{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 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 3 1 }1 1 0 0 0 0 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 "Normal" -1 256 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Special inverse functions .. summ ary and examples" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "load interpolation and function a pproximation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 10 "fcnapprx.m" } {TEXT -1 37 " contains the code for the procedure " }{TEXT 0 5 "remez " }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the o ne that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx .m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "load inverse functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Mapl e m-file " }{TEXT 262 9 "invfcns.m" }{TEXT -1 52 " contains the code f or the special inverse function." }}{PARA 0 "" 0 "" {TEXT -1 123 "It c an be read into a Maple session by a command similar to the one that f ollows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procdrs/invfcns.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "load roo t-finding procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file \+ " }{TEXT 262 7 "roots.m" }{TEXT -1 38 " contains the code for the proc edures " }{TEXT 0 6 "secant" }{TEXT -1 1 " " }{TEXT -1 4 "and " } {TEXT 0 7 "findmin" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a comma nd similar to the one that follows, where the file path gives its loca tion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/ procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "load numerical integration procedures and data" }} {PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 6 "intg.m " }{TEXT -1 5 " and " }{TEXT 262 8 "gkdata.m" }{TEXT -1 67 " contain t he code and data for the numerical integration procedure " }{TEXT 0 5 "GKint" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be read into a Maple session by commands simila r to those that follow, where the file paths give their location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/i ntg.m\";\nread \"K:\\\\Maple/procdrs/gkdata.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "The special inverse func tions: " }{TEXT 0 29 "W,R,S,T,Y,Z,U,P,sP,sM,V,Q,K,M" }{TEXT -1 1 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT 0 1 "W" }{TEXT -1 13 ": inverse of " }{XPPEDIT 18 0 "g(x)=x*ex p(x)" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(x)=x*exp(x)" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expG6#F'F) " }{TEXT -1 6 " .. 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_x@$3-FQ6$FdxFjo/-F]q6$F7FdxF<>Fdx,$FdxF<@$3-FQ6$FdxF_q/-F]q6$F3FdxFfp C%>FgqF\\y>FhsFis@$F_tC$>F`pF\\r[@$F]s>F`pFap@%F`pFgqFVFhtC'>Few-Fgw6% FdpFSFjt>FjwF[x?(F_xF7F7F`xF\\rC%>FdxFex@$Fhx>FdxF_y@$3Fby/FeyFjtC%>Fg qF\\y>FhsFju@$F_tC$>F`pF\\rF]z@$Fav>F`pFap@%F`pF\\vFVFVFV6#Q9inverse~o f~x~->~x*exp(x)FNFNFN" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "plot([x*exp(x),'W(x)',x],x=-1..2.7 ,y=-1..2.7,color=[red,blue,black],\n linestyle=[1$2,2],t hickness=2,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6)-%'CURVESG6%7U7$$!\"\"\"\"!$!3MBWr6WzyO!#=7$$!3 )HLLeH0N>*F-$!3=+mz3s;mOF-7$$!3qm;/m&y<\\)F-$!3)4!G+l\")\\KOF-7$$!3]LL 3nMh-xF-$!3Y%*)*\\Z(yac$F-7$$!3:KLeps@3pF-$!3!Qj^3N]@Y$F-7$$!3Vn;a)p'f $!36te7m.*y#pFbo7$$\"3!z4++vyK[&!#?$\" 3<)*))zxwU8bFho7$$\"3%H,+](z\\q#)Fbo$\"3_m'p]4\"f$)*)Fbo7$$\"3(fm\"HAo HG:F-$\"3u>&G&e\"f1y\"F-7$$\"3ALL$3OZ@O#F-$\"3#[uZhiK:*HF-7$$\"3MLLLV 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" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Summary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matr ix([[g(x), phi(x) = g^(-1)*``(x), phi*`'`(x)], [____________, ________ ________, _____________________], [x*exp(x), W(x), W(x)/(1+W(x))/x], [ x-arctan(x), R(x), 1+1/(R(x)^2)], [x+arctan(x), S(x), (1+S(x)^2)/(2+S( x)^2)], [x*sec(x), T(x), T(x)/x/(1+T(x)*tan(T(x)))], [x*cosh(x), Y(x), 1/(cosh(Y(x))+Y(x)*sinh(Y(x)))], [sinh(x)-x, Z(x), 1/(cosh(Z(x))-1)], [x+tanh(x), U(x), 1/(2-(x-U(x))^2)], [x-tanh(x), P(x), 1/((x-P(x))^2) ], [x+sin(x), sP(x), 1/(1+cos(sP(x)))], [x-sin(x), sM(x), 1/(1-cos(sP( x)))], [exp(x)-1-x-x^2/2, V(x), 2/(2*x+V(x)^2)], [x-arcsinh(x), Q(x), \+ sqrt(1+Q(x)^2)/(sqrt(1+Q(x)^2)-1)], [x+exp(x), K(x), 1/(1+x-K(x))], [d ilog(x), M(x), (1-M(x))/ln(M(x))]]);" "6#-%'matrixG6#727%-%\"gG6#%\"xG /-%$phiG6#F+*&)F),$\"\"\"!\"\"F3-%!G6#F+F3*&F.F3-%\"'G6#F+F37%%-______ ______G%1________________G%6_____________________G7%*&F+F3-%$expG6#F+F 3-%\"WG6#F+*(-FF6#F+F3,&F3F3-FF6#F+F3F4F+F47%,&F+F3-%'arctanG6#F+F4-% \"RG6#F+,&F3F3*&F3F3*$-FT6#F+\"\"#F4F37%,&F+F3-FQ6#F+F3-%\"SG6#F+*&,&F 3F3*$-F[o6#F+FenF3F3,&FenF3*$-F[o6#F+FenF3F47%*&F+F3-%$secG6#F+F3-%\"T G6#F+*(-F\\p6#F+F3F+F4,&F3F3*&-F\\p6#F+F3-%$tanG6#-F\\p6#F+F3F3F47%*&F +F3-%%coshG6#F+F3-%\"YG6#F+*&F3F3,&-F]q6#-F`q6#F+F3*&-F`q6#F+F3-%%sinh G6#-F`q6#F+F3F3F47%,&-F\\r6#F+F3F+F4-%\"ZG6#F+*&F3F3,&-F]q6#-Fer6#F+F3 F3F4F47%,&F+F3-%%tanhG6#F+F3-%\"UG6#F+*&F3F3,&FenF3*$,&F+F3-Fcs6#F+F4F enF4F47%,&F+F3-F`s6#F+F4-%\"PG6#F+*&F3F3*$,&F+F3-F`t6#F+F4FenF47%,&F+F 3-%$sinG6#F+F3-%#sPG6#F+*&F3F3,&F3F3-%$cosG6#-F]u6#F+F3F47%,&F+F3-Fjt6 #F+F4-%#sMG6#F+*&F3F3,&F3F3-Fbu6#-F]u6#F+F4F47%,*-FC6#F+F3F3F4F+F4*&F+ FenFenF4F4-%\"VG6#F+*&FenF3,&*&FenF3F+F3F3*$-Fiv6#F+FenF3F47%,&F+F3-%( arcsinhG6#F+F4-%\"QG6#F+*&-%%sqrtG6#,&F3F3*$-Fgw6#F+FenF3F3,&-F[x6#,&F 3F3*$-Fgw6#F+FenF3F3F3F4F47%,&F+F3-FC6#F+F3-%\"KG6#F+*&F3F3,(F3F3F+F3- F]y6#F+F4F47%-%&dilogG6#F+-%\"MG6#F+*&,&F3F3-Fhy6#F+F4F3-%#lnG6#-Fhy6# F+F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 29 "W,R,S,T,Y,Z,U,P,sP,sM,V,Q,K,M" }{TEXT -1 11 " :examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "p hi(x)=S(x)*K(x)" "6#/-%$phiG6#%\"xG*&-%\"SG6#F'\"\"\"-%\"KG6#F'F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(phi(x),x)=Diff(S(x),x)*K(x)+S(x )*Diff(K(x),x)" "6#/-%%DiffG6$-%$phiG6#%\"xGF*,&*&-F%6$-%\"SG6#F*F*\" \"\"-%\"KG6#F*F2F2*&-F06#F*F2-F%6$-F46#F*F*F2F2" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(1+S(x)^2)/(2+S(x) ^2)*K(x)+S(x)/(1+x-K(x))" "6#/%!G,&*(,&\"\"\"F(*$-%\"SG6#%\"xG\"\"#F(F (,&F.F(*$-F+6#F-F.F(!\"\"-%\"KG6#F-F(F(*&-F+6#F-F(,(F(F(F-F(-F56#F-F3F 3F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "phi := x -> S(x)*K(x);\nD(phi)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGf*6#%\"xG6\"6$%)operatorG%&arro wGF(*&-%\"SG6#9$\"\"\"-%\"KGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&\"\"\"F&*$)-%\"SG6#%\"xG\"\"#F&F&F&,&F-F&F'F&!\"\"-%\"KGF+ F&F&*&F)F&,(F&F&F,F&F0F/F/F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([phi(x),D(phi)(x)],x=-2..4,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$!\"#\"\"!$ \"+L&[0V#!\"*7$$!+Q&3Y$>F-$\"+JGKqAF-7$$!+vq@p=F-$\"+%zNq6#F-7$$!+8.K7 =F-$\"+^!*>*)>F-7$$!+^NUbiUCF-$\"+$)331*)!#67$$!*hka I\"F-$\"+z%**oD%Fip7$$\"(XDn%F-$!+t//=8!#77$$\"*!y?#>\"F-$!+S'R_$HFip7 $$\"*4wY_#F-$!+'33$)>&Fip7$$\"*IOTq$F-$!+c7nZjFip7$$\"*4\">)*\\F-$!+04 @DnFip7$$\"*EP/B'F-$!+3-=biFip7$$\"*)o:;vF-$!+f>vE\\Fip7$$\"*%)[op)F-$ !+>.\"*oHFip7$$\"*i%Qq**F-$!+zAH#o(!#87$$\"+RIKH6F-$\"+6o,xPFip7$$\"+^ rZW7F-$\"+AZ+MyFip7$$\"+[n%)o8F-$\"+:g)\\H\"FV7$$\"+5FL(\\\"F-$\"+NU!Q !>FV7$$\"+e6.B;F-$\"+pvuyDFV7$$\"+p3lW_(zC# F-$\"+3yi5rFV7$$\"+b*=jP#F-$\"+]$)H%G)FV7$$\"+4/3(\\#F-$\"+#>`NY*FV7$$ \"+C4JBEF-$\"+7QJx5F-7$$\"+DVsYFF-$\"+(ogG@\"F-7$$\"+>n#f(GF-$\"+#f0EO \"F-7$$\"+!)RO+IF-$\"+S/G9:F-7$$\"+_!>w7$F-$\"+Dt!on\"F-7$$\"+*Q?QD$F- $\"+D5CX=F-7$$\"+5jypLF-$\"+G=C1?F-7$$\"+Ujp-NF-$\"+c#Hz>#F-7$$\"+gEd@ OF-$\"+O5qvBF-7$$\"+4'>$[PF-$\"+-8lrDF-7$$\"+6EjpQF-$\"+ZjBlFF-7$$\"\" %F*$\"+WJnzHF--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F]\\l-F$6$7S7$F($!+ 4L*R]#F-7$F4$!+&e)y\"H#F-7$F>$!+?V:9@F-7$FC$!+`FqA>F-7$FH$!+&RL$Rn&FV7$Fjr$\"*L!QJqFV7$F_s$\"+52 )3O\"FV7$Fds$\"+G7;a>FV7$Fis$\"+Hqp'e#FV7$F_t$\"+JaYRKFV7$Fdt$\"+'GWn! QFV7$Fit$\"+bbT?WFV7$F^u$\"+9k*o0&FV7$Fcu$\"+W^t#o&FV7$Fhu$\"+w[F\"H'F V7$F]v$\"+3hyppFV7$Fbv$\"+r\"e4e(FV7$Fgv$\"+6geL#)FV7$F\\w$\"+oIaB))FV 7$Faw$\"+m*=_Y*FV7$Ffw$\"+e!>k+\"F-7$F[x$\"+UZOo5F-7$F`x$\"+oA7G6F-7$F ex$\"+!e'o*=\"F-7$Fjx$\"+,0!zC\"F-7$F_y$\"+O)RiI\"F-7$Fdy$\"+&[IGO\"F- 7$Fiy$\"+&RtOT\"F-7$F^z$\"+`Paq9F-7$Fcz$\"+(RA,_\"F-7$Fhz$\"+d!Q;d\"F- 7$F][l$\"+)R['>;F-7$Fb[l$\"+o@%)p;F--Fg[l6&Fi[lF]\\lF]\\lFj[l-%+AXESLA BELSG6$Q\"x6\"Q!Fjel-%%VIEWG6$;F(Fb[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The minimum point on the graph of " }{XPPEDIT 18 0 "y=phi(x)" "6#/%\"yG-%$phiG6#%\"xG" } {TEXT -1 14 " occurs where " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "xmin := fsolve(D(phi)(x));\nymin := evalf(evalf(phi(xmin),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xmi nG$\"+:;y\"*[!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%yminG$!+4+BGn!# 6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "The coordinates of the minimum point are approximately (0.4891781615, -0. 06728230009). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "findmin(phi(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+:;y\"*[!#5$!+4+BGn!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 21 " We find the value of " }{XPPEDIT 18 0 "Int(exp(Y(x)),x=-4..4)" "6#-%$I ntG6$-%$expG6#-%\"YG6#%\"xG/F,;,$\"\"%!\"\"F0" }{TEXT -1 26 " (correc t to 10 digits). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 530 "f := x -> exp(Y(x)): # function\na := -4: # \+ lower limit of integral\nb := 4: # upper limit of integral\nclr := whe at: # color for shading\npp := plot([0,f(x)],x=a..b,adaptive=false,num points=20):\nu := op(1,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[po lygonplot]([seq([u[i],v[i],v[i+1],u[i+1]],i=1..19)],\n \+ color=clr,style=patchnogrid):\np2 := plot([[[a,0],[a,f (a)]],[[b,0],[b,f(b)]]],color=black):\np3 := plot(f(x),x=-4.5..4.5,thi ckness=2): # adjust plot range\nplots[display]([p1,p2,p3],labels=[`x`, `y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6(-%)POL YGONSG677&7$$!\"%\"\"!$F*F*7$F($\"+t?8f?!#57$$!+U!o%fN!\"*$\"+C:(*=AF/ 7$F1F+7&F6F07$$!+TN;wJF3$\"+n7*[Q#F/7$F9F+7&F=F87$$!+Tv4XFF3$\"+RI-7EF /7$F@F+7&FDF?7$$!+cT<6BF3$\"+&4h>!HF/7$FGF+7&FKFF7$$!+LJJz=F3$\"+mY(GG $F/7$FNF+7&FRFM7$$!+#oA*y9F3$\"+cD>oPF/7$FUF+7&FYFT7$$!+i:Mk5F3$\"+k3j 0XF/7$FfnF+7&FjnFen7$$!*v6eN'F3$\"+/(>%odF/7$$!+Z<\"eN'F/F+7&FaoF\\o7$ $!*Fe>3#F3$\"+aG6b\")F/7$$!+u#e>3#F/F+7&FjoFeo7$$\"*NDUJ#F3$\"+M39`7F3 7$$\"+Z`A9BF/F+7&FcpF^p7$$\"*G'Q'='F3$\"+`&=Or\"F37$$\"+uiQ'='F/F+7&F \\qFgp7$$\"+OAba5F3$\"+N`y3AF37$FaqF+7&FeqF`q7$$\"+[dD#\\\"F3$\"+l$\\s m#F37$$\"+ZdD#\\\"F3F+7&F\\rFgq7$$\"+'=kS\">F3$\"+lT5zIF37$$\"+&=kS\"> F3F+7&FerF`r7$$\"+dm5(H#F3$\"+%=YKV$F37$FjrF+7&F^sFir7$$\"+07e_FF3$\"+ 6=!\\$QF37$$\"+/7e_FF3F+7&FesF`s7$$\"+\"*yUQJF3$\"+MIohTF37$FjsF+7&F^t Fis7$$\"+9$fse$F3$\"+qj(*GXF37$FatF+7&FetF`t7$$\"\"%F*$\"+c]Tc[F37$Fht F+-%'COLOURG6&%$RGBG$\")#)eq%)!\")Fau$\")h>!\\(Fcu-%&STYLEG6#%,PATCHNO GRIDG-%'CURVESG6$7$F'F,-F^u6&F`uF*F*F*-F[v6$7$F\\uFgtF^v-F[v6%7S7$$!#X !\"\"$\"+Bat2>F/7$$!+7c#QI%F3$\"+i(zQ'>F/7$$!+F`8LTF3$\"+f01;?F/7$$!+ \\i$F3$\"+De5xBF/7$$!+UAy,IF3$\"+G@$3Z#F /7$$!+/AY6GF3$\"+2#oOd#F/7$$!+_Yp:EF3$\"+kc&4p#F/7$$!+'[iKW#F3$\"++(ec !GF/7$$!+3Z9\\AF3$\"+42.]HF/7$$!+i)HU0#F3$\"+h/j9JF/7$$!+#H$Rm=F3$\"+2 6>'H$F/7$$!+#p>ep\"F3$\"+!)[E&[$F/7$$!+#=\"*H\\\"F3$\"+r)Q\"[PF/7$$!+I )o6K\"F3$\"+%H0C,%F/7$$!+(e)H@6F3$\"++))*QQ%F/7$$!*b&zV%*F3$\"+Rre!z%F /7$$!*P8F](F3$\"+nOMb`F/7$$!*6WVl&F3$\"+_5BegF/7$$!*pkds$F3$\"+OsUTqF/ 7$$!*uEZ&>F3$\"+d>n`#)F/7$$!(1BW%F3$\"+,fnb**F/7$$\"*eX)R>F3$\"+&Qu)47 F37$$\"*Fdrm$F3$\"+*[gJT\"F37$$\"*@,F`&F3$\"+!G#>O;F37$$\"*l!**fuF3$\" +8\"fB'=F37$$\"*OnaM*F3$\"+***pk2#F37$$\"+.j(p6\"F3$\"+74UwAF37$$\"+MK `>8F3$\"+6Id!\\#F37$$\"+X'R:]\"F3$\"+MCgwEF37$$\"+Q.(ep\"F3$\"+a=FpGF3 7$$\"+GG'>(=F3$\"+=B6RIF37$$\"+K%yW1#F3$\"+J(z,A$F37$$\"+81iXAF3$\"+6J f'Q$F37$$\"+'Qm\\V#F3$\"+ll%ob$F37$$\"+)['3?EF3$\"+[m)*>PF37$$\"+y+*Q \"GF3$\"+1`e()QF37$$\"+qfa+IF3$\"+Fy$F3$\"+2*Hd?%F37$$ \"+$eI2Q$F3$\"+'4O:O%F37$$\"+l%zYb$F3$\"+%*3t-XF37$$\"+9X/aPF3$\"+dERi YF37$$\"+!**eB$RF3$\"+*)pO.[F37$$\"+8%zC7%F3$\"+Dd*=&\\F37$$\"+<*[WI%F 3$\"+\\-W#4&F37$$\"#XFiv$\"+;q\"=C&F3-F^u6&F`u$\"#5FivF+F+-%*THICKNESS G6#\"\"#-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;FgvFh elF\\gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 57 "An analytical expression can be found fo r this integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(exp(Y(x)),x=-4..4);\nvalue(%);\nevalf(eva lf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#-%\"YG 6#%\"xG/F,;!\"%\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\" \")F&*&,,*$)-%$expG6#-%\"YG6#\"\"%F2F&F&*(\"\"#F&F/F&F+F&F&*(F'F&F/F&) F,F4F&F&F&!\"\"*&F4F&F/F&F&F&F,!\"#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l-w&Q\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Alternatively, a value may be found by numerical int egration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "Int(exp(Y(x)),x=-4..4);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#-%\"YG6#%\"xG/F,;!\"%\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l-w&Q\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 22 "We find the value of " }{XPPEDIT 18 0 "Int(sP(x),x=0..7*Pi/3+sqrt (3)/2)" "6#-%$IntG6$-%#sPG6#%\"xG/F);\"\"!,&*(\"\"(\"\"\"%#PiGF0\"\"$! \"\"F0*&-%%sqrtG6#F2F0\"\"#F3F0" }{TEXT -1 26 " (correct to 10 digits ). 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pju&[RvFin$\"3M+I(*Gc!Q$pFin7$$\"3C+!42vmxu(Fin$\"3I+5*[+\"G_qFin7$$\" 3q**\\7G$4*>zFin$\"3#)**>'>$>karFin7$$\"3S**z_I<&G0)Fin$\"3')***pHSTrB (Fin7$$\"3>**p%e2SH9)Fin$\"3s**\\K-W4&H(Fin7$$\"3U+!fNP-x=)Fin$\"36+gd 8')eCtFin-%'COLOURG6&%$RGBG$\")=THv!\")Fj_uFj_u-%&STYLEG6#%%LINEG-F$6% F&-%&COLORG6&Fi_u$\"\")!\"\"\"\"!Ff`u-F^`u6#%&POINTG-%&TITLEG6#%?contr ibuting~evaluation~pointsG-%+AXESLABELSG6$Q!6\"Fdau-%'SYMBOLG6#%'CIRCL EG-%%VIEWG6$%(DEFAULTGF]bu" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Dnumber~of~function~evaluations~-->~G\"%H5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?abrK!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 22 "We find the va lue of " }{XPPEDIT 18 0 "Int(M(x),x = -1 .. 1);" "6#-%$IntG6$-%\"MG6# %\"xG/F);,$\"\"\"!\"\"F-" }{TEXT -1 26 " (correct to 10 digits). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 538 "f := x -> M(x): # function\na := -1: # lower limit of integral\nb := 1: # upper limit of integral\nclr := COLOR(RGB,1,.8,.9): # color f or shading\npp := plot([0,f(x)],x=a..b,adaptive=false,numpoints=20):\n u := op(1,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[polygonplot]([s eq([u[i],v[i],v[i+1],u[i+1]],i=1..19)],\n \+ color=clr,style=patchnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0], [b,f(b)]]],color=black):\np3 := plot(f(x),x=-1.2..1.7,thickness=2): # \+ adjust plot range\nplots[display]([p1,p2,p3],labels=[`x`,`y`]);" }} {PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6(-%)POLYGONSG677& 7$$!\"\"\"\"!$F*F*7$F($\"+?&o`E#!\"*7$$!+0,n)*))!#5$\"+\\0c)4#F/7$F1F+ 7&F6F07$$!+`)3/%zF3$\"+XOAf>F/7$F9F+7&F=F87$$!+_QuioF3$\"+^)[)3=F/7$$! +`QuioF3F+7&FDF?7$$!+*QNzx&F3$\"+z'3Tm\"F/7$FIF+7&FMFH7$$!+JGG)p%F3$\" +>5`E:F/7$$!+KGG)p%F3F+7&FTFO7$$!+0nI(p$F3$\"+C\"QYS\"F/7$FYF+7&FgnFX7 $$!+0R&3m#F3$\"+vc0%G\"F/7$FjnF+7&F^oFin7$$!+OH&*)e\"F3$\"+8SEl6F/7$$! +PH&*)e\"F3F+7&FeoF`o7$$!*o&*[?&F3$\"+,#GF0\"F/7$$!+%o&*[?&!#6F+7&F^pF io7$$\")Mc&y&F/$\"+Y^yH%*F37$$\"+oLc&y&FapF+7&FhpFcp7$$\"*d'fY:F/$\"+? *)p7&)F37$$\"+olfY:F3F+7&FaqF\\q7$$\"*f!QOEF/$\"+Mf#\\`(F37$$\"+*e!QOE F3F+7&FjqFeq7$$\"*PR1t$F/$\"+@yX5mF37$$\"+o$R1t$F3F+7&FcrF^r7$$\"*Yg^y %F/$\"+4u3tdF37$$\"+j/;&y%F3F+7&F\\sFgr7$$\"*kmFu&F/$\"+gKfd]F37$$\"+U mwUdF3F+7&FesF`s7$$\"*,`9)oF/$\"+]\"4=E%F37$$\"+6IX\")oF3F+7&F^tFis7$$ \"*tpg%yF/$\"+sN,MOF37$$\"+E(pg%yF3F+7&FgtFbt7$$\"*H[\"o*)F/$\"+SZ)o&H F37$$\"+%G[\"o*)F3F+7&F`uF[u7$$\"\"\"F*$\"+b0d%Q#F37$FeuF+-%&COLORG6&% $RGBGFfu$\"\")F)$\"\"*F)-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6$7$F'F,-%' COLOURG6&F]vF*F*F*-Fgv6$7$FiuFduFjv-Fgv6%7V7$$!#7F)$\"+8j*pe#F/7$$!+`# )yO6F/$\"+KWp#[#F/7$$!+]!)y\"3\"F/$\"+tP%RR#F/7$$!+pX$*>5F/$\"+BYL'H#F /7$$!+L3rw&*F3$\"+>XS+AF/7$$!+]_.d*)F3$\"+u/A2@F/7$$!+hm^#Q)F3$\"+NZ%G -#F/7$$!+Rfj(y(F3$\"+G;]P>F/7$$!+Y%4C<(F3$\"+<2Q^=F/7$$!+#*f:flF3$\"+h !yww\"F/7$$!+N%\\$GfF3$\"+g_y$o\"F/7$$!+yYts`F3$\"+RYs6;F/7$$!+dSCZZF3 $\"+K?jK:F/7$$!+7^=>TF3$\"+#za`X\"F/7$$!+UR$R^$F3$\"+M,*GQ\"F/7$$!+'*y IkHF3$\"+QIx=8F/7$$!+l$\\2J#F3$\"+V**fW7F/7$$!*c*4d\\.AF37$$\"+\"4]U4\"F/$\"+a?;/>F37$$\"+9zpc6F/$\"+.NM3;F37$$\"+!fUo@ \"F/$\"+1#G1M\"F37$$\"+4#\\$y7F/$\"+`Ej%3\"F37$$\"+@lMR8F/$\"+\")G.!\\ )Fap7$$\"+ \+ " 0 "" {MPLTEXT 1 0 42 "Int(M(x),x=-1..1);\nvalue(%);\nevalf[11](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"MG6#%\"xG/F);!\"\"\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&-%\"MG6#\"\"\"F(-%#lnG6#F%F( !\"\"F%F(-%&dilogGF+F(*&-F&6#F,F(-F*6#F0F(F(F0F,-F.F3F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\",a94u;#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, a value may be found by \+ numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "GKint(M(x),x=-1..1,info=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptive~10-21~node~Gauss-Kronrod~qu adratureG" }}{PARA 13 "" 1 "" {GLPLOT2D 403 303 303 {PLOTDATA 2 "6(-%' CURVESG6%777$$!3e**\\)4k8v&**!#=$\"3))**>4%e*zeA!#<7$$!3V**><&Gl!R(*F* $\"3?+!)\\k_>DAF-7$$!3u**H(*otT*H*F*$\"3w**z*G`D%e@F-7$$!3o****)omL1l) F*$\"3)***49(4\")>1#F-7$$!3!)***\\W!ea5yF*$\"38+]4,0vS>F-7$$!35++*Ho&4 %z'F*$\"3++?M1P\\*z\"F-7$$!3K+!GsW.ci&F*$\"3%)**>br*3Vk\"F-7$$!3H+?HTR &RL%F*$\"3%***>(**[S:[\"F-7$$!3%***4=!fO]%HF*$\"3)***H1tQb;8F-7$$!3#** *f\")*QV()[\"F*$\"31++$o)=Ya6F-7$$\"\"!Fgn$\"\"\"Fgn7$$\"3#***f\")*QV( )[\"F*$\"3)****HN1;ic)F*7$$\"3%***4=!fO]%HF*$\"3w**f[)R'QosF*7$$\"3H+? HTR&RL%F*$\"3K+?oxw*\\7'F*7$$\"3K+!GsW.ci&F*$\"3\")**>es_&G9&F*7$$\"35 ++*Ho&4%z'F*$\"3()**pOKnv?VF*7$$\"3!)***\\W!ea5yF*$\"3)***H+zIQcOF*7$$ \"3o****)omL1l)F*$\"3t**pL0TqUJF*7$$\"3u**H(*otT*H*F*$\"3w***>\"eT)yw# F*7$$\"3V**><&Gl!R(*F*$\"3$***H3nGtCDF*7$$\"3e**\\)4k8v&**F*$\"3'****) )QL\"=2CF*-%'COLOURG6&%$RGBG$\")=THv!\")F`rF`r-%&STYLEG6#%%LINEG-F$6%F &-%&COLORG6&F_r$\"\")!\"\"FgnF\\s-Fdr6#%&POINTG-%&TITLEG6#%?contributi ng~evaluation~pointsG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q!6\"F]t-%%V IEWG6$%(DEFAULTGFbt" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$% Dnumber~of~function~evaluations~-->~G\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+X\"4u;#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 22 "We find the va lue of " }{XPPEDIT 18 0 "Int(1/M(x),x = 1/2 .. 1);" "6#-%$IntG6$*&\" \"\"F'-%\"MG6#%\"xG!\"\"/F+;*&F'F'\"\"#F,F'" }{TEXT -1 26 " (correct \+ to 10 digits). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 544 "f := x -> 1/M(x): # function\na := -1: # lowe r limit of integral\nb := 1: # upper limit of integral\nclr := aquamar ine: # color for shading\npp := plot([0,f(x)],x=a..b,adaptive=false,nu mpoints=20):\nu := op(1,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[p olygonplot]([seq([u[i],v[i],v[i+1],u[i+1]],i=1..19)],\n \+ color=clr,style=patchnogrid):\np2 := plot([[[a,0],[a, f(a)]],[[b,0],[b,f(b)]]],color=black):\np3 := plot(f(x),x=-1.2..1.2,th ickness=2,color=brown): # adjust plot range\nplots[display]([p1,p2,p3] ,labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6(-%)POLYGONSG677&7$$!\"\"\"\"!$F*F*7$F($\"+BGH9W!#57$$!+ 0,n)*))F/$\"+i57$$!+*QNzx&F/$\"+6HA4gF/7$FH F+7&FLFG7$$!+JGG)p%F/$\"+%R+3b'F/7$$!+KGG)p%F/F+7&FSFN7$$!+0nI(p$F/$\" +$Qr#>rF/7$FXF+7&FfnFW7$$!+0R&3m#F/$\"+4V#yy(F/7$FinF+7&F]oFhn7$$!+OH& *)e\"F/$\"+(=YFep7$$\"+UmwUdF/F+7&FesF`s7$$\"*,`9)oFep$\"+ b8UYBFep7$$\"+6IX\")oF/F+7&F^tFis7$$\"*tpg%yFep$\"+&QyQ$Fep7$$\"+%G[\"o*)F/F+7&F`uF[u7 $$\"\"\"F*$\"+Dsi$>%Fep7$FeuF+-%'COLOURG6&%$RGBG$\")p:#R%!\")$\")`B)e) F`v$\")fqkdF`v-%&STYLEG6#%,PATCHNOGRIDG-%'CURVESG6$7$F'F,-F[v6&F]vF*F* F*-Fjv6$7$FiuFduF]w-Fjv6%7V7$$!#7F)$\"+in[lQF/7$$!+IooZ6Fep$\"+y\")>** RF/7$$!+?%p@5\"Fep$\"+w\\'47%F/7$$!+L.)40\"Fep$\"+LUKkUF/7$$!+5$>X***F /$\"+Se%fT%F/7$$!+XVo\"[*F/$\"+>SjuXF/7$$!+&o?i+*F/$\"+qXx*fF/$\"+&\\:q!f F/7$$!++j%zZ&F/$\"+H-%H:'F/7$$!+!y[q(\\F/$\"+PgL/kF/7$$!+Xe=AXF/$\"+CQ 'ek'F/7$$!+?)48)RF/$\"+)))R3&pF/7$$!+!)o6BNF/$\"+(=kbA(F/7$$!+l&H,*HF/ $\"+M(Hdc(F/7$$!+![X$=DF/$\"+2U%p)yF/7$$!+lNs+?F/$\"+$QTJE)F/7$$!*5Dy] \"Fep$\"+g\"omk)F/7$$!)DPN**Fep$\"+ilwv!*F/7$$!)Yg7_Fep$\"+>VT)\\*F/7$ $!(:Y=\"Fep$\"+lV;))**F/7$$\");#H<&Fep$\"+#z4Q0\"Fep7$$\")g3z(*Fep$\"+ qnZ06Fep7$$\"**pQv9Fep$\"+.@pl6Fep7$$\"*%3L*)>Fep$\"+IwEL7Fep7$$\"*jC@ \\#Fep$\"+&y,`I\"Fep7$$\"*[.'yHFep$\"+&fx7Q\"Fep7$$\"*Hb(=NFep$\"+&HMQ Z\"Fep7$$\"*s0T+%Fep$\"+gTOl:Fep7$$\"*!4KAXFep$\"+j/5t;Fep7$$\"*)3!>* \\Fep$\"+x'Q4y\"Fep7$$\"*#eF0bFep$\"+P7l6>Fep7$$\"*k@$))fFep$\"+;8y[?F ep7$$\"*qVK\\'Fep$\"+\"QV#4AFep7$$\"*I(*o)pFep$\"+lm*fQ#Fep7$$\"*)oq.v Fep$\"+hN1'f#Fep7$$\"*#fX,!)Fep$\"+MHQFGFep7$$\"*@w/^)Fep$\"+v'R'*4$Fe p7$$\"*c\"G:!*Fep$\"+C-L8MFep7$$\"*CX\"z%*Fep$\"+!oM*[PFep7$$\"+P&y5+ \"Fep$\"+#*Gs.UFep7$$\"+k!H'[5Fep$\"+P![7p%Fep7$$\"+a%yR2\"Fep$\"+EZ() *)\\Fep7$$\"+WyK*4\"Fep$\"+4pu?`Fep7$$\"+W/fB6Fep$\"+@RQscFep7$$\"+WI& y9\"Fep$\"+3dvjgFep7$$\"+Al#R<\"Fep$\"+.6HOlFep7$$\"#7F)$\"+!QKF2(Fep- F[v6&F]v$\")#)eqkF`v$\"))eqk\"F`vF]hl-%*THICKNESSG6#\"\"#-%+AXESLABELS G6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;FfwFeglF[il" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "An analytical expression can be found for this integral. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(1/M(x),x=-1..1);\nvalue(%);\nevalf[11](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'-%\"MG6#%\"xG!\"\"/F+;F,F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"\"#F&*$)-%#lnG6#-%\"MG6#F &F'F&F&F&-%&dilogGF,F&*&#F&F'F&*$)-F+6#-F.6#!\"\"F'F&F&F:-F1F7F:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\",q`3Kp#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, a value ma y be found by numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "GKint(1/M(x),x=-1..1,info= 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptive~10-21~node~Gauss-Kro nrod~quadratureG" }}{PARA 13 "" 1 "" {GLPLOT2D 403 303 303 {PLOTDATA 2 "6(-%'CURVESG6%777$$!3e**\\)4k8v&**!#=$\"3%)**HUk-8FWF*7$$!3V**><&Gl !R(*F*$\"3!)**4:Lw)R\\%F*7$$!3u**H(*otT*H*F*$\"37+!y1s1Ij%F*7$$!3o**** )omL1l)F*$\"33+q>#)\\q\\[F*7$$!3!)***\\W!ea5yF*$\"3y***)3Uek_^F*7$$!35 ++*Ho&4%z'F*$\"3;+]&*H'=rb&F*7$$!3K+!GsW.ci&F*$\"3_+q(\\4#e\"3'F*7$$!3 H+?HTR&RL%F*$\"3y**HF`6t\\nF*7$$!3%***4=!fO]%HF*$\"3]+Sm#R(e&f(F*7$$!3 #***f\")*QV()[\"F*$\"3m**4Z%HW?m)F*7$$\"\"!Ffn$\"\"\"Ffn7$$\"3#***f\") *QV()[\"F*$\"3-+SiylPn6!#<7$$\"3%***4=!fO]%HF*$\"3!***zn'=@eP\"F^o7$$ \"3H+?HTR&RL%F*$\"3(***>D!o`Ej\"F^o7$$\"3K+!GsW.ci&F*$\"33++d^^WW>F^o7 $$\"35++*Ho&4%z'F*$\"3:+!43S4WJ#F^o7$$\"3!)***\\W!ea5yF*$\"3x**pY*=V\\ t#F^o7$$\"3o****)omL1l)F*$\"3%***>/CJ(>=$F^o7$$\"3u**H(*otT*H*F*$\"3B+ +%H'z'Gh$F^o7$$\"3V**><&Gl!R(*F*$\"3u**\\)[2:3'RF^o7$$\"3e**\\)4k8v&** F*$\"3!)**ziniBaTF^o-%'COLOURG6&%$RGBG$\")=THv!\")F`rF`r-%&STYLEG6#%%L INEG-F$6%F&-%&COLORG6&F_r$\"\")!\"\"FfnF\\s-Fdr6#%&POINTG-%&TITLEG6#%? contributing~evaluation~pointsG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q! 6\"F]t-%%VIEWG6$%(DEFAULTGFbt" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P&3Kp#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 6" }{TEXT 265 25 " .. Maclaurin series for \+ " }{XPPEDIT 18 0 "M(x)" "6#-%\"MG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-x+x^2/4-x^3/72+x^4/576+31/86400*x^ 5+149/1036800*x^6+18037/304819200*x^7+125603/4877107200*x^8+15378773/1 316818944000*x^9+` . . . `;" "6#,8\"\"\"F$%\"xG!\"\"*&F%\"\"#\"\"%F&F$ *&F%\"\"$\"#sF&F&*&F%F)\"$w&F&F$*(\"#JF$\"&+k)F&F%\"\"&F$*(\"$\\\"F$\" (+o.\"F&F%\"\"'F$*(\"&P!=F$\"*+#>[IF&F%\"\"(F$*(\"'.c7F$\"++s5x[F&F%\" \")F$*(\")t(y`\"F$\".+S%*=oJ\"F&F%\"\"*F$%(~.~.~.~GF$" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "series(M(x),x,10);\np := unapply(convert(%,polynom),x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"\"\"!!\"\"F%#F%\"\" %\"\"##F'\"#s\"\"$#F%\"$w&F)#\"#J\"&+k)\"\"&#\"$\\\"\"(+o.\"\"\"'#\"&P !=\"*+#>[I\"\"(#\"'.c7\"++s5x[\"\")#\")t(y`\"\".+S%*=oJ\"\"\"*-%\"OG6# F%\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,6\"\"\"F-9$!\"\"*&#F-\"\"%F-*$)F.\"\"#F-F-F-*&#F-\"#sF-* $)F.\"\"$F-F-F/*&#F-\"$w&F-*$)F.F2F-F-F-*&#\"#J\"&+k)F-*$)F.\"\"&F-F-F -*&#\"$\\\"\"(+o.\"F-*$)F.\"\"'F-F-F-*&#\"&P!=\"*+#>[IF-*$)F.\"\"(F-F- F-*&#\"'.c7\"++s5x[F-*$)F.\"\")F-F-F-*&#\")t(y`\"\".+S%*=oJ\"F-*$)F.\" \"*F-F-F-F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "plot([M(x),p(x)],x=-5..4,y=-1..11,color=[red,b lue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 487 487 487 {PLOTDATA 2 "6&-%'CUR VESG6$7N7$$!\"&\"\"!$\"+Huqq9!\")7$$!+7c#Q![!\"*$\"+,9/t8F-7$$!+F`8LYF 1$\"+l4#=H\"F-7$$!+\\i\"*\\F17$$!+#p>e>#F1$\"+U\"**)yXF17$$!+#=\"*H*>F 1$\"+)4gq6%F17$$!+I)o6#=F1$\"+sY;\\PF17$$!+(e)H@;F1$\"+C\"*>ZLF17$$!+b &zVW\"F1$\"+W]\"R,$F17$$!+P8F]7F1$\"+_8srEF17$$!+6WVl5F1$\"+(36zO#F17$ $!*pkds)F1$\"+yI-t?F17$$!*uEZ&pF1$\"+[dU@=F17$$!*1BW/&F1$\"+tK%*p:F17$ $!*Ua,1$F1$\"+&=R)H8F17$$!*tUGL\"F1$\"+m)ex8\"F17$$\")@,F`F1$\"+,APu%* !#57$$\"*l!**fCF1$\"+fxH*o(F`t7$$\"*OnaM%F1$\"+='**e6'F`t7$$\"*.j(phF1 $\"+n0@_ZF`t7$$\"*MK`>)F1$\"+@38\"F1$\"+tj/K9F`t7$$\"+GG'>P\"F1$\"+sx-2t!#67$$\"+I1Ao9F1$\"+98,ZT Fdv7$$\"+K%yWc\"F1$\"+wyKX:Fdv7$$\"+/77(e\"F1$\"+*QXC.\"Fdv7$$\"+xRw4; F1$\"+\"e0ao&!#77$$\"+k`3@;F1$\"+2Sc\"f$Fiw7$$\"+]nSK;F1$\"+)4$G'p\"Fi w7$$\"+Vu1Q;F1$\"+!Hre])!#87$$\"+O\"GPk\"F1$\"+f\"pF?\"Fix7$%*undefine dGF`y-%'COLOURG6&%$RGBG$\"*++++\"F-$F*F*Fgy-F$6$7eo7$F($!3Ye;RGcYx6!#< 7$$!3))*\\i:?ya(\\F^z$!3lPRg=-\\?f!#=7$$!3u**\\7.k&4&\\F^z$!3VtW'z`&QG M!#>7$$!3h*\\(o/YVE\\F^z$\"3k1]nCpEo\\Fdz7$$!3[***\\i!G\">!\\F^z$\"3r* HT(o!\\A+\"F^z7$$!3A**\\P4#pG&[F^z$\"3V-,jbyuR>F^z7$$!3%)****\\7c#Q![F ^z$\"3s')GERcO&y#F^z7$$!3%*\\P4TI:hZF^z$\"3=v!Gs&zh^MF^z7$$!3,+vop/[=Z F^z$\"3#)*)eu9rcdSF^z7$$!35]7G)*y!en%F^z$\"3^$=DnR_rg%F^z7$$!3=+](oKNJ j%F^z$\"3%pz8@@'4/^F^z7$$!3]]PMdb9&e%F^z$\"3-3A0)**QWg&F^z7$$!3&**\\7y ybr`%F^z$\"3wYD0D#[t/'F^z7$$!3S\\7G=g;*[%F^z$\"3N'f*QO$GsV'F^z7$$!3s** *\\([i#[mRF^z$\"3,[8[bcCH#)F^z7$$!3D+++vu!>#RF^z$\"3/3D&*G$[ \"R#)F^z7$$!3++](ovKt(QF^z$\"34nea].\\M#)F^z7$$!31+]PzO-&y$F^z$\"3spgm (H3T=)F^z7$$!35+](=g9Fp$F^z$\"39i0&*GnD(3)F^z7$$!3E+](=U[sf$F^z$\"3+>) \\S==x%zF^z7$$!3)***\\(=C#y,NF^z$\"3Y:x@/ZXwxF^z7$$!3;+]P/AY6LF^z$\"3[ \"ei8\\VuO(F^z7$$!3x****\\_Yp:JF^z$\"3'G3OQ7)H*)oF^z7$$!34++D'[iK%HF^z $\"3s\")p.b3yXkF^z7$$!3u****\\2Z9\\FF^z$\"3%*p$oCG!3TfF^z7$$!3=++]i)HU b#F^z$\"3O,/_Om8UaF^z7$$!3u****\\#H$RmBF^z$\"31Ix9'Q1q(\\F^z7$$!39+](= p>e>#F^z$\"3eU&pR)=#>d%F^z7$$!3'*****\\#=\"*H*>F^z$\"3YjPU0GG9TF^z7$$! 3'*******H)o6#=F^z$\"3YiIp[#))zu$F^z7$$!31+](oe)H@;F^z$\"3ekFYK;\"oM$F ^z7$$!3E+++b&zVW\"F^z$\"3O+4]npy8IF^z7$$!3e**\\(oLr-D\"F^z$\"3+#>Q8O*o rEF^z7$$!3c**\\i5WVl5F^z$\"3K.r)4D/zO#F^z7$$!3z#***\\(okds)Fdz$\"3q7F? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 537 "f := x -> M(cos(x)): # function\na := 0: # lower limit of integral\nb := 2*Pi: # upper limit of integral\nclr := wheat: # color for shading\npp := plot([0,f(x)],x =a..b,adaptive=false,numpoints=40):\nu := op(1,op(1,pp)): v := op(1,op (2,pp)):\np1 := plots[polygonplot]([seq([u[i],v[i],v[i+1],u[i+1]],i=1. .39)],\n color=clr,style=patchnogrid): \np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\np3 := \+ plot(f(x),x=0..7,thickness=2,color=red): # adjust plot range\nplots[di splay]([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6(-%)POLYGONSG6K7&7$$\"\"!F)F(7$F($\"5$>TazYb qXQ#!#?7$$\"5%4o)y^xog&o\"F-$\"5[Tl)Gb=;.Y#F-7$F/F(7&F3F.7$$\"5eL-_v3W C_JF-$\"5#\\(\\3()Gv$=l#F-7$F6F(7&F:F57$$\"5.&o`iMqJ;![F-$\"5#)Gb%>b10 V,$F-7$F=F(7&FAF<7$$\"5W**yb\"y?_>Y'F-$\"5Qv#QB+-Aya$F-7$FDF(7&FHFC7$$ \"5ss=Sj)\\\"Q9\")F-$\"5()=:>HE%=(fUF-7$FKF(7&FOFJ7$$\"5nCqa*o(\\RY'*F 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45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "An estimate for the value of this integral may be found by numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "GKint(M(sin(x)),x=0..2*Pi,in fo=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Madaptive~10-21~node~Gauss- Kronrod~quadratureG" }}{PARA 13 "" 1 "" {GLPLOT2D 412 358 358 {PLOTDATA 2 "6(-%'CURVESG6%7L7$$\"3M+++o;utm!#?$\"3d+!))GQuL$**!#=7$$ \"3C++!f\"Hv)4%!#>$\"3?+5a[vU%f*F-7$$\"3'****HMis/5\"F-$\"3h****))3)>< $*)F-7$$\"3$****\\&z'z&>@F-$\"3%****y1[Tc+)F-7$$\"3w***f!3m=RMF-$\"3%* **z\\QPN2pF-7$$\"3_++ADF#e.&F-$\"3R**>M)yj>u&F-7$$\"3a+!Rq.'GroF-$\"3! ****feCj0j%F-7$$\"34++ZOV?+*)F-$\"33+]36!#<$ \"3!***\\puG+oHF-7$$\"3)****o>1XpL\"FU$\"3>+5h]t%3`#F-7$$\"3/+]zEjzq:F U$\"3-++oa0d%Q#F-7$$\"3')**4i\"fZY!=FUFen7$$\"3-+!3Ef,M.#FU$\"3m**fpuG +oHF-7$$\"3;+I%*>Ad^AFU$\"3#)**fu&F-7$$\"3)***Ry#*Rn(z#FU$\"3X****\\QPN2p 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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+x8,tq!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "The cycloid as a function of one variable " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The \+ curve traced out by a point on the rim of a wheel of radius 1 rolling \+ along the " }{TEXT 264 1 "x" }{TEXT -1 28 " axis without slipping is a " }{TEXT 260 7 "cycloid" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 82 "If the point starts off at the origin, the parametric equations of the curve are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([x=t-sin(t),``],[y=1-cos(t),``])" "6#-%*PIECEWISEG6$7$/%\"xG, &%\"tG\"\"\"-%$sinG6#F*!\"\"%!G7$/%\"yG,&F+F+-%$cosG6#F*F/F0" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The parameter " }{TEXT 263 1 "t" }{TEXT -1 47 " can be eliminated \+ with the aid of the inverse " }{XPPEDIT 18 0 "h^(-1)*``(x)" "6#*&)%\"h G,$\"\"\"!\"\"F'-%!G6#%\"xGF'" }{TEXT -1 17 " of the function " } {XPPEDIT 18 0 "h(t)=t-sin(t)" "6#/-%\"hG6#%\"tG,&F'\"\"\"-%$sinG6#F'! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The equation of \+ the cycloid is then: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=1-cos(h^(-1)*``(x))" "6#/%\"yG,&\"\"\"F&-%$cosG6#*&)%\"hG,$F&! \"\"F&-%!G6#%\"xGF&F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 2 "sM" }{TEXT -1 38 " can be used for the in verse function " }{XPPEDIT 18 0 "h^(-1)*``(x)" "6#*&)%\"hG,$\"\"\"!\" \"F'-%!G6#%\"xGF'" }{TEXT -1 88 " in numerical calculations. For examp le, it can be used to plot the graph of the curve. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "c := x -> \+ 1-cos(sM(x));\nplot(c(x),x=-Pi..3*Pi,color=[red,magenta],numpoints=60, thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&\"\"\"F--%$cosG6#-%#sMG6#9$!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 816 152 152 {PLOTDATA 2 "6&-%'CURVESG6$7]q7$$!+ZE fTJ!\"*$\"\"#\"\"!7$$!+w6F*7$$!+/(\\(=HF*$\"+\">'y$*>F* 7$$!+yC!=#GF*$\"+<%*=()>F*7$$!+__&[s#F*$\"+m?@y>F*7$$!+g1!o]#F*$\"+J!* >\\>F*7$$!+i1I(G#F*$\"+&ePt!>F*7$$!+4R%)o?F*$\"+lr]_=F*7$$!+MlIm=F*$\" +p%[#*y\"F*7$$!+93fc;F*$\"+A]?5.*!#5$\"+!)*GkG\"F* 7$$!*_eD0)F*$\"+ULZ57F*7$$!+bZ-]pFgo$\"+k<1<6F*7$$!**4\\ZeF*$\"+Ej&Q, \"F*7$$!+?,VSZFgo$\"+2EJ#)*)Fgo7$$!*DpLj$F*$\"+!emAn(Fgo7$$!+q4^mDFgo$ \"+gk]4iFgo7$$!*p_'*\\\"F*$\"+&oU%QWFgo7$$!+GzC:5Fgo$\"+=U\"=Y$Fgo7$$! +];V3`!#6$\"+QQ*pF#Fgo7$$!+QyS')GFer$\"+fn>H:Fgo7$$!*DSQk%Fer$\"*SIUd% Fgo7$$\"+)yRw&>Fer$\"+7uu%=\"Fgo7$$\")OmzVF*$\"+J\\&)3?Fgo7$$\"+y/(R, \"Fgo$\"+09-fMFgo7$$\"+&fu**e\"Fgo$\"+rla0YFgo7$$\"+7(yf;#Fgo$\"+#fS.f &Fgo7$$\"*$G)>u#F*$\"+'*)>rY'Fgo7$$\"+5\\)yr$Fgo$\"+qozyxFgo7$$\"**py$ p%F*$\"+;\">0$*)Fgo7$$\"+:J**GeFgo$\"+F..75F*7$$\"*C*>kpF*$\"+W6K=6F*7 $$\"+5!\\!pzFgo$\"+!30P?\"F*7$$\"*y)*Q(*)F*$\"+$>(4#G\"F*7$$\"+qd)y6\" F*$\"+J#=MV\"F*7$$\"+,+&yK\"F*$\"+y$*Gb:F*7$$\"+^c#pa\"F*$\"+M$oHm\"F* 7$$\"+&[0\"[F*$\"+B/l@=F*7$$\"+Un]!>#F*$\"+O_ q%)=F*7$$\"+A&>nQ#F*$\"+gB!z#>F*7$$\"++Zj)f#F*$\"+W,#H'>F*7$$\"+9QcF*7$$\"+FNlCHF*$\"+'R6T*>F*7$$\"+TKuJIF*$\"+f7\\)*>F*7$$\"+ X!e`8$F*$\"+T^****>F*7$$\"+]G(*QKF*$\"+.W\"))*>F*7$$\"+C%>SN$F*$\"+XSN %*>F*7$$\"+)*f1pMF*$\"+b]c')>F*7$$\"+E\\\"en$F*$\"+!*36k>F*7$$\"+ONc'* QF*$\"+bM)y#>F*7$$\"+(e%f'4%F*$\"+9Xt$)=F*7$$\"+g8G:VF*$\"+%*\\_A=F*7$ $\"+r&[5_%F*$\"+6*y[,EFgo7$$\"+h/\\5jF*$\"+\"z%[u9Fgo7$$\"+\\)4e L'F*$\"+Wh9kAFgo7$$\"+B'[kQ'F*$\"+e!)o*\\$Fgo7$$\"+)R(3PkF*$\"+&zo<^%F go7$$\"+!Rq]a'F*$\"+PS:(G'Fgo7$$\"+#Q`Il'F*$\"+?_paxFgo7$$\"+$)oScnF*$ \"+d3@t*)Fgo7$$\"+$Qg(foF*$\"+;@w05F*7$$\"+k6_upF*$\"+clz86F*7$$\"+V>G *3(F*$\"+L@:67F*7$$\"+[jl)=(F*$\"+yW7)G\"F*7$$\"+`2.)G(F*$\"+u28f8F*7$ $\"+LCh8vF*$\"+;p=,:F*7$$\"+Lc?>xF*$\"+$pp2h\"F*7$$\"+$H#4WzF*$\"+k&y> r\"F*7$$\"+V98Q\")F*$\"+q$H`y\"F*7$$\"+$R>\"f$)F*$\"+c/]a=F*7$$\"+VGes &)F*$\"+g\"*z2>F*7$$\"+Lo!fy)F*$\"+5$Q&[>F*7$$\"+`gW)**)F*$\"+#Q$>x>F* 7$$\"+yy`+\"*F*$\"+C'Ho)>F*7$$\"+.(HE?*F*$\"+7\\#Q*>F*7$$\"+BQq8$*F*$ \"+?uX)*>F*7$$\"+UzxC%*F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F-F-F_jl-% +AXESLABELSG6$Q\"x6\"Q!Fdjl-%*THICKNESSG6#F,-%%VIEWG6$;$!+aEfTJF*$\"+i zxC%*F*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 116 "We can solve a root-finding problem such as that \+ of finding the point of intersection of the cycloid with the curve " } {XPPEDIT 18 0 "y=sqrt(9-2*x)" "6#/%\"yG-%%sqrtG6#,&\"\"*\"\"\"*&\"\"#F *%\"xGF*!\"\"" }{TEXT -1 2 ". 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" }} {PARA 0 "" 0 "" {TEXT -1 7 "Hence: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx=``(dy/dt)/``(dx/dt)" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&-%!G6#*&F%F&%#dtGF(F&-F+6#*&F'F&F.F(F(" }{XPPEDIT 18 0 "``=sin(t)/ (1-cos(t))" "6#/%!G*&-%$sinG6#%\"tG\"\"\",&F*F*-%$cosG6#F)!\"\"F/" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=s in(h^(-1)*``(x))/(1-cos(h^(-1)*``(x))" "6#/%!G*&-%$sinG6#*&)%\"hG,$\" \"\"!\"\"F--F$6#%\"xGF-F-,&F-F--%$cosG6#*&)F+,$F-F.F--F$6#F1F-F.F." } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Diff(cos(sM(x)),x)=diff(cos(sM(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%$cosG6#-%#sMG6#%\"xGF-,$*&-%$si nGF)\"\"\",&F2F2F'!\"\"F4F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The derivative of the cycloid function can be p lotted as follows. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 4 "mean" } {TEXT -1 21 " or average value of " }{XPPEDIT 18 0 "c(x)" "6#-%\"cG6#% \"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "1/Pi;" "6 #*&\"\"\"F$%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(c(x),x = 0 \+ .. Pi);" "6#-%$IntG6$-%\"cG6#%\"xG/F);\"\"!%#PiG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 54 "This average value may be computed using \+ the function " }{XPPEDIT 18 0 "sM(x)" "6#-%#sMG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c := x -> 1-cos(sM(x));\n1/Pi*Int(c(x),x=0..Pi);\nval ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,&\"\"\"F--%$cosG6#-%#sMG6#9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,&\"\"\"F(-%$cosG6#-%#sMG6#%\"xG!\"\"/F /;\"\"!%#PiG*$F4F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The integ ral can also be evaluated numerically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "1/Pi*Int(c(x),x=0..Pi); \nevalf(evalf(%,14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,& \"\"\"F(-%$cosG6#-%#sMG6#%\"xG!\"\"/F/;\"\"!%#PiG*$F4F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+++++:!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Alternatively, the integral " }{XPPEDIT 18 0 "Int(c(x),x = 0 .. Pi) = Int(1-cos(g^(-1)*``(x)),x = 0 .. Pi);" "6#/-%$IntG6$-%\" cG6#%\"xG/F*;\"\"!%#PiG-F%6$,&\"\"\"F2-%$cosG6#*&)%\"gG,$F2!\"\"F2-%!G 6#F*F2F:/F*;F-F." }{TEXT -1 62 " can be determined analytically by mea ns of the substitution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x=t-sin(t)" "6#/%\"xG,&%\"tG\"\"\"-%$sinG6#F&!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1-cos(g^(-1)*``(x)),x = 0 .. Pi); " "6#-%$IntG6$,&\"\"\"F'-%$cosG6#*&)%\"gG,$F'!\"\"F'-%!G6#%\"xGF'F//F3 ;\"\"!%#PiG" }{TEXT -1 14 " " }{XPPEDIT 18 0 "PIECEWISE([ x = g(t), `` = t-sin(t)],[g^(-1)*``(x) = t, ``],[dx/dt = g*`'`(t), `` \+ = 1-cos(t)],[`when `*x = 0, t = 0],[`when `*x = Pi, t = Pi]);" "6# -%*PIECEWISEG6'7$/%\"xG-%\"gG6#%\"tG/%!G,&F,\"\"\"-%$sinG6#F,!\"\"7$/* &)F*,$F0F4F0-F.6#F(F0F,F.7$/*&%#dxGF0%#dtGF4*&F*F0-%\"'G6#F,F0/F.,&F0F 0-%$cosG6#F,F47$/*&%(when~~~GF0F(F0\"\"!/F,FN7$/*&FMF0F(F0%#PiG/F,FS" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = Int((1-cos(t))*g*`'`(t),t = 0 .. Pi);" "6#/%!G-%$IntG6$*(,&\"\"\"F*- %$cosG6#%\"tG!\"\"F*%\"gGF*-%\"'G6#F.F*/F.;\"\"!%#PiG" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int((1-cos(t)) ^2,t = 0 .. Pi)" "6#/%!G-%$IntG6$*$,&\"\"\"F*-%$cosG6#%\"tG!\"\"\"\"#/ F.;\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "The ave rage value of " }{XPPEDIT 18 0 "1-cos(g^(-1)*``(x));" "6#,&\"\"\"F$-%$ cosG6#*&)%\"gG,$F$!\"\"F$-%!G6#%\"xGF$F," }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$%#PiG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1-cos(g^(-1)*``(x)),x = 0 .. Pi) = 1/Pi;" "6#/-%$IntG6$,&\"\"\"F(-%$cosG6#*&)%\"gG,$F(!\"\"F(-%!G6 #%\"xGF(F0/F4;\"\"!%#PiG*&F(F(F8F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "In t((1-cos(t))^2,t = 0 .. Pi)" "6#-%$IntG6$*$,&\"\"\"F(-%$cosG6#%\"tG!\" \"\"\"#/F,;\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "1/Pi*Int((1-cos(t))^2,t= 0..Pi);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$*$),& \"\"\"F*-%$cosG6#%\"tG!\"\"\"\"#F*/F.;\"\"!%#PiG*$F4F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Since the cycloid function " }{XPPEDIT 18 0 "c(x)" "6#-%\"cG6#%\"x G" }{TEXT -1 7 " is an " }{TEXT 260 4 "even" }{TEXT -1 35 " function, \+ its Fourier series is a " }{TEXT 260 13 "cosine series" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c+a[1]*cos(x)+a[ 2]*cos(2*x)+a[3]*cos(3*x)+` . . . `;" "6#,,%\"cG\"\"\"*&&%\"aG6#F%F%-% $cosG6#%\"xGF%F%*&&F(6#\"\"#F%-F+6#*&F1F%F-F%F%F%*&&F(6#\"\"$F%-F+6#*& F8F%F-F%F%F%%(~.~.~.~GF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "We have already determined the constant term " }{XPPEDIT 18 0 "c=3 /2" "6#/%\"cG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The coefficient " }{XPPEDIT 18 0 "a[k]" "6#&%\"aG6#%\" kG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "cos(k*x)" "6#-%$cosG6#*&%\"kG\" \"\"%\"xGF(" }{TEXT -1 14 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[k]=2/Pi" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#Pi G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(c(x)*cos(k*x),x=0..Pi)" "6 #-%$IntG6$*&-%\"cG6#%\"xG\"\"\"-%$cosG6#*&%\"kGF+F*F+F+/F*;\"\"!%#PiG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " } {XPPEDIT 18 0 "x=t-sin(t)" "6#/%\"xG,&%\"tG\"\"\"-%$sinG6#F&!\"\"" } {TEXT -1 18 " as before gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[k] = 2/Pi;" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1-cos(t))^2*cos(k*(t-sin(t))),t =0..Pi)" "6#-%$IntG6$*&,&\"\"\"F(-%$cosG6#%\"tG!\"\"\"\"#-F*6#*&%\"kGF (,&F,F(-%$sinG6#F,F-F(F(/F,;\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "For example, a numerical approximation for " } {XPPEDIT 18 0 "a[2]" "6#&%\"aG6#\"\"#" }{TEXT -1 23 " can be obtained \+ from: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[2]=2/Pi" "6#/&%\"aG6#\"\"#*&F'\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(c(x)*cos(2*x),x = 0 .. Pi);" "6#-%$IntG6$*&-%\"cG6#%\"xG\"\"\"-%$ cosG6#*&\"\"#F+F*F+F+/F*;\"\"!%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(eval f[14](2/Pi*NCint(c(x)*cos(2*x),x=0..Pi)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+\"z2*QA!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "Alternatively, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[2] := 2/Pi;" "6#>&%\"aG6#\"\"#*&F'\" \"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((1-cos(t))^2*cos(2 *(t-sin(t))),t = 0 .. Pi);" "6#-%$IntG6$*&,&\"\"\"F(-%$cosG6#%\"tG!\" \"\"\"#-F*6#*&F.F(,&F,F(-%$sinG6#F,F-F(F(/F,;\"\"!%#PiG" }{TEXT -1 2 " . " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "2/Pi*Int((1-cos(t))^2* cos(2*(t-sin(t))),t=0..Pi);\nevalf(evalf(%,14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$,$-%$IntG6$*&),&\"\"\"F+-%$cosG6#%\"tG!\"\"\"\"#F+-F- 6#,&*&F1F+F/F+F0*&F1F+-%$sinGF.F+F+F+/F/;\"\"!%#PiG*&F1F+F " 0 "" {MPLTEXT 1 0 97 "for k from 1 to 100 do\n a[k] := evalf(2/Pi*Int((1-cos(t))^2*cos(k*(t-sin(t))),t=0..Pi));\nen d do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The graph of the corre sponding truncated Fourier cosine series is plotted below. 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