{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 1 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 1 2 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 2 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 14 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 14 0 0 0 0 0 2 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 128 0 128 1 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 128 0 128 1 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 128 0 128 1 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 128 0 128 1 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 128 0 128 1 1 1 0 0 0 0 0 0 0 0 }{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 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 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 6 6 1 0 1 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 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -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 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 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 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 257 24 "Calculus Single Variable " }{TEXT 260 1 "\n" }{TEXT 259 35 "Brian E. Blank and Steven G. Krantz " }{TEXT 261 3 " \n\n" }{TEXT 262 11 "Section 5.1" }{TEXT 263 1 "\n" } {TEXT 256 46 "Introduction to Integration - The Area Problem" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "1. Sigma Notation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The expression" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ " }{XPPEDIT 18 0 "Sum(a[j],j = M .. N);" "6#-%$Sum G6$&%\"aG6#%\"jG/F);%\"MG%\"NG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "means" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 " " }{XPPEDIT 18 0 "a[M]+a[M+1]+` ... `+a[N-1]+a[N];" "6#,,&%\"aG6#%\"MG\"\"\"&F%6#, &F'F(F(F(F(%&~...~GF(&F%6#,&%\"NGF(F(!\"\"F(&F%6#F0F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "For example" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " \+ " }{XPPEDIT 18 0 "Sum(j^2,j = 3 .. 10) = 3^2+4^2+5^2+6^2+7^2+ 8^2+9^2+10^2;" "6#/-%$SumG6$*$%\"jG\"\"#/F(;\"\"$\"#5,2*$F,F)\"\"\"*$ \"\"%F)F0*$\"\"&F)F0*$\"\"'F)F0*$\"\"(F)F0*$\"\")F)F0*$\"\"*F)F0*$F-F) F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "When we can find an explicit formula that allows us to express a sum " }{XPPEDIT 18 0 "Sum(a[j],j = M .. N); " "6#-%$SumG6$&%\"aG6#%\"jG/F);%\"MG%\"NG" }{TEXT -1 53 " without us ing sigma notation it is said to be a " }{TEXT 264 11 "closed form" }{TEXT -1 70 " formula. Unfortunately, closed form expressions are re latively rare." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Here are three closed for m summation formulas that have been known from ancient times:" }} {PARA 0 "" 0 "" {TEXT -1 26 "The first two are called " }{TEXT 271 9 "power-sum" }{TEXT -1 13 " formulas. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }{TEXT 268 8 " " }{XPPEDIT 269 0 "Sum(j,j = 1 .. N) = N*(N+1)/2;" "6#/-%$SumG6$%\"jG/F';\"\"\"%\"NG*(F+F*,&F+F*F* F*F*\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " " }{XPPEDIT 267 0 "Sum(j^2,j = 1 . . N) = N*(N+1)*(2*N+1)/6;" "6#/-%$SumG6$*$%\"jG\"\"#/F(;\"\"\"%\"NG**F -F,,&F-F,F,F,F,,&*&F)F,F-F,F,F,F,F,\"\"'!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "In 1713 Jakob Bernoulli publish ed a closed form expression that works for for all power sums " } {XPPEDIT 18 0 "Sum(j^p,j = 1 .. N);" "6#-%$SumG6$)%\"jG%\"pG/F';\"\"\" %\"NG" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 21 " a positive integer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " \+ " }{TEXT 265 4 " " }{XPPEDIT 266 0 "Sum(r^j,j = 0 .. N) = ` \+ `*(1-r^(N+1))/(1-r);" "6#/-%$SumG6$)%\"rG%\"jG/F);\"\"!%\"NG*(%$~~~G \"\"\",&F0F0)F(,&F-F0F0F0!\"\"F0,&F0F0F(F4F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This last formula is said to be the sum of a " }{TEXT 270 16 "geome tric series" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here is a simpl e but interesting series that has a closed form. The function " } {XPPEDIT 18 0 "F;" "6#%\"FG" }{TEXT -1 41 " is defined on the nonnega tive integers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " " }{TEXT 272 4 " " }{XPPEDIT 257 0 "Sum(F(j)-F(j-1),j = 1 .. N) = F(N)-F(0);" "6#/-%$SumG6$,&-%\"FG 6#%\"jG\"\"\"-F)6#,&F+F,F,!\"\"F0/F+;F,%\"NG,&-F)6#F3F,-F)6#\"\"!F0" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "This typ e of sum is called a " }{TEXT 273 10 "collapsing" }{TEXT -1 4 " or " } {TEXT 274 11 "telescoping" }{TEXT -1 5 " sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "2. Retracing Arc himedes" }}{PARA 0 "" 0 "" {TEXT -1 29 "\nLet us calculate the area \+ " }{XPPEDIT 18 0 "A(a,b);" "6#-%\"AG6$%\"aG%\"bG" }{TEXT -1 23 " un der the curve " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" } {TEXT -1 1 " " }{MPLTEXT 1 0 1 " " }{TEXT -1 20 "over the interval \+ " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 11 " of the \+ " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 23 "-axis. \nNotice that \n \n" }{MPLTEXT 1 0 18 " " }{XPPEDIT 18 0 "A(a,b) = A(0 ,b)-A(0,a);" "6#/-%\"AG6$%\"aG%\"bG,&-F%6$\"\"!F(\"\"\"-F%6$F,F'!\"\" " }{MPLTEXT 1 0 4 " " }{TEXT -1 82 "\n\nso we need only solve our p roblem under the assumption that the left endpoint is" }{MPLTEXT 1 0 2 " 0" }{TEXT -1 57 ". \n(This type of simplification of a problem is \+ called " }{TEXT 258 15 "range reduction" }{TEXT -1 73 ". It is a ve ry important technique in mathematics and computer science.)" }}{PARA 0 "" 0 "" {TEXT -1 108 "\n\nWe will use inscribed rectangles; here is \+ a picture that shows eight inscribed rectangles (with b = 2). \n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "with(student): #This packag e holds the required command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "leftbox(x^2, x = 0..2, 8);" }}{PARA 13 "" 1 "" {GLPLOT2D 337 223 223 {PLOTDATA 2 "6--%)POLYGONSG6$7&7$$\"\"!F)F(F'7$$\"+++++D!#5F(F *-%&COLORG6&%$RGBG$\"\"(!\"\"$\"\"*F4F2-F$6$7&F*7$F+$\"++++]i!#67$$\"+ ++++]F-F;7$F?F(F.-F$6$7&FA7$F?F+7$$\"+++++vF-F+7$FGF(F.-F$6$7&FI7$FG$ \"++++DcF-7$$\"\"\"F)FN7$FQF(F.-F$6$7&FS7$FQFQ7$$\"++++]7!\"*FQ7$FYF(F .-F$6$7&Ffn7$FY$\"+++]i:Fen7$$\"+++++:FenF[o7$F^oF(F.-F$6$7&F`o7$F^o$ \"++++]AFen7$$\"++++]$\"3'fth!>PY+>!#?7$ $\"3'pmm;H[D:)F\\q$\"374dzkVSYmF_q7$$\"3LLLLe0$=C\"!#=$\"3Ry/hNJ9U:F\\ q7$$\"3ILLL3RBr;Fhq$\"3Q3JOwF-$z#F\\q7$$\"3Ymm;zjf)4#Fhq$\"3pU9lin5/WF \\q7$$\"3=LL$e4;[\\#Fhq$\"3!\\2H?N2TA'F\\q7$$\"3p****\\i'y]!HFhq$\"3'p y7`.#[R%)F\\q7$$\"3,LL$ezs$HLFhq$\"3V1NO@BZ36Fhq7$$\"3_****\\7iI_PFhq$ \"364mB\">!)zS\"Fhq7$$\"3#pmmm@Xt=%Fhq$\"3_@TN'*fQ`+&Fhq$\"3*=/Z7,,>]#Fhq7$$\"3()* *****\\Z/NaFhq$\"3Oc-XV6(R&HFhq7$$\"3'*******\\$fC&eFhq$\"34U-MW!G^U$F hq7$$\"3ELL$ez6:B'Fhq$\"3m(*4;ER<$)QFhq7$$\"3Smmm;=C#o'Fhq$\"3+j3kpbBl WFhq7$$\"3-mmmm#pS1(Fhq$\"3!GXE/Y2,*\\Fhq7$$\"3]****\\i`A3vFhq$\"3@D)3 %4[MPcFhq7$$\"3slmmm(y8!zFhq$\"3l$3L9kyJC'Fhq7$$\"3V++]i.tK$)Fhq$\"3>R HTH&RM%pFhq7$$\"39++](3zMu)Fhq$\"33$)\\NbE%[k(Fhq7$$\"3#pmm;H_?<*Fhq$ \"3Fvn5CVl7%)Fhq7$$\"3emm;zihl&*Fhq$\"3-Me-![,,:*Fhq7$$\"39LLL3#G,***F hq$\"3;PP*=;m-)**Fhq7$$\"3uep1$Fgx7$$\"3%*******p(G**y\"Fgx$\"3c7Pn,]%Q?$Fgx7$$\"3lmm;9 @BM=Fgx$\"3[>.k[uSkLFgx7$$\"3ELLL`v&Q(=Fgx$\"3!fQ%=I@M6NFgx7$$\"30++DO l5;>Fgx$\"3!G(*f#eUYrOFgx7$$\"3/++v.Uac>Fgx$\"3`@xA@_1GQFgx7$Fbp$\"\"% F)-%'COLOURG6&F1$\"*++++\"!\")F(F(-%*THICKNESSG6#Fcp-%&STYLEG6#%%LINEG -%+AXESLABELSG6$Q\"x6\"Q!F\\al-%%VIEWG6$;F(Fbp%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}} {PARA 0 "" 0 "" {TEXT -1 61 "\nThe approximation is better with 20 i nscribed rectangles.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "le ftbox(x^2, x = 0..2, 20);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "69-%'CURVESG6&7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3'fth!>P Y+>!#?7$$\"3'pmm;H[D:)F-$\"374dzkVSYmF07$$\"3LLLLe0$=C\"!#=$\"3Ry/hNJ9 U:F-7$$\"3ILLL3RBr;F9$\"3Q3JOwF-$z#F-7$$\"3Ymm;zjf)4#F9$\"3pU9lin5/WF- 7$$\"3=LL$e4;[\\#F9$\"3!\\2H?N2TA'F-7$$\"3p****\\i'y]!HF9$\"3'py7`.#[R %)F-7$$\"3,LL$ezs$HLF9$\"3V1NO@BZ36F97$$\"3_****\\7iI_PF9$\"364mB\">!) zS\"F97$$\"3#pmmm@Xt=%F9$\"3_@TN'*fQ`+&F9$\"3*=/Z7,,>]#F97$$\"3()******\\Z/NaF9$\"3Oc -XV6(R&HF97$$\"3'*******\\$fC&eF9$\"34U-MW!G^U$F97$$\"3ELL$ez6:B'F9$\" 3m(*4;ER<$)QF97$$\"3Smmm;=C#o'F9$\"3+j3kpbBlWF97$$\"3-mmmm#pS1(F9$\"3! 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urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}}}{PARA 0 "" 0 "" {TEXT -1 82 "\n\nThe approximation is much better with 60 (for exampl e) inscribed rectangles.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "leftbox(x^2, x = 0..2, 60);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6[o-%)POLYGONSG6$7&7$$\"\"!F)F(F'7$$\"+LLLLL!#6F( F*-%&COLORG6&%$RGBG$\"\"(!\"\"$\"\"*F4F2-F$6$7&F*7$F+$\"+66666!#77$$\" +nmmmmF-F;7$F?F(F.-F$6$7&FA7$F?$\"+WWWWWF=7$$\"+++++5!#5FF7$FIF(F.-F$6 $7&FL7$FI$FJF-7$$\"+LLLL8FKFQ7$FSF(F.-F$6$7&FU7$FS$\"+yxxxFe^l7$$\" +LLLL9Fe^lFgel7$FjelF(F.-F$6$7&F\\fl7$Fjel$\"+WWWa?Fe^l7$$\"+nmmm9Fe^l Fafl7$FdflF(F.-F$6$7&Fffl7$Fdfl$\"+666^@Fe^l7$$\"+++++:Fe^lF[gl7$F^glF (F.-F$6$7&F`gl7$F^gl$\"++++]AFe^l7$$\"+LLLL:Fe^lFegl7$FhglF(F.-F$6$7&F jgl7$Fhgl$\"+666^BFe^l7$$\"+nmmm:Fe^lF_hl7$FbhlF(F.-F$6$7&Fdhl7$Fbhl$ \"+WWWaCFe^l7$$FbsFe^lFihl7$F\\ilF(F.-F$6$7&F]il7$F\\il$\"++++gDFe^l7$ 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mm(y8!zFgam$\"3l$3L9kyJC'Fgam7$$\"3V++]i.tK$)Fgam$\"3>RHTH&RM%pFgam7$$ \"39++](3zMu)Fgam$\"33$)\\NbE%[k(Fgam7$$\"3#pmm;H_?<*Fgam$\"3Fvn5CVl7% )Fgam7$$\"3emm;zihl&*Fgam$\"3-Me-![,,:*Fgam7$$\"39LLL3#G,***Fgam$\"3;P P*=;m-)**Fgam7$$\"3uep1$Ffhm7$$\"3%*******p(G**y\"Ffh m$\"3c7Pn,]%Q?$Ffhm7$$\"3lmm;9@BM=Ffhm$\"3[>.k[uSkLFfhm7$$\"3ELLL`v&Q( =Ffhm$\"3!fQ%=I@M6NFfhm7$$\"30++DOl5;>Ffhm$\"3!G(*f#eUYrOFfhm7$$\"3/++ v.Uac>Ffhm$\"3`@xA@_1GQFfhm7$Fa`m$\"\"%F)-%'COLOURG6&F1$\"*++++\"!\")F (F(-%*THICKNESSG6#Fb`m-%&STYLEG6#%%LINEG-%+AXESLABELSG6$Q\"x6\"Q!F[an- %%VIEWG6$;F(Fa`m%(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" "Cu rve 13" "Curve 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" "Cur ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3 8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" " Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv e 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57 " "Curve 58" "Curve 59" "Curve 60" "Curve 61" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "In general, we divide the inter val " }{XPPEDIT 18 0 "[0, b];" "6#7$\"\"!%\"bG" }{TEXT -1 9 " into \+ " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 42 " equal subintervals, e ach having length " }{XPPEDIT 18 0 "Delta = b/N;" "6#/%&DeltaG*&%\"bG \"\"\"%\"NG!\"\"" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Delta := b/N; " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&DeltaG*&%\"bG\"\"\"%\"NG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 21 "\nFor each integer " }{XPPEDIT 18 0 "j = 1,2,`... `,N;" "6&/%\"jG\"\"\"\"\"#%$...G%\"NG" }{TEXT -1 17 " we will let \+ " }{XPPEDIT 18 0 "x[j];" "6#&%\"xG6#%\"jG" }{TEXT -1 37 " denote the left endpoint of the " }{XPPEDIT 18 0 "j*`'th`;" "6#*&%\"jG\"\"\"%$ 'thGF%" }{TEXT -1 15 " subinterval: " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x[j] = (j-1)*Delta;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#%\"jG*(,&F'\"\"\"F*!\"\"F*%\" bGF*%\"NGF+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The sum of the areas of the ins cribed rectangles is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "areaApproximationEquation := Sum( x[j]^2 *Delta, j = 1..N) = factor(sum( ((j-1)*b/N)^2*Delta, j = 1..N));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%:areaApproximationEquationG/-%$SumG6 $*(&%\"xG6#%\"jG\"\"#%\"bG\"\"\"%\"NG!\"\"/F-;F0F1,$*,\"\"'F2F/\"\"$,& *&F.F0F1F0F0F0F2F0,&F1F0F0F2F0F1!\"#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "For ease \+ of evaluation, let us make a function out of our approximation:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A := unapply(rhs(areaApproxi mationEquation), N); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGf*6#%\" NG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"'F/**%\"bG\"\"$,&*&\"\"#F/9 $F/F/F/!\"\"F/,&F7F/F/F8F/F7!\"#F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "for N from 500 to 5000 \+ by 500 do evalf(A(N)) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$ \"+++MBL!#5\"\"\")%\"bG\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&$\"++]LGL!#5\"\"\")%\"bG\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&$\"+T2+IL!#5\"\"\")%\"bG\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"+]P$3L$!#5\"\"\")%\"bG\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"++OLJL!#5\"\"\")%\"bG\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"+_omJL!#5\"\"\")%\"bG\"\"$F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&$\"+)*[!>L$!#5\"\"\")%\"bG\"\"$F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"+QM3KL!#5\"\"\")%\"bG\"\"$F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"+0BAKL!#5\"\"\")%\"bG\"\"$F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"++MLKL!#5\"\"\")%\"bG\" \"$F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The numerical evidence so far obta ined suggests that, at least approximately," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ " }{XPPEDIT 18 0 "A(0,b) = b^3/3;" "6#/-%\"AG6$\" \"!%\"bG*&F(\"\"$F*!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "\nIt appears th at as the value of the number " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 57 " of subintervals increases, the rectangle approximation " } {XPPEDIT 18 0 "A(N);" "6#-%\"AG6#%\"NG" }{TEXT -1 13 " to the area " } {XPPEDIT 18 0 "A(0,b);" "6#-%\"AG6$\"\"!%\"bG" }{TEXT -1 66 " under th e curve gets more accurate. Let us see what happens as " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 22 " tends to infinity.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "N;\nN := 'N'; # restores the \"li teral\" value to N, which last was 5500\nN;\nlimit(A(N), N = infinity) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%+b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"NG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"%\"bGF%\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Here is the behind-t he-scenes background. In order to obtain an explicit formula for " } {XPPEDIT 18 0 "A(N);" "6#-%\"AG6#%\"NG" }{TEXT -1 77 ", Maple has to compute the sum of the squares of the integers from 1 to " } {XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 4 ": \n\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "Sum(j^2, j = 1..N) = factor(sum(j^2, j = 1..N) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$)%\"jG\"\"#\"\"\"/F) ;F+%\"NG,$**\"\"'!\"\"F.F+,&F.F+F+F+F+,&*&F*F+F.F+F+F+F+F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "\nwhich i s a formula that was known to the ancient Greeks. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Let us conclude by explicitly stating the solution to the probl em we set ourselves:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 33 " " }{XPPEDIT 18 0 "A(a, b) = b^3/3-a^3/3;" "6#/-%\"AG6$%\"aG%\"bG,&*&F(\"\"$F+!\"\"\"\"\"*&F'F +F+F,F," }{MPLTEXT 1 0 5 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 " 3. A Look Forward" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "If " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F' \"\"#" }{TEXT -1 11 ", then " }{XPPEDIT 18 0 "F(x) = x^3/3" "6#/-% \"FG6#%\"xG*&F'\"\"$F)!\"\"" }{TEXT -1 28 " is an antiderivative of \+ " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 77 ". The area \+ formula obtained in the first section can be restated as follows:" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "The area under the graph of " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\" xG" }{TEXT -1 27 " and over the interval " }{XPPEDIT 18 0 "[a, b]; " "6#7$%\"aG%\"bG" }{TEXT -1 8 " is " }{XPPEDIT 18 0 "F(b)-F(a);" "6#,&-%\"FG6#%\"bG\"\"\"-F%6#%\"aG!\"\"" }{TEXT -1 8 " where " } {XPPEDIT 18 0 "F;" "6#%\"FG" }{TEXT -1 27 " is an antiderivative of \+ " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 235 " Because of the connect ion between the antiderivative and the area problem, a connection that we have observed in only one case so far but which is, in fact, gener al, we denote the area under the graph of a positive-valued function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 27 " and over the interv al " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 5 " by" } }{PARA 0 "" 0 "" {TEXT -1 30 " \n " } {XPPEDIT 18 0 "Int(f(x),x = a .. b);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\" aG%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The formula that we have observed for " }{XPPEDIT 18 0 "f(x) = x ^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 75 " can be written as\n \n \n " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) = F(b)-F(a);" "6#/-%$IntG6$-%\"fG6#%\"xG/F* ;%\"aG%\"bG,&-%\"FG6#F.\"\"\"-F16#F-!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ " }{XPPEDIT 18 0 "F(x) = Int(f(x),x);" "6#/-% \"FG6#%\"xG-%$IntG6$-%\"fG6#F'F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "4. Fermat's Near Miss " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 23 " is a constant and " }{XPPEDIT 18 0 "f(x) = x^s;" "6#/-%\"fG6#%\"xG)F'%\"sG" }{TEXT -1 75 ", then t he formula\n\n \n " }{XPPEDIT 18 0 "Int(f(x),x = a .. b) = F(b)-F(a);" "6#/-%$IntG6$-%\"fG 6#%\"xG/F*;%\"aG%\"bG,&-%\"FG6#F.\"\"\"-F16#F-!\"\"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "where" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 " " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F(x) = Int(f(x),x);" "6#/-%\"FG6#%\"xG-%$IntG6$-%\"fG6#F'F'" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }{XPPEDIT 18 0 "Int(x^s,x = a .. b) = b^(s+1)/(s+1)-a^( s+1)/(s+1);" "6#/-%$IntG6$)%\"xG%\"sG/F(;%\"aG%\"bG,&*&)F-,&F)\"\"\"F2 F2F2,&F)F2F2F2!\"\"F2*&)F,,&F)F2F2F2F2,&F)F2F2F2F4F4" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Fermat proved this formula when " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 27 " is a rational number: " }{XPPEDIT 18 0 "s = p/q;" "6#/%\"sG*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 18 " for intege rs " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 9 " and " } {XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 92 ". \nFermat's proof of this formula is based on an ingenious exhaustion of the area under " } {XPPEDIT 18 0 "y = x^(p/q);" "6#/%\"yG)%\"xG*&%\"pG\"\"\"%\"qG!\"\"" } {TEXT -1 378 " by an infinite sequence of rectangles with bases having lengths that form a geometric progression. His clever calculation inv olves the closed form expression for a geometric series, as given in S ection 1 above. If you are interested, the details of Fermat's proof m ay be found on pages 414 and 415. \n\nFermat does not seem to have no ticed the role that the indefinite integral " }{XPPEDIT 18 0 "int(f(x) ,x);" "6#-%$intG6$-%\"fG6#%\"xGF)" }{TEXT -1 59 " plays in calculatin g the value of the definite integral " }{XPPEDIT 18 0 "int(f(x),x = a .. b);" "6#-%$intG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 150 " . \+ If he had made the connection, he would have had a greater claim to th e introduction of calculus, now generally attributed to Leibniz and Ne wton." }}{PARA 0 "" 0 "" {TEXT -1 89 " \n \nThe Maple function presen ted here will illustrate Fermat's unique exhaustion scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 548 "FermatsRectangularApproximation := proc()\nlocal p, q, r, N, k, x, rect, fnPlot:\np := args[1]:\nq := ar gs[2]:\nr := args[3]:\nN := args[4]:\n for k from 0 to N do \n x [k] := r^(k*q):\n if k > 0 then\n rect[k] := plottools[recta ngle]([x[k],sqrt(x[k])], [x[k-1],0],\n \+ color=COLOR(RGB,.7,.7,.85)):\n end if:\n end do:\nfnPl ot := plot(x^(p/q), x=0..1, thickness=2, color=MAROON, \+ tickmarks=[2,2], numpoints=700):\nreturn plots[display]( fnPlot,seq(rect[k],k=1..N));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "FermatsRectangularApproximation(1, 2, 0.9, 30);" }} {PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6D-%'CURVESG6%7hfm 7$$\"\"!F)F(7$$\"3yKI7'G+o\\\"!#?$\"3i<))4G.&)oQ!#>7$$\"3aW5t6#e\"*z#F -$\"3[-%)=Zrq!H&F07$$\"31H1W3fzjUF-$\"3u;f:6[xHlF07$$\"3P]]`u@9QdF-$\" 3CA'p-f_]d(F07$$\"3uzf=I6[0sF-$\"3U'*[sY0^)[)F07$$\"3#*\\]`u@*ec)F-$\" 3U[Vp=0@b#*F07$$\"3'o`vG,>X(**F-$\"35lA2!Q^s)**F07$$\"3Vy[T!HKJ9\"F0$ \"3%4a\"R[O\"FT7$$\"3G-<>M*z#p:F0$\"3'Hm@\"Q!4FD\"FT7$$\"3&)ze6 \"y!R<d#\\58FT7$$\"39DxS&y4h'=F0$\"3'y,`#*GcgO\"FT7$$\"3\" =H/I5G%4?F0$\"3jLx+\")Ha<9FT7$$\"3Tu@_l[dR@F0$\"3[RAko&GFY\"FT7$$\"3s9 .Aa>L%H#F0$\"3w.UuO_q9:FT7$$\"3e$Qoy]JaU#F0$\"3#)eN!Hpztb\"FT7$$\"39Ac $\\XJzd#F0$\"3RZ1N,Qf0;FT7$$\"3'3t6#[(>Hr#F0$\"3?fh%\\L%4Z;FT7$$\"3q%= H/IB5'GF0$\"3QbaeqfX\"p\"FT7$$\"3;*Rzb!H0-IF0$\"39uZ,HLkKFT7$$\"3')3f8&f=c&QF0$\"3!Gqlc\"Hdj>FT7$$\"3B V.Co(pE+%F0$\"3!3^F2Ln1+#FT7$$\"3e90O_1`YTF0$\"3jB%>G=.j.#FT7$$\"37Ac$ \\XBdG%F0$\"3#eA.E*))>q?FT7$$\"3g-@ZILFSWF0$\"3'=;B!3c>2@FT7$$\"3eN]_n K9zXF0$\"3=.T9%G$*)R@FT7$$\"3P0O_mlTFZF0$\"3o#\\$)RIiU<#FT7$$\"37f;;8U xh[F0$\"3vkG]JJ%\\?#FT7$$\"3ag?W4Em3]F0$\"3*Hg,w:/!QAFT7$$\"3%e^hI9to9 &F0$\"3Mr935@noAFT7$$\"3%p#*[U(HM\"H&F0$\"3B4mRK>H+BFT7$$\"3_$f:4k)eKa F0$\"3)*3eV>9zIBFT7$$\"3k(\\[Rwf/e&F0$\"3'3x)*pn*HiBFT7$$\"3yT!H.BxGs& F0$\"3U8`@)e`AR#FT7$$\"3y!f8&f'>&oeF0$\"3a#))*pnF]ACFT7$$\"3T;7&e4cH,' F0$\"3GV=joH8_CFT7$$\"3E6vD!=yc9'F0$\"3W:#Gu$y/zCFT7$$\"3)pmmmm#z(H'F0 $\"3xm\\vH.a4DFT7$$\"3\"H(4owc%QV'F0$\"3duIpYE]ODFT7$$\"3Sf'=f#FT7$$\"35kU)*)G9>(o F0$\"3b[@QKOV@EFT7$$\"3ZU(>Qn5a+(F0$\"3o^+zMOxYEFT7$$\"3+0Qmk-$p:(F0$ \"3Q7'yN'RCvEFT7$$\"3'3yYF#R-&H(F0$\"3U#4Ie>I4q#FT7$$\"3%QV.CowgW(F0$ \"3H&Rq>-](GFFT7$$\"3!4G(4o'4kd(F0$\"3'34B(oy__FFT7$$\"3N*)=KDP%[s(F0$ \"3a8Tl3.OzFFT7$$\"31&e4npB#oyF0$\"3#fp%3=a.0GFT7$$\"3#[hI9+5:,)F0$\"3 /2[zGFT7$$\"3@*Qs1Zv'R%)F0$\"3'460HFT7$$\"3YS#oxVx4e)F0$ \"3#=))*HW0LHHFT7$$\"342\\V/6pH()F0$\"3[b=Un]gaHFT7$$\"3?5r(Ry7V'))F0$ \"3k>]9z&*HxHFT7$$\"3zxZMT\\48!*F0$\"3:&)Q#pp\"=-IFT7$$\"3%>T))=K-c:*F 0$\"3E)>!*feAe-$FT7$$\"3Kc#za$Gw(H*F0$\"3S\\5w;LA\\IFT7$$\"3Qb%=H/]iW* F0$\"3qtEqQ_ZtIFT7$$\"39E\")o\"=7Ie*F0$\"35+)Hd1Wc4$FT7$$\"3`\"=F!>`2B (*F0$\"3fTx8>Y==JFT7$$\"3S^gBl;sx)*F0$\"3U\"p!y!H%)G9$FT7$$\"3U!Qmk7u< +\"FT$\"3k**yMd:3lJFT7$$\"3@%))=Kv$4;5FT$\"3kGHr^Di(=$FT7$$\"3sa#yZWc1 .\"FT$\"3pP6UZQQ5KFT7$$\"3k;;8#RUS/\"FT$\"3F*e\"z$[k6B$FT7$$\"3es2ay8J e5FT$\"3/\")f!e!)pJD$FT7$$\"35ay\\[u:X9'[F$FT7$$\"3]$RvF%* =w3\"FT$\"3ak!3_#o!zH$FT7$$\"3r'y]bo&)45\"FT$\"3f#\\&\\M/6=LFT7$$\"3/= F!>83k6\"FT$\"3Y>kI<%p7M$FT7$$\"3_2`r+dKI6FT$\"3WSu[y;.iLFT7$$\"3XgAe2 k4W6FT$\"3A_7b>%\\CQ$FT7$$\"3=yZMTV!*e6FT$\"3bP%**3ToUS$FT7$$\"3K)y^i< sP<\"FT$\"30%*)zq'e.EMFT7$$\"3Yozd/(*H(=\"FT$\"3CsMic:sXMFT7$$\"3-r'p( Q`m,7FT$\"3!Q'pA\"\\/lY$FT7$$\"3P$H0Pz%o:7FT$\"3I3(=v;nm[$FT7$$\"30E#e 2.#)3B\"FT$\"3\\>SEQGR3NFT7$$\"3l?W4m=.W7FT$\"3(3MJ9f$3FNFT7$$\"3g=H/I \\8f7FT$\"3#yT!QJ\"H%[NFT7$$\"3'Qy[TgKLF\"FT$\"3EQ>AH'y$oNFT7$$\"3_3g? 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