{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 207 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 208 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 209 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 210 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 211 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle9" -1 212 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle10" -1 213 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle11" -1 214 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle13" -1 216 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle14" -1 217 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle15" -1 218 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle16" -1 219 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "_cstyle17" -1 220 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle18" -1 221 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle20" -1 223 "MS Serif" 1 14 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle21" -1 224 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" 208 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 208 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 208 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "_cstyle38" -1 262 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle6" -1 207 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle9" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_psty le10" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle11" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle12" -1 211 1 {CSTYLE "" -1 -1 "Cour ier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 } {PSTYLE "_pstyle13" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle14" -1 213 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle16" -1 215 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle17" -1 216 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_ pstyle18" -1 217 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle20" -1 219 1 {CSTYLE "" -1 -1 "MS Serif" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle21" -1 220 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 207 "" 0 "" {TEXT 207 31 "Integrating a Discrete \+ Function" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{PARA 209 "" 0 "" {TEXT 209 78 "2004 Autar Kaw, Loubna Guennoun, University of South Flo rida, kaw@eng.usf.edu," }}{PARA 209 "" 0 "" {TEXT 209 35 "http://numer icalmethods.eng.usf.edu" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{PARA 208 "" 0 "" {TEXT 208 128 "NOTE: This worksheet demonstrates the use o f Maple to illustrate the integration of a discrete function using dif ferent methods." }}}{SECT 0 {PARA 210 "" 0 "" {TEXT 210 23 "Section I: Introduction" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{PARA 208 "" 0 "" {TEXT 208 337 "The following worksheet will illustrate how to integrat e a discrete function. This is usually done either by calculating the \+ area of the trapezoids defined by the data points or by integrating an interpolant. So here we show the answer by trapezoidal rule as well a s with 4 types of interpolation. For more info on interpolation see th e " }{URLLINK 17 "topic" 4 "http://numericalmethods.eng.usf.edu/mws/ge n/05inp/index.html" "" }{TEXT 208 37 ". The following outlines this sh eet. " }}{PARA 208 "" 0 "" {TEXT 208 8 " " }}{PARA 208 "" 0 "" {TEXT 208 27 " 1. Trapezoidal Rule" }}{PARA 208 "" 0 "" {TEXT 208 67 " 2. Polynomial Interpolation\n 3. Spline Interpo lation" }}{PARA 208 "" 0 "" {TEXT 208 32 " a. Linear sp line" }}{PARA 208 "" 0 "" {TEXT 208 35 " b. Quadratic s pline" }}{PARA 208 "" 0 "" {TEXT 208 31 " c. Cubic spli ne" }}{PARA 208 "" 0 "" {TEXT 208 1 " " }}{PARA 0 "" 0 "" {TEXT 262 7 "[click " }{URLLINK 17 "here" 4 "numericalmethods.eng.usf.edu/mws/gen/ 07int/mws_gen_int_txt_trapdiscrete.doc" "" }{TEXT 262 20 " for textboo k notes]" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 211 8 "restart;" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{SECT 0 {PARA 210 "" 0 "" {TEXT 210 16 "Section II: Data" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{PARA 208 "" 0 "" {TEXT 208 114 "This section is the only s ection where the user may interact with the program. The user may ente r any set of data " }{TEXT 213 1 "X" }{TEXT 208 6 ", and " }{TEXT 213 1 "Y" }{TEXT 208 274 ", and the lower and upper limit for the function to be integrated. By entering this data, the program will calculate t he average value of the integral using the trapezoidal rule, polynomia l interpolation, and spline interpolation. The X data needs to be in \+ ascending order." }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "The following is the " }{TEXT 259 1 "y" }{TEXT -1 4 " vs " }{TEXT 258 1 "x" }{TEXT -1 34 " data. The data is presen ted as " }{TEXT 260 5 "(x,y)" }{TEXT -1 15 " pairs of data." }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 216 63 "\nXY:=[[0,320],[10,120], [15,620],[20,7720],[22.5,320],[30,620]];" }{TEXT 208 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#XYG7(7$\"\"!\"$?$7$\"#5\"$?\"7$\"#:\"$?'7$\"# ?\"%?x7$$\"$D#!\"\"F(7$\"#IF." }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }} {EXCHG {PARA 208 "" 0 "" {TEXT 208 32 "The lower limit of the integral " }{TEXT 213 4 "a. " }{TEXT 208 10 "Note that " }{TEXT 256 1 "a" } {TEXT 208 51 " is within the minimum and the maximum of X values." }}} {EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 7 "a:=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"\"!\"\"" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 32 "The upper limit of the integral " }{TEXT 261 1 "b" } {TEXT 208 13 ". Note that " }{TEXT 257 1 "b" }{TEXT 208 51 " is withi n the minimum and the maximum of X values." }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 8 "b:=16.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG\"#I" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 208 109 "This is the end o f the user's section. All information must be entered before proceedin g to the next section." }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}}{SECT 0 {PARA 215 "" 0 "" {TEXT 217 38 "Section III: Checking for Valid Inpu ts" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 12 "n:=nops(XY);" }{MPLTEXT 1 219 92 "\nX:=array(1..n): \nY:=array(1..n):\nfor i from 1 to n do\nX[i]:=XY[i,1]:\nY[i]:=XY[i,2] :\nend do:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"'" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 212 "" 0 "" {TEXT 212 54 "The first check is to make sure that the low er limit " }{TEXT 220 1 "a" }{TEXT 212 73 " of the integral is equal \+ or greater than the first input of the X array." }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 199 "lower_limit:=proc(a,xone,xn)\nlocal iflag: \nif a>=xone and a<=xn then\niflag:=OK:\nRETURN (iflag);\nelse\niflag: =NotOK;\nprintf(\"a can not be less than X1 or greater then Xn\"):\nRE TURN (iflag);\nfi;\nend proc:" }{MPLTEXT 1 219 0 "" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 34 "chec k_1:=lower_limit(a,X[1],X[n]);" }{MPLTEXT 1 219 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(check_1G%#OKG" }}}{PARA 212 "" 0 "" {TEXT 212 0 " " }}{PARA 212 "" 0 "" {TEXT 212 54 "The second check is to make sure t hat the upper limit " }{TEXT 220 1 "b" }{TEXT 212 69 " of the integral is equal or less than the last unput of the X array." }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 199 "upper_limit:=proc(b,xone,xn)\nlocal \+ iflag:\nif b>=xone and b<=xn then\niflag:=OK:\nRETURN (iflag);\nelse\n iflag:=NotOK;\nprintf(\"b can not be less than X1 or greater than Xn\" ):\nRETURN (iflag);\nfi;\nend proc:" }{MPLTEXT 1 219 0 "" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 34 "check_2:=upper_limit(b,X[1],X[n]);" }{MPLTEXT 1 219 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(check_2G%#OKG" }}}{PARA 212 "" 0 " " {TEXT 212 0 "" }}{PARA 212 "" 0 "" {TEXT 212 89 "The third check is \+ to make sure that the user put enough data in both the X and Y arrays. " }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 220 "n_data:=proc(nx,ny) \nlocal iflag:\nif nx=ny then\niflag:=OK:\nRETURN (iflag);\nelse\nifla g:=NotOK;\nprintf(\"The number of data on the X array has to be equal \+ to the number of data on the Y array\"):\nRETURN (iflag);\nfi;\nend pr oc:" }{MPLTEXT 1 219 0 "" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 33 "check_3:=n_data(nops(X), nops(Y));" }{MPLTEXT 1 219 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(c heck_3G%#OKG" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{PARA 212 "" 0 "" {TEXT 212 77 "The last check is to make sure the data in the X array i s in ascending order." }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 226 "x_asc:=proc(x)\nlocal i,iflag:\nfor i from 2 by 1 to n do\nif X[i ]>X[i-1] then\niflag:=OK:\nRETURN (iflag);\nelse\niflag:=NotOK;\nprint f(\"The data in the X array has to be in an ascending order\"):\nRETUR N (iflag);\nfi;\nend do;\nend proc:" }{MPLTEXT 1 219 0 "" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{PARA 212 "" 0 "" {TEXT 212 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 18 "check_4:=x_asc(X);" } {MPLTEXT 1 219 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(check_4G%#OKG " }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{PARA 212 "" 0 "" {TEXT 212 163 "The following spreadsheet displays the results of checking the va lidity of the user's input. If all the check's results is \"OK\", then all the input data are valid." }}{PARA 212 "" 0 "" {TEXT 212 0 "" }} {PARA 211 "> " 0 "" {MPLTEXT 1 211 15 "with( Spread ):" }}{PARA 211 "> " 0 "" {MPLTEXT 1 211 33 "EvaluateSpreadsheet(Input Check):" }} {EXCHG {PARA 212 "" 0 "" {SPREADSHEET {NAME "Input Check" } {ROWHEIGHTS } {COLWIDTHS 1 270 2 106 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }468 187 187 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Check" 20 "6#%&CheckG" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Results" 20 "6#%(ResultsG" }0 }{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Lower*limit*check" 20 "6#*(%&LowerG\"\"\"%&limitGF%%&checkGF%" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "check_1" 20 "6#%#OKG" }0 }{CELL 3 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Upper*limit*check" 20 "6#*(%&UpperG\"\"\"%&limitGF%%&checkGF%" }0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "check_2" 20 "6#%#OKG" }0 }{CELL 4 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Number*of*data*check " 20 "6#**%'NumberG\"\"\"%#ofGF%%%dataGF%%&checkGF%" }0 }{CELL 4 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "check_3" 20 "6#%#OKG" } 0 }{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Ascending* order*check" 20 "6#*(%*AscendingG\"\"\"%&orderGF%%&checkGF%" }0 } {CELL 5 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "check_4" 20 " 6#%#OKG" }0 }}{TEXT 212 0 "" }}}{PARA 212 "" 0 "" {TEXT 212 0 "" }} {PARA 212 "" 0 "" {TEXT 212 0 "" }}{PARA 212 "" 0 "" {TEXT 212 0 "" }} }{SECT 0 {PARA 210 "" 0 "" {TEXT 210 43 "Section IV: Integrating a Dis crete Function" }}{SECT 0 {PARA 217 "" 0 "" {TEXT 221 16 "Trapezoidal \+ Rule" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 552 "trap:=proc(a,b,X,Y,n)\nlocal S_trap,S_1,S_2,Y_a,Y_ b,l_index,u_index,i:\n\nl_index:=1:\nu_index:=n:\n\nS_1:=0:\nS_2:=0:\n \nfor i from 1 by 1 to n do\n\nif a>=X[i] then\nl_index:=i:\nY_a:=Y[i] +(Y[i+1]-Y[i])/(X[i+1]-X[i])*(a-X[i]):\nS_1:=(X[l_index+1]-a)*(Y[l_ind ex+1]+Y_a)/2.0:\nend if:\n\nif b>X[i] then\nu_index:=i:\nY_b:=Y[i]+(Y[ i+1]-Y[i])/(X[i+1]-X[i])*(b-X[i]):\nS_2:=(b-X[u_index])*(Y_b+Y[u_index ])/2.0:\nend if:\n\nend do:\n\nS_trap:=S_1+S_2:\nfor i from (l_index+1 ) by 1 to (u_index-1) do\nS_trap:=S_trap+(X[i+1]-X[i])*(Y[i+1]+Y[i])/2 .0:\nend do:\n\nreturn(S_trap):\nend proc:\n" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 24 "S_trap: =trap(a,b,X,Y,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'S_trapG$\"+++J WQ!\"&" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 209 "trap:=x->spline(X,Y,x,linear):\nplot([XY,trap], X[1]..X[n],style=[POINT,LINE],symbol=CIRCLE,symbolsize=15,thickness=2, title=\"Trapezoidal Rule\",color=[BLACK,ORANGE],legend=[\"Center of Ho les\",\"Trapezoidal Rule\"]);" }{MPLTEXT 1 219 0 "" }}{PARA 13 "" 1 " " {GLPLOT2D 381 297 297 {PLOTDATA 2 "6)-%'CURVESG6&7(7$$\"\"!F)$\"$?$F )7$$\"#5F)$\"$?\"F)7$$\"#:F)$\"$?'F)7$$\"#?F)$\"%?xF)7$$\"3+++++++]A!# ;F*7$$\"#IF)F4-%'COLOURG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-%'LEGENDG6#Q 0Center~of~Holes6\"-F$6&7]p7$F)F*7$$\"+]i9Rl!#5$\"+vq@pI!\"(7$$\"+WA)G A\"!\"*$\"+^NUbHFY7$$\"+Qeui=Fgn$\"+K3XFGFY7$$\"+i3&o]#Fgn$\"+G)H')p#F Y7$$\"+pX*y9$Fgn$\"+'3@/d#FY7$$\"+WTAUPFgn$\"+r^b^CFY7$$\"+%*zhdVFgn$ \"+,kZGBFY7$$\"+%>fS*\\Fgn$\"+h\")=,AFY7$$\"+>$f%GcFgn$\"+O\"3V2#FY7$$ \"+Dy,\"G'Fgn$\"+NkzV>FY7$$\"+7iUW\"FY7$$\" +%pnsM*Fgn$\"+hkaI8FY7$$\"+0_J&o*Fgn$\"+fp$HE\"FY7$$\"+siL-5!\")$\"*si LA\"!\"'7$$\"+JL(4.\"Fcs$\"*JL(4:Ffs7$$\"+!R5'f5Fcs$\"*!R5'z\"Ffs7$$\" +/QBE6Fcs$\"*/QBY#Ffs7$$\"+:o?&=\"Fcs$\"*:o?0$Ffs7$$\"+a&4*\\7Fcs$\"*a &4*p$Ffs7$$\"+j=_68Fcs$\"*j=_J%Ffs7$$\"+Wy!eP\"Fcs$\"*Wy!e\\Ffs7$$\"+U C%[V\"Fcs$\"*UC%[bFfs7$$\"+O3om9Fcs$\"*O3o'eFfs7$$\"+J#>&)\\\"Fcs$\"*J #>&='Ffs7$$\"+v.fJ:Fcs$\"*K$eo5!\"&7$$\"+>:mk:Fcs$\"*d$>Q:F^w7$$\"+[+X $f\"Fcs$\"*o!*p%>F^w7$$\"+w&QAi\"Fcs$\"*y(ybBF^w7$$\"+v4L`;Fcs$\"*%)*H (z#F^w7$$\"+uLU%o\"Fcs$\"*\">\")QKF^w7$$\"+k[a;%=Fcs$\"*>r_Z&F^w7$$\"+MaKs=Fcs$\"*;@q!fF^w7$$\"+A\\31 >Fcs$\"**eS'Q'F^w7$$\"+6W%)R>Fcs$\"*k!zloF^w7$$\"+8)y,(>Fcs$\"*9RlH(F^ w7$$\"+:K^+?Fcs$\"*%)3[q(F^w7$$\"+Suq;?Fcs$\"*yfaA(F^w7$$\"+k;!H.#Fcs$ \"*v5hu'F^w7$$\"+))e4\\?Fcs$\"*shnE'F^w7$$\"+7,Hl?Fcs$\"*o7uy&F^w7$$\" +'[k*z?Fcs$\"*@^IN&F^w7$$\"+g)QY4#Fcs$\"*u*o=\\F^w7$$\"+MKJ4@Fcs$\"*FG V[%F^w7$$\"+4w)R7#Fcs$\"*xm*\\SF^w7$$\"+w0.S@Fcs$\"*&\\4vNF^w7$$\"+WN2 c@Fcs$\"*5B-5$F^w7$$\"+6l6s@Fcs$\"*F^`i#F^w7$$\"+y%f\")=#Fcs$\"*Xz/:#F ^w7$$\"+gYD.AFcs$\"*1iOq\"F^w7$$\"+T)\\$=AFcs$\"*rWoD\"F^w7$$\"+A]WLAF cs$\")NF+\")F^w7$$\"+/-a[AFcs$\")'*4KOF^w7$$\"+cD^]AFcs$\"+C-0-KFY7$$ \"+3\\[_AFcs$\"+K'R*4KFY7$$\"+gsXaAFcs$\"+S!Hy@$FY7$$\"+6'HkD#Fcs$\"+W %=dA$FY7$$\"+9VPgAFcs$\"+cs\\TKFY7$$\"+=!>VE#Fcs$\"+sgFdKFY7$$\"+E%3AF #Fcs$\"+/P$))G$FY7$$\"+Ly4!G#Fcs$\"+K8R?LFY7$$\"+[m(eH#Fcs$\"+#f1NQ$FY 7$$\"+ial6BFcs$\"+[=iYMFY7$$\"+i@OtBFcs$\"+['[Mp$FY7$$\"+fL'zV#Fcs$\"+ OM&=&RFY7$$\"+!*>=+DFcs$\"+gzs+UFY7$$\"+E&4Qc#Fcs$\"++\"Q_X%FY7$$\"+%> 5pi#Fcs$\"+!ySwq%FY7$$\"+bJ*[o#Fcs$\"+?EdR\\FY7$$\"+r\"[8v#Fcs$\"+!o#R 0_FY7$$\"+Ijy5GFcs$\"+?`9VaFY7$$\"+/)fT(GFcs$\"+?#Rmp&FY7$$\"+1j\"[$HF cs$\"+?_ERfFY7$FA$\"+++++iFY-FC6&FE$\")+++!)Fcs$\")Vyg>FcsFegl-FG6#%%L INEG-FK6#Q1Trapezoidal~RuleFN-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!FNF dhl-%&TITLEGF[hl-%'SYMBOLG6$%'CIRCLEGF3-%%VIEWG6$;F(F@%(DEFAULTG" 1 2 4 1 15 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Center of Holes" "Trapezoidal Rule" }}}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}{PARA 212 " " 0 "" {TEXT 212 0 "" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}}{SECT 0 {PARA 217 "" 0 "" {TEXT 221 24 "Polynomial Interpolation" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 26 "poly_inter:=interp(X,Y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+poly_interG,.*&$\"+KLLxG!#5\"\"\")% \"xG\"\"&F*F**&$\"+KLLCA!\")F*)F,\"\"%F*!\"\"*&$\"+****frG!\"&F*F,F*F* *&$\"+jm;:h!\"(F*)F,\"\"$F*F**&$\"+jm;_q!\"'F*)F,\"\"#F*F4\"$?$F*" }}} {PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 36 "S_polyinter:=int(poly_inter,x=a..b);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%,S_polyinterG$\"+$e " 0 "" {MPLTEXT 1 218 227 "p oly_inter:=x->interp(X,Y,x):\nplot([XY,poly_inter],X[1]..X[n],style=[P OINT,LINE],symbol=CIRCLE,symbolsize=15,thickness=2,title=\"Polynomial \+ Interpolation\",color=[BLACK,RED],legend=[\"Center of Holes\",\"Polyno mial Interpolation\"]);" }{MPLTEXT 1 219 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 445 289 289 {PLOTDATA 2 "6)-%'CURVESG6&7(7$$\"\"!F)$\"$?$F)7 $$\"#5F)$\"$?\"F)7$$\"#:F)$\"$?'F)7$$\"#?F)$\"%?xF)7$$\"3+++++++]A!#;F *7$$\"#IF)F4-%'COLOURG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-%'LEGENDG6#Q0C enter~of~Holes6\"-F$6&7`p7$F)F*7$$\"+ilyM;!#5$\"+!*yjG[!\"'7$$\"+DJdpK FV$\"+\"fUh(*)FY7$$\"+)ofV!\\FV$\"+_xzx7!\"&7$$\"+]i9RlFV$\"+$HB\\i\"F ^o7$$\"+XV)RQ*FV$\"+xb_a@F^o7$$\"+WA)GA\"!\"*$\"+lS&ff#F^o7$$\"+TS\"Ga \"F\\p$\"+uF&f*HF^o7$$\"+Qeui=F\\p$\"+Ur>.LF^o7$$\"+]$)z%=#F\\p$\"+hZ6 GNF^o7$$\"+i3&o]#F\\p$\"+1ZDxOF^o7$$\"+*y6rm#F\\p$\"+tc&es$F^o7$$\"+;F PFGF\\p$\"+KnwePF^o7$$\"+zJ]2HF\\p$\"+_(R'pPF^o7$$\"+UOj()HF\\p$\"+I\" [px$F^o7$$\"+1TwnIF\\p$\"++u!3y$F^o7$$\"+pX*y9$F\\p$\"+M8L\"y$F^o7$$\" +ipZ'H$F\\p$\"+[h&Qx$F^o7$$\"+c$f]W$F\\p$\"+`^*fv$F^o7$$\"+]**e$F^o 7$$\"+%*zhdVF\\p$\"+XOKdMF^o7$$\"+%>fS*\\F\\p$\"+M\"4p5$F^o7$$\"+>$f%G cF\\p$\"+NIr(o#F^o7$$\"+Dy,\"G'F\\p$\"+([m#>AF^o7$$\"+70mNFY7$$\"+a&4*\\7Fjw$!+Af!Q]$FY7$$\"+3dr!G\" Fjw$!+TUn=LFY7$$\"+j=_68Fjw$!+[-I?IFY7$$\"+Wy!eP\"Fjw$!+tniy?FY7$$\"+U C%[V\"Fjw$!*sQ*p!*FY7$$\"+J#>&)\\\"Fjw$\"+P^CMeF]x7$$\"+>:mk:Fjw$\"++Q %=Fjw$\"+!4l'GyFY7 $$\"+MaKs=Fjw$\"+sgkc!)FY7$$\"+A\\31>Fjw$\"+H'Q)y\")FY7$$\"+6W%)R>Fjw$ \"+I)GQ:)FY7$$\"+8)y,(>Fjw$\"+Go.**zFY7$$\"+:K^+?Fjw$\"+oa29xFY7$$\"+k ;!H.#Fjw$\"+K!QAE(FY7$$\"+7,Hl?Fjw$\"+%z?bl'FY7$$\"+4w)R7#Fjw$\"+XLJg^ FY7$$\"+y%f\")=#Fjw$\"+M&*3eHFY7$$\"+/-a[AFjw$\"+S^$e(QF]x7$$\"+ial6BF jw$!+e<,PFFY7$$\"+i@OtBFjw$!+/B\\6hFY7$$\"+fL'zV#Fjw$!*D-!=)*F^o7$$\"+ !*>=+DFjw$!+yeDN8F^o7$$\"+E&4Qc#Fjw$!+g)ptm\"F^o7$$\"+%>5pi#Fjw$!+UljO >F^o7$$\"+u;!fl#Fjw$!+q`\\I?F^o7$$\"+bJ*[o#Fjw$!+!H!>+@F^o7$$\"+4p],FF jw$!+RZiF@F^o7$$\"+j17=FFjw$!+np\"\\9#F^o7$$\"+SvUEFFjw$!+*\\\"\\\\@F^ o7$$\"+N_FY7$FA$\"++%****>'F]x-FC6&FE$\" *++++\"FjwF(F(-FG6#%%LINEG-FK6#Q9Polynomial~InterpolationFN-%*THICKNES SG6#\"\"#-%+AXESLABELSG6$Q!FNFail-%&TITLEGFhhl-%'SYMBOLG6$%'CIRCLEGF3- %%VIEWG6$;F(F@%(DEFAULTG" 1 2 4 1 15 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Center of Holes" "Polynomial Interpolation" }}}}{PARA 212 "" 0 "" {TEXT 212 0 "" }}}{SECT 0 {PARA 217 "" 0 "" {TEXT 221 20 " Spline Interpolation" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{SECT 0 {PARA 219 "" 0 "" {TEXT 223 13 "Linear Spline" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 36 "linear_ spline:=spline(X,Y,x,linear);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.li near_splineG-%*PIECEWISEG6'7$,&\"$?$\"\"\"*&\"#?F+%\"xGF+!\"\"2F.\"#57 $,&\"$!))F/*&\"$+\"F+F.F+F+2F.\"#:7$,&\"&!o?F/*&\"%?9F+F.F+F+2F.F-7$,& $\"++++#p'!\"&F+*&$\"++++gH!\"'F+F.F+F/2F.$\"$D#F/7$,&$\"+++++e!\"(F/* &$\"+++++S!\")F+F.F+F+%*otherwiseG" }}}{PARA 208 "" 0 "" {TEXT 208 0 " " }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 36 "S_linear:=int(linear_spline,x=a..b);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%)S_linearG$\"+++JWQ!\"&" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 234 "linear _spline:=x->spline(X,Y,x,linear):\nplot([XY,linear_spline],X[1]..X[n], style=[POINT,LINE],symbol=CIRCLE,symbolsize=15,thickness=2,title=\"Lin ear Spline Interpolation\",color=[BLACK,GREEN],legend=[\"Center of Hol es\",\"Linear Spline\"]);" }{MPLTEXT 1 219 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 408 285 285 {PLOTDATA 2 "6)-%'CURVESG6&7(7$$\"\"!F)$\"$?$F)7 $$\"#5F)$\"$?\"F)7$$\"#:F)$\"$?'F)7$$\"#?F)$\"%?xF)7$$\"3+++++++]A!#;F *7$$\"#IF)F4-%'COLOURG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-%'LEGENDG6#Q0C enter~of~Holes6\"-F$6&7]p7$F)F*7$$\"+]i9Rl!#5$\"+vq@pI!\"(7$$\"+WA)GA \"!\"*$\"+^NUbHFY7$$\"+Qeui=Fgn$\"+K3XFGFY7$$\"+i3&o]#Fgn$\"+G)H')p#FY 7$$\"+pX*y9$Fgn$\"+'3@/d#FY7$$\"+WTAUPFgn$\"+r^b^CFY7$$\"+%*zhdVFgn$\" +,kZGBFY7$$\"+%>fS*\\Fgn$\"+h\")=,AFY7$$\"+>$f%GcFgn$\"+O\"3V2#FY7$$\" +Dy,\"G'Fgn$\"+NkzV>FY7$$\"+7iUW\"FY7$$\"+% pnsM*Fgn$\"+hkaI8FY7$$\"+0_J&o*Fgn$\"+fp$HE\"FY7$$\"+siL-5!\")$\"*siLA \"!\"'7$$\"+JL(4.\"Fcs$\"*JL(4:Ffs7$$\"+!R5'f5Fcs$\"*!R5'z\"Ffs7$$\"+/ QBE6Fcs$\"*/QBY#Ffs7$$\"+:o?&=\"Fcs$\"*:o?0$Ffs7$$\"+a&4*\\7Fcs$\"*a&4 *p$Ffs7$$\"+j=_68Fcs$\"*j=_J%Ffs7$$\"+Wy!eP\"Fcs$\"*Wy!e\\Ffs7$$\"+UC% [V\"Fcs$\"*UC%[bFfs7$$\"+O3om9Fcs$\"*O3o'eFfs7$$\"+J#>&)\\\"Fcs$\"*J#> &='Ffs7$$\"+v.fJ:Fcs$\"*K$eo5!\"&7$$\"+>:mk:Fcs$\"*d$>Q:F^w7$$\"+[+X$f \"Fcs$\"*o!*p%>F^w7$$\"+w&QAi\"Fcs$\"*y(ybBF^w7$$\"+v4L`;Fcs$\"*%)*H(z #F^w7$$\"+uLU%o\"Fcs$\"*\">\")QKF^w7$$\"+k[a;%=Fcs$\"*>r_Z&F^w7$$\"+MaKs=Fcs$\"*;@q!fF^w7$$\"+A\\31>F cs$\"**eS'Q'F^w7$$\"+6W%)R>Fcs$\"*k!zloF^w7$$\"+8)y,(>Fcs$\"*9RlH(F^w7 $$\"+:K^+?Fcs$\"*%)3[q(F^w7$$\"+Suq;?Fcs$\"*yfaA(F^w7$$\"+k;!H.#Fcs$\" *v5hu'F^w7$$\"+))e4\\?Fcs$\"*shnE'F^w7$$\"+7,Hl?Fcs$\"*o7uy&F^w7$$\"+' [k*z?Fcs$\"*@^IN&F^w7$$\"+g)QY4#Fcs$\"*u*o=\\F^w7$$\"+MKJ4@Fcs$\"*FGV[ %F^w7$$\"+4w)R7#Fcs$\"*xm*\\SF^w7$$\"+w0.S@Fcs$\"*&\\4vNF^w7$$\"+WN2c@ Fcs$\"*5B-5$F^w7$$\"+6l6s@Fcs$\"*F^`i#F^w7$$\"+y%f\")=#Fcs$\"*Xz/:#F^w 7$$\"+gYD.AFcs$\"*1iOq\"F^w7$$\"+T)\\$=AFcs$\"*rWoD\"F^w7$$\"+A]WLAFcs $\")NF+\")F^w7$$\"+/-a[AFcs$\")'*4KOF^w7$$\"+cD^]AFcs$\"+C-0-KFY7$$\"+ 3\\[_AFcs$\"+K'R*4KFY7$$\"+gsXaAFcs$\"+S!Hy@$FY7$$\"+6'HkD#Fcs$\"+W%=d A$FY7$$\"+9VPgAFcs$\"+cs\\TKFY7$$\"+=!>VE#Fcs$\"+sgFdKFY7$$\"+E%3AF#Fc s$\"+/P$))G$FY7$$\"+Ly4!G#Fcs$\"+K8R?LFY7$$\"+[m(eH#Fcs$\"+#f1NQ$FY7$$ \"+ial6BFcs$\"+[=iYMFY7$$\"+i@OtBFcs$\"+['[Mp$FY7$$\"+fL'zV#Fcs$\"+OM& =&RFY7$$\"+!*>=+DFcs$\"+gzs+UFY7$$\"+E&4Qc#Fcs$\"++\"Q_X%FY7$$\"+%>5pi #Fcs$\"+!ySwq%FY7$$\"+bJ*[o#Fcs$\"+?EdR\\FY7$$\"+r\"[8v#Fcs$\"+!o#R0_F Y7$$\"+Ijy5GFcs$\"+?`9VaFY7$$\"+/)fT(GFcs$\"+?#Rmp&FY7$$\"+1j\"[$HFcs$ \"+?_ERfFY7$FA$\"+++++iFY-FC6&FEF($\"*++++\"FcsF(-FG6#%%LINEG-FK6#Q.Li near~SplineFN-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!FNFbhl-%&TITLEG6#Q< Linear~Spline~InterpolationFN-%'SYMBOLG6$%'CIRCLEGF3-%%VIEWG6$;F(F@%(D EFAULTG" 1 2 4 1 15 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Cent er of Holes" "Linear Spline" }}}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}} {SECT 0 {PARA 219 "" 0 "" {TEXT 223 16 "Quadratic Spline" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 42 "quadratic_spline:=spline(X,Y,x,quadratic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1quadratic_splineG-%*PIECEWISEG6(7$,&$\"$?$\"\"!\"\" \"*&$\"3?+++:zGul!#ed[6#!#:F-F3F-!\"\"*&$\"3/+++tjGsF!#;F-),&F3F-\"#5FAF4F-FA 2F3$\"3+++++++]7FE7$,($\"+o)Gb!>!\"&FA*&$\"3#******>\"fo68!#9F-F3F-F-* &$\"3&******z(>dBLF@F-),&F3F-\"#:FAF4F-F-2F3$\"3+++++++] " 0 "" {MPLTEXT 1 218 42 "S_quadratic:=int(quadrat ic_spline,x=a..b);" }{MPLTEXT 1 219 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,S_quadraticG$\"+*3-)fJ!\"&" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 248 "quadratic_spline:=x->spline(X,Y,x,quadratic):\nplo t([XY,quadratic_spline],X[1]..X[n],style=[POINT,LINE],symbol=CIRCLE,sy mbolsize=15,thickness=2,title=\"Quadratic Spline Interpolation\",color =[BLACK,BLUE],legend=[\"Center of Holes\",\"Quadratic Spline\"]);" } {MPLTEXT 1 219 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 430 262 262 {PLOTDATA 2 "6)-%'CURVESG6&7(7$$\"\"!F)$\"$?$F)7$$\"#5F)$\"$?\"F)7$$\" #:F)$\"$?'F)7$$\"#?F)$\"%?xF)7$$\"3+++++++]A!#;F*7$$\"#IF)F4-%'COLOURG 6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-%'LEGENDG6#Q0Center~of~Holes6\"-F$6& 7ip7$F)F*7$$\"+]i9Rl!#5$\"+S>6GK!\"(7$$\"+WA)GA\"!\"*$\"+'f9$)H$FY7$$ \"+Qeui=Fgn$\"+#4;\"GMFY7$$\"+i3&o]#Fgn$\"+d![Jh$FY7$$\"+pX*y9$Fgn$\"+ #)>Y^QFY7$$\"+WTAUPFgn$\"+f\"z17%FY7$$\"+%*zhdVFgn$\"+d1Q[WFY7$$\"+%>f S*\\Fgn$\"+U'o'R[FY7$$\"+>$f%GcFgn$\"+!pWs9&FY7$$\"+Dy,\"G'Fgn$\"+Mf\" 3B&FY7$$\"+7BY#FY 7$$\"+siL-5!\")$\"+a)R/:\"FY7$$\"+!R5'f5F^s$!+^B&=f\"F^s7$$\"+/QBE6F^s $!+i#G9\">FY7$$\"+:o?&=\"F^s$!+jzznOFY7$$\"+%=ev@\"F^s$!+LA@8ZFY7$$\"+ a&4*\\7F^s$!+\\ml;eFY7$$\"+JEJl7F^s$!*H))zF'!\"'7$$\"+3dr!G\"F^s$!*>;; e'F_u7$$\"+'y=hH\"F^s$!*OPvs'F_u7$$\"+j=_68F^s$!*!>v:nF_u7$$\"+eLfF8F^ s$!+lULNlFY7$$\"+a[mV8F^s$!+\"pDK='FY7$$\"+\\jtf8F^s$!+bfUfcFY7$$\"+Wy !eP\"F^s$!+x^$R'\\FY7$$\"+V^K09F^s$!+T3MRKFY7$$\"+UC%[V\"F^s$!*p,gN*FY 7$$\"+J#>&)\\\"F^s$\"+_$)\\1gFY7$$\"+v.fJ:F^s$\"+YT`n5F_u7$$\"+>:mk:F^ s$\"+3$=rg\"F_u7$$\"+[+X$f\"F^s$\"+3d,O@F_u7$$\"+w&QAi\"F^s$\"+hK+?FF_ u7$$\"+v4L`;F^s$\"+kag7MF_u7$$\"+uLU%o\"F^s$\"+'3o%pTF_u7$$\"+k[a;%=F^s$\"+1!H/%zF_u7$$\"+MaKs=F^s$ \"+a6X^#)F_u7$$\"+1`w!)=F^s$\"+Ex$zI)F_u7$$\"+y^?*)=F^s$\"+)=a9N)F_u7$ $\"+]]k(*=F^s$\"+S0+#Q)F_u7$$\"+A\\31>F^s$\"+$zw&*R)F_u7$$\"+%zCX\">F^ s$\"+DH=/%)F_u7$$\"+mY'H#>F^s$\"+[*=eR)F_u7$$\"+QXSJ>F^s$\"+q[[u$)F_u7 $$\"+6W%)R>F^s$\"+j1=S$)F_u7$$\"+8)y,(>F^s$\"+]@!)4\")F_u7$$\"+:K^+?F^ s$\"+gU(=r(F_u7$$\"+k;!H.#F^s$\"+&y\\?5(F_u7$$\"+7,Hl?F^s$\"+Jq@,jF_u7 $$\"+g)QY4#F^s$\"+?Oe5aF_u7$$\"+4w)R7#F^s$\"+FA6jVF_u7$$\"+w0.S@F^s$\" +-BdcPF_u7$$\"+WN2c@F^s$\"+2yqwJF_u7$$\"+6l6s@F^s$\"+UbmBEF_u7$$\"+y%f \")=#F^s$\"+laW(4#F_u7$$\"+T)\\$=AF^s$\"+5@\"*z6F_u7$$\"+/-a[AF^s$\"+* p^Od$FY7$$\"+Ly4!G#F^s$!+)[p*3SFY7$$\"+ial6BF^s$!+9VPb5F_u7$$\"+7)3DM# F^s$!+L#4\\f\"F_u7$$\"+i@OtBF^s$!+&3S_.#F_u7$$\"+gFm0CF^s$!+$QK**Q#F_u 7$$\"+fL'zV#F^s$!+Tb*ej#F_u7$$\"+;!=NX#F^s$!+@Xb:FF_u7$$\"+uE2pCF^s$!+ !G***pFF_u7$$\"+.+&oZ#F^s$!+=+DF^s$!+)4YK!GF_u7$$\"+Ka83DF^s$!+zsb &z#F_u7$$\"+u))3;DF^s$!+)>w7y#F_u7$$\"+;B/CDF^s$!+OGSgFF_u7$$\"+ed*>`# F^s$!+.s$Ht#F_u7$$\"+UE!za#F^s$!+L\"H#eEF_u7$$\"+E&4Qc#F^s$!+y>:dDF_u7 $$\"+g)f`f#F^s$!+&[_'yAF_u7$$\"+%>5pi#F^s$!+8Bn'*=F_u7$$\"+bJ*[o#F^s$! +H8@v6F_u7$$\"+r\"[8v#F^s$!*/#\\y\\F_u7$$\"+Ijy5GF^s$!*C)*)HFFY7$$\"+/ )fT(GF^s$\"+bd)oL$FY7$$\"+b!)[/HF^s$\"+h1k]XFY7$$\"+1j\"[$HF^s$\"+RPzJ aFY7$$\"+`\"3u'HF^s$\"+N%[z+'FY7$FAF4-FC6&FEF(F($\"*++++\"F^s-FG6#%%LI NEG-FK6#Q1Quadratic~SplineFN-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!FNF[ \\m-%&TITLEG6#Q?Quadratic~Spline~InterpolationFN-%'SYMBOLG6$%'CIRCLEGF 3-%%VIEWG6$;F(F@%(DEFAULTG" 1 2 4 1 15 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Center of Holes" "Quadratic Spline" }}}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 219 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 0 "" }}}}{SECT 0 {PARA 219 "" 0 "" {TEXT 223 12 "Cubic \+ Spline" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 34 "cubic_spline: =spline(X,Y,x,cubic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-cubic_spli neG-%*PIECEWISEG6'7$,*$\"$?$\"\"!\"\"\"*&$\"3'******4r!Q&=#!#:F-%\"xGF -F-*&$\"3Maj&fJ_a=)!#IF-)F2\"\"#F-F-*&$\"3#)*****4r!Q&Q#!# 8Uh:(!#;F-),&F2F-FAF?F8F-F?*&$\"3)******H*H`>QFMF-)FOF>F-F-2F2\"#:7$,* $\"+&3RfT#!\"&F?*&$\"3)********Qf>l\"!#9F-F2F-F-*&$\"3^F-F?2F2\"#?7$,*$\"+WG_tQFZF-* &$\"3/+++A9w]:FhnF-F2F-F?*&$\"3#*H`N!p7>9\"FhnF-),&F2F-FcoF?F8F-F?*&$ \"3%******\\.pGJ#F1F-)F_pF>F-F-2F2$\"$D#F?7$,*$\"+CZJ5mFZF-*&$\"3()*** **HV&pBHFhnF-F2F-F?*&$\"36iT%H'3RFfF1F-),&F2F-$FfpF?F?F8F-F-*&$\"37+++ QfRMEFMF-)FbqF>F-F?%*otherwiseG" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" } }{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 34 "S_cubic:=int(cubic_spli ne,x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(S_cubicG$\"+.BrZE!\" &" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 218 230 "cubic_spline:=x->spline(X,Y,x,cubic):\nplot([XY,cu bic_spline],X[1]..X[n],style=[POINT,LINE],symbol=CIRCLE,symbolsize=15, thickness=2,title=\"Cubic Spline Interpolation\",color=[BLACK,YELLOW], legend=[\"Center of Holes\",\"Cubic spline\"]);" }{MPLTEXT 1 219 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 402 252 252 {PLOTDATA 2 "6)-%'CURVESG6&7( 7$$\"\"!F)$\"$?$F)7$$\"#5F)$\"$?\"F)7$$\"#:F)$\"$?'F)7$$\"#?F)$\"%?xF) 7$$\"3+++++++]A!#;F*7$$\"#IF)F4-%'COLOURG6&%$RGBGF(F(F(-%&STYLEG6#%&PO INTG-%'LEGENDG6#Q0Center~of~Holes6\"-F$6&7^p7$F)F*7$$\"+]i9Rl!#5$\"+%[ #QAY!\"(7$$\"+WA)GA\"!\"*$\"+h1%)GeFY7$$\"+Qeui=Fgn$\"+3Aj;rFY7$$\"+i3 &o]#Fgn$\"+q_j-$)FY7$$\"+pX*y9$Fgn$\"+d/FN$*FY7$$\"+WTAUPFgn$\"+tx!G, \"!\"'7$$\"+%*zhdVFgn$\"+5X#\\2\"F^p7$$\"+%>fS*\\Fgn$\"+V5G96F^p7$$\"+ >$f%GcFgn$\"+nBqC6F^p7$$\"+Dy,\"G'Fgn$\"+c%f:5\"F^p7$$\"+7FY7$$\"+/QBE6F_s$!+q\"*zYaFY7$$\"+: o?&=\"F_s$!+1GNM!)FY7$$\"+%=ev@\"F_s$!+u\")Ho!*FY7$$\"+a&4*\\7F_s$!+ec @I(*FY7$$\"+JEJl7F_s$!+())G@*)*FY7$$\"+3dr!G\"F_s$!+?DuV**FY7$$\"+'y=h H\"F_s$!+w:ow)*FY7$$\"+j=_68F_s$!*1rDo*F^p7$$\"+a[mV8F_s$!*0k:$))F^p7$ $\"+Wy!eP\"F_s$!*S=ZJ(F^p7$$\"+V^K09F_s$!*N(4q_F^p7$$\"+UC%[V\"F_s$!*x e3a#F^p7$$\"+O3om9F_s$\"*l$GQ7F^p7$$\"+J#>&)\\\"F_s$\"*/\"[cfF^p7$$\"+ v.fJ:F_s$\"+H1W)=\"F^p7$$\"+>:mk:F_s$\"+^;>o=F^p7$$\"+[+X$f\"F_s$\"+qI >7DF^p7$$\"+w&QAi\"F_s$\"+O2R)=$F^p7$$\"+v4L`;F_s$\"+57yORF^p7$$\"+uLU %o\"F_s$\"+mgn%o%F^p7$$\"+k[a;tF^p7$$\"+10#>% =F_s$\"+$)el]xF^p7$$\"+MaKs=F_s$\"+D*=l1)F^p7$$\"+y^?*)=F_s$\"+(y1a=)F ^p7$$\"+A\\31>F_s$\"+PG(*f#)F^p7$$\"+%zCX\">F_s$\"+]skz#)F^p7$$\"+mY'H #>F_s$\"+Gd0(G)F^p7$$\"+QXSJ>F_s$\"+jI!=G)F^p7$$\"+6W%)R>F_s$\"+=T\\j# )F^p7$$\"+8)y,(>F_s$\"*85Q3)!\"&7$$\"+:K^+?F_s$\"+97,7xF^p7$$\"+k;!H.# F_s$\"+!4(R%4(F^p7$$\"+7,Hl?F_s$\"+$=.^G'F^p7$$\"+g)QY4#F_s$\"+%4ocU&F ^p7$$\"+4w)R7#F_s$\"+.*QE[%F^p7$$\"+WN2c@F_s$\"+$)eR(R$F^p7$$\"+y%f\") =#F_s$\"+*))3+I#F^p7$$\"+T)\\$=AF_s$\"+2zR(H\"F^p7$$\"+/-a[AF_s$\"*%e0 GOF^p7$$\"+Ly4!G#F_s$!+m@#*p]FY7$$\"+ial6BF_s$!+X(oME\"F^p7$$\"+7)3DM# F_s$!+jGF)*=F^p7$$\"+i@OtBF_s$!+[s9MCF^p7$$\"+fL'zV#F_s$!+,.EcKF^p7$$ \"+uE2pCF_s$!+@IF=+DF_s$!+'\\oqp$F^p7$$\"+u))3;DF_s$!+@* f\"fPF^p7$$\"+ed*>`#F_s$!+iu*=!QF^p7$$\"+UE!za#F_s$!+6t\"f#QF^p7$$\"+E &4Qc#F_s$!+dc&=$QF^p7$$\"+$p%ezDF_s$!+tQ^?QF^p7$$\"+g)f`f#F_s$!+<]j#z$ F^p7$$\"+F]86EF_s$!+#fR)[PF^p7$$\"+%>5pi#F_s$!+M\"[(*o$F^p7$$\"+u;!fl# F_s$!+VPLVNF^p7$$\"+bJ*[o#F_s$!+rY?^LF^p7$$\"+r\"[8v#F_s$!+]75fFF^p7$$ \"+Ijy5GF_s$!+t[3\"3#F^p7$$\"+/)fT(GF_s$!+&f)fU7F^p7$$\"+1j\"[$HF_s$!* ^.qk$F^p7$FA$\")*****>'F\\^l-FC6&FE$\"*++++\"F_sFhglF(-FG6#%%LINEG-FK6 #Q-Cubic~splineFN-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!FNFghl-%&TITLEG 6#Q;Cubic~Spline~InterpolationFN-%'SYMBOLG6$%'CIRCLEGF3-%%VIEWG6$;F(F@ %(DEFAULTG" 1 2 4 1 15 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "C enter of Holes" "Cubic spline" }}}}{PARA 212 "" 0 "" {TEXT 212 0 "" }} {PARA 208 "" 0 "" {TEXT 208 0 "" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" } }{PARA 208 "" 0 "" {TEXT 208 0 "" }}}{PARA 208 "" 0 "" {TEXT 208 0 "" }}}{SECT 0 {PARA 210 "" 0 "" {TEXT 210 21 "Section V: Conclusion" }} {PARA 220 "" 0 "" {TEXT 224 0 "" }}{PARA 211 "> " 0 "" {MPLTEXT 1 211 15 "with( Spread ):" }}{PARA 211 "> " 0 "" {MPLTEXT 1 211 39 "Evaluate Spreadsheet(Discrete Function):" }}{EXCHG {PARA 208 "" 0 "" {SPREADSHEET {NAME "Discrete Function" } {ROWHEIGHTS 1 50 2 84 3 50 4 50 5 50 6 50 7 50 8 50 9 50 10 50 11 50 12 50 13 50 14 50 15 50 16 50 17 50 18 50 19 50 20 50 21 50 22 50 23 50 24 50 25 50 26 50 27 50 28 50 29 50 30 50 31 50 32 50 33 50 34 50 35 50 36 50 37 50 38 50 39 50 40 50 41 50 42 50 43 50 44 50 45 50 46 50 47 50 48 50 49 50 50 50 51 50 52 50 53 50 54 50 55 50 56 50 57 50 58 50 59 50 60 50 61 50 62 50 63 50 64 50 65 50 66 50 67 50 68 50 69 50 70 50 71 50 72 50 73 50 74 50 75 50 76 50 77 50 78 50 79 50 80 50 81 50 82 50 83 50 84 50 85 50 86 50 87 50 88 50 89 50 90 50 91 50 92 50 93 50 94 50 95 50 96 50 97 50 98 50 99 50 } {COLWIDTHS 1 185 2 260 3 211 4 113 5 144 6 107 7 100 8 100 9 100 10 100 11 100 12 100 13 100 14 100 15 100 16 100 17 100 18 100 19 100 20 100 21 100 22 100 23 100 24 100 25 100 26 100 27 100 28 100 29 100 30 100 31 100 32 100 33 100 34 100 35 100 36 100 37 100 38 100 39 100 40 100 41 100 42 100 43 100 44 100 45 100 46 100 47 100 48 100 49 100 50 100 51 100 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }858 214 214 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 } {R5MATHOBJ "Method*of*Integration" 20 "6#*(%'MethodG\"\"\"%#ofGF%%,Int egrationGF%" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 } {R5MATHOBJ "Trapezoidal*Rule" -1 "6#*&%,TrapezoidalG\"\"\"%%RuleGF%" } 0 }{CELL 1 3 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Polynomial *Interpolation" -1 "6#*&%+PolynomialG\"\"\"%.InterpolationGF%" }0 } {CELL 1 4 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Linear*spline " -1 "6#*&%'LinearG\"\"\"%'splineGF%" }0 }{CELL 1 5 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Quadratic*Spline" -1 "6#*&%*QuadraticG\" \"\"%'SplineGF%" }0 }{CELL 1 6 {CELLOPTS 0 -1 -1 0 0 255 255 255 } {R5MATHOBJ "Cubic*spline" -1 "6#*&%&CubicG\"\"\"%'splineGF%" }0 } {CELL 2 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Average*Value " -1 "6#*&%(AverageG\"\"\"%&ValueGF%" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "S_trap" -1 "6#$\"+++JWQ!\"&" }0 }{CELL 2 3 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "S_polyinter" -1 "6#$ \"+$e " 0 "" {MPLTEXT 1 211 13 "with (plots);" }}{PARA 208 "" 0 "" {TEXT 208 0 "" } }{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+ redefined\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7fn%(animateG%*animate3 dG%-animatecurveG%&arrowG%-changecoordsG%,complexplotG%.complexplot3dG %*conformalG%,conformal3dG%,contourplotG%.contourplot3dG%*coordplotG%, coordplot3dG%-cylinderplotG%,densityplotG%(displayG%*display3dG%*field plotG%,fieldplot3dG%)gradplotG%+gradplot3dG%,graphplot3dG%-implicitplo tG%/implicitplot3dG%(inequalG%,interactiveG%2interactiveparamsG%-listc ontplotG%/listcontplot3dG%0listdensityplotG%)listplotG%+listplot3dG%+l oglogplotG%(logplotG%+matrixplotG%)multipleG%(odeplotG%'paretoG%,plotc ompareG%*pointplotG%,pointplot3dG%*polarplotG%,polygonplotG%.polygonpl ot3dG%4polyhedra_supportedG%.polyhedraplotG%'replotG%*rootlocusG%,semi logplotG%+setoptionsG%-setoptions3dG%+spacecurveG%1sparsematrixplotG%+ sphereplotG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 214 603 "trap:=x->spline(X,Y,x,linear): \npoly_inter:=x->interp(X,Y,x):\nlinear_spline:=x->spline(X,Y,x,linear ):\nquadratic_spline:=x->spline(X,Y,x,quadratic):\ncubic_spline:=x->sp line(X,Y,x,cubic):\nplot([XY,trap,poly_inter,linear_spline,quadratic_s pline,cubic_spline],X[1]..X[n],style=[POINT,LINE,LINE,LINE,LINE,LINE], symbol=CIRCLE,symbolsize=15,thickness=2,title=\"Comparison of Trapezoi dal Rule, Polynomial Interpolation, and Spline Interpolation\",color=[ BLACK,ORANGE,RED,GREEN,BLUE,YELLOW],legend=[\"Center of Holes\",\"Trap ezoidal Rule\",\"Polynomial Interpolation\",\"Linear Spline\",\"Quadra tic Spline\",\"Cubic spline\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 886 482 482 {PLOTDATA 2 "6--%'CURVESG6&7(7$$\"\"!F)$\"$?$F)7$$\"#5F)$\"$? \"F)7$$\"#:F)$\"$?'F)7$$\"#?F)$\"%?xF)7$$\"3+++++++]A!#;F*7$$\"#IF)F4- %'COLOURG6&%$RGBGF(F(F(-%&STYLEG6#%&POINTG-%'LEGENDG6#Q0Center~of~Hole s6\"-F$6&7]p7$F)F*7$$\"+]i9Rl!#5$\"+vq@pI!\"(7$$\"+WA)GA\"!\"*$\"+^NUb HFY7$$\"+Qeui=Fgn$\"+K3XFGFY7$$\"+i3&o]#Fgn$\"+G)H')p#FY7$$\"+pX*y9$Fg n$\"+'3@/d#FY7$$\"+WTAUPFgn$\"+r^b^CFY7$$\"+%*zhdVFgn$\"+,kZGBFY7$$\"+ %>fS*\\Fgn$\"+h\")=,AFY7$$\"+>$f%GcFgn$\"+O\"3V2#FY7$$\"+Dy,\"G'Fgn$\" +NkzV>FY7$$\"+7iUW\"FY7$$\"+%pnsM*Fgn$\"+hk aI8FY7$$\"+0_J&o*Fgn$\"+fp$HE\"FY7$$\"+siL-5!\")$\"*siLA\"!\"'7$$\"+JL (4.\"Fcs$\"*JL(4:Ffs7$$\"+!R5'f5Fcs$\"*!R5'z\"Ffs7$$\"+/QBE6Fcs$\"*/QB Y#Ffs7$$\"+:o?&=\"Fcs$\"*:o?0$Ffs7$$\"+a&4*\\7Fcs$\"*a&4*p$Ffs7$$\"+j= _68Fcs$\"*j=_J%Ffs7$$\"+Wy!eP\"Fcs$\"*Wy!e\\Ffs7$$\"+UC%[V\"Fcs$\"*UC% [bFfs7$$\"+O3om9Fcs$\"*O3o'eFfs7$$\"+J#>&)\\\"Fcs$\"*J#>&='Ffs7$$\"+v. fJ:Fcs$\"*K$eo5!\"&7$$\"+>:mk:Fcs$\"*d$>Q:F^w7$$\"+[+X$f\"Fcs$\"*o!*p% >F^w7$$\"+w&QAi\"Fcs$\"*y(ybBF^w7$$\"+v4L`;Fcs$\"*%)*H(z#F^w7$$\"+uLU% o\"Fcs$\"*\">\")QKF^w7$$\"+k[a; %=Fcs$\"*>r_Z&F^w7$$\"+MaKs=Fcs$\"*;@q!fF^w7$$\"+A\\31>Fcs$\"**eS'Q'F^ w7$$\"+6W%)R>Fcs$\"*k!zloF^w7$$\"+8)y,(>Fcs$\"*9RlH(F^w7$$\"+:K^+?Fcs$ \"*%)3[q(F^w7$$\"+Suq;?Fcs$\"*yfaA(F^w7$$\"+k;!H.#Fcs$\"*v5hu'F^w7$$\" +))e4\\?Fcs$\"*shnE'F^w7$$\"+7,Hl?Fcs$\"*o7uy&F^w7$$\"+'[k*z?Fcs$\"*@^ IN&F^w7$$\"+g)QY4#Fcs$\"*u*o=\\F^w7$$\"+MKJ4@Fcs$\"*FGV[%F^w7$$\"+4w)R 7#Fcs$\"*xm*\\SF^w7$$\"+w0.S@Fcs$\"*&\\4vNF^w7$$\"+WN2c@Fcs$\"*5B-5$F^ w7$$\"+6l6s@Fcs$\"*F^`i#F^w7$$\"+y%f\")=#Fcs$\"*Xz/:#F^w7$$\"+gYD.AFcs $\"*1iOq\"F^w7$$\"+T)\\$=AFcs$\"*rWoD\"F^w7$$\"+A]WLAFcs$\")NF+\")F^w7 $$\"+/-a[AFcs$\")'*4KOF^w7$$\"+cD^]AFcs$\"+C-0-KFY7$$\"+3\\[_AFcs$\"+K 'R*4KFY7$$\"+gsXaAFcs$\"+S!Hy@$FY7$$\"+6'HkD#Fcs$\"+W%=dA$FY7$$\"+9VPg AFcs$\"+cs\\TKFY7$$\"+=!>VE#Fcs$\"+sgFdKFY7$$\"+E%3AF#Fcs$\"+/P$))G$FY 7$$\"+Ly4!G#Fcs$\"+K8R?LFY7$$\"+[m(eH#Fcs$\"+#f1NQ$FY7$$\"+ial6BFcs$\" +[=iYMFY7$$\"+i@OtBFcs$\"+['[Mp$FY7$$\"+fL'zV#Fcs$\"+OM&=&RFY7$$\"+!*> =+DFcs$\"+gzs+UFY7$$\"+E&4Qc#Fcs$\"++\"Q_X%FY7$$\"+%>5pi#Fcs$\"+!ySwq% FY7$$\"+bJ*[o#Fcs$\"+?EdR\\FY7$$\"+r\"[8v#Fcs$\"+!o#R0_FY7$$\"+Ijy5GFc s$\"+?`9VaFY7$$\"+/)fT(GFcs$\"+?#Rmp&FY7$$\"+1j\"[$HFcs$\"+?_ERfFY7$FA $\"+++++iFY-FC6&FE$\")+++!)Fcs$\")Vyg>FcsFegl-FG6#%%LINEG-FK6#Q1Trapez oidal~RuleFN-F$6&7`pFR7$$\"+ilyM;FV$\"+!*yjG[Ffs7$$\"+DJdpKFV$\"+\"fUh (*)Ffs7$$\"+)ofV!\\FV$\"+_xzx7F^w7$FT$\"+$HB\\i\"F^w7$$\"+XV)RQ*FV$\"+ xb_a@F^w7$Fen$\"+lS&ff#F^w7$$\"+TS\"Ga\"Fgn$\"+uF&f*HF^w7$F[o$\"+Ur>.L F^w7$$\"+]$)z%=#Fgn$\"+hZ6GNF^w7$F`o$\"+1ZDxOF^w7$$\"+*y6rm#Fgn$\"+tc& es$F^w7$$\"+;FPFGFgn$\"+KnwePF^w7$$\"+zJ]2HFgn$\"+_(R'pPF^w7$$\"+UOj() HFgn$\"+I\"[px$F^w7$$\"+1TwnIFgn$\"++u!3y$F^w7$Feo$\"+M8L\"y$F^w7$$\"+ ipZ'H$Fgn$\"+[h&Qx$F^w7$$\"+c$f]W$Fgn$\"+`^*fv$F^w7$$\"+]**e$F^w7$F_p$\"+XOKdMF ^w7$Fdp$\"+M\"4p5$F^w7$Fip$\"+NIr(o#F^w7$F^q$\"+([m#>AF^w7$Fcq$\"+#HT' *z\"F^w7$Fhq$\"+&RU=M\"F^w7$F]r$\"+ev9n\"*Ffs7$Fbr$\"+i$*)*QbFfs7$Fgr$ \"+&=^]s#Ffs7$Fas$\"*ueQ'RFY7$F]t$!+75%yk\"Ffs7$$\"+(4AH4\"Fcs$!+)=Q,Q #Ffs7$Fbt$!+*o2A$HFfs7$$\"+5.sb6Fcs$!+?VFuKFfs7$Fgt$!+05G$[$Ffs7$$\"+% =ev@\"Fcs$!+I>0mNFfs7$F\\u$!+Af!Q]$Ffs7$$\"+3dr!G\"Fcs$!+TUn=LFfs7$Fau $!+[-I?IFfs7$Ffu$!+tniy?Ffs7$F[v$!*sQ*p!*Ffs7$Fev$\"+P^CMeFY7$F`w$\"++ QF^w7$$\"+u;!fl#Fcs$!+q`\\I?F^w 7$Ffel$!+!H!>+@F^w7$$\"+4p],FFcs$!+RZiF@F^w7$$\"+j17=FFcs$!+np\"\\9#F^ w7$$\"+SvUEFFcs$!+*\\\"\\\\@F^w7$$\"+N_Ffs7$FA$\"++%****>'FY-FC6&FE$\"*++++\" FcsF(F(Fggl-FK6#Q9Polynomial~InterpolationFN-F$6&FQ-FC6&FEF(Fi[nF(Fggl -FK6#Q.Linear~SplineFN-F$6&7ipFR7$FT$\"+S>6GKFY7$Fen$\"+'f9$)H$FY7$F[o $\"+#4;\"GMFY7$F`o$\"+d![Jh$FY7$Feo$\"+#)>Y^QFY7$Fjo$\"+f\"z17%FY7$F_p $\"+d1Q[WFY7$Fdp$\"+U'o'R[FY7$Fip$\"+!pWs9&FY7$F^q$\"+Mf\"3B&FY7$Fcq$ \"+Yx&)3^FY7$Fhq$\"+t^Q_ZFY7$F]r$\"++Q(3;%FY7$Fbr$\"+1WQpLFY7$Fgr$\"+' )*>BY#FY7$Fas$\"+a)R/:\"FY7$F]t$!+^B&=f\"Fcs7$Fbt$!+i#G9\">FY7$Fgt$!+j zznOFY7$F_am$!+LA@8ZFY7$F\\u$!+\\ml;eFY7$$\"+JEJl7Fcs$!*H))zF'Ffs7$Fga m$!*>;;e'Ffs7$$\"+'y=hH\"Fcs$!*OPvs'Ffs7$Fau$!*!>v:nFfs7$$\"+eLfF8Fcs$ !+lULNlFY7$$\"+a[mV8Fcs$!+\"pDK='FY7$$\"+\\jtf8Fcs$!+bfUfcFY7$Ffu$!+x^ $R'\\FY7$$\"+V^K09Fcs$!+T3MRKFY7$F[v$!*p,gN*FY7$Fev$\"+_$)\\1gFY7$Fjv$ \"+YT`n5Ffs7$F`w$\"+3$=rg\"Ffs7$Few$\"+3d,O@Ffs7$Fjw$\"+hK+?FFfs7$F_x$ \"+kag7MFfs7$Fdx$\"+'3o%pTFfs7$Fix$\"+Wr')=]Ffs7$F^y$\"+C3&o$fFfs7$Fcy $\"+D#e()y'Ffs7$Fhy$\"+Av2huFfs7$F]z$\"+1!H/%zFfs7$Fbz$\"+a6X^#)Ffs7$$ \"+1`w!)=Fcs$\"+Ex$zI)Ffs7$$\"+y^?*)=Fcs$\"+)=a9N)Ffs7$$\"+]]k(*=Fcs$ \"+S0+#Q)Ffs7$Fgz$\"+$zw&*R)Ffs7$$\"+%zCX\">Fcs$\"+DH=/%)Ffs7$$\"+mY'H #>Fcs$\"+[*=eR)Ffs7$$\"+QXSJ>Fcs$\"+q[[u$)Ffs7$F\\[l$\"+j1=S$)Ffs7$Fa[ l$\"+]@!)4\")Ffs7$Ff[l$\"+gU(=r(Ffs7$F`\\l$\"+&y\\?5(Ffs7$Fj\\l$\"+Jq@ ,jFfs7$Fd]l$\"+?Oe5aFfs7$F^^l$\"+FA6jVFfs7$Fc^l$\"+-BdcPFfs7$Fh^l$\"+2 yqwJFfs7$F]_l$\"+UbmBEFfs7$Fb_l$\"+laW(4#Ffs7$F\\`l$\"+5@\"*z6Ffs7$Ff` l$\"+*p^Od$FY7$F^cl$!+)[p*3SFY7$Fhcl$!+9VPb5Ffs7$$\"+7)3DM#Fcs$!+L#4\\ f\"Ffs7$F]dl$!+&3S_.#Ffs7$$\"+gFm0CFcs$!+$QK**Q#Ffs7$Fbdl$!+Tb*ej#Ffs7 $$\"+;!=NX#Fcs$!+@Xb:FFfs7$$\"+uE2pCFcs$!+!G***pFFfs7$$\"+.+&oZ#Fcs$!+ w7y#Ffs7$$ \"+;B/CDFcs$!+OGSgFFfs7$$\"+ed*>`#Fcs$!+.s$Ht#Ffs7$$\"+UE!za#Fcs$!+L\" H#eEFfs7$F\\el$!+y>:dDFfs7$$\"+g)f`f#Fcs$!+&[_'yAFfs7$Fael$!+8Bn'*=Ffs 7$Ffel$!+H8@v6Ffs7$F[fl$!*/#\\y\\Ffs7$F`fl$!*C)*)HFFY7$Fefl$\"+bd)oL$F Y7$Fhjm$\"+h1k]XFY7$Fjfl$\"+RPzJaFY7$F`[n$\"+N%[z+'FY7$FAF4-FC6&FEF(F( Fi[nFggl-FK6#Q1Quadratic~SplineFN-F$6&7^pFR7$FT$\"+%[#QAYFY7$Fen$\"+h1 %)GeFY7$F[o$\"+3Aj;rFY7$F`o$\"+q_j-$)FY7$Feo$\"+d/FN$*FY7$Fjo$\"+tx!G, \"Ffs7$F_p$\"+5X#\\2\"Ffs7$Fdp$\"+V5G96Ffs7$Fip$\"+nBqC6Ffs7$F^q$\"+c% f:5\"Ffs7$Fcq$\"+2\")f\\5Ffs7$Fhq$\"*X!y@&*Ffs7$F]r$\"*Oc64)Ffs7$Fbr$ \"*j*)oC'Ffs7$Fgr$\"*<;j9%Ffs7$Fas$\"+nP[$3\"FY7$F]t$!+w8ZO>FY7$Fbt$!+ q\"*zYaFY7$Fgt$!+1GNM!)FY7$F_am$!+u\")Ho!*FY7$F\\u$!+ec@I(*FY7$Fh`n$!+ ())G@*)*FY7$Fgam$!+?DuV**FY7$F`an$!+w:ow)*FY7$Fau$!*1rDo*Ffs7$F]bn$!*0 k:$))Ffs7$Ffu$!*S=ZJ(Ffs7$Fjbn$!*N(4q_Ffs7$F[v$!*xe3a#Ffs7$F`v$\"*l$GQ 7Ffs7$Fev$\"*/\"[cfFfs7$Fjv$\"+H1W)=\"Ffs7$F`w$\"+^;>o=Ffs7$Few$\"+qI> 7DFfs7$Fjw$\"+O2R)=$Ffs7$F_x$\"+57yORFfs7$Fdx$\"+mgn%o%Ffs7$Fix$\"+#)e #eV&Ffs7$F^y$\"+nXdVhFfs7$Fcy$\"+)ySIx'Ffs7$Fhy$\"+@3s>tFfs7$F]z$\"+$) el]xFfs7$Fbz$\"+D*=l1)Ffs7$F^fn$\"+(y1a=)Ffs7$Fgz$\"+PG(*f#)Ffs7$F[gn$ \"+]skz#)Ffs7$F`gn$\"+Gd0(G)Ffs7$Fegn$\"+jI!=G)Ffs7$F\\[l$\"+=T\\j#)Ff s7$Fa[l$\"*85Q3)F^w7$Ff[l$\"+97,7xFfs7$F`\\l$\"+!4(R%4(Ffs7$Fj\\l$\"+$ =.^G'Ffs7$Fd]l$\"+%4ocU&Ffs7$F^^l$\"+.*QE[%Ffs7$Fh^l$\"+$)eR(R$Ffs7$Fb _l$\"+*))3+I#Ffs7$F\\`l$\"+2zR(H\"Ffs7$Ff`l$\"*%e0GOFfs7$F^cl$!+m@#*p] FY7$Fhcl$!+X(oME\"Ffs7$Fgjn$!+jGF)*=Ffs7$F]dl$!+[s9MCFfs7$Fbdl$!+,.EcK Ffs7$F\\\\o$!+@IF'F^w-FC6&FEFi[nFi[nF(Fggl -FK6#Q-Cubic~splineFN-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!FNFgap-%&TI TLEG6#Q]pComparison~of~Trapezoidal~Rule,~Polynomial~Interpolation,~and ~Spline~InterpolationFN-%'SYMBOLG6$%'CIRCLEGF3-%%VIEWG6$;F(F@%(DEFAULT G" 1 2 4 1 15 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Center of \+ Holes" "Trapezoidal Rule" "Polynomial Interpolation" "Linear Spline" " Quadratic Spline" "Cubic spline" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "3 10 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }