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First we extend the area problem and the idea of using approximating rectangles for a continuous function which is \+ not necessarily positive over the interval [a,b]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart; wi th(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f:=1+5*x;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&\"\"\"F&%\"xG\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plot(f,x=-2..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 211 189 189 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"# \"\"!$!\"*F*7$$!1++]P&3Y$>!#:$!1,+](oUIn)F07$$!1+]iv@oF07$$!1+]i!3%f+:F0$!1+]7./(H]'F07$$!1+]7oS:P9F0$!1+]iS .x&='F07$$!1++]<#)*=P\"F0$!1++](3\"\\feF07$$!1++vG3U98F0$!1++vVT5sbF07 $$!1++]-\\r\\7F0$!1++]7Xd[_F07$$!1++](GVZ=\"F0$!1++]PkrB\\F07$$!1++](4 J@7\"F0$!1,+]([b1h%F07$$!1+]iIKFl5F0$!1**\\7`hOEVF07$$!1+++vsjw**!#;$! 1++]P'=$))RF07$$!1++++h*QS*Ffp$!1+++]![>q$F07$$!1++Dc>mP()Ffp$!1,]7y4$ )oLF07$$!1,++]=$z9)Ffp$!1+++Df'R2$F07$$!1***\\iX/4](Ffp$!1+]7GAX]FF07$ $!1***\\(o8y%)oFfp$!1+]P%o!RUCF07$$!1****\\i:#>C'Ffp$!1++D\"yg47#F07$$ !1***\\7ev:l&Ffp$!1**\\i!z(yD=F07$$!1++](o2[,&Ffp$!1++vVQS2:F07$$!1++D 1[Q`VFfp$!1+]7.Cpw6F07$$!1++vVUhxPFfp$!1-+v=72))))Ffp7$$!1****\\iiwbJF fp$!1(***\\78$)ydFfp7$$!1,++]kL8DFfp$!10++]AomDFfp7$$!1,+]7U%[)=Ffp$\" 1b***\\P*ydd!#<7$$!1++Dccuw7Ffp$\"1***\\(=$\"1*****\\2mD+\"F07$$\"10++]7,Hl Fjt$\"1++]i0XE8F07$$\"1***\\P4w)R7Ffp$\"1**\\(o/Q*>;F07$$\"1-++vZf\")= Ffp$\"1,+](Q(zS>F07$$\"1)**\\P/-a[#Ffp$\"1**\\(=-,FC#F07$$\"1++v=Yb;JF fp$\"1+]P4tFeDF07$$\"1*****\\i@Ot$Ffp$\"1****\\73\"o'GF07$$\"1***\\PfL 'zVFfp$\"1**\\(oz;)*=$F07$$\"1-+++*>=+&Ffp$\"1,++]*44]$F07$$\"1++]i_4Q cFfp$\"1++DJw/>QF07$$\"1-+vV>5piFfp$\"1,](=(4bMTF07$$\"1,++]:$*[oFfp$ \"1,++vdYCWF07$$\"1,+]7<[8vFfp$\"1++Dc3ucZF07$$\"1++++L'y5)Ffp$\"1+++] ;$R0&F07$$\"1,+vV!)fT()Ffp$\"1+](=-*zq`F07$$\"1,+DcI;[$*Ffp$\"1,]7G:3u cF07$$\"\"\"F*$\"\"'F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$ Q\"x6\"%!G-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 407 "As we can see, the fucntion is not positive over the entire inter val and if we \"owned the land between the graph and the x axis\" we w ould be thinking of the area of two triangles one below the x axis fro m -2 to -.2; the other above the axis from -.2 to 1. the first has are a 1.8 *9*.5 and the other 1.2*6*.5 for a total of 11.7 square units. 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\\F,7$$\"+++++zFcamF\\eo7$F_eoF*Fb\\l-Fg[l6$7&Faeo7$F_eo$\"+++++^F,7$$ \"+++++#)FcamFfeo7$FieoF*Fb\\l-Fg[l6$7&F[fo7$Fieo$\"++++]_F,7$$\"+++++ &)FcamF`fo7$FcfoF*Fb\\l-Fg[l6$7&Fefo7$Fcfo$\"+++++aF,7$$\"+++++))FcamF jfo7$F]goF*Fb\\l-Fg[l6$7&F_go7$F]go$\"++++]bF,7$$\"+++++\"*FcamFdgo7$F ggoF*Fb\\l-Fg[l6$7&Figo7$Fggo$\"+++++dF,7$$\"+++++%*FcamF^ho7$FahoF*Fb \\l-Fg[l6$7&Fcho7$Faho$\"++++]eF,7$$\"+++++(*FcamFhho7$F[ioF*Fb\\l-Fg[ l6$7&F]io7$F[ioFezFbz7$FczF*Fb\\l-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$; F(Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "C urve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26 " "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "C urve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45 " "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "C urve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64 " "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "C urve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83 " "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "C urve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" }}} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$SumG6$,&!\"*\"\"\"-%\"iG6#,&!\"#F )F+#\"\"$\"$+\"#F0\"#?/F+;F)F1F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$! ++++vU!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 599 "We see that the an swer is nowhere close to 11.7 and is in fact negative. Looking at the \+ approximating rectangles shows us that in the sum when we take the hei ght to be f(x*) for some x* in the subinterval , this value will be ne gative when x is between -2 and .2. So we guess that this process will be yielding the negative of the area of the first triangle + the area of the second triangle. That is 1.8*(-9)*.5 + 1.2*(4).5= -8.1 + 3.6=- 4.5. This answer seems more consistent with the -4.275 above. This is \+ considered a \"Net Area\" ie Area above x axis - Area belowbelow x axi s. We switch to n=1000." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " rightbox(f(x),x=-2..1,1000); rightsum(f(x),x=-2..1,1000); evalf(%);" } }{PARA 13 "" 1 "" {GLPLOT2D 216 138 138 {PLOTDATA 2 "6gin-%'CURVESG6&7 S7$$!\"#\"\"!$!\"*F*7$$!+Q&3Y$>F,$!+!pUIn)F,7$$!+w@oF,7$$!+\"3%f+:F,$!+0/( H]'F,7$$!+oS:P9F,$!+S.x&='F,7$$!+=#)*=P\"F,$!+!4\"\\feF,7$$!+H3U98F,$! +XT5sbF,7$$!+.\\r\\7F,$!+:Xd[_F,7$$!+)GVZ=\"F,$!+SkrB\\F,7$$!+)4J@7\"F ,$!+!\\b1h%F,7$$!+JKFl5F,$!+bhOEVF,7$$!*FPm(**F,$!+N'=$))RF,7$$!*5'*QS *F,$!+]![>q$F,7$$!*'>mP()F,$!+!)4$)oLF,7$$!*&=$z9)F,$!+Df'R2$F,7$$!*Y/ 4](F,$!+IAX]FF,7$$!*P\"y%)oF,$!+&o!RUCF,7$$!*c@>C'F,$!+!yg47#F,7$$!*ev :l&F,$!+!z(yD=F,7$$!*p2[,&F,$!+XQS2:F,7$$!*\"[Q`VF,$!+0Cpw6F,7$$!*C9wx $F,$!*?r!))))F,7$$!*Emd:$F,$!*IJ)ydF,7$$!*XOL^#F,$!*D#omDF,7$$!*@W[)=F ,$\")&*yddF,7$$!*mXnF\"F,$\"*qrih$F,7$$!)*eb,'F,$\"*b?A*pF,7$$\"':K^F, $\"+vgc-5F,7$$\")8,HlF,$\"+l0XE8F,7$$\"*4w)R7F,$\"+X!Q*>;F,7$$\"*y%f\" )=F,$\"+!R(zS>F,7$$\"*/-a[#F,$\"+?5qUAF,7$$\"*ial6$F,$\"+5tFeDF,7$$\"* j@Ot$F,$\"+:3\"o'GF,7$$\"*fL'zVF,$\"+&z;)*=$F,7$$\"*!*>=+&F,$\"+]*44]$ F,7$$\"*E&4QcF,$\"+Iw/>QF,7$$\"*%>5piF,$\"+q4bMTF,7$$\"*bJ*[oF,$\"+vdY CWF,7$$\"*r\"[8vF,$\"+b3ucZF,7$$\"*Ijy5)F,$\"+];$R0&F,7$$\"*/)fT()F,$ \"+?!*zq`F,7$$\"*1j\"[$*F,$\"+I:3ucF,7$$\"\"\"F*$\"\"'F*-%'COLOURG6&%$ RGBG$\"*++++\"!\")F*F*-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"#-%)POLYGON SG6$7&7$F(F*7$F($!++++&)*)F,7$$!++++(*>F,F\\\\l7$F_\\lF*-%&COLORG6&Fjz $\"\"(!\"\"$\"\"*Fg\\lFe\\l-Fg[l6$7&Fa\\l7$F_\\l$!++++q*)F,7$$!++++%*> F,F^]l7$Fa]lF*Fb\\l-Fg[l6$7&Fc]l7$Fa]l$!++++b*)F,7$$!++++\"*>F,Fh]l7$F [^lF*Fb\\l-Fg[l6$7&F]^l7$F[^l$!++++S*)F,7$$!++++))>F,Fb^l7$Fe^lF*Fb\\l -Fg[l6$7&Fg^l7$Fe^l$!++++D*)F,7$$!++++&)>F,F\\_l7$F__lF*Fb\\l-Fg[l6$7& Fa_l7$F__l$!++++5*)F,7$$!++++#)>F,Ff_l7$Fi_lF*Fb\\l-Fg[l6$7&F[`l7$Fi_l $!++++&*))F,7$$!++++z>F,F``l7$Fc`lF*Fb\\l-Fg[l6$7&Fe`l7$Fc`l$!++++!))) 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\\ilF*Fb\\l-Fg[l6$7&F]\\il7$F[\\il$\"++++qBF,7$$\"++++SFF[]yFb\\il7$Fe \\ilF*Fb\\l-Fg[l6$7&Fg\\il7$Fe\\il$\"++++&Q#F,7$$\"++++qFF[]yF\\]il7$F _]ilF*Fb\\l-Fg[l6$7&Fa]il7$F_]il$F`hclF,7$$FbjelF[]yFf]il7$Fh]ilF*Fb\\ l-Fg[l6$7&Fi]il7$Fh]il$\"++++:CF,7$$\"++++IGF[]yF^^il7$Fa^ilF*Fb\\l-Fg [l6$7&Fc^il7$Fa^il$\"++++ICF,7$$\"++++gGF[]yFh^il7$F[_ilF*Fb\\l-Fg[l6$ 7&F]_il7$F[_il$\"++++XCF,7$$\"++++!*GF[]yFb_il7$Fe_ilF*Fb\\l-Fg[l6$7&F g_il7$Fe_il$\"++++gCF,7$$\"++++?HF[]yF\\`il7$F_`ilF*Fb\\l-Fg[l6$7&Fa`i l7$F_`il$\"++++vCF,7$$\"++++]HF[]yFf`il7$Fi`ilF*Fb\\l-Fg[l6$7&F[ail7$F i`il$\"++++!\\#F,7$$\"++++!)HF[]yF`ail7$FcailF*Fb\\l-Fg[l6$7&Feail7$Fc ail$\"++++0DF,7$$\"++++5IF[]yFjail7$F]bilF*Fb\\l-Fg[l6$7&F_bil7$F]bil$ \"++++?DF,7$$\"++++SIF[]yFdbil7$FgbilF*Fb\\l-Fg[l6$7&Fibil7$Fgbil$\"++ ++NDF,7$$\"++++qIF[]yF^cil7$FacilF*Fb\\l-Fg[l6$7&Fccil7$Facil$FihclF,7 $$F\\[flF[]yFhcil7$FjcilF*Fb\\l-Fg[l6$7&F[dil7$Fjcil$\"++++lDF,7$$\"++ ++IJF[]yF`dil7$FcdilF*Fb\\l-Fg[l6$7&Fedil7$Fcdil$\"++++!e#F,7$$\"++++g JF[]yFjdil7$F]eilF*Fb\\l-Fg[l6$7&F_eil7$F]eil$\"++++&f#F,7$$\"++++!>$F []yFdeil7$FgeilF*Fb\\l-Fg[l6$7&Fieil7$Fgeil$\"++++5EF,7$$\"++++?KF[]yF ^fil7$FafilF*Fb\\l-Fg[l6$7&Fcfil7$Fafil$\"++++DEF,7$$\"++++]KF[]yFhfil 7$F[gilF*Fb\\l-Fg[l6$7&F]gil7$F[gil$\"++++SEF,7$$\"++++!G$F[]yFbgil7$F egilF*Fb\\l-Fg[l6$7&Fggil7$Fegil$\"++++bEF,7$$\"++++5LF[]yF\\hil7$F_hi lF*Fb\\l-Fg[l6$7&Fahil7$F_hil$\"++++qEF,7$$\"++++SLF[]yFfhil7$FihilF*F b\\l-Fg[l6$7&F[iil7$Fihil$\"++++&o#F,7$$\"++++qLF[]yF`iil7$FciilF*Fb\\ l-Fg[l6$7&Feiil7$Fciil$FbiclF,7$$Ff[flF[]yFjiil7$F\\jilF*Fb\\l-Fg[l6$7 &F]jil7$F\\jil$\"++++:FF,7$$\"++++IMF[]yFbjil7$FejilF*Fb\\l-Fg[l6$7&Fg jil7$Fejil$\"++++IFF,7$$\"++++gMF[]yF\\[jl7$F_[jlF*Fb\\l-Fg[l6$7&Fa[jl 7$F_[jl$\"++++XFF,7$$\"++++!\\$F[]yFf[jl7$Fi[jlF*Fb\\l-Fg[l6$7&F[\\jl7 $Fi[jl$\"++++gFF,7$$\"++++?NF[]yF`\\jl7$Fc\\jlF*Fb\\l-Fg[l6$7&Fe\\jl7$ Fc\\jl$\"++++vFF,7$$\"++++]NF[]yFj\\jl7$F]]jlF*Fb\\l-Fg[l6$7&F_]jl7$F] ]jl$\"++++!z#F,7$$\"++++!e$F[]yFd]jl7$Fg]jlF*Fb\\l-Fg[l6$7&Fi]jl7$Fg]j l$\"++++0GF,7$$\"++++5OF[]yF^^jl7$Fa^jlF*Fb\\l-Fg[l6$7&Fc^jl7$Fa^jl$\" ++++?GF,7$$\"++++SOF[]yFh^jl7$F[_jlF*Fb\\l-Fg[l6$7&F]_jl7$F[_jl$\"++++ NGF,7$$\"++++qOF[]yFb_jl7$Fe_jlF*Fb\\l-Fg[l6$7&Fg_jl7$Fe_jl$F[jclF,7$$ F`\\flF[]yF\\`jl7$F^`jlF*Fb\\l-Fg[l6$7&F_`jl7$F^`jl$\"++++lGF,7$$\"+++ +IPF[]yFd`jl7$Fg`jlF*Fb\\l-Fg[l6$7&Fi`jl7$Fg`jl$\"++++!)GF,7$$\"++++gP F[]yF^ajl7$FaajlF*Fb\\l-Fg[l6$7&Fcajl7$Faajl$\"++++&*GF,7$$\"++++!z$F[ ]yFhajl7$F[bjlF*Fb\\l-Fg[l6$7&F]bjl7$F[bjl$\"++++5HF,7$$\"++++?QF[]yFb bjl7$FebjlF*Fb\\l-Fg[l6$7&Fgbjl7$Febjl$\"++++DHF,7$$\"++++]QF[]yF\\cjl 7$F_cjlF*Fb\\l-Fg[l6$7&Facjl7$F_cjl$\"++++SHF,7$$\"++++!)QF[]yFfcjl7$F icjlF*Fb\\l-Fg[l6$7&F[djl7$Ficjl$\"++++bHF,7$$\"++++5RF[]yF`djl7$Fcdjl F*Fb\\l-Fg[l6$7&Fedjl7$Fcdjl$\"++++qHF,7$$\"++++SRF[]yFjdjl7$F]ejlF*Fb \\l-Fg[l6$7&F_ejl7$F]ejl$\"++++&)HF,7$$\"++++qRF[]yFdejl7$FgejlF*Fb\\l -Fg[l6$7&Fiejl7$Fgejl$\"\"$F*7$$FdeelF[]yF^fjl7$FafjlF*Fb\\l-Fg[l6$7&F bfjl7$Fafjl$\"++++:IF,7$$\"++++ISF[]yFgfjl7$FjfjlF*Fb\\l-Fg[l6$7&F\\gj l7$Fjfjl$\"++++IIF,7$$\"++++gSF[]yFagjl7$FdgjlF*Fb\\l-Fg[l6$7&Ffgjl7$F dgjl$\"++++XIF,7$$\"++++!4%F[]yF[hjl7$F^hjlF*Fb\\l-Fg[l6$7&F`hjl7$F^hj l$\"++++gIF,7$$\"++++?TF[]yFehjl7$FhhjlF*Fb\\l-Fg[l6$7&Fjhjl7$Fhhjl$\" ++++vIF,7$$\"++++]TF[]yF_ijl7$FbijlF*Fb\\l-Fg[l6$7&Fdijl7$Fbijl$\"++++ !4$F,7$$\"++++!=%F[]yFiijl7$F\\jjlF*Fb\\l-Fg[l6$7&F^jjl7$F\\jjl$\"++++ 0JF,7$$\"++++5UF[]yFcjjl7$FfjjlF*Fb\\l-Fg[l6$7&Fhjjl7$Ffjjl$\"++++?JF, 7$$\"++++SUF[]yF][[m7$F`[[mF*Fb\\l-Fg[l6$7&Fb[[m7$F`[[m$\"++++NJF,7$$ \"++++qUF[]yFg[[m7$Fj[[mF*Fb\\l-Fg[l6$7&F\\\\[m7$Fj[[m$F\\[dlF,7$$Fb]f lF[]yFa\\[m7$Fc\\[mF*Fb\\l-Fg[l6$7&Fd\\[m7$Fc\\[m$\"++++lJF,7$$\"++++I VF[]yFi\\[m7$F\\][mF*Fb\\l-Fg[l6$7&F^][m7$F\\][m$\"++++!=$F,7$$\"++++g VF[]yFc][m7$Ff][mF*Fb\\l-Fg[l6$7&Fh][m7$Ff][m$\"++++&>$F,7$$\"++++!R%F []yF]^[m7$F`^[mF*Fb\\l-Fg[l6$7&Fb^[m7$F`^[m$\"++++5KF,7$$\"++++?WF[]yF g^[m7$Fj^[mF*Fb\\l-Fg[l6$7&F\\_[m7$Fj^[m$\"++++DKF,7$$\"++++]WF[]yFa_[ m7$Fd_[mF*Fb\\l-Fg[l6$7&Ff_[m7$Fd_[m$\"++++SKF,7$$\"++++![%F[]yF[`[m7$ F^`[mF*Fb\\l-Fg[l6$7&F``[m7$F^`[m$\"++++bKF,7$$\"++++5XF[]yFe`[m7$Fh`[ mF*Fb\\l-Fg[l6$7&Fj`[m7$Fh`[m$\"++++qKF,7$$\"++++SXF[]yF_a[m7$Fba[mF*F b\\l-Fg[l6$7&Fda[m7$Fba[m$\"++++&G$F,7$$\"++++qXF[]yFia[m7$F\\b[mF*Fb \\l-Fg[l6$7&F^b[m7$F\\b[m$Fe[dlF,7$$F\\^flF[]yFcb[m7$Feb[mF*Fb\\l-Fg[l 6$7&Ffb[m7$Feb[m$\"++++:LF,7$$\"++++IYF[]yF[c[m7$F^c[mF*Fb\\l-Fg[l6$7& F`c[m7$F^c[m$\"++++ILF,7$$\"++++gYF[]yFec[m7$Fhc[mF*Fb\\l-Fg[l6$7&Fjc[ m7$Fhc[m$\"++++XLF,7$$\"++++!p%F[]yF_d[m7$Fbd[mF*Fb\\l-Fg[l6$7&Fdd[m7$ Fbd[m$\"++++gLF,7$$\"++++?ZF[]yFid[m7$F\\e[mF*Fb\\l-Fg[l6$7&F^e[m7$F\\ e[m$\"++++vLF,7$$\"++++]ZF[]yFce[m7$Ffe[mF*Fb\\l-Fg[l6$7&Fhe[m7$Ffe[m$ \"++++!R$F,7$$\"++++!y%F[]yF]f[m7$F`f[mF*Fb\\l-Fg[l6$7&Fbf[m7$F`f[m$\" ++++0MF,7$$\"++++5[F[]yFgf[m7$Fjf[mF*Fb\\l-Fg[l6$7&F\\g[m7$Fjf[m$\"+++ +?MF,7$$\"++++S[F[]yFag[m7$Fdg[mF*Fb\\l-Fg[l6$7&Ffg[m7$Fdg[m$\"++++NMF ,7$$\"++++q[F[]yF[h[m7$F^h[mF*Fb\\l-Fg[l6$7&F`h[m7$F^h[m$F^\\dlF,7$$Ff ^flF[]yFeh[m7$Fgh[mF*Fb\\l-Fg[l6$7&Fhh[m7$Fgh[m$\"++++lMF,7$$\"++++I\\ F[]yF]i[m7$F`i[mF*Fb\\l-Fg[l6$7&Fbi[m7$F`i[m$\"++++![$F,7$$\"++++g\\F[ ]yFgi[m7$Fji[mF*Fb\\l-Fg[l6$7&F\\j[m7$Fji[m$\"++++&\\$F,7$$\"++++!*\\F []yFaj[m7$Fdj[mF*Fb\\l-Fg[l6$7&Ffj[m7$Fdj[m$\"++++5NF,7$$\"++++?]F[]yF [[\\m7$F^[\\mF*Fb\\l-Fg[l6$7&F`[\\m7$F^[\\m$\"++++DNF,7$$\"++++]]F[]yF e[\\m7$Fh[\\mF*Fb\\l-Fg[l6$7&Fj[\\m7$Fh[\\m$\"++++SNF,7$$\"++++!3&F[]y F_\\\\m7$Fb\\\\mF*Fb\\l-Fg[l6$7&Fd\\\\m7$Fb\\\\m$\"++++bNF,7$$\"++++5^ F[]yFi\\\\m7$F\\]\\mF*Fb\\l-Fg[l6$7&F^]\\m7$F\\]\\m$\"++++qNF,7$$\"+++ +S^F[]yFc]\\m7$Ff]\\mF*Fb\\l-Fg[l6$7&Fh]\\m7$Ff]\\m$\"++++&e$F,7$$\"++ ++q^F[]yF]^\\m7$F`^\\mF*Fb\\l-Fg[l6$7&Fb^\\m7$F`^\\m$Fg\\dlF,7$$F`_flF []yFg^\\m7$Fi^\\mF*Fb\\l-Fg[l6$7&Fj^\\m7$Fi^\\m$\"++++:OF,7$$\"++++I_F []yF__\\m7$Fb_\\mF*Fb\\l-Fg[l6$7&Fd_\\m7$Fb_\\m$\"++++IOF,7$$\"++++g_F []yFi_\\m7$F\\`\\mF*Fb\\l-Fg[l6$7&F^`\\m7$F\\`\\m$\"++++XOF,7$$\"++++! H&F[]yFc`\\m7$Ff`\\mF*Fb\\l-Fg[l6$7&Fh`\\m7$Ff`\\m$\"++++gOF,7$$\"++++ ?`F[]yF]a\\m7$F`a\\mF*Fb\\l-Fg[l6$7&Fba\\m7$F`a\\m$\"++++vOF,7$$\"++++ ]`F[]yFga\\m7$Fja\\mF*Fb\\l-Fg[l6$7&F\\b\\m7$Fja\\m$\"++++!p$F,7$$\"++ ++!Q&F[]yFab\\m7$Fdb\\mF*Fb\\l-Fg[l6$7&Ffb\\m7$Fdb\\m$\"++++0PF,7$$\"+ +++5aF[]yF[c\\m7$F^c\\mF*Fb\\l-Fg[l6$7&F`c\\m7$F^c\\m$\"++++?PF,7$$\"+ +++SaF[]yFec\\m7$Fhc\\mF*Fb\\l-Fg[l6$7&Fjc\\m7$Fhc\\m$\"++++NPF,7$$\"+ +++qaF[]yF_d\\m7$Fbd\\mF*Fb\\l-Fg[l6$7&Fdd\\m7$Fbd\\m$F`]dlF,7$$Fj_flF []yFid\\m7$F[e\\mF*Fb\\l-Fg[l6$7&F\\e\\m7$F[e\\m$\"++++lPF,7$$\"++++Ib F[]yFae\\m7$Fde\\mF*Fb\\l-Fg[l6$7&Ffe\\m7$Fde\\m$\"++++!y$F,7$$\"++++g bF[]yF[f\\m7$F^f\\mF*Fb\\l-Fg[l6$7&F`f\\m7$F^f\\m$\"++++&z$F,7$$\"++++ !f&F[]yFef\\m7$Fhf\\mF*Fb\\l-Fg[l6$7&Fjf\\m7$Fhf\\m$\"++++5QF,7$$\"+++ +?cF[]yF_g\\m7$Fbg\\mF*Fb\\l-Fg[l6$7&Fdg\\m7$Fbg\\m$\"++++DQF,7$$\"+++ +]cF[]yFig\\m7$F\\h\\mF*Fb\\l-Fg[l6$7&F^h\\m7$F\\h\\m$\"++++SQF,7$$\"+ +++!o&F[]yFch\\m7$Ffh\\mF*Fb\\l-Fg[l6$7&Fhh\\m7$Ffh\\m$\"++++bQF,7$$\" ++++5dF[]yF]i\\m7$F`i\\mF*Fb\\l-Fg[l6$7&Fbi\\m7$F`i\\m$\"++++qQF,7$$\" ++++SdF[]yFgi\\m7$Fji\\mF*Fb\\l-Fg[l6$7&F\\j\\m7$Fji\\m$\"++++&)QF,7$$ \"++++qdF[]yFaj\\m7$Fdj\\mF*Fb\\l-Fg[l6$7&Ffj\\m7$Fdj\\m$Fi]dlF,7$$Fd` flF[]yF[[]m7$F][]mF*Fb\\l-Fg[l6$7&F^[]m7$F][]m$\"++++:RF,7$$\"++++IeF[ ]yFc[]m7$Ff[]mF*Fb\\l-Fg[l6$7&Fh[]m7$Ff[]m$\"++++IRF,7$$\"++++geF[]yF] \\]m7$F`\\]mF*Fb\\l-Fg[l6$7&Fb\\]m7$F`\\]m$\"++++XRF,7$$\"++++!*eF[]yF g\\]m7$Fj\\]mF*Fb\\l-Fg[l6$7&F\\]]m7$Fj\\]m$\"++++gRF,7$$\"++++?fF[]yF a]]m7$Fd]]mF*Fb\\l-Fg[l6$7&Ff]]m7$Fd]]m$\"++++vRF,7$$\"++++]fF[]yF[^]m 7$F^^]mF*Fb\\l-Fg[l6$7&F`^]m7$F^^]m$\"++++!*RF,7$$\"++++!)fF[]yFe^]m7$ Fh^]mF*Fb\\l-Fg[l6$7&Fj^]m7$Fh^]m$\"++++0SF,7$$\"++++5gF[]yF__]m7$Fb_] mF*Fb\\l-Fg[l6$7&Fd_]m7$Fb_]m$\"++++?SF,7$$\"++++SgF[]yFi_]m7$F\\`]mF* Fb\\l-Fg[l6$7&F^`]m7$F\\`]m$\"++++NSF,7$$\"++++qgF[]yFc`]m7$Ff`]mF*Fb \\l-Fg[l6$7&Fh`]m7$Ff`]m$Fb^dlF,7$$F^aflF[]yF]a]m7$F_a]mF*Fb\\l-Fg[l6$ 7&F`a]m7$F_a]m$\"++++lSF,7$$\"++++IhF[]yFea]m7$Fha]mF*Fb\\l-Fg[l6$7&Fj a]m7$Fha]m$\"++++!3%F,7$$\"++++ghF[]yF_b]m7$Fbb]mF*Fb\\l-Fg[l6$7&Fdb]m 7$Fbb]m$\"++++&4%F,7$$\"++++!>'F[]yFib]m7$F\\c]mF*Fb\\l-Fg[l6$7&F^c]m7 $F\\c]m$\"++++5TF,7$$\"++++?iF[]yFcc]m7$Ffc]mF*Fb\\l-Fg[l6$7&Fhc]m7$Ff c]m$\"++++DTF,7$$\"++++]iF[]yF]d]m7$F`d]mF*Fb\\l-Fg[l6$7&Fbd]m7$F`d]m$ \"++++STF,7$$\"++++!G'F[]yFgd]m7$Fjd]mF*Fb\\l-Fg[l6$7&F\\e]m7$Fjd]m$\" ++++bTF,7$$\"++++5jF[]yFae]m7$Fde]mF*Fb\\l-Fg[l6$7&Ffe]m7$Fde]m$\"++++ qTF,7$$\"++++SjF[]yF[f]m7$F^f]mF*Fb\\l-Fg[l6$7&F`f]m7$F^f]m$\"++++&=%F ,7$$\"++++qjF[]yFef]m7$Fhf]mF*Fb\\l-Fg[l6$7&Fjf]m7$Fhf]m$F[_dlF,7$$Fha flF[]yF_g]m7$Fag]mF*Fb\\l-Fg[l6$7&Fbg]m7$Fag]m$\"++++:UF,7$$\"++++IkF[ ]yFgg]m7$Fjg]mF*Fb\\l-Fg[l6$7&F\\h]m7$Fjg]m$\"++++IUF,7$$\"++++gkF[]yF ah]m7$Fdh]mF*Fb\\l-Fg[l6$7&Ffh]m7$Fdh]m$\"++++XUF,7$$\"++++!\\'F[]yF[i ]m7$F^i]mF*Fb\\l-Fg[l6$7&F`i]m7$F^i]m$\"++++gUF,7$$\"++++?lF[]yFei]m7$ Fhi]mF*Fb\\l-Fg[l6$7&Fji]m7$Fhi]m$\"++++vUF,7$$\"++++]lF[]yF_j]m7$Fbj] mF*Fb\\l-Fg[l6$7&Fdj]m7$Fbj]m$\"++++!H%F,7$$\"++++!e'F[]yFij]m7$F\\[^m F*Fb\\l-Fg[l6$7&F^[^m7$F\\[^m$\"++++0VF,7$$\"++++5mF[]yFc[^m7$Ff[^mF*F b\\l-Fg[l6$7&Fh[^m7$Ff[^m$\"++++?VF,7$$\"++++SmF[]yF]\\^m7$F`\\^mF*Fb \\l-Fg[l6$7&Fb\\^m7$F`\\^m$\"++++NVF,7$$\"++++qmF[]yFg\\^m7$Fj\\^mF*Fb \\l-Fg[l6$7&F\\]^m7$Fj\\^m$Fd_dlF,7$$FbbflF[]yFa]^m7$Fc]^mF*Fb\\l-Fg[l 6$7&Fd]^m7$Fc]^m$\"++++lVF,7$$\"++++InF[]yFi]^m7$F\\^^mF*Fb\\l-Fg[l6$7 &F^^^m7$F\\^^m$\"++++!Q%F,7$$\"++++gnF[]yFc^^m7$Ff^^mF*Fb\\l-Fg[l6$7&F h^^m7$Ff^^m$\"++++&R%F,7$$\"++++!z'F[]yF]_^m7$F`_^mF*Fb\\l-Fg[l6$7&Fb_ ^m7$F`_^m$\"++++5WF,7$$\"++++?oF[]yFg_^m7$Fj_^mF*Fb\\l-Fg[l6$7&F\\`^m7 $Fj_^m$\"++++DWF,7$$\"++++]oF[]yFa`^m7$Fd`^mF*Fb\\l-Fg[l6$7&Ff`^m7$Fd` ^m$\"++++SWF,7$$\"++++!)oF[]yF[a^m7$F^a^mF*Fb\\l-Fg[l6$7&F`a^m7$F^a^m$ \"++++bWF,7$$\"++++5pF[]yFea^m7$Fha^mF*Fb\\l-Fg[l6$7&Fja^m7$Fha^m$\"++ ++qWF,7$$\"++++SpF[]yF_b^m7$Fbb^mF*Fb\\l-Fg[l6$7&Fdb^m7$Fbb^m$\"++++&[ %F,7$$\"++++qpF[]yFib^m7$F\\c^mF*Fb\\l-Fg[l6$7&F^c^m7$F\\c^m$F\\aclF,7 $$F^felF[]yFcc^m7$Fec^mF*Fb\\l-Fg[l6$7&Ffc^m7$Fec^m$\"++++:XF,7$$\"+++ +IqF[]yF[d^m7$F^d^mF*Fb\\l-Fg[l6$7&F`d^m7$F^d^m$\"++++IXF,7$$\"++++gqF []yFed^m7$Fhd^mF*Fb\\l-Fg[l6$7&Fjd^m7$Fhd^m$\"++++XXF,7$$\"++++!4(F[]y F_e^m7$Fbe^mF*Fb\\l-Fg[l6$7&Fde^m7$Fbe^m$\"++++gXF,7$$\"++++?rF[]yFie^ m7$F\\f^mF*Fb\\l-Fg[l6$7&F^f^m7$F\\f^m$\"++++vXF,7$$\"++++]rF[]yFcf^m7 $Fff^mF*Fb\\l-Fg[l6$7&Fhf^m7$Fff^m$\"++++!f%F,7$$\"++++!=(F[]yF]g^m7$F `g^mF*Fb\\l-Fg[l6$7&Fbg^m7$F`g^m$\"++++0YF,7$$\"++++5sF[]yFgg^m7$Fjg^m F*Fb\\l-Fg[l6$7&F\\h^m7$Fjg^m$\"++++?YF,7$$\"++++SsF[]yFah^m7$Fdh^mF*F b\\l-Fg[l6$7&Ffh^m7$Fdh^m$\"++++NYF,7$$\"++++qsF[]yF[i^m7$F^i^mF*Fb\\l -Fg[l6$7&F`i^m7$F^i^m$Fe`dlF,7$$FdcflF[]yFei^m7$Fgi^mF*Fb\\l-Fg[l6$7&F hi^m7$Fgi^m$\"++++lYF,7$$\"++++ItF[]yF]j^m7$F`j^mF*Fb\\l-Fg[l6$7&Fbj^m 7$F`j^m$\"++++!o%F,7$$\"++++gtF[]yFgj^m7$Fjj^mF*Fb\\l-Fg[l6$7&F\\[_m7$ Fjj^m$\"++++&p%F,7$$\"++++!R(F[]yFa[_m7$Fd[_mF*Fb\\l-Fg[l6$7&Ff[_m7$Fd [_m$\"++++5ZF,7$$\"++++?uF[]yF[\\_m7$F^\\_mF*Fb\\l-Fg[l6$7&F`\\_m7$F^ \\_m$\"++++DZF,7$$\"++++]uF[]yFe\\_m7$Fh\\_mF*Fb\\l-Fg[l6$7&Fj\\_m7$Fh \\_m$\"++++SZF,7$$\"++++![(F[]yF_]_m7$Fb]_mF*Fb\\l-Fg[l6$7&Fd]_m7$Fb]_ m$\"++++bZF,7$$\"++++5vF[]yFi]_m7$F\\^_mF*Fb\\l-Fg[l6$7&F^^_m7$F\\^_m$ \"++++qZF,7$$\"++++SvF[]yFc^_m7$Ff^_mF*Fb\\l-Fg[l6$7&Fh^_m7$Ff^_m$\"++ ++&y%F,7$$\"++++qvF[]yF]__m7$F`__mF*Fb\\l-Fg[l6$7&Fb__m7$F`__m$F^adlF, 7$$F^dflF[]yFg__m7$Fi__mF*Fb\\l-Fg[l6$7&Fj__m7$Fi__m$\"++++:[F,7$$\"++ ++IwF[]yF_`_m7$Fb`_mF*Fb\\l-Fg[l6$7&Fd`_m7$Fb`_m$\"++++I[F,7$$\"++++gw F[]yFi`_m7$F\\a_mF*Fb\\l-Fg[l6$7&F^a_m7$F\\a_m$\"++++X[F,7$$\"++++!p(F []yFca_m7$Ffa_mF*Fb\\l-Fg[l6$7&Fha_m7$Ffa_m$\"++++g[F,7$$\"++++?xF[]yF ]b_m7$F`b_mF*Fb\\l-Fg[l6$7&Fbb_m7$F`b_m$\"++++v[F,7$$\"++++]xF[]yFgb_m 7$Fjb_mF*Fb\\l-Fg[l6$7&F\\c_m7$Fjb_m$\"++++!*[F,7$$\"++++!y(F[]yFac_m7 $Fdc_mF*Fb\\l-Fg[l6$7&Ffc_m7$Fdc_m$\"++++0\\F,7$$\"++++5yF[]yF[d_m7$F^ d_mF*Fb\\l-Fg[l6$7&F`d_m7$F^d_m$\"++++?\\F,7$$\"++++SyF[]yFed_m7$Fhd_m F*Fb\\l-Fg[l6$7&Fjd_m7$Fhd_m$\"++++N\\F,7$$\"++++qyF[]yF_e_m7$Fbe_mF*F b\\l-Fg[l6$7&Fde_m7$Fbe_m$FgadlF,7$$FhdflF[]yFie_m7$F[f_mF*Fb\\l-Fg[l6 $7&F\\f_m7$F[f_m$\"++++l\\F,7$$\"++++IzF[]yFaf_m7$Fdf_mF*Fb\\l-Fg[l6$7 &Fff_m7$Fdf_m$\"++++!)\\F,7$$\"++++gzF[]yF[g_m7$F^g_mF*Fb\\l-Fg[l6$7&F `g_m7$F^g_m$\"++++&*\\F,7$$\"++++!*zF[]yFeg_m7$Fhg_mF*Fb\\l-Fg[l6$7&Fj g_m7$Fhg_m$\"++++5]F,7$$\"++++?!)F[]yF_h_m7$Fbh_mF*Fb\\l-Fg[l6$7&Fdh_m 7$Fbh_m$\"++++D]F,7$$\"++++]!)F[]yFih_m7$F\\i_mF*Fb\\l-Fg[l6$7&F^i_m7$ F\\i_m$\"++++S]F,7$$\"++++!3)F[]yFci_m7$Ffi_mF*Fb\\l-Fg[l6$7&Fhi_m7$Ff i_m$\"++++b]F,7$$\"++++5\")F[]yF]j_m7$F`j_mF*Fb\\l-Fg[l6$7&Fbj_m7$F`j_ m$\"++++q]F,7$$\"++++S\")F[]yFgj_m7$Fjj_mF*Fb\\l-Fg[l6$7&F\\[`m7$Fjj_m $\"++++&3&F,7$$\"++++q\")F[]yFa[`m7$Fd[`mF*Fb\\l-Fg[l6$7&Ff[`m7$Fd[`m$ F`bdlF,7$$FbeflF[]yF[\\`m7$F]\\`mF*Fb\\l-Fg[l6$7&F^\\`m7$F]\\`m$\"++++ :^F,7$$\"++++I#)F[]yFc\\`m7$Ff\\`mF*Fb\\l-Fg[l6$7&Fh\\`m7$Ff\\`m$\"+++ +I^F,7$$\"++++g#)F[]yF]]`m7$F`]`mF*Fb\\l-Fg[l6$7&Fb]`m7$F`]`m$\"++++X^ F,7$$\"++++!H)F[]yFg]`m7$Fj]`mF*Fb\\l-Fg[l6$7&F\\^`m7$Fj]`m$\"++++g^F, 7$$\"++++?$)F[]yFa^`m7$Fd^`mF*Fb\\l-Fg[l6$7&Ff^`m7$Fd^`m$\"++++v^F,7$$ \"++++]$)F[]yF[_`m7$F^_`mF*Fb\\l-Fg[l6$7&F`_`m7$F^_`m$\"++++!>&F,7$$\" ++++!Q)F[]yFe_`m7$Fh_`mF*Fb\\l-Fg[l6$7&Fj_`m7$Fh_`m$\"++++0_F,7$$\"+++ +5%)F[]yF_``m7$Fb``mF*Fb\\l-Fg[l6$7&Fd``m7$Fb``m$\"++++?_F,7$$\"++++S% )F[]yFi``m7$F\\a`mF*Fb\\l-Fg[l6$7&F^a`m7$F\\a`m$\"++++N_F,7$$\"++++q%) F[]yFca`m7$Ffa`mF*Fb\\l-Fg[l6$7&Fha`m7$Ffa`m$FibdlF,7$$F\\fflF[]yF]b`m 7$F_b`mF*Fb\\l-Fg[l6$7&F`b`m7$F_b`m$\"++++l_F,7$$\"++++I&)F[]yFeb`m7$F hb`mF*Fb\\l-Fg[l6$7&Fjb`m7$Fhb`m$\"++++!G&F,7$$\"++++g&)F[]yF_c`m7$Fbc `mF*Fb\\l-Fg[l6$7&Fdc`m7$Fbc`m$\"++++&H&F,7$$\"++++!f)F[]yFic`m7$F\\d` mF*Fb\\l-Fg[l6$7&F^d`m7$F\\d`m$\"++++5`F,7$$\"++++?')F[]yFcd`m7$Ffd`mF *Fb\\l-Fg[l6$7&Fhd`m7$Ffd`m$\"++++D`F,7$$\"++++]')F[]yF]e`m7$F`e`mF*Fb \\l-Fg[l6$7&Fbe`m7$F`e`m$\"++++S`F,7$$\"++++!o)F[]yFge`m7$Fje`mF*Fb\\l -Fg[l6$7&F\\f`m7$Fje`m$\"++++b`F,7$$\"++++5()F[]yFaf`m7$Fdf`mF*Fb\\l-F g[l6$7&Fff`m7$Fdf`m$\"++++q`F,7$$\"++++S()F[]yF[g`m7$F^g`mF*Fb\\l-Fg[l 6$7&F`g`m7$F^g`m$\"++++&Q&F,7$$\"++++q()F[]yFeg`m7$Fhg`mF*Fb\\l-Fg[l6$ 7&Fjg`m7$Fhg`m$FbcdlF,7$$FffflF[]yF_h`m7$Fah`mF*Fb\\l-Fg[l6$7&Fbh`m7$F ah`m$\"++++:aF,7$$\"++++I))F[]yFgh`m7$Fjh`mF*Fb\\l-Fg[l6$7&F\\i`m7$Fjh `m$\"++++IaF,7$$\"++++g))F[]yFai`m7$Fdi`mF*Fb\\l-Fg[l6$7&Ffi`m7$Fdi`m$ \"++++XaF,7$$\"++++!*))F[]yF[j`m7$F^j`mF*Fb\\l-Fg[l6$7&F`j`m7$F^j`m$\" ++++gaF,7$$\"++++?*)F[]yFej`m7$Fhj`mF*Fb\\l-Fg[l6$7&Fjj`m7$Fhj`m$\"+++ +vaF,7$$\"++++]*)F[]yF_[am7$Fb[amF*Fb\\l-Fg[l6$7&Fd[am7$Fb[am$\"++++! \\&F,7$$\"++++!)*)F[]yFi[am7$F\\\\amF*Fb\\l-Fg[l6$7&F^\\am7$F\\\\am$\" ++++0bF,7$$\"++++5!*F[]yFc\\am7$Ff\\amF*Fb\\l-Fg[l6$7&Fh\\am7$Ff\\am$ \"++++?bF,7$$\"++++S!*F[]yF]]am7$F`]amF*Fb\\l-Fg[l6$7&Fb]am7$F`]am$\"+ +++NbF,7$$\"++++q!*F[]yFg]am7$Fj]amF*Fb\\l-Fg[l6$7&F\\^am7$Fj]am$F[ddl F,7$$F`gflF[]yFa^am7$Fc^amF*Fb\\l-Fg[l6$7&Fd^am7$Fc^am$\"++++lbF,7$$\" ++++I\"*F[]yFi^am7$F\\_amF*Fb\\l-Fg[l6$7&F^_am7$F\\_am$\"++++!e&F,7$$ \"++++g\"*F[]yFc_am7$Ff_amF*Fb\\l-Fg[l6$7&Fh_am7$Ff_am$\"++++&f&F,7$$ \"++++!>*F[]yF]`am7$F``amF*Fb\\l-Fg[l6$7&Fb`am7$F``am$\"++++5cF,7$$\"+ +++?#*F[]yFg`am7$Fj`amF*Fb\\l-Fg[l6$7&F\\aam7$Fj`am$\"++++DcF,7$$\"+++ +]#*F[]yFaaam7$FdaamF*Fb\\l-Fg[l6$7&Ffaam7$Fdaam$\"++++ScF,7$$\"++++!G *F[]yF[bam7$F^bamF*Fb\\l-Fg[l6$7&F`bam7$F^bam$\"++++bcF,7$$\"++++5$*F[ ]yFebam7$FhbamF*Fb\\l-Fg[l6$7&Fjbam7$Fhbam$\"++++qcF,7$$\"++++S$*F[]yF _cam7$FbcamF*Fb\\l-Fg[l6$7&Fdcam7$Fbcam$\"++++&o&F,7$$\"++++q$*F[]yFic am7$F\\damF*Fb\\l-Fg[l6$7&F^dam7$F\\dam$FdddlF,7$$FjgflF[]yFcdam7$Feda mF*Fb\\l-Fg[l6$7&Ffdam7$Fedam$\"++++:dF,7$$\"++++I%*F[]yF[eam7$F^eamF* Fb\\l-Fg[l6$7&F`eam7$F^eam$\"++++IdF,7$$\"++++g%*F[]yFeeam7$FheamF*Fb \\l-Fg[l6$7&Fjeam7$Fheam$\"++++XdF,7$$\"++++!\\*F[]yF_fam7$FbfamF*Fb\\ l-Fg[l6$7&Fdfam7$Fbfam$\"++++gdF,7$$\"++++?&*F[]yFifam7$F\\gamF*Fb\\l- Fg[l6$7&F^gam7$F\\gam$\"++++vdF,7$$\"++++]&*F[]yFcgam7$FfgamF*Fb\\l-Fg [l6$7&Fhgam7$Ffgam$\"++++!z&F,7$$\"++++!e*F[]yF]ham7$F`hamF*Fb\\l-Fg[l 6$7&Fbham7$F`ham$\"++++0eF,7$$\"++++5'*F[]yFgham7$FjhamF*Fb\\l-Fg[l6$7 &F\\iam7$Fjham$\"++++?eF,7$$\"++++S'*F[]yFaiam7$FdiamF*Fb\\l-Fg[l6$7&F 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"6#,$-%$SumG6$,&!\"*\"\"\"-%\"iG6#,&!\"#F)F+#\"\"$\"%+5# F0\"$+#/F+;F)F1F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++]xW!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "This seems to support the observat ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 256 179 "If g is continuous (or monotone) on [a,b] then this \+ approximating rectangle sum process will have a limit which we denote \+ as the definite integral of g over [a,b] the notation is " }{XPPEDIT 257 0 "Int(g,x = a .. b);" "6#-%$IntG6$%\"gG/%\"xG;%\"aG%\"bG" }{TEXT 258 182 " and this definite integral can be interpreted as the differe nce of areas where g is above the x axis and below the x axis. We say \+ that g is an integrable function when this happens." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We try this for our fun ction " }{XPPEDIT 18 0 "f (x)=1+5*x" "6#/-%\"fG6#%\"xG,&\"\"\"F)*&\"\" &F)F'F)F)" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(f,x=-2..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"*\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "This is the answer -4.5 that we w ere expecting. We also use the term ...\"integrating f from -2 to 1\" \+ in describing this process of evaluating the definite integral. We can now define an " }{TEXT 259 19 "indefinite integral" }{TEXT 260 0 "" } {TEXT -1 96 " which is a new function, based on this process of integr ating continuous functions g, such as :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " G:=x-> Int( g(t),t=-2 ..x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGR6#%\"xG6\"6$%)operator G%&arrowGF(-%$IntG6$-%\"gG6#%\"tG/F2;!\"#9$F(F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 124 " This is defining a new function G for each x, wh ich says to find its value you integrate g over the interval from -2 t o x. " }{TEXT 261 124 "The interesting fact is that this G is itself a n antiderivative for g. That is G is differentiable and its derivative is g. " }{TEXT 262 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "This is one part of the Fundamental theorem of Calculus. This yie lds a valuable tool in evaluating these definite integrals. " }}{PARA 0 "" 0 "" {TEXT -1 39 " This says that over [a,b] G(b)-G(a) =" } {XPPEDIT 256 0 "Int(g,x = a .. b)" "6#-%$IntG6$%\"gG/%\"xG;%\"aG%\"bG " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 321 " This equati on says that to find the definite integral, first we identify an antid erivative of g over [a, b] then simply evaluate that antiderivative at the two endpoints and subtract. Because of this theorem we can identi fy the process of finding antiderivatives and finding indefinite integ rals. We check this with our " }{XPPEDIT 18 0 "f (x)=1+5*x" "6#/-%\"fG 6#%\"xG,&\"\"\"F)*&\"\"&F)F'F)F)" }{TEXT -1 36 " we know an antideriva tive would be " }{XPPEDIT 18 0 "x+5*x^2/2;" "6#,&%\"xG\"\"\"*&*&\"\"&F %*$)F$\"\"#F%F%F%F+!\"\"F%" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F:=int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FG,&%\"xG\"\"\"*$)F&\"\"#F'#\"\"&F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 " You should note that here we have used the integral comm and " }{TEXT 263 8 "int(f,x)" }{TEXT -1 89 " without specifying the in terval to get this antiderivative and we also gave it a name F." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs(x=-2,F); subs(x=1,F);subs(x=1,F)-subs(x=-2,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"( \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"*\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Here we see that we get the same answer as we \+ expected -4.5. in our previous worksheet we found the area under the c urve " }{XPPEDIT 18 0 "x^2-4*x+5" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F'F%F' !\"\"\"\"&F'" }{TEXT -1 97 " over the interval [0,3] to be 6. We check the process by integrating this expression over [0,3]." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(x ^2-4*x+5,x=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We further check it by finding an antide rivative for the expression and applying the fundamental theorem." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "F:=int(x^2-4*x+5,x); subs(x= 0,F); subs(x=3,F);subs(x=3,F)-subs(x=0,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,(*$)%\"xG\"\"$\"\"\"#F*F)*$)F(\"\"#F*!\"#F(\"\"& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "This shows the antiderivative and its values at 0 and 3 and then substracts giving the same answer of 6 . " }}}}{MARK "16 0 2" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }