ࡱ> 02/M %bjbj== ,LWWQ!^l  """8Z<T*" g*i*i*i*i*i*i*$, .d**$*$$$\g*$g*$$2(*C* 02e*L"2!X3*C*$*0*;*B/$B/C*$ Curve Fitting 5.1 Linear Regression and Matrix Algebra Error is inherent in data. When data exhibits substantial error rigorous techniques must be used to fit the "best" curve to the data. Otherwise prediction of intermediate values, or the derivatives of values, may yield unsatisfactory results. Visual inspection may be used to fit the "best" line through data points, but this method is very subjective. Some criterion must be devised as a basis for the fit. One criterion would be to derive a curve that minimizes the discrepancy between the data points and the curve. Least-squares regression is one technique for accomplishing this objective. It is easiest to interpolate between data points, and to develop correlation, when the dependent variable is linearly related to the independent variable. While individual variables may not be linearly related, they may be grouped together or mathematically manipulated, such as having their log or square root taken, to yield a linear relationship. We wish to fit the "best" straight line to the set of paired data points: (x1,Y1), (x2,Y2), ,(xi,Yi). The mathematical expression for the calculated values is: yi = a1 + a2 xi where yi is the calculated (linear) value approximating the experimental value Yi. The model error, or residual, ei can be represented as ei = Yi - a1 - a2 xi where ei is discrepancy between the measured value Yi and the approximated value yi as predicted by the linear equation.  EMBED CorelDRAW.Graphic.10  Figure 5.1-1. Relationship between the model equation and the data We wish to find values for a1 and a2 to give the "best" fit for all the data. One strategy would be to select values for a1 and a2 to yield a straight line that minimizes the sum of the errors ei's. Since error is undesirable regardless of sign this criterion is inadequate because negative errors can cancel positive errors. The problem may be fixed if one selects a1 and a2 such that the absolute value of the sum of errors is minimized. However one may show that this criterion does not yield a unique "best" fit. A third criterion for fitting the "best" line is the minimax criterion. With this technique one selects a line that minimizes the maximum distance that an individual data point deviates from the calculated line. Unfortunately this strategy gives an undue influence to an outliner, a single point with a large error. A strategy that overcomes the shortcomings of these previous approaches is to minimize the sum of the squares of the errors or residuals, between the measured Yi's and the yi's calculated from the linear model. This criterion has a number of advantages. A unique line results for a given data set. This criterion also leads to the to the most likely a1 and a2 from a statistical standpoint. Regression analysis is used to determine the constants in a relationship between variables. We only consider the simple case where y is a linear function of x. In other words we wish to find an equation y = a1 + a2x to best fit the obtained experimental data xi and Yi. At the values xi, the experimental values Yi are subject to random errors. Lets define ei = Yi - yi to be the difference between the experimental and predicted values. The least-squares criterion requires that S defined by Eq. (5.1-1) be a minimum S = e12 + e22 + ..... + eN2 =  EMBED Equation.3  (5.1-1) or S =  EMBED Equation.3 {Yi - [a1 + a2 (xi)]}2 (5.1-2) Setting the derivative of this sum with respect to each coefficient equal to zero will result in a minimum for the sum. Thus the coefficients a1 and a2 must satisfy the conditions  EMBED Equation.3 = EMBED Equation.3 {-2}{Yi - [a1 + a2 (xi)]} = 0 (5.1-3.a)  EMBED Equation.3 = EMBED Equation.3 {-2(xi)}{Yi - [a1 + a2 (xi)]} = 0 (5.1-3.b) We have two equations in the two unknowns a1 and a2, so we may solve for a unique set of coefficients. Dividing Eqs. (5.1-3.a) and (5.1-3.b) by (-2) and rearranging a1 N + a2 EMBED Equation.3 xi =  EMBED Equation.3 Yi (5.1-4.a) a1 EMBED Equation.3 xi + a2 EMBED Equation.3  xixi =  EMBED Equation.3  xiYi (5.1-4.b) The system can be expressed in the matrix notation A.a = B (5.1-5.a) or  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  (5.1-5.b) The column vector a can be easily solved using the matrix capability of Matlab a = A\B (5.1-6) Example 5.1.1: The following data represent the concentration of reactant A in a constant volume reactor. (Ref. Module 3: Linear Regression by Bequette4 ) Time (min) 0 1 2 3 4 5 CA, kmol/m3 8.47 5.00 2.95 1.82 1.05 0.71 If the reaction is first order, A --> B, determine the reaction rate constant k where rA(kmol/m3.min) = kCA. Solution: The material balance for reactant A in a constant volume batch reactor is  EMBED Equation.3  = -kCA Separating variables and integrating:  EMBED Equation.3  = -kdt ln CA = ln CAo -kt where CAo is the initial concentration of A. For this example: N = 6, y = ln CA, a1 = ln CAo, and x = t. The calculated values CAo and k can be obtained from the solution of Eqs. (5.1-6) by using the matrix algebra capability of Matlab. The coefficient matrix A and the column vector B can be determined by defining a new matrix w w =  EMBED Equation.3  and the transpose of w wT =  EMBED Equation.3  so that A = wT*w and B = wT*Y Matrix A can be obtained by the following Matlab statements: >> f1=ones(6,1); >> f2=[0; 1; 2; 3; 4; 5]; >> w=[f1 f2] w = 1 0 1 1 1 2 1 3 1 4 1 5 >> A=w'*w Note: w' is the Matlab notation for the transpose of w A = 6 15 15 55 The right hand vector B is obtained by the following Matlab statements: >> Ca=[8.47; 5; 2.95; 1.82; 1.05; 0.71]; >> Y=log(Ca) Y = 2.1365e+000 1.6094e+000 1.0818e+000 5.9884e-001 4.8790e-002 -3.4249e-001 >> B=w'*Y B = 5.1329e+000 4.0523e+000 The solution vector a is then >> a=A\B a = 2.1098e+000 -5.0171e-001 and the linear relationship between the variables is ln CA = 2.1098 - 0.50171t The same values for the parameters can also be obtained by using polyfit, a function provided by Matlab, to find the best linear fit of the data. % Matlab program for Example 5.1-1 % Least square curve fitting of ln(Ca) = ln(Cao) - kt % t=[0 1 2 3 4 5]; Ca=[8.47 5 2.95 1.82 1.05 0.71]; Y=log(Ca); ap=polyfit(t,Y,1) ap = -0.5017 2.1098 You should notice that the first element in vector ap is the coefficient of the highest degree term. This is the convention used by Matlab in any polynomial functions. The experimental data and the best fitted line can be plotted by the following Matlab statements >> ycal=polyval(ap,t) ycal = 2.1098e+000 1.6081e+000 1.1063e+000 6.0463e-001 1.0291e-001 -3.9880e-001 >> plot(t,ycal,t,Y,'o') >> ylabel ('ln Ca'); xlabel ('t, min') The parameters ap(1) and ap(2) are converted back to the physical parameters: >> Cao_cal=exp(ap(2)) Cao_cal = 8.2464e+000 k=-ap(1) k = 5.0171e-001 The experimental data and the fitted model can also be compared on a time-concentration plot by the following Matlab statements. The results are presented in Figure 5.1-2 >> t1=0:0.25:5; >> Ca_cal=Cao_cal*exp(-k*t1); >> plot(t1,Ca_cal,t,Ca,'o') >> ylabel('Ca');xlabel('t,min') A crude measure of the how well the data is fitted by a straight line is given by the linear correlation coefficient r, which is defined for two variables t and Y as r =  EMBED Equation.3  where Ct =  EMBED Equation.3  CY =  EMBED Equation.3  Values of r may range from (-1) to (1). The positive value indicates a positive correlation, i.e., the dependent variable is increasing with the independent variable. If |r| is exactly 1, the data is perfectly represented by the straight line.  Figure 5.1-2. Experimental and fitted concentration as a function of time The correlation coefficient for the straight line Y = ln CA = 2.1098 - 0.50171t can be evaluated by the following Matlab statements: % corre.m % Evaluate the linear correlation coefficient r of two vector t and Y % t = [0; 1; 2; 3; 4; 5]; Ca = [8.47; 5; 2.95; 1.82; 1.05; 0.71]; Y = log(Ca); N = length(t); sumt = sum(t); sumY= sum(Y); sumts = t*t; sumYs = Y*Y; sumtY = t*Y; ct = N*sumts - sumt*sumt; cy = N*sumYs - sumY*sumY; r = (N*sumtY - sumt*sumY)/sqrt(ct*cy) The linear correlation coefficient for this example is r = -9.9915e-001 4 Bequette, B.W., Process Dynamics Modeling, Analysis, and Simulation, Prentice Hall, 1998 PAGE  PAGE 5-1 :;/?@BCHIKLSTVW  5 7 : ; ? @ D E G H Q R ~  F G M N CJEH5\j;&(A UV jUH*OJQJ5CJ 5CJ\V:;/02 3 I J  !$`a$$a$d  e!Z]ZH]H$a$Q%%% ) * l m ]^lmBE56$ e!Z]Za$$ e!Z]Za$$`a$$ a$/0`aefjk  "#6789JK^_`acdijnorsvx78KLMNOPcdeflmr j EHUj]ph; CJUVmHnHu j EHUj{ph; CJUVmHnHuCJEH j EHUj9ph; CJUVmHnHu jEHUjoh; CJUVmHnHu jUH*CJEH:rswx{| jEHUjph; CJUVmHnHu jEHUjuh; CJUVmHnHuEHH* jEHUj]ph; CJUVmHnHu jEHUjph; CJUVmHnHu jUCJEH7KL?@TU 5$ @ x e!Z]Za$$ e!Z]Za$$ e!Z]Za$$ e!Z]Za$$ e!Z]Za$ $%89:;=>?A~jqh; CJUVmHnHu jV!EHUjqh; CJUVmHnHu jEHUj#qh; CJUVmHnHuOJQJ jEHU jEHUjuh; CJUVmHnHuCJEH jU jEHUjph; CJUVmHnHu.AKRSUbc  DEXYZ[ijqtxz j'EHUjrh; CJUVmHnHu j%EHUjbrh; CJUVmHnHuCJEHCJEH0J>* CJOJQJ>*CJOJQJOJQJ jU jo#EHU@56BCcdyzij$ @ x e!Z]Za$$ e!Z]Za$   -235:;>?@CKLvw|u 34x !!!CJEHCJ jz,EHUjL((A CJUV jUCJEHOJQJj)CJEHOJQJUj'(A CJUVjCJOJQJU CJOJQJOJQJ=!"+,CDWX\i $ @ x e!Z]Za$ivw &4BPQ[\`n|} <GZ[\auv$a$ 34x y !!!!$ @ x e!Z]Za$!!!!!!!!!!!!!!!!!!!!!""""i#j##%>%O%P%Q%R%%%%%%%%%%%%%%%0JmHnHu0J j0JU0JCJ CJOJQJ5\ j6UOJQJ j%4EHUjVuh; CJUVmHnHu j1EHUj/uh; CJUVmHnHuCJEH jU j.EHUjuth; CJUVmHnHu/!!!""",#-#_#`######$$ $H$U$d$ $ a$$ @ x e!Z]Za$$ @ x e!Z]Za$  d d$$$$$%%=%>%B%P%Q%%%%%%%%%%%&`#$*01h0P/ =!"#$%Dd/d 0  # A2yMTKByUDy`!MMTKBy%*7!xLsޗ?וE B3C:4Ζk̬6d4[֙?tʕ!"qi1mbi&V&]'M&&u4:f(z9GM$qs/\^I1&qw2>y3];Xg2尮99=?f H'g$8N$`p7Ia;ípyE{7{ýq6BƐDږX~쵗ݶ/Ev[鯢^ۄȸ V\Fz?nsd0.!eL[̲? ? ز`.)&i/pK0>I'hϣqsq7E(eǙ6c$b =d6v>aNR-$ ?@ABCDEFGHIJKLMNOPQRTVWXYZ\[]_^`abdcegfhijlkmnoqprstvuwxy{|~}Root Entryn F0*3&Data 'DWordDocumentm,LObjectPoolp,R)0*_10931502676T3H觜R) S)Ole ObjInfoCorelDRAW4= "%(+.147:=@CFIJKNQTWX[^adgjknqruxyz{~Oh+'Oh+'0$ D P \ h tRIFF<CDRAvrsnLISTdoc pfrdmcfg( *4@@?8888 8c1 LastUsed/6a.?AVCLastUsed/6a.?AVCLastUsed/6a.?AVCddd?????@@C:\Corel\WebSite\LastUsed/6a.?AVCDocTemplate@@/6a.?AVCMultiDocTemplate@@/6a.?AVWDrawMultiDocTemplate@@Initializing the App Recovery systemPLGT3280.DLLcoreldrw.exeRAVE.exedrawintl.dll.tlbDrawBrowser\Rave@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta@l2wwww ta(@LIST$fnttfontX@ ArialXwc0X(IqWesternfontX@ AvantGarde Bk BTX(IqWesternfontX@ Arialarde Bk BTX(IqArabicLIST8arrtarrw+(A DD3R3 LISTfiltLIST0filcfild$2@<LIST0filcfild$ dw@<LISTfilcfild LISTotltoutl]d??d@<outl9 d??d@<(A(Aoutl= d??d@<outl=d??d@<outl^d??d@<LISTstlt; `< <^=]P?JJJ@@@(]]]@@@?@B@B@B' -pc9 V? c5  ncd8j8oDefault Artistic Text;<P??p8joDefault Graphic`<=؈q8o;<(?p8j]o`<=Ho`<=o`<=o`<=LISTclolLISTxclo cloa4dBlack@0@<$clof  $`Lcloo ]=LISTuil LISTpageflgsbboxLISTgobjLIST`layrflgs LISTHlgobloda;; (4;GuidesLIST`layrflgsLISTHlgobloda<< (4<dDesktopLIST`layrflgsLISTHlgobloda<< (4<dLayer 1LIST^layrflgsLISTFlgobloda99 (49GridLIST"pageflgsbbox+ )h0_obbx + )h0_+-h_LIST!gobjLISTPlayrflgs LIST8lgobloda,, ,LISTPlayrflgsLIST8lgobloda,, ,dLIST layrflgsLIST8lgobloda,, ,dLISTVobj flgs bboxufbobbx ufbf;busdneLISTlgobloda0HTX\`dd..JotrDX-LIST\trfltrfdPPPu(?,?_YALISTdobj spndqflgs bboxu_#_obbx u_#_u_#_usdnVLISTHlgobloda @htx| (d..>J؈qY' ^ftil0??LIST\trfltrfdPPP) (?,?WAtxsmq?,?WA؈qUS@xitxtjLISTbobj flgs bboxgek_obbx gek_e9_usdnWLISTlgobloda4P\`dhl d..J]@ED=LIST\trfltrfdPPP: (?\?BeYALISTTobj spndpflgs bbox RCexeobbx RCexe RCexeusdnXLIST8lgobloda <`lptx| d..>J؈q ^ftil0??LIST\trfltrfdPPP/ (?̷?@fYAtxsmp?̷?@fYA؈qUS@yitxtjLISTTobj spndpflgs bbox+f.]fobbx +f.]f+f.]fusdnYLIST8lgobloda <`lptx| d..>J؈q ^ftil0??LIST\trfltrfdPPPr(?S?YAtxsmp?S?YA؈qUS@YitxtjLISTTobj spnd@pflgs bbox2Nf}Oeobbx 2Nf}Oe2Nf}OeusdnZLIST8lgobloda <`lptx| d..>J؈qH ^ftil0??LIST\trfltrfdPPP3(?i?}YAtxsm@p?i?}YA؈qUS@eitxtjLISTbobj flgs bbox;sfsfobbx ;sfsf}YALISTtobj flgs bbox )h0_obbx )h0_V-hH_usdncLISTlgobloda4P\`dhl d..J]h~, f<DDf<DDD'Al)9LIST\trfltrfdPPP(?81?)WALISTPlayrflgsLIST8lgobloda,, ,sumi87N)` DRAWBITMAPS R)R)STREAMSLISTSTREAMSLISTDRAWPATTERNS R)R)STREAMSLISTSummaryInformation(_996700140Fвm)вm)Ole College of Engineering CORELDRWCollege of Engineering1@@@TX@TXCDRA FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native d_996700217F))uation Equation.39q7H<~IdI e i  2i=1N " FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native H7,IoI  i=1N " FMicrosoft Equation 3.0 DS Equation Equation.39q70̐II "S"a 1_996700283"F*)*)Ole CompObjfObjInfoEquation Native L_996700253FpB)pB)Ole  CompObj !f FMicrosoft Equation 3.0 DS Equation Equation.39q7,IoI  i=1N " FMicrosoft Equation 3.0 DS EqObjInfo!#Equation Native $H_996700318$Fp )p )Ole &CompObj#%'fObjInfo&)Equation Native *L_996701572O)F0j)0j)uation Equation.39q70,ITI "S"a 2 FMicrosoft Equation 3.0 DS Equation Equation.39qOle ,CompObj(*-fObjInfo+/Equation Native 0H7,ITI  i=1N " FMicrosoft Equation 3.0 DS Equation Equation.39q7,ܤII  i=1N "_996700362E.F0)0)Ole 2CompObj-/3fObjInfo05Equation Native 6H_9967004123F))Ole 8CompObj249f FMicrosoft Equation 3.0 DS Equation Equation.39q7,IoI  i=1N " FMicrosoft Equation 3.0 DS EqObjInfo5;Equation Native <H_9967016548F))Ole >CompObj79?fObjInfo:AEquation Native BH_9967004511@=F`)`)uation Equation.39q7,I`mI  i=1N " FMicrosoft Equation 3.0 DS Equation Equation.39qOle DCompObj<>EfObjInfo?GEquation Native H7dIЊI Nx ii=1N " x ii=1N " x i x ii=1N " [] FMicrosoft Equation 3.0 DS Equation Equation.39q_996700553BF ) )Ole LCompObjACMfObjInfoDOEquation Native Ph_996700599;'GFH)H)Ole RCompObjFHSf7L I~I a 1 a 2 [] FMicrosoft Equation 3.0 DS Equation Equation.39q7ߜJI Y ii=1N " x i YObjInfoIUEquation Native V_996700770LF`p*`p*Ole Y ii=1N " [] FMicrosoft Equation 3.0 DS Equation Equation.39q70kIvI dC A dtCompObjKMZfObjInfoN\Equation Native ]L_996700804JcQF**Ole _CompObjPR`fObjInfoSbEquation Native cX FMicrosoft Equation 3.0 DS Equation Equation.39q7<ċIܡI dC A C A FMicrosoft Equation 3.0 DS Equation Equation.39q_1093150611YVF!*!*Ole eCompObjUWffObjInfoXhp| 1t 1 1t 2 ""1t N [] FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native i_1093150796[F,*,*Ole lCompObjZ\mfObjInfo]oEquation Native p_996701301`FP;*P;*Ole sp|Ժ 11"1t 1 t 2 "t N [] FMicrosoft Equation 3.0 DS Equation Equation.39q7<~IdI Nt iCompObj_atfObjInfobvEquation Native w$_996701487^heFF*F* Y i "t i Y ii=1N " i=1N " i=1N " (C t C Y ) 1/2- FMicrosoft Equation 3.0 DS Equation Equation.39qOle |CompObjdf}fObjInfogEquation Native 7ߔ II t i2 "t ii=1N " () 2i=1N " FMicrosoft Equation 3.0 DS Equation Equation.39q7ߘ4{I`mI NY i2_996701526jFU*U*Ole CompObjikfObjInfolEquation Native 1TableB/SummaryInformation(oDocumentSummaryInformation80 "Y ii=1N " () 2i=1N "Oh+'0|  8 D P\dltCurve Fittingourv tknguyentinkngkng Normal.dotn tknguyentn3ngMicrosoft Word 9.0@[@{GN+,u]}y:S5hԀb|fz6zGEDd B  S A? 2[%WՄX,7Ky`!/%WՄX,8xcdd`` @bD"L1JE `x A?dm@[pC0&dT1Tf: `g11W&@\ F\N %O?odYo 9~br<K &1BMr`6nr0#o盀3 "E@1{Ĥ\Y\ːzf:&?Dd B  S A? 2[`y[C\C2}0>W\)xTzӭOؕ xR%~y1\Qbp6Wi1g8c sa.kw$Wo$ߛ|390ރu8DO7o!y6C9':~yVÆHkVK}9uN-sKNZ4a2+M>oWC6b#X`K{,ʫ£5k4Y=+?]Vb#Ӹ'#nx6F1޹onZ>m ڙDŽR0>DdB  S A ?  2t:bJϯ`{2@_!y`!Wt:bJϯ`{2@@ `\%xJAgfOI.`atZX)Dȁ\G{ k|_JmXvf.D.$ބ4sE6mst{PIt r!pmZi(J6KDY[\@uYr99sT=|KgY ^=zo:> `$^\Jn7_?vҨ>S>ᗔz`3{KگĈg\q|č]T4OS}hˤ="fZх`N`DdDlB  S A? 2a.zA";ICmP=;&y`!5.zA";ICmPxQjA}3kB$*H*ҀXk~@bqgJ'v_ yT{| bۄA;ųIMQ9\|RcĮZU_lc/2]QAPXg^^\7fF~8 Y+i0 q,-oed%#ԇx1:FV}wV<݁Tg.^Fb›J#֮Js=DdDB  S A? 2gwUxEC2(y`!;wUxE@ xQNBQ ڃ.$8&ae@oý@B؜|ŁWpb3KspIO6ןC( "+ʉ2E,%[)dXWnU@E&zR)n,رPiYyλͯ}&QMxǫ`S4 :vɷ6{3>xɽBzOi=C/ Fv#-\w"12N=X~HS>lDd J  C A ? "2ss_V@7|u/*y`!ss_V@7|u@ 8PVxڥKP]ҚƂEDb'Ef; :6P1Vl?ѹY,"_p.">x>{{wH),$DY0"T8.$ƙ5)y0duvZ쌊ICT ÓOq8IDȔQ`^~mlMD0QX`!g̘<Թe7tX{wSs9oܱ$? _SFͯx}V Q}.(]t]yD`t5n7֕rW&0TcNA2+qpTˌlr^i Vb*2\ lOj4 1yDdJ  C A? "2MG- $5(K9,y`!MG- $5(K9 F`\}xcdd`` @c112BYL%bpu Qp()JR"?[XA*|,bÍ"9,ǨƷ!6P?gq@̅Qo |.hjxrF&&\p,&'DdB  S A? 2#XR1;q-x7/y`!XR1;q-x" `Phn x/A߼۽ݻc Q]Bq~+FG6A;q\'z?@TZQ( k* ̼#a6>37Ղ ZweZF#F)u䦛Z4C7Ԅu{5rIrq-W*\x,C1Xj}JןR{(f ;lG[0VaqNzk4egFWKޟQDk>f4&}+8tgLW0e· 3OPV+BƏ:zԼ횼K~/7l3 ɷI# NSfW˧;Hl7+ gS #j"dbyDd|B  S A? 2 b۹QMNGZAd1y`! b۹QMNGZAdd` xJAϜى&`bXحFK1 #,p1.`VN} J㥴unua9?gH.@Ȁxo #7$D P̓i9EKa > Biuysl}1)X4wJ=!" y8 Ŷ]U'rl9.7|kϿ}쟍:~1ʯh?y\hviL<*%\f^]yCTSZFݯ`ȱLdDg zUYI}E {}lo#~z͌wԡ-S*uڲIZ=A-MχpA(y#DdB  S A? 2k.I2@^i4y`!k.I2@^t xxcdd`` @b1##X`=F !#T57LVVKoxZA@'ۢ?5vmA[ց0yklBu()}Uhq@3x 1f؀&QE ufTzp~y}F;v 3hor&2hIf4ڽh]h9юC{ h&'}ϡ~_@c~FwFftObtOES'NΨS='=wu~Z09,~.iAb^vRv#K 9; lEqWwĿvzX/~{4fO?m)}w~ѯS!9?b*budM0Dz)mNũG}"I1UU~O{mZ1󰈥 :5L3YΩ>F uw?a52Mzop@IAAz nIT =S`/[,+,o {*lTU#[m}5D2RP{z]«MJżV 6U˩z2NY/6M;^2tߦ3nS'kЬӦVkoc7Mo{0Ǩ=ò9}eMP=jۆpR}x_;w8waM>o/I!|W|"|7|CbWSP!XѕaM^nZM1an,LTg,0rZo:7moZo:7m:an,o,D d=`=`u|T~&u?s,2&q|5|V}/̱\a,/zMJe&/k7Ӧ[R:LTg,`.M]]vi 27#͍+"&|u&|ub'c9ˠvxi-K;|ݴN.U÷ښs,"eU 8(~h EenoWqTgx&sd7~h'#o =Ӧf&7 [MyvwHPC|| =a e1O~ϷSogo|;|; coovMjM& MjM& 7ȷF]v1kb|'@N>_65[6i]94oAi{\{<y|)/LO3nMvk6 ]o.} f772772 eoSO1q~0ښW[jn;+=$gss'Ysin{jC۾[}kIM(")"-1 oĸxM[b|Wlq%~yjgLZe|U4gĀZ6>(MJ3U͛K¨М=F94_W`MsĚuҎXZ:ۦPOZ1;ϳ̣ߵ՞"RjK܌ߓ&g%-fvw:P)ː=+ )|{w,i*'z;ov-sY,'uտe|j۲s]egk>ފ AXd92 ,AvVGj+>NJ%;Uc_bfǒ|MI?n(_#=IoM$Wm K#G왑TtqO-X^і,[6#lmFڌaksrVUڜues0EAB WB~3[C'D<~Hii֦MOV6i%z҂d )4VhРOABG#EQ;zjD[ 6lm&L`ks6RʊDjUh*ToB[C#pk+eYi_'(=}lm|ckӦ~%zK%gИO1Bcl| =vDύW-67Z\m54F/(SeQJFOPƧPPPƧPv+Jl(zꦢGK]R/>QNn݇mcj}"^,4ibk֦ɧM[uWn,\^m(kejC)^ H\6 O94[lmN9"6'6y9[lm9H<)|Fo`kӦM[6 lmDҦFiSWPS x->_696ɳɳɳɳ5DfeAmlm>mlmlm>mlmFffUAm|}Գ_gkH$6ww7z=%햛޾ߞMΨeFzh`F/ DE8$EZV{GeE>+1J^xg9l},gp.ٌ7ĶCzJnPy%8%2sWZbZ"_ż[}B{ (^ۻw(qwC|SW(Cs$lIpZGG9jbϖZ}.TzR|]̶՚97۝xcPoeC/s@x՜.+,0 hp  Cal Poly Pomona:! Curve Fitting Title  FMicrosoft Word Document MSWordDocWord.Document.89q i8@8 NormalCJ_HaJmH sH tH Z@Z Heading 1$ & F<@&5CJ KH OJQJ\^JaJ \@\ Heading 2$ & F<@& 56CJOJQJ\]^JaJV@V Heading 3$ & F<@&5CJOJQJ\^JaJJ@J Heading 4$ & F<@&5CJ\aJL@L Heading 5 & F<@&56CJ\]aJF@F Heading 6 & F<@&5CJ\aJ8@8 Heading 7 & F<@&>@> Heading 8 & F<@&6]L @L Heading 9 & F<@&CJOJQJ^JaJ<A@< Default Paragraph Font, @, Footer  !&)@& Page Number4O4 time$dha$ OJQJaJNC@"N Body Text Indent$`a$ OJQJaJ<&@1< Footnote ReferenceCJEH>@B> Footnote TextCJOJQJaJ!]`!L:;/023IJ)*l m ] ^ l m   B E 56KL?@TU 56BCcdyzij!"+,CDWX\ivw &4BPQ[\`n|} <GZ[\auv 34xy,-_`  H U d !!=!>!B!P!Q!!!!!!!!!! 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000@0 0@0@0@0@0@0@0 0 r!%# 5i!d$% !"$%%" 6 8 J ^ ` 7KMOce$8:DXZ !:::::::::::::::::::::: !! OLE_LINK2!!RW46FHPRel      / 0 ` a e f j k   c d r s lm{|[^ =>?@8>_begmoqsuw*-agnt06ZazGIJQ\^ms !'CEMO;Aegd h s w Q!S![!!!!@C46JO _ ` m o     B D ABdr_begz79"$D{  >BGI\^<GCF|~ L P Y ` d h >!?!Q!!!!333333333333333333333333333333333333333333333333333333333333333333333 tknguyenIC:\Documents and Settings\tknguyen\Desktop\swim2001\egr511\Notes\Chap.doctknguyenJC:\Documents and Settings\tknguyen\Desktop\swim2001\egr511\Notes\Ch5-1.doctknguyenaC:\Documents and Settings\tknguyen\Application Data\Microsoft\Word\AutoRecovery save of Ch5-1.asdtknguyenaC:\Documents and Settings\tknguyen\Application Data\Microsoft\Word\AutoRecovery save of Ch5-1.asdtknguyenJC:\Documents and Settings\tknguyen\Desktop\swim2001\egr511\Notes\Ch5-1.docB[  ^`o( Chapter ^`o(^`o(^`o(^`o(^`o(^`o(^`o(^`o(BO!Q!!!@ ! @UnknownG: Times New Roman5Symbol3& : Arial3TimesCMChicagoArialMMMonacoCourier NewY New YorkTimes New Roman"qhqƢq&x:!20!Q!2Q Curve FittingtknguyentknguyenCompObjj