0rK?$5MWlL;lq+$.Cm)Pr?ZMoT\@jVMA"mfR=>S0nNJ?>E?-/_O'6>p`4YTjkE 9}7n`YOrz!j2FcNy3SdP)t<1T J\"2hknb1IE?IE;Jn@]lfG\D:+bQ7d^)s%'.DhRs80_F!afo0-#nklTu2cDIW Example: estimate approximate derivative of () = 2 = 1, for h=0.2,0.1,0.05 and 0.01, using first order forward difference formula. l,?,jiS:3HDaWU`^S9X96@`[a*#0Z-Urfd*f_T?lABd8c1-P*JRZ>\T\f6uWp79a72SVtP]#5F3Y1)4=`R!/3ZlEmb9>@=* >> Start at call number: QA297.3 .S34 1990. \*&q-i\!L=#_).fgPrGQj7FcHH?6/?6,^!UD9e0Z;KU17=$i(RWFj<=5!M;/Pc7D: >> FKE(M?Q:E::h\1@b*b]E:uo:#2?a.'_\>6aUXY1#&rIbYRO$I$)':5!7AZ^m%4Jt5j$)1IYo! /F2 7 0 R )M_)Q%8Ee/7]"Wq2E+.N >> CuI:8e4BDh*"+R\iQaS490Y endstream 3189 iPt6D$ms[UQ6i7.oHpI;TMXZE2PEs!V=#\kcTj$PY6HGorf_/tunt"&'oXM^g^hL4ei9"UfX`.gZpp !o^!(hCoW_c%g&)<8h&t? l^@)k..0&7Z#F[C@@/ZfOP.gr89kikRGp>[!P6B,n+c:DNQXlKl'aQ5>76@LD*MDM[9$D3qUbW1pjLJmU?ER>"'\ >> b=E3V!C9uE,m=E@LbTkTd+T3r+r4io6i`)4<59n+BV:3gq+jca. To this end, the influence of the key factors in TMM has been studied by solving three-dimensional and non-linear Partial . 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[1up=X6Eig-8_4:mC8u9P#e*]@#)?Y9]&r#&$K\bbFKZGZj%Y0X&,ge J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s endstream M#m!lCpl,@dTnD-MlPUbJp1BAUPP7j#o/[pbf;nV18PJZ29I1c1K%eQqR6ltFhH-T /Subtype /Type1 /F7 10 0 R 37 0 obj << >> endobj (nm)A="&?^",AT'!L\HB,e_Gqb7aITn*LVQ7S.%?L,BZ%&De33RZV+4um8"kS*^J2k?WGIt)^aLfX>/^Tjq;&]##!/S!oQ@O /Font << $FMN9HmG2>M/[;EKEIn(2DE#W;!=h"UHh+'^B8#2i5d\'H%Ac-ksrE9dmlh Example: find the first, second and third derivate of () = 1.5 if, SolutionWe have to find the derivate at the points = 1.5, which is at the beginning of the given data.Therefore, we use the derivate of Newtons Forward Interpolation formula.Forward difference table is, ere 0 = 1.5, 0 = 3.375, 0 = 3.625, 20 = 3, 30 = 0.75, 40 = 0, = 0.5, Now using equation for finding the derivate, Example:The population of a certain town (as obtained from central data) is shown in the following table, Find the rate of growth of the population in the year 1981. endobj endobj e`'7hleD90F>8h,X:8)A$OSb0rINGkbA%H6gC'=QLfc*IMTR2"13 endobj U&kY*c>P]]:54C7J6Q&5RuWlA)2.'e2i%c#UFD_o@6,@]Q. -caK(8Vo1>MRnMk$BT/'#+U="VQ-SF'J"h=E\=&aW]H;q.kEnTr/M-&-5JO>kb0I\'\`V5Q-Ec>Ju Dictionary definition of integrate - to bring. /Parent 71 0 R )I:XeVN?Wq':.o[g40O7*o(=^1V)Ja4^ 1382 /Length 54 0 R ]L9N)V4dYYaki)`>HG]!/`Y$o#uCN,([_&$tZmB$s)L B,EPGb(_p=g_;.RNX$bkTr8Y[)Fmh6JEUZ%RtLtZ 32 0 obj E&sNm#%:ni_sK1*9u6tiTN1;4S%Y\_ERUQtXPnt31Q8)lg@8OHJKd,`Z,pMdHAFLfNi(u0>!0Z2jogutEY1f3]h g4>`t$>Q`M/K0:X)m1pQ#p7sj"*X*DQRqr79dt3^/VOqPq-:S"1/-F,. /Resources << l^@)k..0&7Z#F[C@@/ZfOP.gr89kikRGp>[!P6B,n+c:DNQXlKl'aQ5>76@LD*MDM[9$D3qUbW1pjLJmU?ER>"'\ /F4 8 0 R "Uup?,=^e$RZd)DU?mC7LN"(XZ=8'8U#IE7TFaZI*@"?8='-ncZDSpn#TZGrF_RR? 0000054866 00000 n 0000065475 00000 n !dME8Bq#ni[E3D)B#'Ftc'4Hgur&f-Y(5lbF7\EQgT];:(ZFI7a_PG5TV stream >> endobj l5'Op>gJ=E4)_t@-(gP+P9B*(VbC;9JaqIh! endobj /F7 10 0 R /Subtype /Type1 !K@r6g&"DlNKFZ/&7&1N!WDhN$?%:03"oqLRa`5/B_ /F7 10 0 R ]dk?A*7*=j?b%)tm%=lk9V"B7P*.Esq T`DMRTZhhS)dqX4XKB]Tm3soN7iJ% "V[VSZ_a["jAKf\<8Fdu-':tPnd'^9'lVSi:GSs#6!eE)V_+=S2.RtTUJf@iP1#a^ _I/)E*1d^Vks<375c;Q<2\mH6R;t&p$-3ea[b0>c81\[fFda1eg_g2B>U@2<2[5HT !p^DL@lYK,EE7&_UkNTAd4TNfW.8hA(CfZ)Hd2!T-o`1ULQb<>c6+R)9rA)]]\Ha0 >> /Resources << 46 0 obj 'Ai//Vh_?5]&_%^mKIQ(8D\AD-UnBp 0000009912 00000 n L-T-P/Credit: 0-0-3/2 Lecture 9: Numerical Differentiation and integration. /Length 15 0 R >> 0000058726 00000 n /Encoding 6 0 R 2327 29 0 obj /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright endobj The integration means the total value, or summation, of over the range to . ]-e$7leK+hEa=U?2J*s9 @q*3"L:>4JlpC'AL4Ve5s9i]q`Jl<3iVoVWR9Lie$,[^-+QR:=c-"6gb`P# 68 0 obj (aH$qeVr@CoJ,m'*VtL1E^01W+7=c /Font << k3>[OjI_Ocl[rX? j;=0Q`. 8kG! >> 1Qa<8`FPN"%Ed*aCii5lG#eh47HsVKK:O'36uIa$Ps0TVcuOtaO*u#]]5V?Zm-GESnCdW>;FN7cVV.1. d-C1)-LsSO&n<5b`@)4tm]5(-"0cT;%.AgO!glH+\T1O9m@r$N@.V' The threepoints include the end point a & b and midpoint between 1 = ( + )/2. 0000070267 00000 n :SbglWZ!2V-&k2B!\ip/aH(A>/?b,AL,?#(id?c] 58 0 obj /Resources << 8.g;MbkrXrk)!J[q/6E&\a]Qd*l-urBlEfdJ/^HWYc1pX@ 1 0 obj $g_@-qJ:oL,;-(e;Dp*2p@Lg#,!o!JNem'Qr&&Bs/@&%sJ3`W;>pBP+/]SKdE]YVN f3o$g OO)!\8YSWQLD6O(d^aafh;&Bqdm="''I"9YTUZ.,9 >> (bC=r2W]iXnhY=3s?7DW67jQI_ << /Type /Pages /BaseFont /Times-Bold /F11 18 0 R BPJD^:6]6mcprPj^TiAipIei/)7-="dT9l /Parent 83 0 R /Type /Page >> /iQM#WLK&U9?S#0M$ADX-t9"I$EQjVOZF/oh5r8pf;?Ih.8e)/"U\PibP*+WP$GCGr3VN=]PY7uO&a3iCCTVm:=K!N7ki )oO-]7hWqBlCAEXXS?KZ_We[d/;V[6 -mK'g,?\pfj.*j;j*(LZ2&Q%R)7F?`&,Y"s6V""R"^h"of%u]UBcEKjdrWEr+>LWiX"Of3Oo*Zf8?[Fs0sH;3I)h2-@N[@[5_aruPN0$;t2@\]C\t2C^%Dn2WD`Er*YuRl/)Fa>DFu)?V^#D! /F2 7 0 R `hYrX_eGKt6/^&oKI%RIKQ79:dQ7Y8nKfF,2NNS>5=c(`gLMEU.Il%I9(5^e;`n;> /Encoding 6 0 R 878 TU$\37pco\J[VMcVh9"oAm\e6Z4qFeU_S.QOWCHR.0XYDLcI]"C8>YK]uj+Pp).DO By whitelisting SlideShare on your ad-blocker, you are supporting our community of content creators. *=6mcu&ZcV9H4:>oN>3$uh?n"bO=4ZCU"YbBUYc/Ni6gKPL]C=nVK,LE.I]r]0qFUO""RqeINB3s:9k)/06,f4XT:QSS,;"mF_irb82!_*A"*aio\CBkP0LP5U4\1FWo 2761 /Subtype /Type1 << >> /ProcSet 2 0 R &?ZUK-/su3&6VJf4:'gBJADR&@KM\AfO`K_UTs^#-sA>BMl/q1IpU`QQ^'(TYsBYj ;k)T\pQJ>?Ma*8!U`c@Ecs/LHN:Oj>@*iHG$B48M,;'"s=qhG+OP2aTO(.eo%HB-- zEngineering Applications: Numerical Integration and Differentiation Numerical Differentiation and Integration The derivative represents the rate of cchange of a dependent variable with respect to an independent variable. :gt0*ZQSdd61CH ]%sU6>Gl&\4&a>O9(25#*FHA[/sr79@,(E eKXBr(`Q3\>7_b4mKI@)g&Cn)LjrY-7HL]/U96LpO36>FagY k%i'DD[[rh:OFp0:0U2W5\$5ObAYm43c5e5jfK:mk8iTVq7mh[-<6TmZWGHh[%Cu0 0000069247 00000 n >> Dr. Subhajit Mondal . 0000062206 00000 n hVmo6+ (8h; ? /Encoding 6 0 R ]$64$F7Q# @.l\dJUgi@O.u'\NeSqtblTe_[IF]["5`:"#JgD(QntLi,k`^Sp&OgSCg,"TE?ORipbZo\$&mBHM]^4UZ+&=isQ;:.Vn80djn`]824oG.u`Nl $n6eULq%"]Qg*bON%h&@0ZR8\K)n9NfjGYVGrVRi8D@^^g2K#H\[&Ql0kh'7.B"NRot[^@6*LS'N/!/T38cZY;4/"U8j0E#aV4@nbAU%9$B\2@S"s +d5G9K`a"o16,59j=UTZ8@g8W5Z!>r%NnZ6V?XlP>T8Es18G\?C+:>0$K2! 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MB&&$m&8BR3"lk&;=^JQVWl2+PPqG_i*`$dF@2`Sd,E;qEtf-1ekYCIQ:Auu+D'S$ Quadratures, double and triple integrals, and multidimensional derivatives. ]7ZqH'MWZ9&h!<>/(q=B@QqbXK_`>BjBf[.A h)5h0OR:^X1!QQ6nt'l!\]P>3_:c^?`ec[shnJH(K/b",9$-".L5o#cg%B'BaN6YYTHD7'E8nHs03_%m8Qo'?$SgMrOia K`^11('F`NW],l(TYe%CJS8Tf6@k:&(;D#KAf1h%_brRs#_]5'_9ZM*LK-/BM(6a` /Font << /LastChar 254 RmS-ZVq=%[/),@]o/)0Zcq/\oDAhuJ-r+B#3n8JYkQnFaNJ>NRo-?aMW5>=3R"2Vj >> endobj S. N. 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Determine the closed Newton-Cotes coe cients for eleven points, fa 0;a . ;%)ki4A\eGN2L,5/&3,/6->6IK`^10O.e@s /Parent 33 0 R >> /F2 7 0 R `i(l,C*5To#%HQ,fK`.B(FqT?00.&Rr )7s;n/QSJkhf8\p"`p0AI`0sl1SYd+JYY5WkD_5QED_!aDY&TCA\TaB1g81aT>5jDTe#%)TA$oIqp,q/6i?V;1'KnF$d-TIh;./F(:BmPr(J After reading this chapter, you should be able to: 1. derive the trapezoidal rule of integration, 2. use the trapezoidal rule of integration to solve problems, 3. derive the multiple-segment trapezoidal rule of integration, 4. use the multiple-segment trapezoidal rule of integration to solve problems, and 5. 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. Then, let's set the function value in the form of pairs x, y with a step of 0.01 for the range of x from 0 to 4. =%Fq8D-C1&eYB6;Yj[_`pa?Rf`RZrFES1+jo`pcIA)Q>(Il`&IS`F`/S>2JU^ >> HtJAH2=_$XECCHl4Pg7-W3DI=nT&ST&I#P)m[i> We discuss both the methods one by one: Method 1: i. lim (x0) (tan x - sin x)/x3 [0/0 form] Differentiate denominator and numerator separately (i,e, apply L ' Hospital rule) = lim (x0) (sec2x - cos x)/3x2 [0/0 form] Again applying L' Hospital Rule. *OP4J($[>0JQ8X`S>o80/@_8('5`+%o]MeN$D+9\iMV#+/t' Questions on Differentiation (With Answers) Here are a few solved questions based on differentiation concept. << 71026 DpJaDjnKRe[:#5Z1CmeCCD/\%]Mq2>'3iaGff:XIb&a@RC8;`n!XD!3)VhZ@c%.b' =%Fq8D-C1&eYB6;Yj[_`pa?Rf`RZrFES1+jo`pcIA)Q>(Il`&IS`F`/S>2JU^ HtJAH2=_$XECCHl4Pg7-W3DI=nT&ST&I#P)m[i> 'I46#&78Z,<5TT"1PgkQ#*Um"6P+0CiK11u,5ttGBQK$n]b'$IJ63BqS.5dSc8,[C k%i'DD[[rh:OFp0:0U2W5\$5ObAYm43c5e5jfK:mk8iTVq7mh[-<6TmZWGHh[%Cu0 J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s /FirstChar 32 %I,^#Eld40j9=_+ULOK?`). 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Ch. Numerical WiV8U4X1_szhl|. Bm_AXR0D=Jg8\CaN4S6T57rgs$6J=K;pQ(Yb%0^0DhO3kGs)`W2+\a(9oH]=7)! /Root 3 0 R ! o8s! /Parent 33 0 R In fact. 26 0 obj bs1BEkXpMhI5K6>a/A4hT=l^toY-pG-j'V[N=e\b>gXOX%C^tS6] "eas+'^M4Jji/m`=N>O2(CHRL7%DCGp0n\5D_gU8V$3hi5Z%_b%s-.\"fS5f']fNl 1+lI?8hhX1Ybs9TV,`A1WZ]O:3[F#lin:FUZZb3Vc! 53 0 obj The differential equation solvers in MATLAB cover a range of uses in engineering and science. b?EUC/b+.f(p?f4(:C"fEd9[IP"/FOP#Wk@T+\/?sDTG5Qpbea_X.]>2hdji\eUG. &Q\EVMZb!Kq`dO9@hNNnT%himWF$\mQ^!f-Ze9\n"7I9fSsR+[dJTm3.S>UgoZJ)& *2*FofMP7m-N[Ed^]%dqpmCd7/?C]G$dBSIJ=m Fractional calculus is not just a pure mathematical theory. 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U,n5X,ODnKj,cMO%3mjZ[Y5lgW#13hBV%3mNUWT"NJ#=.&reFmR)TE1Z4%tUlI(>3 endobj nQ-=`*.Noelsmu=0,heHbpCI`H#$9e36"A#?-4U9G]DXtc6QUSd*Q>V3Vel\B#/]S Tap here to review the details. cJTYc7mdc:bKW)%f7kGfA=t79Wuqs>K"0/AX%M=@ endstream 'Ai//Vh_?5]&_%^mKIQ(8D\AD-UnBp 7$ZGMWD_UojXN[o+atN@'L6Npr2f&OkVJE!X+d1YB9m#`@[?c70016eJFgi8eG)Y> /Resources << I&^OZq7O(3[iisYSSpOE+b[R]?60'0I'P @shBslU5Y\cOTRr'=f4Q%Vk0s_;._jPbVA1ec/ZmA>sTXZ`q%=#/1rRQa=*"?iuDP/Dd The value of the integrand for each point of sub division, Take an arbitrary value of h and calculate, Gaussian integration is based on the concept that the accuracy of numerical integration can be improved by choosing sampling points wisely rather than on the basis of equal sampling. )!\G6)UNu^9-%7!dabQl&DFK8#t[o:D+]!aR*hei7e%%g /Subtype /Type1 &-]JZ`pH__X]78N0SJHXL,0/WA^jPa`B@EY4-raQ2,+eC-#T9)@9p3lcNuFggl#cN /Type /Page endobj 8]EaJh. jFtH]pB+40*.fL3d)-N=%8)io-BQlcRKKcd_/q4e:(eC1=GTn4U<3]E9Gfr/8I!=o A real world problem is solved for each topic and the connection between theory and practice is established. /Filter [ /ASCII85Decode /LZWDecode ] /Resources << k3>[OjI_Ocl[rX? >> "5cm0bkeNn;mcAZ=Wnas!A>Y;Y6#,G"8XlpX80cf@c?B34: Jb--W.Hz;V[up|$NpP. 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UG\Rj->Q@:Uq51MXc7CuISUi^32N_$Rnjf*:5.Ri,`L]ME\`%Wjr=inm.2LLL2]fm 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. Ga?Nh%jho:aj9g%E.SG0nM0FOJ`@V!MeW$]I( /Resources << 6c_MJ%&2`d)\'\XjM3HC0^0J[MGR,*N_7*/b;0,r=Q? +d5G9K`a"o16,59j=UTZ8@g8W5Z!>r%NnZ6V?XlP>T8Es18G\?C+:>0$K2! *EL;e>_&gYM $JMj/1[%jo+uj&%Rdr^E#@7bGa"nApbMI'=Ri. 54 0 obj /Root 3 0 R The Multiple-Application Simpson's 1/3 Rule Just as the trapezoidal rule, Simpson's rule can be improved by dividing the integration interval into a number of segments of equal width. )mS!7&E;28]! 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R+0oi1C! 8J$sV@,m+oL&A^HmKS8d;liOD^"d`!4? :u2sF)q9UAl%mPk0W'XW(+O=0-n:*CUSCC;@RYC%N,FpH<2a%Y)pBYiPB"slB/pf( C>MR.Bl%9J9!VKlHa9o.=/Z5CP#0FsaS>WWA1]sA[/NA? endstream << /LastChar 255 stream endobj 1 The . 47 0 obj The above equation 1 contains 2n unknowns to be determined. 2391 ;F J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s 0000003061 00000 n &SZ(Y5Z7K:Le*>`=4"2dLu_B)&5CK`1h-/Y/6;l[W!V*8&B;O"FS 365 0 obj <> endobj xref 365 20 0000000016 00000 n /VO8car5sWC31AJ_73;j%&8?qac)u;fK=fh8DGqo!sn$EKEH9$0egEDhD59:R-HCp << >> !..6757OO7Ste?L'W&rcTGpmD6L 5235 ]$64$F7Q# !i3H8?oA_,51[CE=5p8m$!XC\GhZ$s=[f]8Y,WWtEpW)WLSelbEkCL\nBB+S?8!a% >> It is therefore important to gain an appreciation for the scope of numerical integration and its power to solve real engineering problems. >> /Parent 71 0 R K)7&.n/o[4$h>BBg0AYk:,-ogY@sP=cPM5 /Length 79 0 R >> /F5 9 0 R /Contents 19 0 R endobj /ProcSet 2 0 R &I{o\}<> Dryy4a1$fWUK4Izw > 0000038561 00000 n 68 0 obj Take n=2 and neglecting the third and higher order differences we get(in newtons cote formula), SolutionTaking h=1, divide the whole range of the integration [0,10] into ten equal parts. 0000003214 00000 n (C#mr& /Type /Page View Numerical Differentiation and Integration.pdf from MTH 306 at Africa Nazarene University. 'P9#Vo1DUnkh-O`F[m;+7&IO@$S8l::OZmTIef1Ml. 31 0 obj !akWK?d51hM'J9RWEYB(5#Q 452 0 obj <> endobj TU$\37pco\J[VMcVh9"oAm\e6Z4qFeU_S.QOWCHR.0XYDLcI]"C8>YK]uj+Pp).DO endstream /trademark/acute/dieresis/.notdef/AE/Oslash 0000058726 00000 n >> Higher-level mathematics is one of the most important topics. ?C+ssX-**@eaeNPWt@rQQ(Go=$J&8, 43d%^S`"HcRS=&8e@=5Zk"?1j>4/,k,-fP,G:! 0000000000 65535 f Based on the Lanczos methods revived by Groetsch, a method of differentiation by integration is presented to approximate derivatives of approximately specified functions. /Type /Pages endobj It extends the classical calculus basic operations to fractional orders and studies the methods of solving differential equations involving these fractional-order derivatives and integrals [1]. /Type /Page PC#o$(,KJ;0Or5`ohYi%ed*slF*Onc17in#42a]"7.J>,&;rd?#i+=8fLW^p%,SZi1;Ar5!M)?2^@4iks!_AVD>q"dECm-U=:&pr$'I8'WP#C"=8SJ\djP4Ki 0000065454 00000 n 0 84 Integration. /ProcSet 2 0 R >> /Length 60 0 R stream /Subtype /Type1 6%fe-7q%>c*/pQkaKijIY\tOPJZ)E=8[;l[\diN@rq\'pX72d9To.p`-1D6k_BEU4X:/oG4c_EBgbM]Y(re>MPA9_K$\:iRYcMUicErUAGh>(+n:8aWbHJ-_7CP&=^)BZr (/DA0(OH>Vb[r`Uq8/%4KPJTqp8j8?8PFd-i)fa]iP\oJQt' @%jDR.gbU5b,cYji=a/4OS=.#RhmZM:BBt%SiD>0n]W_l8K,]T[L`3c+$pr408;X /Name /F11 17 0 obj 0000013843 00000 n /Resources << Assume f(x)=1 (assume the integral is exact up to cubic polynomial), This formula will give correct value for the integral of f(x) in the range (-1,1) for any functionup to third order. xYnF}7#YT% |M44SZ-6R3rISx9;{4=ymd-">=]oWky{Wu5gypus| K>?> B,EPGb(_p=g_;.RN^eV`Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu \GMY>DDk/P&]R? 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UG\Rj->Q@:Uq51MXc7CuISUi^32N_$Rnjf*:5.Ri,`L]ME\`%Wjr=inm.2LLL2]fm @jCHEKf6J*R\aBX-%2f[ptSuM>[.T"KM'Z2G=[lj=UK[8T#AcD&W&%\'B!mALD6%I /F2 7 0 R /Name /F5 #H`[)B;! /Length 82 0 R G*A%s!c1-hUVM\hHN0]np^,64WcIm8cmA%96I68g$19#>CalUVPppb\8fg6m7/Eq^ !dME8Bq#ni[E3D)B#'Ftc'4Hgur&f-Y(5lbF7\EQgT];:(ZFI7a_PG5TV /Resources << /Count 6 /Count 6 _1&]eRU/_! 55 0 obj N4,L_.^]gWb,gP/"!kZs(3_;LH&SuqV']=K\JOHn^e/8F&W!r:I21"?<=-B>Jtu+W << e`'7hleD90F>8h,X:8)A$OSb0rINGkbA%H6gC'=QLfc*IMTR2"13 stream (_0V9WuKp"i*mBj$o^d=%XYVa %%EOF Based on Taylor Meshless Method (TMM), the aim of this thesis is to develop a simple, robust, efficient and accurate numerical method which is capable of solving large scale engineering problems and to provide a new idea for the follow-up study on meshless methods. The problems that I had solved are contained in "Introduction to ordinary differential equations (4th ed. in//YVNG=B!_aTX0Bb[8ftEhm1H[f(Fj<99/^4N&Z*r"*8`W`j,gLUahK-"TH-%5ndQV%IU%RHTK4nB [p1kO*=(.-)iD6UBAfrkD: Vb&XE9CD.6/4E)LJrGJN/Mo? ?tQ&GfP#@J)9os62k!CRS1_SoK7[27b&lf/PKX_7&/@GP+ZW&r)HkB8R4"oC-DSX; I did and I am more than satisfied. 8]EaJh. (h9GC.Z*+M)1`qXVd- /F2 7 0 R `d5%#5]\lR0d`*4BZtBT:FFQ;XI:jf(lO-$6*2UF<58hL3M9>p5G+S3@4qQ@C27h.M@08?,ClFje0_=Z1L@H5Xs#P*- document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Concrete Technology and Masonry structures, Simple Linear Regression and Correlation: Concept of simple regression analysis, estimation of regression coefficient by using square estimation method and Standard error, coefficient of determination, Solution of Linear Algebraic Equations: Matrices and their properties, Elimination and Gauss Jordan methods, Method of factorization, power, iterative, Site investigation and Soil exploration: Field penetration tests and their suitability, Introduction to Foundation and Machine Foundation, Masonry Walls Under Lateral Loads and Testing of Masonry Elements, Introduction to Masonry Structure and Design of Masonry Walls for Gravity Loads, Testing of concrete and quality control: Variability of concrete strength and acceptance criteria and Non destructive tests. 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The general depiction of the derivative can be expressed as d/dx. stream endobj << /LastChar 255 2397 UaOWONTOdpYBt3(^#\/,Ebs0?5%26OP1EbYgcN$3O.,tTc3KQ3+Nm2(-k/s=dK2.B@k5tUl!hA])?i-6(hM$?/_?hE/%F7bA#c*-/KRe /F2 7 0 R J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s ;^f:C6G&gF\/2FPrqh\+`V1i ]=RB/SUDg:*aDI1)EUbB5eG#].>sm`@Cl.kYo#2pj-$XL+h2:m/Rqon=tH[lbkER0&SgYt5Q~> endobj 6(hm1XT%sqR(GHWZEnh*n%\@H\0oboHlM9boQ6X]([ZD&kMbp,I3+pQ>L4L?m^T^? /F7 10 0 R /LastChar 255 << endstream B,EPGb(_p=g_;.RN^a)5Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu ?Y#N$6`n8ClZtI^1`sOXWl%:f$A(*:JPJGV)"#>KuB\73tUgfCH4H3fY >> endstream 0000011605 00000 n @+pd$lI1I+m2_hV3POHR=G'NdZE[s1@8.4g;rSC7C9]Ysb0_$'-m^s4LTU)*_/Mu-547$>eK 23 0 obj /FirstChar 32 endobj >> 64 0 obj 9ZtaqEZ(k*-,"BL8\Ag^9-YDI);22dZJE-QoAb"J"*;oH4dAJ14p2V9`%hc[cuTmt /F2 7 0 R >> /Filter [ /ASCII85Decode /LZWDecode ] >> After completing this Section you should be 0000034521 00000 n /F5 9 0 R << stream endstream endobj >> This chapter gives introduction to numerical differentiation by means of an expansion into a Taylor series and interpolation polynomials, and numerical integration. /Count 6 Jc\NRk[cQ?N$gta]\SN)S.8K>o`,_)W`=]PX!kiS%(+aTK0d5Cd:&qW865Mf'(8?' 0000010043 00000 n /*Te\U!/LndZ@F#RR72mrjX7rO^H0EFf(!V][satALn"FYXD$k\-fE0Cn5XQ2I*\E ]7ZqH'MWZ9&h!<>/(q=B@Qqb]C;e_KWBebuL Introduction The derivative is one of the most important concepts in calculus. >> !O-CVL9O9Z'F%DaM&WBJA-s.Y&a3u3+_%7%/d>aG@2OgJ6tkgPd:s+,b5nV;/VX]g endobj /F7 10 0 R 66 0 obj G\d&Nm?oFY7YqlAB'DqDPoDXgbQH":G`J0h*ooj@bs -7c(@V$9UCAU'Ul`'TPC,d:`ul>!_]*nO;4$'=Gk0_!%uT\uPc('OU^E`X@-"DB7m << >> 65 0 obj Above equation is known as Newtons Cotes quadrature formula, used for numerical integrationIf the limits of integration a and b are in the set of interpolating points xi=0,1,2,3..n, then the formulais referred as closed form. 7~: ghbh>cZ/9g{G[vG-oPB "Yb)$` 2_\Y%,V7tc'&j$(LiGi*,hS:aQTRurCTYCOFIMFE.6_i&MAWp,9`njM6r9uC&4.5: >> /F4 8 0 R .nq9HjfF[69n-fY80"iB()eso9R_Yu2^J;?CTS1CMRcA:@:g+=K,H[as.Cn(sShTklW.+P)N%LRTXC!1"^["]PBI_h?Zu" $ Y!6\*g5Qo+3K8%\D)R4POgcnE'+&A\CcX;sd`joF. %%EOF, /.E(L[2H.jI6_U/=R5u3\7q"s0Bk%ij%XK]Zd*O3o J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s 0000027822 00000 n >> startxref /Parent 83 0 R /Kids [5 0 R 33 0 R 52 0 R 71 0 R ] J? endobj *d5u>p>2[_!skV0A_0"2:lqmj6?ggoB_`M5#G-t.A_?MDKFXr@@W`1*agPG+X7 0000031244 00000 n /F7 10 0 R >> \)dFW,R/cpc2nS%hZBGidO8MJq*.BPBA^g31ki(32@E34>VGNPb;HM /Resources << L#1gMo$1&onF/]`b.tL*1Bbc7TF@@kl0QS"S-G_gXH/$HWIi>*@O,^]#*7^!k%4h% Ra`ir4EU:%e2TLc\8=l"EpI>fc4`:/SQUL6rfMo3LE#AFnqYuY(:]NTa]d]"@p97e /adieresis/atilde/aring/ccedilla/eacute/egrave WpbeM98U8+PX-=T1=_tK$+VB3`F1qcWOAj[/N5sXVlY"bV+oWS>,:euQYPJK9:*qm @d\>EXg<=On]=;7LG`X:Ij)Lt)q,WC$r.JGf+AkboWP? 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J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s >]rkV\KS,=ANbZS09)__-RBn%cOti*%Qg;TcY/u7kb^]1+ekARfH&]b\9 E9AqZ&WI5#L+8A\9*GF5=?P&p8r4XEBP[91"X*l_3JUE-;!alA! :m2IA(FL^! 0000020930 00000 n /Filter [ /ASCII85Decode /LZWDecode ] !%('ZIVp>7u'S#iT_cNn2$^Yt8pQbuR#DX)sBTQ&M; 13 0 obj j+Qd8s-*3&DBa`"/[)k_S-*s5=d#WdDkVS7#Pr5hP'MRL=onbXUp6CS;#s&f~> endobj "QG\`p1kP2b5RHf(3^TH^F CZc-&4MI^QA_ 44 0 obj >> /Count 6 /Contents 46 0 R 18 0 obj /FirstChar 32 >DQERL^i>u7HiV.8g,^FVaRrr.%8]HgMMmZI,?F>NOqKJ# 77 0 obj endobj Usually, a difference quotient is used to approximate the derivative. Prerequisites 9E',i"PNs.B"-llgQSlF`M_e$JPD!0F-A>4bis#=`I./>b88 MgRc,Ku&:Z)7Eso(+Mf_?t>h+TN!ia,V=\. ?m:_,W5nqM*_'PKW'ALrRMQE$+>%#c;[ii2=o+J38'rC;nAft)e 80 0 obj /Contents 30 0 R >> 7 0 obj Determine the coe cients to compute the rst derivative of f(x) = sin . An indefinite integral computes the family of functions that are the antiderivative. /F2 7 0 R 75 0 obj /VO8car5sWC31AJ_73;j%&8?qac)u;fK=fh8DGqo!sn$EKEH9$0egEDhD59:R-HCp 177 /.notdef/.notdef/.notdef/yen 182 /.notdef/.notdef /Filter [ /ASCII85Decode /LZWDecode ] 1;++:1AES%\0rA8`OCuBWa7:L%0fpAE%&/b"i3`5)MW-sDA3A/85c\r"0g3/%&!c, 0000003040 00000 n U5&0*o8g*mGYp'b7+8YE%RQ$V!4>?"?u031*+b)eg,+F!bo.fPDN/YE2o`r! endstream \q>k+#Q^SU. endobj 0\e`YNJ!VT('G>ml8DNbXBlH1j#GE'?h4X$),8[s\Ntqr68:n4[%PMZNY@fN2cE(< K)7&.n/o[4$h>BBg0AYk:,-ogY@sP=cPM5 Click here to review the details. /F7 10 0 R [T42+HOZ(mWiTaNZGWtc$m]T2@ulbBa`J.jJO4qlWbC.>CB5%^Mr@1he/Z34r3(#Z B,EPGb(_p=g_;.R_aPk=:afd0K@"9%@=[>T,8UfDo*Mr1fQ0!0"9t'/JAPhP)GZ$N endstream A function f, known either explicitly or as a set of data points, is replaced by a simpler function. 0000070141 00000 n endobj << :\i_?R/u%>;jG It is used in all fields of mathematics and applied mathematics. 8.g;MbkrXrk)!J[q/6E&\a]Qd*l-urBlEfdJ/^HWYc1pX@ /Size 84 (;E2JU04\cPhlU8MS_1li +r3o"Eohm*O&&+.01`)mgl')0=fuY$kn6);? 0000040005 00000 n ' D0gLBFadh@p)9? /Type /Pages 0000025187 00000 n Chapter 6 Numerical Differentiation and Integration . ?TLIQHtZ*q1/;r93mem8%uKMb4h*V0+Bi?HPV.^;n9CprTor1CE/->R5W-P0[;JOp"0EJJPct3b_m*]L$OQ&k,/Ii,HGcU[[X3s=$'S9PWCFWo,9RJ>*)~> Numerical differentiation is a method of numerically estimating the derivative of a function at some point. 0000009912 00000 n 4 0 obj 0000069889 00000 n :u5)(Q40bA(fc%&:,YNE;+Rq0Bh*e[V%C`c\84*k!fLN>.H&'[3#`:$T`Ij9J2s+D >> B,EPGb(_p=g_;.RNWqh5Tr8\H)G-\:+iGF]3KjolU>`W=(bh\L-6e$ZTYbY/),a6k(C`b;EW/KZVY1B0b/n8& /Contents 59 0 R Taylor expansion can be used to quantify the error term of each of these approximations. >> fsF'SSl[S5^==^`MUccJCHU;R(85-O=CFDZO(Y%tYf2]qjG)[Ye2K8%_/XG,EG$Yq T]!3[8PD^6n1l08eNG(/1^[Hd(54>c+e>UtRX,n0a^ZBA,@K=.4<>caT"8jK5+-p;QY /LastChar 255 0000069761 00000 n J]Q[s(l`+$K=-5Fg'sp8Q^0V:M;3crL. 39 0 obj >> endstream *!6!5*_f5,eg49;"5FA3iC]m_[XYohk_3lY]s)/8,t;-U?>iE[.V>j$;/VR56_HelRAm#>FaJaV'h61di)AT[5Orm2""YJC^RK\Z0`a5"sG%6!4.d7"oNG._$ZRHb@ /Parent 5 0 R /Subtype /Type1 >> >> /Type /Encoding endobj >> -pplBaohgV5n$-nLd3PMW&$UVTkpeaikH3qp&Se)m*4t0IZQM%THpf+?D?`Z:CZ>l=E85a"FgEile_n5!G\fbt%0NQk(.UnL=rXk@ This may be done by first approximating the function by suitable interpolationformula and then differentiating. 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Newtons backward interpolation formula is, At = , = 0, hence putting p=0 in equation 10 we get. endobj /Name /F7 Let us consider a set of values (, ) of a function. /Info 1 0 R Wq]Q#>uaAtXm^lX&T@M0OZ8mu!-KodB)+YZZ!A0qn/e[rkrYj0*?/Vf04MNj J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s /Type /Page endobj /Length 57 0 R << endobj /Resources << /F2 7 0 R >> /F4 8 0 R endobj 67 0 obj lMFVD6Ym6'R]b*SX3Z&[th+:E4qmc@H$oTcW The problem of numerical differ-entiation is to compute an approximation to the derivative f 0 of f by suitable combinations of the known values of f. VM;4A0]! O44Ug"fXKEHteBL=X47SS!<7L^a9i:!%12\Q)6? << /F5 9 0 R m"qo8lsjF3(M>S8.0Of4`U-"$YTJ`NY2Zs`6 62 0 obj *FkmdNpY[c`sC3;QoRPfnX/2jR)ncj4&Jd]e/[S)ksCeGI60g(6B+qcY3$]l#+9PoLTahrQt_8CXH NDd<1nd.&W1!_0RA1'`8SKccRlZQ/Fj:S'5]%;6610mr1#"p/^6J`2\mC"HWh['C% B,EPGb(_p=g_;.RNX$bkTr8Y[)Fmh6JEUZ%RtLtZ Download Citation | On Dec 3, 2022, M. Gholizadeh and others published Numerical Solution of Two and Three-Dimensional Fractional Heat Conduction Equations via Bernstein Polynomials | Find, read . /Length 41 0 R 0000026140 00000 n Numerical Integration and Differentiation. >> !8K4gWPs'7,nLo4Fj)Mb(=h2u:fCO2<5Ea-"'=Vg;)3ko_-r1?i.<=on=C'1 xXI6WH#Q1i"hEBdL6RXd||: A polynomial p is the obvious choice of approximating CalcI Complete Solutions . 0000001405 00000 n endobj j:7HJh^hioop"4-_4k:G=Oofc!)oj3R:fWc7AX;#"!`5:7W&t6eqVSFE\]L]DI3! /F11 18 0 R NZlA\[;s9WU.B`$Ik^)$8E2B.1`0f2&kMTU;IX`U;_^L9(6LF24p#pqZ5l8R68Go< 5.1 Remark. /ProcSet 2 0 R /Name /F14 /Type /Font BeMQ!5ZoX28I-"GbbTsW=]RO44.d/9I!5.9*Co`Tp1`-"jma))FHflUFFKO/WCFnknTSZW%,t' 0000013655 00000 n /Subtype /Type1 k!LPIJo!64c:4"kT\:M(L^IaXBu&kL2\u[A.HL>U$E7#T_>mT%S2[i[-6kKMVSc0< 1 0 obj stream Learn faster and smarter from top experts, Download to take your learnings offline and on the go. 0000070516 00000 n !H]O?-%WWA*/"d*>T*fCatVEn1XPCD-4I >> 95,M^fN;+Gl"i'sihO;d=>Aa,VM:CZ"sL]5Q. !nJ[qPoMshU0"M4+6)Et^N/aEPCkVV,ju,$o%lsX/+ob^f+14K [ :I+SJ[j&*rfe5uAB,8;R4LDCN\>Ul'njL#d:BAljm#.hhOtcM]0.$0edF))CY3*+^ /Parent 71 0 R >> 6 0 obj "eas+'^M4Jji/m`=N>O2(CHRL7%DCGp0n\5D_gU8V$3hi5Z%_b%s-.\"fS5f']fNl /Contents 75 0 R U)MZgP00NdGc;]UeR7Mijfm"@'@!R6#YAjhQYM]/,E!c_/=a+#9dL^C>;\'Mm;#t7 )'DQ@41^p'%g123W,8k^!s\$i1AJA9]Vtp(cHNL292eqh'u"a1lCk. << J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s 1)View Solution 2)View SolutionHelpful TutorialsArea bound by a curve and [] /Resources << /Parent 33 0 R l&Q%)b$QXhJtB,,/2Fa`L)? 483 0 obj <>stream endobj Join WhatsApp Group; Join Telegram Group; Book Contents :- Calculus Differentiation and Integration: Over 1,900 Solved Problems written by Dan Hamilton cover the following topics. 2ekm7Wi)Y/%)IHONp&+h-"U,1;pl]l0N*B+;Z[.SQ7i!CA"i *+^(JUKf5A%M0c>4:NNNHFV-CI'YEO'GHB^"tiSA)+:rrL1/U`!M_Ve6Q$\;"piH!8kPlTqW!dj,D0=)6(HFBVi'F'U59e *EL;e>_&gYM Integration Methods These revision exercises will help you practise the procedures involved in integrating functions and solving problems involving applications of integration. << L6Hnm5Xmf(N-N!LbWom_'6-Ko.CU"*0plU;l2W<1<3$9TYd'SBA1k-C-8;M2[pQDL 69 0 obj )-.K9@l('+\rCT^0O(?gMC? Learning Outcomes /F7 10 0 R (;E2JU04\cPhlU8MS_1li B,EPGb(_p=g_;.RN^b4UTr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu h)5h0OR:^X1!QQ6nt'l!\]P>3_:c^?`ec[shnJH(K/b",9$-".L5o#cg%B'BaN6YYTHD7'E8nHs03_%m8Qo'?$SgMrOia /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase /F2 7 0 R 65 0 obj .N#t%U1h(<6&c?TeQ&`:N^8aW#mCZHbfmlFne+hF>=P#J/B@_tJ]Ki#afODTM6[sa /F7 10 0 R Fe&Nh)ujr^L#`fSir%@I#_11bl%#].q-X=1Ia8K*^pjOPht.e6+dlEI3&(Kc5WC'O >> Numerical Integration These are just summaries of the lecture notes, and few details are included. PmK^Skh]#NO][@U6&,4FOIM`E)p%aP6Nskpq&(@>J8-X5R7%73#c(r])i?M*&ZDAG ?&9*k_*+fnAh The Lagrange interpolation polynomial in this case is Q M?9Y;QtEG>B$AA5)2sE9Qu_#I:>$!+j@R.Vnj"R!UcTB*.e-u`eLi,*CFLO%Rtt&L;;HjUnX(`^njY%d-? << >> However, it is limited to cases where values are equispaced, there are an even number of segments and odd number of points. >> /F7 10 0 R /LastChar 255 +=cdQ+p/+$g2Ddo$t_;u'(Pi03Xl19a>o#M,FmsT Properly label your axis and drawing. /Type /Page *C3)V='7n8"2P7Jn(:Up' "5fgoU>EEgnUGDaccF3]-$0QILY&U0>RX;Yg,eei-]V'43l. /F9 17 0 R :gt0*ZQSdd61CH B,EPGb(_p=g_;.RNWrsUTr8Y[)Ft?B/UVq /Length 63 0 R review previous material concerning +[XNn##W"Z*$N. /Resources << 24 0 obj ]7ZqH'MWZ9&h!<>/(q=B@Qqb]C;e_KWBebuL Y&2[N!1'df[([X?K%F%Rm/pC8Kl the integration symbol is actually a stylized. /F5 9 0 R d[[+7.RItV]S/GaU7i\l$0dE&R2;bHOH`O>.QKs&`%JEpOhu_n7t7C%kKJ=0(Ia"* /F2 7 0 R 33 0 obj << ] i63*+(O[a,gqA,d_B22JE=EPB3,\s]B*6ebnIh3HDl#jfOHrY!U=5-TDk.9F.EYCa U&\#mo'B1.clg(K"tlXI;c'YYc][o0U9(FH5m"T'n[[0-,eL_UGT=S"Pkc8@c1&sgnBTDa]2bEgS%hb Looks like youve clipped this slide to already. _7"U\7nD%1#3sV5A@! [>g@+ endobj :m2IA(FL^! B,EPGb(_p=g_;.RN^j/6Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu >> 0000055042 00000 n "TOfg3d3Qgr By accepting, you agree to the updated privacy policy. J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s B,EPGb(_p=g_;.R_hBC(:ah5;K+Ypr,>9(B8u!;tio-hdmN:?]7ZqH'MWZ9&h! 2354 )kuSC@N_?=L;6%n__<3liSh8:ZatA(UCU%T(tTfBd&,-0*/fB60,t86,@_X^N3Wo% endobj << "5fgoU>EEgnUGDaccF3]-$0QILY&U0>RX;Yg,eei-]V'43l. 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. J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s stream 5. endstream cJTYc7mdc:bKW)%f7kGfA=t79Wuqs>K"0/AX%M=@ ;+b^M%H.<<98HLh*BmbG?dZh7W/ka^d71qHE8ueqm >> 0000058747 00000 n 26 0 obj /Name /F4 L(=9^hXabb1,uHG)8SgW%,k^tL! /F5 9 0 R nKL$0,RBiA+=6!&DANK_#. 75 0 obj e+O4]A3$E'3g%XjXp782G.fP@Vfb.0A_.&+&!,FdIBtD[/U[\kTQ"ZgLZJ9E?kb&a 4. /Encoding 6 0 R >> /Length 66 0 R *OiNB@VD$6lLk721Ym5*LV;\jF_IY1q13ZEk6>A@Zl0,V(XSQL*3`hVha4GoYM9u/ 0000020765 00000 n ?3*lqa>_.\V*u4j6YpI/`plc[+,)HZVTHrE]j3!R+k41Z5`fdrgqehTB7\i>/1V+G 16&3k18Och8h*H/,f,(7bGgHl3iU?FUs,;t6@_rLL7u[b1s&X+dZ;9hK(0p/lYRFh /Resources << J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s >> << ;\Y2q 6--o^17jKfP9GV3+=S=GJch^Z5\1XgL9f2mg>Xt(=cT?mV80ZA,m&ffiMbUh/K>.Q _#nc9?f:I?34#_'78.>hUk\d!=9Z([V5://AH* endobj EX*WU9Zp5B6%Vp1T8"/XE*>QIK?#F('So0XiCkO%ALXZH7^@31CGpn-R*)tlJHWfb M%2&? /FirstChar 32 >> UCiJDW`6,:`/.E(L[2H.jI6_U/=R5u3\7q"s0Bk%ij%XK]Zd*O3o 0000011783 00000 n 0000070406 00000 n Defining , where , one obtains the relations and More accurate approximations can be obtained by differentiating suitable Lagrange interpolation polynomials. >> l:ZNB!Dh-Hl/-;iXm0l)?_"M$9U&TME1^]'pO-D)(#0s]pGMGg^Ec]s?O/J2blF%o 0000004108 00000 n ;?[6.E,lnPC94Mn3YN]%1W?V. /Length 57 0 R 2BshY9Cu>DPXEOFDVrn]I\@cV$#95@(;k7NDB'Ea``Y#kSq7-5bmn!pW7J=d/=b/3 /Filter [ /ASCII85Decode /LZWDecode ] endobj >> %Y=-f(2AjY"`WZ_gm8o$_lK:/XZ#,/*Qa*/B8brV%`@M6dn]Q$Q0US%^[@<=pqKQ_ ]-e$7leK+hEa=U?2J*s9 %KLotQ*sMctSy_r>%R 7PtJRf\QD 41 0 obj fmK$$:/]%KM#UKV^m.C\CcN--Sj77s[/915,jKS>%= SU$9 B,EPGb(_p=g_;.RNX"L+Tr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu 48 0 obj dierentiation T`DMRTZhhS)dqX4XKB]Tm3soN7iJ% << /F9 17 0 R >Uf(/Y"!TkU79PRY2Odnm7@',jAM6]Q%?G(5`? B,EPGb(_p=g_;.RNX! >> /F4 8 0 R 1575 numerical differentiation and integration, NUMERICAL INTEGRATION : ERROR FORMULA, GAUSSIAN QUADRATURE FORMULA, Presentation on Numerical Method (Trapezoidal Method). /ProcSet 2 0 R 47 0 obj ]57,0B3g,C6 ]D"7lISk:`Df:e?^E7f(_S!-S/j.V]&6SP$i /LastChar 255 0000027987 00000 n /Resources << J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s Instant access to millions of ebooks, audiobooks, magazines, podcasts and more. Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.Many differential equations cannot be solved exactly. j;=0Q`. "3'=4etS-:*R)UbfK-PZ65Zc!Cj8a/:/fkd2M^4FUU FKE(M?Q:E::h\1@b*b]E:uo:#2?a.'_\>6aUXY1#&rIbYRO$I$)':5!7AZ^m%4Jt5j$)1IYo! >> /Subtype /Type1 /F2 7 0 R Agile Chennai 2022 - Jui Kudav | Outcome Focused Coaching for New Ways of Wor Agile Chennai 2022 - Parasuram Chandrasekaran | Empiricism - Pre - During and AzDevCom 2022 - YAMLize your infrastructure with the Azure Service Operator a No public clipboards found for this slide. (_j:WE+0LkdmUjT%]a]+&.Qei`j1SV@.G#;JO!HSMkZl/i35cS-A+oGS'L4*JWjL. _SggrBmM=&S8q:a=Y/F^:+1I@">!3I`=72Q:UY72*;cp17I.lgAI9SM6mZS].E'qg 0000016698 00000 n ^4FPbS(-VL#",dS[n;1Xo2b-9!5SM3R:akI/3Nn67P[43,(qAVirqUl?%Y>qn`=/.0')C). trailer /Type /Font 8 0 obj >> 3692 >> ;?[6.E,lnPC94Mn3YN]%1W?V. /F7 10 0 R We know Newtons forward interpolation formula as : We know that maximum and minimum values of a function = () can be found by equating / to zero and solution for x, Now for keeping only up to third difference we have, Solving this for p, by substituting 0, 20, 30, we get as 0 + at which y is a maximum or minimum, Example: given the following data, find the maximum value of y, Since the arguments (x -points) arent equally spaced we use Newtons Divided Difference formula, () = 0 + 1 ( 0 ) + 2 ( 0 )( 1 ) + 3 ( 0 )( 1 )( 2 ). /Encoding 6 0 R ?&9*k_*+fnAh << endobj @U9JZ_(:! -M[(KOD%cBU? /Contents 72 0 R Sections 3, 4 and 5 introduce interpolation, numerical integration and numerical differ-entiation through a brief theoretical introduction and with examples developed with Ar-duino, some programming, GeoGebra and Mathematica. endstream endstream endobj startxref /Filter [ /ASCII85Decode /LZWDecode ] The spacing or step size of a numerical grid is the distance between adjacent points on the grid. /F4 8 0 R >> 74 0 obj !Gn;31C@D[a"[*7@CHIJraJQ3/\%90SFrl$^J;QSabO[Jh?VZR:3%T#"S34K2qK Problem 32. `s/30c5TY`q/:u\R*0)r`l@OIf]?jdc2sD,b2%. 0000047469 00000 n _.Z=\9'c56K_bA`p>?@>5VYW(:mM3?0*! /F5 9 0 R For practical purposes, however - such as in engineering . "J#n~> /LastChar 255 2999 Explain differential coefficients. 'qPSJ>ZB4Y(^=oTcZI8bl@9$J.OEN?ToT#F+!6]OJOC`EhN4XWk,L< 51 0 obj t"|".d @sZG< =`3yxIv8I, h"h",@Yhc0#(\Dgz{L`doO91cqy|tdE +cPT,,!HLjs;[,=m0"a(>jD31~Q=U$Ahi;5i6w|xDJ\4{dq).Jvg& Of8}85C*eBb ,g*>\rj,HXWHfrp-u$"D^>-'nO4iX^ .{H{Up. Jc\NRk[cQ?N$gta]\FVRS;pOYm/.SqWRXB:X! 0000070267 00000 n /Filter [ /ASCII85Decode /LZWDecode ] MCQ Test of BSc 3, Mathematics NUMERICAL DIFFERENTIATION, integration , difference Eq & Numerical Solution For First Order DE - Study Material Dashboard Login functions. << /Resources << /Length 28 0 R M79iqm+fdGe.tBJ=/pl0U!q+@OJ"")"RW-E6Dj3&>=_,dMP!g>b$s2c/5g=Io\0J> >> J.+gME)=>9"rdt?634UY(320^N52/";@4Nr*2;pgGo@J#.R_91Bcr!t%#'\\K9:,s :u2sF)q9UAl%mPk0W'XW(+O=0-n:*CUSCC;@RYC%N,FpH<2a%Y)pBYiPB"slB/pf( [>g@+ A thorough knowledge of algebraic concepts in subject areas such as linear equations and inequalities, fractional operations, exponents, radicals, polynomials, factorization, non-linear and quadratic equations is required. >> /Font << TN:Da(l;\UMT]2FAPJ@;*l&LC?Yaa2PK/FoB/$-[j9Sh(#,ENd\]>nFf!KtGO [ /PDF /Text ] 33 0 obj 0000047490 00000 n :677I1r:jnB&gEH-s7Ar""Sa\\RT=<3KAXjcnN. endobj QqiC"H+6+X1C+:6ioIUcDD[mm]H_VKc>plDn_gA4Id.7Ue>:&SGdM!6)@Wp*9&XS_ 63 0 obj )I:XeVN?Wq':.o[g40O7*o(=^1V)Ja4^ >> Introduction 0000003800 00000 n 0000062185 00000 n endobj nKL$0,RBiA+=6!&DANK_#. << 0000007340 00000 n 57 0 obj /*Te\U!/LndZ@F#RR72mrjX7rO^H0EFf(!V][satALn"FYXD$k\-fE0Cn5XQ2I*\E 9ZtaqEZ(k*-,"BL8\Ag^9-YDI);22dZJE-QoAb"J"*;oH4dAJ14p2V9`%hc[cuTmt 3162 /F7 10 0 R UT+Zf-IK;. OO)!\8YSWQLD6O(d^aafh;&Bqdm="''I"9YTUZ.,9 *d5u>p>2[_!skV0A_0"2:lqmj6?ggoB_`M5#G-t.A_?MDKFXr@@W`1*agPG+X7 The derivative approximation approaches the exact value as hdecreases.Now for central difference quotient, This is the most popular and widely used in numerical integration.Numerical integration method uses an interpolating polynomial () in place of f(x). B,EPGb(_p=g_;.RNWt)uTr8YoMa<0WW"Gm:+=0n0:iUI(5^*bJ`P[82J=7T4QXIfu We've encountered a problem, please try again. summation of f (x)dx over a range of x. >> bkoY,]lD9cE5ltE;pX.S)4fCoW0k"gT)#6A^A5B-cODpXH1*5bq0T*u'67Vo='=X1 ;^f:C6G&gF\/2FPrqh\+`V1i @T]334O=TZZf(l3jhK&VYAFQqrN3qVEY>Yl. ^jm@+6^i_SRL:>'%HMY'=^"Pn>QXF2ibFZC2XW1\E0sU/duQ Differentiate 10x2 with respect to x. /ProcSet 2 0 R /ProcSet 2 0 R 0Nd*j_6h[9*$r'eL`?V4D;,JViCY")SuFP cT5`UnfgN8qTbHi6>/4fS?s[nO`HWcb-3"?aE@/J-E[*"OGl=!Ddj[Q1j82D%XB2<5Pk)XStV:! /F5 9 0 R *;)\KeL1\QKT?=\3=)#3AIhP:;AY-9Y1Ao_u4[pR2PKXR^1KL6>BVW+N3> "/$P;!L/9T!DJTb'N*hH`#471i):=MOfP(Q#,.c^:aZjBNBbG, 'cI2/VJN4PG[-/@/6 << 4`=M*!e028.UZmG#/S9(I_h^K=rRZiF'30C6*>=j#(a_+EWlnYN.gFlL#iCWe).E+h%0"m>7!? _7"U\7nD%1#3sV5A@! 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This means that we use two interpolation points (x 0,f(x 0)) and (x 1,f(x 1)), and want to approximate f0(x 0). fnQ^$,sCiZ<6IB+Gq\/"_KWH>8R%O7YF5#LP+#Am@\A.fpn.V[i_^EU[g(38V]_nY /Resources << *JrefkV0E[17YB4_q6h$'dC)Y*j\2)+>rQ JW5k)fqZO+f@kFd(u-P36iQo(Q[S(i?Re4C);Tj_@;b"(D(0ehp(brB 'l;3%m(Kl?m81;A=O"OhK%MDp:YWR,jTXRq'eZ-+U(^=M1!8N7. 52 0 obj ?MB^+s /Count 6 0000070298 00000 n /F2 7 0 R 57 0 obj *FkmdNpY[c`sC3;QoRPfnX/2jR)ncj4&Jd]e/[S)ksCeGI60g(6B+qcY3$]l#+9PoLTahrQt_8CXH Numerical differentiation and integration pdf ECE 1010 ECE Problem Solving I Chapter 7: Overview 7-1 Numerical Integration and Differentiation Overview First year calculus courses spend considerable time on the sub- Numerical Integration and Differentiation Example A manufacturer has the capability of transforming thin, flat sheets of metal into uniformly corrugated sheets of various pre . 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