{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 "Help Normal" -1 30 "Times" 1 12 0 0 0 0 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 "Red Emphasis" -1 259 "Times" 1 12 255 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 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 267 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emp hasis" -1 268 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 258 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 262 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 279 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 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 1 0 1 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 "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 Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 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 "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {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 25 "Half-range Fourier series" }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{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 52 "load Fourier series and Fourier transform procedures" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 268 9 "fourier .m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 13 " FourierSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command si milar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procd rs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 33 "Half-range sine and cosine series" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Suppose that we are given a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[0,L]" "6#7$\" \"!%\"LG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 23 " can be extended to an " }{TEXT 260 4 "eve n" }{TEXT -1 26 " function on the interval " }{XPPEDIT 18 0 "[-L,L]" " 6#7$,$%\"LG!\"\"F%" }{TEXT -1 32 " by reflecting its graph in the " } {TEXT 280 1 "y" }{TEXT -1 38 " axis.\nThis is equivalent to defining \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "-L<=x" "6#1,$%\"LG!\"\"%\"xG" }{XPPEDIT 18 0 "``<0" "6# 2%!G\"\"!" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x)=f(-x)" "6#/-%\"fG6# %\"xG-F%6#,$F'!\"\"" }{TEXT -1 2 ". 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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 358 269 269 {PLOTDATA 2 "62-%'CURVESG6&7S7$$\"\"!F)$ \"3/+++++++=!#<7$$\"3ELLL$e=HG)!#>$\"3KB.(=\"eG)z\"F,7$$\"3mmmTvT)*[:! #=$\"3e'G$*Rg8Sz\"F,7$$\"3ALL$31y%fBF6$\"3%HassgYhy\"F,7$$\"3aLL$eUW`< $F6$\"39HJ,dS+v(4Pr[u\"F,7$$\"3,++ve%\\'>bF6$\"3$oN,C\"zuD6\"F,$\"3.2>A&G\\9_\"F,7$$\"3?L$37C()R=\"F,$\"3-Y!y%*e<')[ \"F,7$$\"3smm;Xfip7F,$\"3%f[Sh0>$[9F,7$$\"3pmmmgJE,:F,$\"3KpZiq\"R ZL\"F,7$$\"3!***\\()o(=Ke\"F,$\"3/&G3(Q*)y$H\"F,7$$\"33+]iE5Eh;F,$\"3' *Q'G2KH[D\"F,7$$\"3tmmTN**oU)>F,$\"3 <8WK_s=+6F,7$$\"3++]7Hb$[0#F,$\"3!*4\"=Y#4Kn5F,7$$\"3=LL3SHgL@F,$\"3gc >#**)*=K.\"F,7$$\"3\"******HQx\\@#F,$\"3![*\\0/4G(***F67$$\"3c***\\(*R 'e%H#F,$\"3Y)f&*4O`))o*F67$$\"3++](o@7;P#F,$\"3x46>ReAQ$)F67$$\"3Qm;aD4:[GF,$\"3_nm$)Q)HP@)F67$$\"33 +]P=p4GHF,$\"3)>))Q7!*=N6)F67$$\"3Umm;R(ei+$F,$\"3%>D9%G$=d/)F67$$\"3c **\\(=#p3)3$F,$\"3U.;)4Xbr+)F67$$\"3OLLL(=(*o;$F,$\"32MK:82g,!)F67$$\" 3MLLem?\\ZKF,$\"3\"4Xi?_5!G!)F67$$\"3cm;az&>uK$F,$\"3E$H>z(43'3)F67$$ \"37+++jY'3S$F,$\"3M9?pGc6n\")F67$$\"3emm\"p,T][$F,$\"3%o$4Y5Y+#H)F67$ $\"3%HLL8NH.c$F,$\"3EI=p=L)>V)F67$$\"3\"***\\()=CgSOF,$\"3Qs[>Bcr4')F6 7$$\"3t**\\7()RV(4Pr[u\"F,7$$!3,++ve%\\'>bF6$!3$oN,C\"zuD6\"F,$!3.2>A&G\\9_\"F,7$$!3?L$37C()R=\"F,$!3-Y!y%*e<')[\"F,7 $$!3smm;Xfip7F,$!3%f[Sh0>$[9F,7$$!3pmmmgJE,:F,$!3KpZiq\"RZL\"F,7$$!3!* **\\()o(=Ke\"F,$!3/&G3(Q*)y$H\"F,7$$!33+]iE5Eh;F,$!3'*Q'G2KH[D\"F,7$$! 3tmmTN**oU)>F,$!3<8WK_s=+6F,7$$!3++]7Hb$[0 #F,$!3!*4\"=Y#4Kn5F,7$$!3=LL3SHgL@F,$!3gc>#**)*=K.\"F,7$$!3\"******HQx \\@#F,$!3![*\\0/4G(***F67$$!3c***\\(*R'e%H#F,$!3Y)f&*4O`))o*F67$$!3++] (o@7;P#F,$!3x46>ReAQ$)F67$$!3Qm;aD4:[GF, $!3_nm$)Q)HP@)F67$$!33+]P=p4GHF,$!3)>))Q7!*=N6)F67$$!3Umm;R(ei+$F,$!3% >D9%G$=d/)F67$$!3c**\\(=#p3)3$F,$!3U.;)4Xbr+)F67$$!3OLLL(=(*o;$F,$!32M K:82g,!)F67$$!3MLLem?\\ZKF,$!3\"4Xi?_5!G!)F67$$!3cm;az&>uK$F,$!3E$H>z( 43'3)F67$$!37+++jY'3S$F,$!3M9?pGc6n\")F67$$!3emm\"p,T][$F,$!3%o$4Y5Y+# H)F67$$!3%HLL8NH.c$F,$!3EI=p=L)>V)F67$$!3\"***\\()=CgSOF,$!3Qs[>Bcr4') F67$$!3t**\\7()RVF'F0F'F1F'F2F3F'F'%(~.~.~.~GF'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 10 " where " }{XPPEDIT 18 0 "c = 1/L;" "6#/%\"cG*&\"\"\"F& %\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. L);" "6# -%$IntG6$-%\"fG6#%\"xG/F);\"\"!%\"LG" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 5 "and " }{XPPEDIT 18 0 "a[k] = 2/L;" "6#/&%\"aG6#%\"kG *&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x)*cos (k*Pi*x/L),x = 0 .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$cosG6#**% \"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;\"\"!F2" }{TEXT -1 3 ". " }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 31 "half-range Fourier sine se ries " }{TEXT -1 14 "of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[0,L] " "6#7$\"\"!%\"LG" }{TEXT -1 4 " is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = Sum(b[k]*sin(k*Pi*x/L),k = 1 .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$*&&%\"bG6#%\"kG\"\"\"-%$sinG6#**F/F0%#PiGF0F 'F0%\"LG!\"\"F0/F/;F0%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ b[1]*sin(Pi*x/L)+b[2]*sin(2*Pi*x/L)+b[3]*sin(3*Pi*x/L)+` . . . `;" "6# /%!G,**&&%\"bG6#\"\"\"F*-%$sinG6#*(%#PiGF*%\"xGF*%\"LG!\"\"F*F**&&F(6# \"\"#F*-F,6#**F6F*F/F*F0F*F1F2F*F**&&F(6#\"\"$F*-F,6#**F=F*F/F*F0F*F1F 2F*F*%(~.~.~.~GF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 9 " where " }{XPPEDIT 18 0 "b[k] = 2/L;" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"%\"LG!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*Pi*x/L),x = 0 .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG \"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;\"\"!F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT 264 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x)=x^2, 0 <=x" "6$/-% \"fG6#%\"xG*$F'\"\"#1\"\"!F'" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\" " }{TEXT -1 7 ", find " }}{PARA 0 "" 0 "" {TEXT -1 25 " (a) the cosi ne series," }}{PARA 0 "" 0 "" {TEXT -1 23 " (b) the sine series," }} {PARA 0 "" 0 "" {TEXT -1 29 " (c) the Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 29 "(a) The constant term in the " } {TEXT 268 13 "cosine series" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c = Int(x^2,x = 0 .. 1);" "6#/%\"cG-%$ IntG6$*$%\"xG\"\"#/F);\"\"!\"\"\"" }{TEXT -1 4 " = " }{XPPEDIT 18 0 " x^3/3" "6#*&%\"xG\"\"$F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWI SE([1 , ``],[0 ,`` ]) = 1/3" "6#/-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F)*& F(F(\"\"$!\"\"" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "The coefficients of the cosine terms in half-range cosine series" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 51 "The coefficients of the cosine terms are given by: " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "a[k] = 2*Int(f(x)*cos*k* Pi*x,x = 0 .. 1);" "6#/&%\"aG6#%\"kG*&\"\"#\"\"\"-%$IntG6$*,-%\"fG6#% \"xGF*%$cosGF*F'F*%#PiGF*F2F*/F2;\"\"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Int(x^2*cos*k*Pi*x,x = 0 .. 1);" "6#/%!G*&\"\"#\"\"\"-%$IntG6$*,%\"xGF&%$cosGF'%\"kGF'%#Pi GF'F,F'/F,;\"\"!F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "No w " }{XPPEDIT 18 0 "Int(x^2*cos*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/- %$IntG6$*,%\"xG\"\"#%$cosG\"\"\"%\"kGF+%#PiGF+F(F+F(-F%6$*&%\"uGF+-%!G 6#*&%#dvGF+%#dxG!\"\"F+F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = \+ x^2;" "6#/%\"uG*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = s in*k*Pi*x/(k*Pi);" "6#/%\"vG*,%$sinG\"\"\"%\"kGF'%#PiGF'%\"xGF'*&F(F'F )F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using \+ the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)% #dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 2*``(x^2*sin*k*Pi*x/(k*Pi));" "6#/&%\"aG6#%\"kG*&\"\"#\" \"\"-%!G6#*.%\"xGF)%$sinGF*F'F*%#PiGF*F/F**&F'F*F1F*!\"\"F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%* PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Int(2*x*``(sin*k*Pi*x/(k*Pi)),x = 0 .. 1);" "6#,$*&\"\"#\"\"\"-% $IntG6$*(F%F&%\"xGF&-%!G6#*,%$sinGF&%\"kGF&%#PiGF&F+F&*&F1F&F2F&!\"\"F &/F+;\"\"!F&F&F4" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "0-4/(k*Pi);" "6#,&\"\"!\"\"\"*&\"\"%F%*&%\"kGF%%#PiG F%!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*sin*k*Pi,x = 0 .. 1); " "6#-%$IntG6$**%\"xG\"\"\"%$sinGF(%\"kGF(%#PiGF(/F';\"\"!F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x*sin*k*Pi*x,x) = Int(u*``(dv/dx),x); " "6#/-%$IntG6$*,%\"xG\"\"\"%$sinGF)%\"kGF)%#PiGF)F(F)F(-F%6$*&%\"uGF) -%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 " u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k *Pi*x/(k*Pi);" "6#/%\"vG,$*,%$cosG\"\"\"%\"kGF(%#PiGF(%\"xGF(*&F)F(F*F (!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using \+ the integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)% #dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a[k] = -4/( k*Pi)" "6#/&%\"aG6#%\"kG,$*&\"\"%\"\"\"*&F'F+%#PiGF+!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-x*cos*k*Pi*x/(k*Pi));" "6#-%!G6#,$*.%\"xG \"\"\"%$cosGF)%\"kGF)%#PiGF)F(F)*&F+F)F,F)!\"\"F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6 %7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 "+ " }{XPPEDIT 18 0 "4/(k*Pi) " "6#*&\"\"%\"\"\"*&%\"kGF%%#PiGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-cos*k*Pi*x/(k*Pi),x = 0 .. 1);" "6#-%$IntG6$,$*,%$cosG\"\"\"% \"kGF)%#PiGF)%\"xGF)*&F*F)F+F)!\"\"F./F,;\"\"!F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-4/(k*Pi)" "6#/%!G,$*&\"\"%\"\"\"*&%\"kGF(%#PiGF(!\" \"F," }{XPPEDIT 18 0 "``(-x*cos*k*Pi*x/(k*Pi))+4/(k*Pi)" "6#,&-%!G6#,$ *.%\"xG\"\"\"%$cosGF*%\"kGF*%#PiGF*F)F**&F,F*F-F*!\"\"F/F**&\"\"%F**&F ,F*F-F*F/F*" }{XPPEDIT 18 0 "``(-sin*k*Pi*x/(k^2*Pi^2));" "6#-%!G6#,$* ,%$sinG\"\"\"%\"kGF)%#PiGF)%\"xGF)*&F*\"\"#F+F.!\"\"F/" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWIS EG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=`` (4/(k^2*Pi^2))*cos*k*Pi-0" "6#/%!G,&**-F$6#*&\"\"%\"\"\"*&%\"kG\"\"#%# PiGF.!\"\"F+%$cosGF+F-F+F/F+F+\"\"!F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(4/(k^2*Pi^2))*(-1)^k;" "6#/%!G*&-F$6#*&\"\"%\"\"\"*&%\"kG\"\"#%# PiGF-!\"\"F*),$F*F/F,F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([ 4/(k^2*Pi^2), `if k is even`],[-4/(k^2*Pi^2), `if k is odd`]);" "6#/%! G-%*PIECEWISEG6$7$*&\"\"%\"\"\"*&%\"kG\"\"#%#PiGF.!\"\"%-if~k~is~evenG 7$,$*&F*F+*&F-F.F/F.F0F0%,if~k~is~oddG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [a[k], `|`, -4/Pi^2, 1/(Pi^2 ), -4/9/Pi^2, 1/(4*Pi^2), -4/25/Pi^2, 1/(9*Pi^2), -4/49/Pi^2, 1/(16*Pi ^2)]])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"' \"\"(\"\")7,&%\"aG6#F(F),$*&F-F**$%#PiGF+!\"\"F:*&F*F**$F9F+F:,$*(F-F* \"\"*F:*$F9F+F:F:*&F*F**&F-F**$F9F+F*F:,$*(F-F*\"#DF:*$F9F+F:F:*&F*F** &F?F**$F9F+F*F:,$*(F-F*\"#\\F:*$F9F+F:F:*&F*F**&\"#;F**$F9F+F*F:" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "interface(showassumed=0): k := 'k': assume(k,in teger):\na[k]=2*Int(x^2*cos(k*Pi*x),x=0..1);\n``=value(rhs(%));\naa := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],[a[k],`|` ,seq(aa(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%#k |irG,$*&\"\"#\"\"\"-%$IntG6$*&)%\"xGF*F+-%$cosG6#*(F'F+%#PiGF+F1F+F+/F 1;\"\"!F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"%\"\"\"%# k|irG!\"#%#PiGF*)!\"\"F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7$7,%#k|irG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&% \"aG6#F(F),$*&F-F*%#PiG!\"#!\"\"*&F*F**$)F8F+F*F:,$*(F-F*\"\"*F:F8F9F: ,$*&F*F**&F-F*F=F*F:F*,$*(F-F*\"#DF:F8F9F:,$*&F*F**&F@F*F=F*F:F*,$*(F- F*\"#\\F:F8F9F:,$*&F*F**&\"#;F*F=F*F:F*Q(pprint06\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The Fouri er half-range cosine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x) = 1/3;" "6#/-%\"FG6#%\"xG*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(``(4*(-1)^k/(k^2*Pi^2))*cos*k*Pi*x,k = 1 .. \+ infinity);" "6#-%$SumG6$*,-%!G6#*(\"\"%\"\"\"),$F,!\"\"%\"kGF,*&F0\"\" #%#PiGF2F/F,%$cosGF,F0F,F3F,%\"xGF,/F0;F,%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/3-4*cos*Pi*x/(Pi^2)+cos*2*Pi*x/(Pi^2)-4*cos*3*Pi *x/(9*Pi^2)+cos*4*Pi*x/(4*Pi^2)-4*cos*5*Pi*x/(25*Pi^2)+` . . .`;" "6#/ %!G,0*&\"\"\"F'\"\"$!\"\"F'*,\"\"%F'%$cosGF'%#PiGF'%\"xGF'*$F-\"\"#F)F )*,F,F'F0F'F-F'F.F'*$F-F0F)F'*.F+F'F,F'F(F'F-F'F.F'*&\"\"*F'*$F-F0F'F) F)*,F,F'F+F'F-F'F.F'*&F+F'*$F-F0F'F)F'*.F+F'F,F'\"\"&F'F-F'F.F'*&\"#DF '*$F-F0F'F)F)%'~.~.~.GF'" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The following pictures compare graphs of some truncated series with t he graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "f := x -> x^2:\nf_ := x -> f(x-2*floor((x+1)/2) ):\nFS := (x,n) -> 1/3+sum(4*(-1)^k/(k^2*Pi^2)*cos(k*Pi*x),k=1..n);\np lot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)],x=-1..3,\n color =[black,red,blue,green,magenta,coral],linestyle=[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF), &#\"\"\"\"\"$F/-%$SumG6$,$**)!\"\"%\"kGF/F8!\"#%#PiGF9-%$cosG6#*(F8F/F :F/9$F/F/\"\"%/F8;F/9%F/F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 502 241 241 {PLOTDATA 2 "6*-%'CURVESG6%7]p7$$!\"\"\"\"!$\"\"\"F*7$$!3ommm;p0k& *!#=$\"3c]R_q%=r9*F07$$!3PLLLLQ6G\"*F0$\"3dO\"Ha@YAL)F07$$!3a++](3-)[( )F0$\"3iOWi'z`Tl(F07$$!3immmT.\\p$)F0$\"3'=;Dz&o$[+(F07$$!3`++]7Y\"H%z F0$\"30.m9a#*)*3jF07$$!3LLLL$))Qj^(F0$\"3ee3\"4-N&\\cF07$$!3#RLLL`Np3( F0$\"3)HEi`_lC-&F07$$!3ULLL$=Kvl'F0$\"3!*4>@xMFKWF07$$!3'*****\\7(p,B' F0$\"3;L_lk9]\")QF07$$!3hnmmTs!G!eF0$\"3u6RR)=dsO$F07$$!3i+++Dve1aF0$ \"3,j&[l')=J#HF07$$!3iLLL3yO5]F0$\"3>(Hyuby.^#F07$$!3i+++vE%)*=%F0$\"3 /;^7k\"yav\"F07$$!3)RLL$3WDTLF0$\"3#Qp?@5)R;6F07$$!3'4++]d(Q&\\#F0$\"3 G&Qk%\\\"fpA'!#>7$$!3:mmmm&4`i\"F0$\"3F'=)\\(=J;k#Fcp7$$!3p*****\\(p7U 7F0$\"3%*\\E-A%zGa\"Fcp7$$!3GKLLLQW*e)Fcp$\"3m+))fOX&yP(!#?7$$!3+qmm;a rvUFcp$\"3j.@VKUFcq7$$\"3Q(*******\\*3q)Fcp$\"3#e\\-, Qd0d(Fcq7$$\"3!)*******4/vG\"F0$\"3o4o^2omd;Fcp7$$\"3!********p=\\q\"F 0$\"3l*o4OxZn!HFcp7$$\"3_mmm\"fBIY#F0$\"3oll58_[mgFcp7$$\"3yKLLLO[kLF0 $\"3S'y'*=,v>8\"F07$$\"3.KLLL&Q\"GTF0$\"3ZZ\"R]x_Tq\"F07$$\"3+*****\\s ]k,&F0$\"3s,`j(yxk^#F07$$\"3Alm;HTg4aF0$\"3^.(HMo\"QEHF07$$\"3WJLLLvv- eF0$\"3%Gnl!*\\*>nLF07$$\"3:mm;H\"4TB'F0$\"3%p\"fVj;T')QF07$$\"3'3++]s gam'F0$\"3*Qv^wmOGW%F07$$\"3c+++]%4i2(F0$\"3^Jp-=SF2]F07$$\"3G+++v\"ep [(F0$\"3-M*>9Faag&F07$$\"30nm;z8`:zF0$\"3I;sX,PcliF07$$\"3#QLLLe/TM)F0 $\"3cM/wH\"3C'pF07$$\"3[LL$3dowt)F0$\"35jGP0_oMwF07$$\"39LLLeDBJ\"*F0$ \"3)3nOM!3%zL)F07$$\"3s****\\([Wdb*F0$\"3Aj'Q5FD78*F07$$\"3Immm;kD!)** F0$\"3=<;C9=bg**F07$$\"3[mmT+07U5!#<$\"3Yu,Ad8Lv\"*F07$$\"3Mmm;f`@'3\" Fgx$\"3cY!H#)pB+N)F07$$\"32LLLj+gC6Fgx$\"3?X+;\"\\]Km(F07$$\"3y****\\n Z)H;\"Fgx$\"3]%HqV*\\%f+(F07$$\"37LLe*HTW?\"Fgx$\"3Xfl3ek8HjF07$$\"3Ym mmJy*eC\"Fgx$\"3'HN]G!3q'o&F07$$\"3;LL$e[E()G\"Fgx$\"3G'**\\&>,5f]F07$ $\"3')******R^bJ8Fgx$\"3#y>/'3`=oWF07$$\"3s****\\AYXt8Fgx$\"3nyhm+6fDR F07$$\"3f*****\\5a`T\"Fgx$\"3+:NSD#3\"=MF07$$\"3k***\\(3S*eX\"Fgx$\"3% \\XTrH80'HF07$$\"3o****\\7RV'\\\"Fgx$\"3%)z0![/)yNDF07$$\"3k*****\\@fk e\"Fgx$\"3;lTe3)f,r\"F07$$\"3/LLL`4Nn;Fgx$\"3lBC[#)Qb16F07$$\"3#****** *\\,s`$=Fgx$\"3P*z$HF^&H#GFcp7$$\"3? 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" }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" } {TEXT -1 12 " we have " }{XPPEDIT 18 0 "1/3+``;" "6#,&*&\"\"\"F%\" \"$!\"\"F%%!GF%" }{XPPEDIT 18 0 "Sum(``(4*(-1)^k/(k^2*Pi^2))*cos(k*Pi) ,k = 1 .. infinity) = 1" "6#/-%$SumG6$*&-%!G6#*(\"\"%\"\"\"),$F-!\"\"% \"kGF-*&F1\"\"#%#PiGF3F0F--%$cosG6#*&F1F-F4F-F-/F1;F-%)infinityGF-" } {TEXT -1 15 ", \nthat is, " }{XPPEDIT 18 0 "Sum(``(4/(k^2*Pi^2)),k \+ = 1 .. infinity) = 2/3;" "6#/-%$SumG6$-%!G6#*&\"\"%\"\"\"*&%\"kG\"\"#% #PiGF/!\"\"/F.;F,%)infinityG*&F/F,\"\"$F1" }{TEXT -1 10 " or " } {XPPEDIT 18 0 "4/Pi^2" "6#*&\"\"%\"\"\"*$%#PiG\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/k^2,k = 1 .. infinity) = 2/3" "6#/-%$SumG6$* &\"\"\"F(*$%\"kG\"\"#!\"\"/F*;F(%)infinityG*&F+F(\"\"$F," }{TEXT -1 36 ".\nThis establishes the formula: " }{XPPEDIT 18 0 "Sum(1/(k^2) ,k = 1 .. infinity) = Pi^2/6;" "6#/-%$SumG6$*&\"\"\"F(*$%\"kG\"\"#!\" \"/F*;F(%)infinityG*&%#PiGF+\"\"'F," }{TEXT -1 4 ".\n " }}{PARA 0 "" 0 "" {TEXT -1 68 "Truncated Fourier cosine series can be obtained usin g the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 " " {TEXT -1 17 "The extension of " }{XPPEDIT 18 0 "f(x)=x^2, x>=0" "6$/ -%\"fG6#%\"xG*$F'\"\"#1\"\"!F'" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 7 " to an " }{TEXT 267 3 "odd" }{TEXT -1 26 " function on th e interval " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "g(x) = \+ PIECEWISE([-x^2, -1 <= x and x < 0],[x^2, 0 <= x and x <= 1]);" "6#/-% \"gG6#%\"xG-%*PIECEWISEG6$7$,$*$F'\"\"#!\"\"31,$\"\"\"F/F'2F'\"\"!7$*$ F'F.31F5F'1F'F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The g raph of the odd extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG " }{TEXT -1 27 " can be plotted as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "g := x -> piecewi se(x<0,-x^2,x^2):\n'g(x)'=g(x);\nplot(g(x),x=-1..1,color=COLOR(RGB,.8, 0,1),thickness=2,tickmarks=[4,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"gG6#%\"xG-%*PIECEWISEG6$7$,$*$)F'\"\"#\"\"\"!\"\"2F'\"\"!7$F-%*ot herwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 346 333 333 {PLOTDATA 2 "6(-%'C URVESG6#7S7$$!\"\"\"\"!F(7$$!3ommm;p0k&*!#=$!3c]R_q%=r9*F.7$$!3wKL$3`0n(F.7$$!3:mmm\"4m(G $)F.$!32V'p4YMo$pF.7$$!3\"QLL3i.9!zF.$!3Z6=$z\"z@ViF.7$$!3\"ommT!R=0vF .$!3(4CONayFj&F.7$$!3u****\\P8#\\4(F.$!30z7`y3zL]F.7$$!3+nm;/siqmF.$!3 qSopHns\\WF.7$$!3[++](y$pZiF.$!335mBmxO.RF.7$$!33LLL$yaE\"eF.$!3m(y?Ic &pyLF.7$$!3hmmm\">s%HaF.$!3ej\"3!Go\"z%HF.7$$!3Q+++]$*4)*\\F.$!3mUqC6( *4)\\#F.7$$!39+++]_&\\c%F.$!3Mc-XV;)Q3#F.7$$!31+++]1aZTF.$!3?U-MW$4-s \"F.7$$!3umm;/#)[oPF.$!3(3L%\\M.:?9F.7$$!3hLLL$=exJ$F.$!3@IvIO>v+6F.7$ $!3*RLLLtIf$HF.$!3&y?J4F*o>')!#>7$$!3]++]PYx\"\\#F.$!3he#)3W3%*3iF[q7$ $!3EMLLL7i)4#F.$!3!\\_(*436US%F[q7$$!3c****\\P'psm\"F.$!31!QHT/)yzFF[q 7$$!3')****\\74_c7F.$!3KK)\\N![%)y:F[q7$$!3)3LLL3x%z#)F[q$!3M!=Wt2u\\& o!#?7$$!3KMLL3s$QM%F[q$!3a8,Dp@*o)=Fer7$$!3]^omm;zr)*!#@$!3+4t+rqAX(*! #C7$$\"3%pJL$ezw5VF[q$\"3Q>$f!R?Fe=Fer7$$\"3s*)***\\PQ#\\\")F[q$\"3UZs D4'35k'Fer7$$\"3GKLLe\"*[H7F.$\"3&e?f/fV;^\"F[q7$$\"3I*******pvxl\"F.$ \"3a([5:F?#[FF[q7$$\"3#z****\\_qn2#F.$\"3um(3N\"e(HJ%F[q7$$\"3U)***\\i &p@[#F.$\"3!RV,qtl6;'F[q7$$\"3B)****\\2'HKHF.$\"3;&Rg9Fg$)f)F[q7$$\"3E lmmmZvOLF.$\"3CrsGPKR86F.7$$\"3i******\\2goPF.$\"3+c+Hh^B?9F.7$$\"3UKL $eR<*fTF.$\"3[xc,u7\\Ix^Bqij(F.7 $$\"3[++]i`1h\"*F.$\"31F(*fd=^#R)F.7$$\"3W++]P?Wl&*F.$\"3]:sFP\"o(\\\" *F.7$$\"\"\"F*Ffz-%*AXESTICKSG6$\"\"%F[[l-%+AXESLABELSG6$Q\"x6\"Q!F`[l -%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBG$\"\")F)$F*F*Ffz-%%VIEWG6$;F(Ffz% (DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "The coeffic ients of the sine terms in half-range sine series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "The coeffic ients of the sine terms are given by: " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "b[k] = 2*Int(f(x)*sin*k*Pi*x,x = 0 .. 1);" "6 #/&%\"bG6#%\"kG*&\"\"#\"\"\"-%$IntG6$*,-%\"fG6#%\"xGF*%$sinGF*F'F*%#Pi GF*F2F*/F2;\"\"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Int(x^2*sin*k*Pi*x,x = 0 .. 1);" "6#/%!G*&\" \"#\"\"\"-%$IntG6$*,%\"xGF&%$sinGF'%\"kGF'%#PiGF'F,F'/F,;\"\"!F'F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "In t(x^2*sin*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*,%\"xG\"\"#%$ sinG\"\"\"%\"kGF+%#PiGF+F(F+F(-F%6$*&%\"uGF+-%!G6#*&%#dvGF+%#dxG!\"\"F +F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x^2;" "6#/%\"uG*$%\"xG \"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*k*Pi*x/(k*Pi);" "6 #/%\"vG,$*,%$cosG\"\"\"%\"kGF(%#PiGF(%\"xGF(*&F)F(F*F(!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration b y parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(d u/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG, &*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 10 ", w e have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[k] = 2* ``((-x^2*cos*k*Pi*x)/(k*Pi));" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"-%!G6#*&, $*,%\"xGF)%$cosGF*F'F*%#PiGF*F1F*!\"\"F**&F'F*F3F*F4F*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWIS EG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Int (2*x*``(-cos*k*Pi*x/(k*Pi)),x = 0 .. 1);" "6#,$*&\"\"#\"\"\"-%$IntG6$* (F%F&%\"xGF&-%!G6#,$*,%$cosGF&%\"kGF&%#PiGF&F+F&*&F2F&F3F&!\"\"F5F&/F+ ;\"\"!F&F&F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -2*cos*k*Pi/(k*Pi)+4/(k*Pi);" "6#/%!G,&*,\"\"#\"\" \"%$cosGF(%\"kGF(%#PiGF(*&F*F(F+F(!\"\"F-*&\"\"%F(*&F*F(F+F(F-F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*cos*k*Pi*x,x = 0 .. 1);" "6#-%$In tG6$*,%\"xG\"\"\"%$cosGF(%\"kGF(%#PiGF(F'F(/F';\"\"!F(" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(2/(k*Pi))*( -1)^(k+1)+4/(k*Pi)" "6#/%!G,&*&-F$6#*&\"\"#\"\"\"*&%\"kGF+%#PiGF+!\"\" F+),$F+F/,&F-F+F+F+F+F+*&\"\"%F+*&F-F+F.F+F/F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x*cos*k*Pi*x,x = 0 .. 1);" "6#-%$IntG6$*,%\"xG\"\" \"%$cosGF(%\"kGF(%#PiGF(F'F(/F';\"\"!F(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x*cos*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*,%\"xG \"\"\"%$cosGF)%\"kGF)%#PiGF)F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG! \"\"F)F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"x G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*k*Pi*x/(k*Pi);" "6#/%\" vG*,%$sinG\"\"\"%\"kGF'%#PiGF'%\"xGF'*&F(F'F)F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration by parts f ormula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x); " "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)% \"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 9 ", we have" } }{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[k] = ``(2/(k*Pi))*(-1)^(k+1)+4/(k* Pi);" "6#/&%\"bG6#%\"kG,&*&-%!G6#*&\"\"#\"\"\"*&F'F/%#PiGF/!\"\"F/),$F /F2,&F'F/F/F/F/F/*&\"\"%F/*&F'F/F1F/F2F/" }{XPPEDIT 18 0 "``(x*sin*k*P i*x/(k*Pi));" "6#-%!G6#*.%\"xG\"\"\"%$sinGF(%\"kGF(%#PiGF(F'F(*&F*F(F+ F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0 , ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{XPPEDIT 18 0 "-4/(k*Pi);" "6#,$*&\"\"%\"\"\"*&%\"kGF&%#PiGF&!\"\"F*" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(sin*k*Pi*x/(k*Pi),x = 0 .. 1);" "6#-%$IntG6$*,%$ sinG\"\"\"%\"kGF(%#PiGF(%\"xGF(*&F)F(F*F(!\"\"/F+;\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(2/(k*Pi))*(-1)^(k+1)+0-4/(k*Pi);" "6#/%!G,( *&-F$6#*&\"\"#\"\"\"*&%\"kGF+%#PiGF+!\"\"F+),$F+F/,&F-F+F+F+F+F+\"\"!F +*&\"\"%F+*&F-F+F.F+F/F/" }{XPPEDIT 18 0 "``(-cos*k*Pi*x/(k^2*Pi^2)); " "6#-%!G6#,$*,%$cosG\"\"\"%\"kGF)%#PiGF)%\"xGF)*&F*\"\"#F+F.!\"\"F/" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]); " "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = ``(2/(k*Pi))*(-1)^(k+1)+``(4/(k^3*Pi^3))*(cos*k*Pi -1);" "6#/%!G,&*&-F$6#*&\"\"#\"\"\"*&%\"kGF+%#PiGF+!\"\"F+),$F+F/,&F-F +F+F+F+F+*&-F$6#*&\"\"%F+*&F-\"\"$F.F9F/F+,&*(%$cosGF+F-F+F.F+F+F+F/F+ F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(2/(k*Pi))*(-1)^(k+1)+``(4/(k^ 3*Pi^3))*((-1)^k-1);" "6#/%!G,&*&-F$6#*&\"\"#\"\"\"*&%\"kGF+%#PiGF+!\" \"F+),$F+F/,&F-F+F+F+F+F+*&-F$6#*&\"\"%F+*&F-\"\"$F.F9F/F+,&),$F+F/F-F +F+F/F+F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = PIECEWISE([-2/(k*Pi), \+ `if k is even`],[2/(k*Pi)-8/(k^3*Pi^3), `if k is odd`]);" "6#/%!G-%*PI ECEWISEG6$7$,$*&\"\"#\"\"\"*&%\"kGF,%#PiGF,!\"\"F0%-if~k~is~evenG7$,&* &F+F,*&F.F,F/F,F0F,*&\"\")F,*&F.\"\"$F/F9F0F0%,if~k~is~oddG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [b[ k], `|`, 2/Pi-8/Pi^3, -1/Pi, 2/3/Pi-8/27/Pi^3, -1/(2*Pi), 2/5/Pi-8/125 /Pi^3, -1/(3*Pi), 2/7/Pi-8/343/Pi^3, -1/(4*Pi)]])" "6#-%'matrixG6#7$7, %\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F),&* &F+F*%#PiG!\"\"F**&F1F**$F8F,F9F9,$*&F*F*F8F9F9,&*(F+F*F,F9F8F9F**(F1F *\"#FF9*$F8F,F9F9,$*&F*F**&F+F*F8F*F9F9,&*(F+F*F.F9F8F9F**(F1F*\"$D\"F 9*$F8F,F9F9,$*&F*F**&F,F*F8F*F9F9,&*(F+F*F0F9F8F9F**(F1F*\"$V$F9*$F8F, F9F9,$*&F*F**&F-F*F8F*F9F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "interface(showass umed=0): k := 'k': assume(k,integer):\nb[k]=2*Int(x^2*sin(k*Pi*x),x=0. .1);\n``=expand(value(rhs(%)));\nbb := unapply(rhs(%),k): k := 'k':\nm atrix([[k,`|`,seq(k,k=1..8)],[b[k],`|`,seq(bb(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$*&\"\"#\"\"\"-%$IntG6$*&)% \"xGF*F+-%$sinG6#*(F'F+%#PiGF+F1F+F+/F1;\"\"!F+F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,(*(\"\"%\"\"\"%#k|irG!\"$%#PiGF*!\"\"**F'F(F)F*F +F*)F,F)F(F(**\"\"#F(F)F,F+F,F.F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7$7,%#k|irG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\" \")7,&%\"bG6#F(F),&*&F1F*%#PiG!\"$!\"\"*&F+F*F8F:F*,$*&F*F*F8F:F:,&*(F 1F*\"#FF:F8F9F:*(F+F*F,F:F8F:F*,$*&F*F**&F+F*F8F*F:F:,&*(F1F*\"$D\"F:F 8F9F:*(F+F*F.F:F8F:F*,$*&F*F**&F,F*F8F*F:F:,&*(F1F*\"$V$F:F8F9F:*(F+F* F0F:F8F:F*,$*&F*F**&F-F*F8F*F:F:Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The Fourier hal f-range sine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " F(x) = Sum(``(``(2/(k*Pi))*(-1)^(k+1)+``(4/(k^3*Pi^3))*((-1)^k-1)),k = 1 .. infinity);" "6#/-%\"FG6#%\"xG-%$SumG6$-%!G6#,&*&-F,6#*&\"\"#\"\" \"*&%\"kGF4%#PiGF4!\"\"F4),$F4F8,&F6F4F4F4F4F4*&-F,6#*&\"\"%F4*&F6\"\" $F7FBF8F4,&),$F4F8F6F4F4F8F4F4/F6;F4%)infinityG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sin*k*Pi*x" "6#**%$sinG\"\"\"%\"kGF%%#PiGF%%\"xGF%" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`` = ``(2/Pi-8/(Pi^3))*sin*Pi*x-sin*2*Pi*x/Pi+``(2/3/Pi -8/27/(Pi^3))*sin*3*Pi*x-sin*4*Pi*x/(2*Pi)+``(2/5/Pi-8/125/(Pi^3))*sin *5*Pi*x-sin*6*Pi*x/(3*Pi)+``(2/7/Pi-8/343/(Pi^3))*sin*7*Pi*x-sin*8*Pi* x/(4*Pi)+` . . . `;" "6#/%!G,4**-F$6#,&*&\"\"#\"\"\"%#PiG!\"\"F,*&\"\" )F,*$F-\"\"$F.F.F,%$sinGF,F-F,%\"xGF,F,*,F3F,F+F,F-F,F4F,F-F.F.*,-F$6# ,&*(F+F,F2F.F-F.F,*(F0F,\"#FF.*$F-F2F.F.F,F3F,F2F,F-F,F4F,F,*,F3F,\"\" %F,F-F,F4F,*&F+F,F-F,F.F.*,-F$6#,&*(F+F,\"\"&F.F-F.F,*(F0F,\"$D\"F.*$F -F2F.F.F,F3F,FFF,F-F,F4F,F,*,F3F,\"\"'F,F-F,F4F,*&F2F,F-F,F.F.*,-F$6#, &*(F+F,\"\"(F.F-F.F,*(F0F,\"$V$F.*$F-F2F.F.F,F3F,FRF,F-F,F4F,F,*,F3F,F 0F,F-F,F4F,*&F?F,F-F,F.F.%(~.~.~.~GF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The following pictures compare graphs of some truncated series with the graph of the functio n " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 285 "g : = x -> piecewise(x<0,-x^2,x^2):\ng_ := x -> g(x-2*floor((x+1)/2)):\nFS := (x,n) ->sum(((2/(k*Pi))*(-1)^(k+1)+(4/(k^3*Pi^3))*((-1)^k-1))*sin( k*Pi*x),k=1..n);\nplot([g_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)] ,x=-1..4,\n color=[black,red,blue,green,magenta,coral],linestyle=[3, 1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%)o peratorG%&arrowGF)-%$sumG6$*&,&**\"\"#\"\"\"%\"kG!\"\"%#PiGF6)F6,&F5F4 F4F4F4F4**\"\"%F4F5!\"$F7F<,&)F6F5F4F4F6F4F4F4-%$sinG6#*(F5F4F7F49$F4F 4/F5;F49%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 603 293 293 {PLOTDATA 2 "6*-%'CURVESG6%7`p7$$!\"\"\"\"!F(7$$!3kLL$e9r]X*!#=$!3S1qFPq$)R*)F.7$$ !3Fnmm\"HU,\"*)F.$!3I$pudcj!RzF.7$$!3d**\\P4E+O%)F.$!3mz\"QD+9m6(F.7$$ 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1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Truncated Fourier sine series can be obtained using the procedure " } {TEXT 0 13 "FourierSeries" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 267 "f := x -> x^2:\n' f(x)'=f(x);\nFourierSeries(f(x),x=0..1,type=sin,numterms=10):\nF := un apply(%,x):\n'F(x)'=F(x);\ng := x -> piecewise(x<0,-x^2,x^2):\ng_ := x -> g(x-2*floor((x+1)/2)):\nplot([g_(x),F(x)],x=-1..4,color=[black,COL OR(RGB,.8,0,1)],linestyle=[3,1],numpoints=100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$)F'\"\"#\"\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,6**\"\"#\"\"\",&\"\"%!\"\"*$)%#PiGF*F+F+ F+F1!\"$-%$sinG6#*&F1F+F'F+F+F+*&F1F.-F46#,$*(F*F+F1F+F'F+F+F+F.*&#F* \"#FF+*(,&F-F.*&\"\"*F+F0F+F+F+F1F2-F46#,$*(\"\"$F+F1F+F'F+F+F+F+F+*&# F+F*F+*&F1F.-F46#,$*(F-F+F1F+F'F+F+F+F+F.*&#F*\"$D\"F+*(,&F-F.*&\"#DF+ F0F+F+F+F1F2-F46#,$*(\"\"&F+F1F+F'F+F+F+F+F+*&#F+FGF+*&F1F.-F46#,$*(\" \"'F+F1F+F'F+F+F+F+F.*&#F*\"$V$F+*(,&F-F.*&\"#\\F+F0F+F+F+F1F2-F46#,$* (\"\"(F+F1F+F'F+F+F+F+F+*&#F+F-F+*&F1F.-F46#,$*(\"\")F+F1F+F'F+F+F+F+F .*&#F*\"$H(F+*(,&F-F.*&\"#\")F+F0F+F+F+F1F2-F46#,$*(FBF+F1F+F'F+F+F+F+ F+*&#F+FZF+*&F1F.-F46#,$*(\"#5F+F1F+F'F+F+F+F+F." }}{PARA 13 "" 1 "" {GLPLOT2D 596 257 257 {PLOTDATA 2 "6&-%'CURVESG6%7fr7$$!\"\"\"\"!F(7$$ !33TTTTTer%*!#=$!3y%y)y914r*)F.7$$!3&)GGGGB\"=,*F.$!3TQ!>S9w77)F.7$$!3 #)QQQQ3v%\\)F.$!3[$HA1=zg@(F.7$$!3aHHHH>EuzF.$!3]fqpJ`))ejF.7$$!358888 oCcuF.$!35'3LQlh&fbF.7$$!3o*)*)*)*[!)f(pF.$!3qM^az.Vm[F.7$$!3sIIIIDpyk F.$!3oG/A!pXt>%F.7$$!3I6666mRkfF.$!3oqLY$p-ub$F.7$$!3WYXXX+v^aF.$!3#[O 6e&y:sHF.7$$!3wRSSS+VC\\F.$!3RtLGA6+DCF.7$$!3q#HHHHm*fWF.$!3qXlSL*H\"* )>F.7$$!3@@@@@,4PRF.$!3U2YDiy1]:F.7$$!3lppppp17MF.$!3;1(p05?U;\"F.7$$! 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F.7$$\"35...G%>F[$F]z$!3\"=I2Kp*3a0HF.7$Ff^m$!3]^B 3=o%y#=F.7$$\"3Y\"===G2tb$F]z$!3VI&=yI5hh\"F.7$$\"35XXXXmspNF]z$!3a\"R bG?RpY\"F.7$$\"3u3444g9#e$F]z$!3+5'3/f!4&Q\"F.7$F[_m$!3;10(f5+[O\"F.7$ $\"30FFF-$**)>OF]z$!3QQ^LIOKW9F.7$F`_m$!3CM+[v&\\B_\"F.7$$\"39FFF-+meO F]z$!3s8M(z(G8,:F.7$$\"3cssssn3sOF]z$!3!f\"e!G87uT\"F.7$$\"3+===VN^&o$ F]z$!3E(GE0r@;F\"F.7$Fe_m$!3mbK@Y3iw5F.7$$\"3+111\"QiNs$F]z$!3wcax]o)) )p'F\\p7$Fj_m$!3vr[4Cs&pU$F\\p7$$\"3gxx-M2EaPF]z$!3.\">6eFv8!HF\\p7$$ \"3522d>qLgPF]z$!3&Q8m*Q7a5DF\\p7$$\"3gOO60LTmPF]z$!3SDw:WaHdAF\\p7$$ \"3klll!f*[sPF]z$!3#yc>+;&HR@F\\p7$$\"3=CCuh@k%y$F]z$!3QUO,[]suAF\\p7$ F_`m$!3G@NoR\"*>/GF\\p7$$\"3JJJJJcy4QF]z$!3(Gi+UN!*\\h$F\\p7$$\"3WzzzH lxAQF]z$!3'ft([7x8VWF\\p7$$\"3dFGGGuwNQF]z$!3+ndG*Q!Ha]F\\p7$Fd`m$!3Qd #e:T@??X4shQF]z$!3Oo]UJNKg\\F\\p7$$\"3YjjjjNouQF]z$!3@ T'4^X(=VTF\\p7$$\"35222#=Yw)QF]z$!3![,%>%Gw1!HF\\p7$Fi`m$!3T6dLJ?[/9F \\p7$$\"3K:::S#yQ#RF]z$\"3(e'yqcpG17F\\p7$F^am$\"3a?qmHSJ!o#F\\p7$$\"3 7&[[)f2OgRF]z$\"3\"3q<:qsys#F\\p7$$\"3%**)*)*)RQdtRF]z$\"3:8Iuc\\5-AF \\p7$$\"3u%\\\\*>py')RF]z$\"3N&)Q*e7]]A\"F\\p7$Fcam$\"3)z=-,u\"G'\\\"F dbm-%&COLORG6&Fiam$\"\")F)Feam$\"\"\"F*-F[bm6#Ffes-%+AXESLABELSG6$Q\"x 6\"Q!F]fs-%%VIEWG6$;F(Fcam%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(c) The constant term in the " }{TEXT 268 14 "Fourier \+ series" }{TEXT -1 4 " is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "c = Int(x^2,x = 0 .. 1);" "6#/%\"cG-%$IntG6$*$%\"xG\"\" #/F);\"\"!\"\"\"" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "x^3/3" "6#*&%\"xG \"\"$F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1 , ``],[0 ,` ` ]) = 1/3" "6#/-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F)*&F(F(\"\"$!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "The coefficients of the cosine terms in the Fourier se ries" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 73 "The coefficients of the cosine terms in the Fourier ser ies are given by: " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "a[k] = 2*Int(f(x)*cos*2*k*Pi*x,x = 0 .. 1);" "6#/&%\"aG6#%\"kG*& \"\"#\"\"\"-%$IntG6$*.-%\"fG6#%\"xGF*%$cosGF*F)F*F'F*%#PiGF*F2F*/F2;\" \"!F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = 2*Int(x^2*cos*2*k*Pi*x,x = 0 .. 1);" "6#/%!G*&\"\" #\"\"\"-%$IntG6$*.%\"xGF&%$cosGF'F&F'%\"kGF'%#PiGF'F,F'/F,;\"\"!F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "In t(x^2*cos*2*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*.%\"xG\"\"# %$cosG\"\"\"F)F+%\"kGF+%#PiGF+F(F+F(-F%6$*&%\"uGF+-%!G6#*&%#dvGF+%#dxG !\"\"F+F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x^2;" "6#/%\"uG* $%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = sin*2*k*Pi*x/(2*k *Pi);" "6#/%\"vG*.%$sinG\"\"\"\"\"#F'%\"kGF'%#PiGF'%\"xGF'*(F(F'F)F'F* F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using t he integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) \+ = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%# dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[k] = 2*``(x^2*sin*2*k*Pi*x/(2*k*Pi));" "6#/&%\"aG6#%\"kG*&\"\" #\"\"\"-%!G6#*0%\"xGF)%$sinGF*F)F*F'F*%#PiGF*F/F**(F)F*F'F*F1F*!\"\"F* " }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]) ;" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-2*Int(2*x*``(sin*2*k*Pi*x/(2*k*Pi)),x = 0 .. 1);" "6#, $*&\"\"#\"\"\"-%$IntG6$*(F%F&%\"xGF&-%!G6#*.%$sinGF&F%F&%\"kGF&%#PiGF& F+F&*(F%F&F1F&F2F&!\"\"F&/F+;\"\"!F&F&F4" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "0-2/(k*Pi);" "6#,&\"\"!\"\"\" *&\"\"#F%*&%\"kGF%%#PiGF%!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int( x*sin*2*k*Pi*x,x = 0 .. 1);" "6#-%$IntG6$*.%\"xG\"\"\"%$sinGF(\"\"#F(% \"kGF(%#PiGF(F'F(/F';\"\"!F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int (x*sin*2*k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*.%\"xG\"\"\"%$ sinGF)\"\"#F)%\"kGF)%#PiGF)F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\" \"F)F(" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*2*k*Pi*x/(2*k*Pi);" "6# /%\"vG,$*.%$cosG\"\"\"\"\"#F(%\"kGF(%#PiGF(%\"xGF(*(F)F(F*F(F+F(!\"\"F ." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the int egration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v- Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\" \"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a[k] = -2/(k*Pi );" "6#/&%\"aG6#%\"kG,$*&\"\"#\"\"\"*&F'F+%#PiGF+!\"\"F." }{TEXT -1 2 " " }{XPPEDIT 18 0 "``(-x*cos*2*k*Pi*x/(2*k*Pi));" "6#-%!G6#,$*0%\"xG \"\"\"%$cosGF)\"\"#F)%\"kGF)%#PiGF)F(F)*(F+F)F,F)F-F)!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*P IECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 "+ " }{XPPEDIT 18 0 "2/(k*Pi);" "6#*&\"\"#\"\"\"*&%\"kGF%%#PiGF%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(-cos*2*k*Pi*x/(2*k*Pi),x = 0 .. 1);" "6#-%$IntG6$,$ *.%$cosG\"\"\"\"\"#F)%\"kGF)%#PiGF)%\"xGF)*(F*F)F+F)F,F)!\"\"F//F-;\" \"!F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(k^2*Pi^2)+2/(k*Pi);" "6#/% !G,&*&\"\"\"F'*&%\"kG\"\"#%#PiGF*!\"\"F'*&F*F'*&F)F'F+F'F,F'" } {XPPEDIT 18 0 "``(-sin*2*k*Pi*x/(4*k^2*Pi^2));" "6#-%!G6#,$*.%$sinG\" \"\"\"\"#F)%\"kGF)%#PiGF)%\"xGF)*(\"\"%F)*$F+F*F)F,F*!\"\"F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*P IECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(k^2*Pi^2);" "6#/%!G*&\"\" \"F&*&%\"kG\"\"#%#PiGF)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [a[k], `|`, 1/(Pi^2), 1/(4*Pi^2), 1/(9*Pi^2), 1/(16*Pi^2), 1/(25*Pi^2), 1/(36*Pi^2), 1/(49*Pi^2), 1/(64 *Pi^2)]])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\" \"'\"\"(\"\")7,&%\"aG6#F(F)*&F*F**$%#PiGF+!\"\"*&F*F**&F-F**$F8F+F*F9* &F*F**&\"\"*F**$F8F+F*F9*&F*F**&\"#;F**$F8F+F*F9*&F*F**&\"#DF**$F8F+F* F9*&F*F**&\"#OF**$F8F+F*F9*&F*F**&\"#\\F**$F8F+F*F9*&F*F**&\"#kF**$F8F +F*F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "interface(showassumed=0): k := 'k': assu me(k,integer):\na[k]=2*Int(x^2*cos(2*k*Pi*x),x=0..1);\n``=value(rhs(%) );\naa := unapply(rhs(%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)], [a[k],`|`,seq(aa(k),k=1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6#%#k|irG,$*&\"\"#\"\"\"-%$IntG6$*&)%\"xGF*F+-%$cosG6#,$**F*F+F'F+ %#PiGF+F1F+F+F+/F1;\"\"!F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G *&\"\"\"F&*&)%#k|irG\"\"#F&)%#PiGF*F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%#k|irG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\" &\"\"'\"\"(\"\")7,&%\"aG6#F(F)*&F*F**$)%#PiGF+F*!\"\",$*&F*F**&F-F*F8F *F:F*,$*&F*F**&\"\"*F*F8F*F:F*,$*&F*F**&\"#;F*F8F*F:F*,$*&F*F**&\"#DF* F8F*F:F*,$*&F*F**&\"#OF*F8F*F:F*,$*&F*F**&\"#\\F*F8F*F:F*,$*&F*F**&\"# kF*F8F*F:F*Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The coefficients of the sine terms in the Fourier series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 71 "The coefficients of the sine terms in the Fourier series \+ are given by: " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 " b[k] = 2*Int(f(x)*sin*2*k*Pi*x,x = 0 .. 1);" "6#/&%\"bG6#%\"kG*&\"\"# \"\"\"-%$IntG6$*.-%\"fG6#%\"xGF*%$sinGF*F)F*F'F*%#PiGF*F2F*/F2;\"\"!F* F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 2*Int(x^2*sin*2*k*Pi*x,x = 0 .. 1);" "6#/%!G*&\"\"#\"\"\"-%$In tG6$*.%\"xGF&%$sinGF'F&F'%\"kGF'%#PiGF'F,F'/F,;\"\"!F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x^2*sin*2 *k*Pi*x,x) = Int(u*``(dv/dx),x);" "6#/-%$IntG6$*.%\"xG\"\"#%$sinG\"\" \"F)F+%\"kGF+%#PiGF+F(F+F(-F%6$*&%\"uGF+-%!G6#*&%#dvGF+%#dxG!\"\"F+F( " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x^2;" "6#/%\"uG*$%\"xG\" \"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v = -cos*2*k*Pi*x/(2*k*Pi);" "6#/%\"vG,$*.%$cosG\"\"\"\"\"#F(%\"kGF(%#PiGF(%\"xGF(*(F)F(F*F(F+F(!\" \"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the \+ integration by parts formula: " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u *v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG !\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" } {TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "b[k] = 2*``((-x^2*cos*2*k*Pi*x)/(2*k*Pi));" "6#/&%\"bG6 #%\"kG*&\"\"#\"\"\"-%!G6#*&,$*.%\"xGF)%$cosGF*F)F*F'F*%#PiGF*F1F*!\"\" F**(F)F*F'F*F3F*F4F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, `` ],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "-2*Int(2*x*``(-cos*2*k*Pi*x/(2*k*Pi)),x = 0 .. 1);" "6#,$*&\"\"#\"\"\"-%$IntG6$*(F%F&%\"xGF&-%!G6#,$*.%$cosGF &F%F&%\"kGF&%#PiGF&F+F&*(F%F&F2F&F3F&!\"\"F5F&/F+;\"\"!F&F&F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/(k *Pi)+2/(k*Pi);" "6#/%!G,&*&\"\"\"F'*&%\"kGF'%#PiGF'!\"\"F+*&\"\"#F'*&F )F'F*F'F+F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*cos*2*k*Pi*x,x = 0 \+ .. 1);" "6#-%$IntG6$*.%\"xG\"\"\"%$cosGF(\"\"#F(%\"kGF(%#PiGF(F'F(/F'; \"\"!F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "Int(x*cos*2*k*Pi*x,x) = In t(u*``(dv/dx),x);" "6#/-%$IntG6$*.%\"xG\"\"\"%$cosGF)\"\"#F)%\"kGF)%#P iGF)F(F)F(-F%6$*&%\"uGF)-%!G6#*&%#dvGF)%#dxG!\"\"F)F(" }{TEXT -1 8 ", \+ where " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v = sin*2*k*Pi*x/(2*k*Pi);" "6#/%\"vG*.%$sinG\"\"\"\"\" #F'%\"kGF'%#PiGF'%\"xGF'*(F(F'F)F'F*F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Hence, using the integration by parts formula: \+ " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-% $IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)- F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 10 ", we have:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[k] = -1/(k*Pi)+2/(k*Pi);" "6#/&%\"bG6#%\" kG,&*&\"\"\"F**&F'F*%#PiGF*!\"\"F-*&\"\"#F**&F'F*F,F*F-F*" }{XPPEDIT 18 0 "``(x*sin*2*k*Pi*x/(2*k*Pi));" "6#-%!G6#*0%\"xG\"\"\"%$sinGF(\"\" #F(%\"kGF(%#PiGF(F'F(*(F*F(F+F(F,F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\" %!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-2/(k*Pi);" "6#,$* &\"\"#\"\"\"*&%\"kGF&%#PiGF&!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(sin*2*k*Pi*x/(2*k*Pi),x = 0 .. 1);" "6#-%$IntG6$*.%$sinG\"\"\"\"\"# F(%\"kGF(%#PiGF(%\"xGF(*(F)F(F*F(F+F(!\"\"/F,;\"\"!F(" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/(k*Pi)+0+ 2/(k*Pi);" "6#/%!G,(*&\"\"\"F'*&%\"kGF'%#PiGF'!\"\"F+\"\"!F'*&\"\"#F'* &F)F'F*F'F+F'" }{XPPEDIT 18 0 "``(-cos*2*k*Pi*x/(4*k^2*Pi^2));" "6#-%! G6#,$*.%$cosG\"\"\"\"\"#F)%\"kGF)%#PiGF)%\"xGF)*(\"\"%F)*$F+F*F)F,F*! \"\"F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/(k*Pi);" "6 #/%!G,$*&\"\"\"F'*&%\"kGF'%#PiGF'!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [b[k], `|`, -1/Pi, -1/(2* Pi), -1/(3*Pi), -1/(4*Pi), -1/(5*Pi), -1/(6*Pi), -1/(7*Pi), -1/(8*Pi)] ])" "6#-%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\" (\"\")7,&%\"bG6#F(F),$*&F*F*%#PiG!\"\"F9,$*&F*F**&F+F*F8F*F9F9,$*&F*F* *&F,F*F8F*F9F9,$*&F*F**&F-F*F8F*F9F9,$*&F*F**&F.F*F8F*F9F9,$*&F*F**&F/ F*F8F*F9F9,$*&F*F**&F0F*F8F*F9F9,$*&F*F**&F1F*F8F*F9F9" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "interface(showassumed=0): k := 'k': assume(k,integer):\nb[k]= 2*Int(x^2*sin(2*k*Pi*x),x=0..1);\n``=value(rhs(%));\nbb := unapply(rhs (%),k): k := 'k':\nmatrix([[k,`|`,seq(k,k=1..8)],[b[k],`|`,seq(bb(k),k =1..8)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"-%$IntG6$ *&)%\"xGF%F&-%$sinG6#,$**F%F&%\"kGF&%#PiGF&F,F&F&F&/F,;\"\"!F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%\"kG,$*&\"\"\"F**&F'F*%#PiGF *!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7,%\"kG%\"|g rG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F),$*&F*F*%#PiG !\"\"F9,$*&F*F**&F+F*F8F*F9F9,$*&F*F**&F,F*F8F*F9F9,$*&F*F**&F-F*F8F*F 9F9,$*&F*F**&F.F*F8F*F9F9,$*&F*F**&F/F*F8F*F9F9,$*&F*F**&F0F*F8F*F9F9, $*&F*F**&F1F*F8F*F9F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "The Fourier series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "F(x) = 1/3+``;" "6#/-%\"FG6#%\"xG,&*& \"\"\"F*\"\"$!\"\"F*%!GF*" }{XPPEDIT 18 0 "Sum(cos*2*k*Pi*x/(k^2*Pi^2) -sin*2*k*Pi*x/(k*Pi),k = 1 .. infinity);" "6#-%$SumG6$,&*.%$cosG\"\"\" \"\"#F)%\"kGF)%#PiGF)%\"xGF)*&F+F*F,F*!\"\"F)*.%$sinGF)F*F)F+F)F,F)F-F )*&F+F)F,F)F/F//F+;F)%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``=1 /3+cos*2*Pi*x/Pi^2-sin*2*Pi*x/Pi+cos*4*Pi*x/(4*Pi^2)-sin*4*Pi*x/(2*Pi) +cos*6*Pi*x/(9*Pi^2)-sin*6*Pi*x/(3*Pi)+cos*8*Pi*x/(16*Pi^2)-sin*8*Pi*x /(4*Pi)+` . . . `" "6#/%!G,6*&\"\"\"F'\"\"$!\"\"F'*,%$cosGF'\"\"#F'%#P iGF'%\"xGF'*$F-F,F)F'*,%$sinGF'F,F'F-F'F.F'F-F)F)*,F+F'\"\"%F'F-F'F.F' *&F3F'*$F-F,F'F)F'*,F1F'F3F'F-F'F.F'*&F,F'F-F'F)F)*,F+F'\"\"'F'F-F'F.F '*&\"\"*F'*$F-F,F'F)F'*,F1F'F9F'F-F'F.F'*&F(F'F-F'F)F)*,F+F'\"\")F'F-F 'F.F'*&\"#;F'*$F-F,F'F)F'*,F1F'F@F'F-F'F.F'*&F3F'F-F'F)F)%(~.~.~.~GF' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The following pictures compare graphs of some truncated s eries with the graph of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "f := x -> x^2:\nf_ := x -> f(x-flo or(x)):\nFS := (x,n) -> 1/3+sum(cos(2*k*Pi*x)/(k^2*Pi^2)-sin(2*k*Pi*x) /(k*Pi),k=1..n);\nplot([f_(x),FS(x,1),FS(x,2),FS(x,3),FS(x,4),FS(x,5)] ,x=-1.2..2.3,\n color=[black,red,blue,green,magenta,coral],linestyle =[3,1$5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6 $%)operatorG%&arrowGF),&#\"\"\"\"\"$F/-%$sumG6$,&*(-%$cosG6#,$**\"\"#F /%\"kGF/%#PiGF/9$F/F/F/F\"!#<$\"3C,++++++k!#=7$$!3ML$3-)\\&=;\" F*$\"3.Ld,U2([-(F-7$$!3rmmTg*4P7\"F*$\"3k)fHZLU)ywF-7$$!3%)\\il#=?04\" F*$\"3/7ib\"QN:F)F-7$$!3=Le*[SIt0\"F*$\"3eP&Q`lfi)))F-7$$!3_\"zWnrn'Q5 F*$\"31CnG)e(fT#*F-7$$!3')\\PfG]+?5F*$\"3#*p`\")R9!Rg*F-7$$!3#*G#=Xot1 ,\"F*$\"3=ot/&elwy*F-7$$!3?3FWSBM,5F*$\"3OxW#*47n!*G`W5\"!#A7$$!31w1' QKQM%**F-$\"3'yw*o2KA*>$F[o7$$!3\"[(=nj*4,#**F-$\"3Ki\"4X!zT#Q'F[o7$$! 35qUHVKXt)*F-$\"3j>RFj\"39g\"!#@7$$!3]mm\"H_'zE)*F-$\"3A$ovtWW***HF[p7 $$!3-nmm\"fo5X*F-$\"3`j\"[0\"pD8I!#?7$$!3WmmTg1Mv!*F-$\"3'e'oyU*[*\\&) Ffp7$$!3C*\\P%)\\)R,()F-$\"3g\"Qg)feO'o\"!#>7$$!30K$ekLcuK)F-$\"33%Q`k I-uz#Faq7$$!3vm;HK=2MwF-$\"3K6dv%4;wf&Faq7$$!38**\\iSB6;pF-$\"3f@ak&4j .^*Faq7$$!3Sl;H2wfthF-$\"3o1C4Fb8k9F-7$$!3d**\\7GTYLaF-$\"33<@\"p)\\K& 3#F-7$$!3)=LL3(e9sYF-$\"3fJy;iHgQGF-7$$!3t)*\\7.6'oL%F-$\"31h)Hl@9r?$F -7$$!3elmTNjd,SF-$\"37OL)fk3\")f$F-7$$!3zJ$e*)4DTi$F-$\"3A5LPv!y^1%F-7 $$!3-)***\\iQnYKF-$\"3!)pS%>RT2c%F-7$$!3i)****\\xsw'GF-$\"3IAwWk\"4q3& F-7$$!3C****\\(or')[#F-$\"3p5k\"=I0?k&F-7$$!3-****\\7RVB@F-$\"3(p]vILH S?'F-7$$!3#))***\\Ph>ez!4LFz'F-7$$!3JKeRZ__E9F-$\"3%*3TNLpW ]tF-7$$!3!em\"HdV&[4\"F-$\"3vAZy\"*=;IzF-7$$!3&*))\\i!*el/qFaq$\"3Ip5- ES8[')F-7$$!3))>LL3#o21$Faq$\"3_![e.m9sR*F-7$$!3&*=$e9m?bA#Faq$\"33:H# *4)[)f&*F-7$$!3/=Le9JF!R\"Faq$\"3)*pkTO#yQs*F-7$$!3\"eA!GFfp$\"39P$43wNC&y!#B7$$\"3i&o ;H2<2&>Faq$\"3bT'o&)4(H0QF[p7$$\"3W(omm;77i$Faq$\"3f%[,cv<8J\"Ffp7$$\" 3][$3_+yv](Faq$\"3#p\")G/vsjj&Ffp7$$\"3'4+vVQ%RR6F-$\"3Y(GXJc>#)H\"Faq 7$$\"3!R$3-8OS$[\"F-$\"3&QEB\"zi[+AFaq7$$\"3%omm;%GTF=F-$\"3mU#))QpP%R LFaq7$$\"3I-]PM\"yAe#F-$\"3-V7Fj.;omFaq7$$\"3=.]7.%)3,LF-$\"3SQY_k%=(* 3\"F-7$$\"3%*omT5:4^SF-$\"3\"4+xDCM6k\"F-7$$\"3go;a)[G)RZF-$\"3+?\"z+T (fYAF-7$$\"3WLLekVs#[&F-$\"3dr&*zXm-1IF-7$$\"3CM$3FR%QaiF-$\"3-g`<8Ct6 RF-7$$\"3#ymmTb]-f'F-$\"3=2%pmBSJM%F-7$$\"3U,]i:n6EpF-$\"3W0g%eF4rz%F- 7$$\"3MoTNr8'))G(F-$\"3(o*=34+v7`F-7$$\"3ENL3Fgg^wF-$\"3u,)p$zuqaeF-7$ $\"3zn;/^nNE!)F-$\"3N:#*\\p-CUkF-7$$\"3K+++vu5,%)F-$\"3m(3]1ogy0(F-7$$ \"3M,](o%zsn()F-$\"3K#HT]L0to(F-7$$\"3O-+v=%[V8*F-$\"3Tm?^.@jV$)F-7$$ \"3!G]ilds!*[*F-$\"3+aqKO)\\U+*F-7$$\"3D.]PMnzV)*F-$\"3+)>pZTL+p*F-7$$ \"3_!pz;KHI*)*F-$\"3c69'f\"H?(y*F-7$$\"3yxV)*3>EU**F-$\"3OoTo'=d[))*F- 7$$\"3)>sOE?yo'**F-$\"3U,&>16mQ$**F-7$$\"3/l!*G'\\%\\\"***F-$\"3Cnu$pA '*H)**F-7$$\"3\"39%**y5h,5F*$\"3OuR,\"F*$\"3#QfoVZ[u$>F[p7$$\"39]7GVdwB5F*$\"3#=' G5Pb5[cF[p7$$\"30DJ?y(eM/\"F*$\"3!oT3HSl'))=Ffp7$$\"3'***\\78=:j5F*$\" 3i;\\(4]^\"))RFfp7$$\"3?Lek[>a)4\"F*$\"3+H!=Hk:0r*Ffp7$$\"3Zmm;%3KR8\" F*$\"3/.F#pJ!y$z\"Faq7$$\"3[++DJ^]47F*$\"3!H*z+-+C*Q%Faq7$$\"3AL3FWb)z F\"F*$\"3'p8bBG'fFxFaq7$$\"3a++vBF&GN\"F*$\"3!>%zdY/0X7F-7$$\"3imT50pH B9F*$\"3]ewq)p-=z\"F-7$$\"3;+v=s8$p\\\"F*$\"37=:k')ySpCF-7$$\"3ymm\"H_ A*o:F*$\"3G_'=3PGnB$F-7$$\"3L$3FWb1mg\"F*$\"3Q!)*Q(=^rzOF-7$$\"3))*\\P fe!HW;F*$\"3-poF\"f.6:%F-7$$\"3#pTNry%e!o\"F*$\"3e2;![_c>j%F-7$$\"3(RL L$))*yor\"F*$\"3cV#Q\"R[:R^F-7$$\"3wLe9^]*Rv\"F*$\"3c%3C:P&3&o&F-7$$\" 3cL$eR666z\"F*$\"3'Q&RGYzceiF-7$$\"3Q]7.P)>z#=F*$\"3s@&)\\lD^aoF-7$$\" 3?nT5g&GZ'=F*$\"3wi#)fE[bxuF-7$$\"3%Q3-Q&>b)*=F*$\"3CAa\"o8cR2)F-7$$\" 3^++]Z`PK>F*$\"3imuD')yB$p)F-7$$\"3CF*$\"373_leSVe!*F-7$$\"3u Lekt29r>F*$\"3%[GuB-W6V*F-7$$\"3)>H#=I@$3)>F*$\"3sD$zfn;.i*F-7$$\"3C]( =n[B0*>F*$\"3B,ZTxxO6)*F-7$$\"3!['G&eKYH*>F*$\"35NqCgSUf)*F-7$$\"3Ozp) \\;p`*>F*$\"3'ewogu(f2**F-7$$\"3#R4@T+#z(*>F*$\"3I\"*)z[$)))e&**F-7$$ \"3\\3_DV[@+?F*$\"3]Yy&3Vidh%!#D7$$\"3hPM_@010?F*$\"3;Kv\"*3y)3c#F[o7$ $\"3um;z*>1*4?F*$\"365=CJ%zK\")*F[o7$$\"3-+D1f%yX/#F*$\"3M!*pQ7!Rs)>Ff p7$$\"3ILLL=2Dz?F*$\"3#4L\\jNw1G'Ffp7$$\"3$oT&Qy&=i6#F*$\"3_FF>'zv1N\" Faq7$$\"3O+vVQk=`@F*$\"3g3m6#\\3mM#Faq7$$\"37+DccB&RA#F*$\"3?>(*)4!eY: ]Faq7$$\"3#)*************H#F*$\"3'**)*************)Faq-%'COLOURG6&%$RG BG\"\"!Fd^mFd^m-%*LINESTYLEG6#\"\"$-F$6%7gu7$F($\"3=NEDC*RPn'F-7$$!3S$ 3_]uj/>\"F*$\"3!zRh)frammF-7$$!3kmT5!\\F4=\"F*$\"3+hb;f6RZmF-7$$!3))\\ i:N7Rr6F*$\"3SS*e\\nSjh'F-7$F/$\"3)=*4_]r]tlF-7$$!3-+DJqCyU6F*$\"3c`g# [lZJX'F-7$F4$\"3Y[&400R!)G'F-7$F9$\"3\"[$>I4?S,fF-7$F>$\"39BV&z>*[.aF- 7$FC$\"3kTN\\^mb#3&F-7$FH$\"3&H1V2`*fPZF-7$FR$\"3]xd(HibLP%F-7$F]p$\"3 g,.P3w#[*RF-7$$!3?m;HdD$*Q'*F-$\"3'))=BH_EYg$F-7$Fbp$\"30ApDy)\\1@$F-7 $$!3sm;/EY?j#*F-$\"3z![Flg!Q=GF-7$Fhp$\"3]V&e&yyFLCF-7$$!3G#3F%z&p$))) )F-$\"3/uMf`SVi?F-7$F]q$\"3U[2bo'4\"4u5F-7$$!3S**\\P%3k2)zF-$\"3O27X_&)*y&fFaq7$Fhq$\"3? 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N%y49Fjp7$$\"3mz.V9c*H8#F*$!3QE1)pYURH)Fjp7$$\"3.QIWg\"f)Q@F*$!3@#p^kr f$G8Feq7$$\"3S'pbkqAZ9#F*$!3qR5\\Bkj)H\"Feq7$$\"3985[)z\\k:#F*$\"3vOlY 6o,9?Fjp7$Fcam$\"3-c+&\\ad;!GFeq7$$\"3;KD?Tb_y@F*$\"3#)HJ_E'Qe&\\Feq7$ $\"3WM()*=>u))=#F*$\"3=vuU\")=k$>'Feq7$$\"3eNoC<&[S>#F*$\"3SwPfYwigjFe q7$$\"3qO\\fUGA*>#F*$\"3sVXv1qTQiFeq7$$\"3'y.Vz;(R/AF*$\"31KB-BP#o(eFe q7$Fham$\"3eR2*p4$o]`Feq7$$\"3J#R6zo6:A#F*$\"3vp5#e@()G,%Feq7$$\"3d/K1MSFeq7$F]bm$\"3 %*)4'fBY@ceFeq7$$\"3C8\"HQ?**zE#F*$\"35LAU$olu$yFeq7$$\"3evg)e8m'yAF*$ \"3!yj*)[e,dX*Feq7$$\"3)pb9>g**RG#F*$\"3u\\1%y6%Hr**Feq7$$\"3#z.Vz1L$* G#F*$\"3:-h2ju_D5F-7$$\"3')=:(R`mYH#F*$\"39m$**H#ykJ5F-7$Fbbm$\"3dL$f- Q*3>5F--%&COLORG6&Fibm$\"\")!\"\"$FjbmFjbm$\"\"\"Fjbm-F\\cm6#F[_s-%+AX ESLABELSG6$Q\"x6\"Q!Fb_s-%%VIEWG6$;$!#7Fh^s$\"#BFh^s%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 46.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 265 8 "Q uestion" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 7 " Given " } {XPPEDIT 18 0 "f(x) = sin^2*x,0 <= x;" "6$/-%\"fG6#%\"xG*&%$sinG\"\"#F '\"\"\"1\"\"!F'" }{XPPEDIT 18 0 "`` <= Pi;" "6#1%!G%#PiG" }{TEXT -1 7 ", find " }}{PARA 0 "" 0 "" {TEXT -1 45 " (a) the Fourier half-range c osine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 43 " (b) the Fourier half-range sine s eries of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Solution " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 44 "(a) The Fourier half- range cosine series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 31 " is simply the 2-term series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 1/2-cos*2*x/2;" "6#/-%\"fG6#%\"x G,&*&\"\"\"F*\"\"#!\"\"F***%$cosGF*F+F*F'F*F+F,F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 21 "The o dd extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = PIECEWISE([-sin^2*x, -Pi < x and x < 0],[sin^2*x, 0 <= x and x < Pi]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,$*&%$sinG\"\"#F'\"\"\"!\" \"32,$%#PiGF1F'2F'\"\"!7$*&F.F/F'F031F7F'2F'F5" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The graph of the odd extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 27 " can be plotted as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "g := x -> piecewise( x<0,-sin(x)^2,sin(x)^2):\n'g(x)'=g(x);\nplot(g(x),x=-Pi..Pi,color=COLO R(RGB,.4,0,.9),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"g G6#%\"xG-%*PIECEWISEG6$7$,$*$)-%$sinGF&\"\"#\"\"\"!\"\"2F'\"\"!7$F-%*o therwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 429 240 240 {PLOTDATA 2 "6'-%' CURVESG6#7]p7$$!3*****4tk#fTJ!#<$!3s[aIH.eVR!#M7$$!3E[vS9#)=o%F37$$!3#RCl$yf()QIF*$!3yGaS mhO^5!#>7$$!3w\"*pr)3PY+$F*$!3S8!)fzY)R'=F>7$$!3)**z*4R\\0XHF*$!3r!)y7 RxA8QF>7$$!3?3E[*ysa)GF*$!3a;*Q,C`vT'F>7$$!3gX&))>2g9v#F*$!3[Uk+a[OY9! #=7$$!3)4577.flh#F*$!3E2\"zWTOC^#FS7$$!3u\"QM::*H#[#F*$!3Q!R&[I+G_PFS7 $$!3y\\AhcI#yN#F*$!35skfkVr$)\\FS7$$!3&e=d-FN*GAF*$!3;pF*$!3pxnHSLlS&)FS7$$!3(or 4AN*4E=F*$!37([*zNCCi$*FS7$$!3ix)*FS7$$!3gwh=Z!R=n\"F*$!3#>Nf\"p1D)*)*FS7$$!3f9& yM5fzj\"F*$!3G'**\\!*Rf\\&**FS7$$!3e_3xf\"zSg\"F*$!3cd8.?m#*))**FS7$$! 3e!>jg@*>q:F*$!2a;)eMk******F*7$$!31Z:0e,=O:F*$!3)4m$4%*>-))**FS7$$!3b .*R+5h@]\"F*$!3'*o5)>)f'H&**FS7$$!3/g#G?/U\"o9F*$!3_*GJkh!*\\*)*FS7$$! 3`;m,%)H7M9F*$!3I)4NEx*H*FS7$$!3S*4!R#[0R=\"F*$!3m&3Vy([Ow&)FS7$$!3pY@Q qWIU5F*$!33SaCVc]duFS7$$!3Wp3w$R)\\B#*FS$!3%)R7'3mcCN'FS7$$!3EJc8o39Gy FS$!3aRY>\\$fT(\\FS7$$!3'=[BP-8If'FS$!3w\"QcgcbBv$FS7$$!3CB3<@?)yB&FS$ !3Qap9h1k,DFS7$$!3_*)R7$$!3S\"4gMblG)>FS$!35JK7mh\\!)QF>7$$!3qTJY)ocYO\"FS$!3GlH6dhv]= F>7$$!3.w^X*zX7.\"FS$!3#>0d/O.(f5F>7$$!3c.@Z/\"\\$ypF>$!3%eN[\"yf$='[F 37$$!3%oW#R9-CWOF>$!3^t[ot3YF8F37$$!38,z7VKJ,JF3$!3<)y4e*H6='*!#B7$$\" 3!yDZ#y22`JF>$\"3k(RFk3h&Q**!#@7$$\"3p0t!3)GF;mF>$\"3+7nw))G7rVF37$$\" 3ONnL)\\Zz+\"FS$\"3IqsrYA_75F>7$$\"39SFf3xEa8FS$\"3$35\\U0cG#=F>7$$\"3 12**yIK@d>FS$\"3!)Rr4*)*=?y$F>7$$\"3'R2()Hve,c#FS$\"3:%et;PdCT'F>7$$\" 3IcwR>9FS7$$\"3*4*=Jof03_FS$\"3w3hD(\\\\eZ#FS7$$\"3 q]1XIqOClFS$\"3+(f`$)p+go$FS7$$\"3%HoBlmlzz(FS$\"3bVpJ/_)R%\\FS7$$\"3^ u'f#4)z?@*FS$\"3'GA\\k@g9M'FS7$$\"39Hvhw&)FS7$$\"3\\-)REfwoI\"F*$\"36!*>h$Q'[>$*FS7 $$\"3_o_p:s2u8F*$\"3MP=r;@)zh*FS7$$\"3cM2vQyFT9F*$\"3]iLYBb=L)*FS7$$\" 3)G#[O8J*GZ\"F*$\"3\"p*G9gaX/**FS7$$\"3=6*yzQ3X]\"F*$\"3yNS49M7c**FS7$ $\"3[**HfiO7O:F*$\"3ecWO!)G)z)**FS7$$\"3!y32s$*Qxc\"F*$\"3=W.)R_1***** FS7$$\"3(e61(3gy+;F*$\"3EJFv&))35***FS7$$\"3&R90-3LQj\"F*$\"3%[!*G=G;. '**FS7$$\"3-sTq^,)om\"F*$\"37iV2UF'z!**FS7$$\"35+K?Bs#**p\"F*$\"3D4\"> j)o>CZ'eYk>F*$\"3py2+*3l'G&)FS7$$\"3kYuyfix%4#F*$\"3?I B0a$)p'\\(FS7$$\"3T4q))fu.GAF*$\"3#yNLNz$eniFS7$$\"3y?w'**=&>gBF*$\"3w 2l8HM**f\\FS7$$\"3!*>3a=Wj\"[#F*$\"3/0/oi\"=(ePFS7$$\"3A?c*)yu\"3i#F*$ \"35<[M#*oevCFS7$$\"3-i-HnWIXFF*$\"3Ar.#\\wN**[\"FS7$$\"3u=RWhN.yGF*$ \"3![v7h,$)py'F>7$$\"3tX=#4!HbTHF*$\"33!\\7s[)\\[RF>7$$\"3ss(*RSA20IF* $\"3AaC>4@B_=F>7$$\"3;HB\"HM-#RIF*$\"3l`!Qw.>Z/\"F>7$$\"3/')[UXCLtIF*$ \"3IgTspOA_YF37$$\"3#HWPzaiu5$F*$\"3?sXGz=Tk6F37$$\"3!)***\\/l#fTJF*$ \"3op'H^#GIe)*!#N-%+AXESLABELSG6$Q\"x6\"Q!Fbel-%*THICKNESSG6#\"\"#-%&C OLORG6&%$RGBG$\"\"%!\"\"$\"\"!F`fl$\"\"*F^fl-%%VIEWG6$;$!+aEfTJ!\"*$\" +aEfTJFifl%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "The coefficients of the sine terms in half-range sine series" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "The coefficients of the sine terms are given by: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[k]=2/Pi" "6#/&%\"bG6#%\"kG*&\"\"#\"\"\"%#PiG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$**-%\"fG6#%\"xG\"\"\"%$sinGF+%\"kGF+F*F+/F*;\"\"!%#PiG" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin^2*x*sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$*,%$sinG\"\"#%\"xG\" \"\"F'F*%\"kGF*F)F*/F);\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2/Pi;" "6#/%!G*&\"\"#\"\"\"%#PiG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(1/2-cos*2*x/2)*sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$**-%!G6#,&*&\"\"\"F,\"\"#!\"\"F,**%$cosGF,F- F,%\"xGF,F-F.F.F,%$sinGF,%\"kGF,F1F,/F1;\"\"!%#PiG" }{TEXT -1 2 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G* &\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*k*x-cos*2 *x*sin*k*x,x = 0 .. Pi);" "6#-%$IntG6$,&*(%$sinG\"\"\"%\"kGF)%\"xGF)F) *.%$cosGF)\"\"#F)F+F)F(F)F*F)F+F)!\"\"/F+;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6#/%!G *&\"\"\"F&%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*k*x,x = \+ 0 .. Pi)-1/Pi;" "6#,&-%$IntG6$*(%$sinG\"\"\"%\"kGF)%\"xGF)/F+;\"\"!%#P iGF)*&F)F)F/!\"\"F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos*2*x*sin*k *x,x = 0 .. Pi);" "6#-%$IntG6$*.%$cosG\"\"\"\"\"#F(%\"xGF(%$sinGF(%\"k GF(F*F(/F*;\"\"!%#PiG" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 " " {TEXT -1 25 "The first term of (i) is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(sin*k*x,x = 0 .. Pi) = 1/Pi;" "6#/-%$IntG6$*(%$s inG\"\"\"%\"kGF)%\"xGF)/F+;\"\"!%#PiG*&F)F)F/!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(-cos*k*x/k);" "6#-%!G6#,$**%$cosG\"\"\"%\"kGF)%\"xGF )F*!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, `` ],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/Pi;" "6 #/%!G*&\"\"\"F&%#PiG!\"\"" }{XPPEDIT 18 0 "``(-cos*k*Pi/k+1/k);" "6#-% !G6#,&**%$cosG\"\"\"%\"kGF)%#PiGF)F*!\"\"F,*&F)F)F*F,F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-(-1)^k)/ (k*Pi);" "6#/%!G*&,&\"\"\"F'),$F'!\"\"%\"kGF*F'*&F+F'%#PiGF'F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "For the second term of \+ (i) we have, if " }{XPPEDIT 18 0 "k<>2" "6#0%\"kG\"\"#" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\" \"F$%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*k*x*cos*2*x,x \+ = 0 .. Pi) = 1/(2*Pi);" "6#/-%$IntG6$*.%$sinG\"\"\"%\"kGF)%\"xGF)%$cos GF)\"\"#F)F+F)/F+;\"\"!%#PiG*&F)F)*&F-F)F1F)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(sin((k+2)*x)+sin((k-2)*x),x=0..Pi)" "6#-%$IntG6$,&- %$sinG6#*&,&%\"kG\"\"\"\"\"#F-F-%\"xGF-F--F(6#*&,&F,F-F.!\"\"F-F/F-F-/ F/;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "( using \+ the formula: " }{XPPEDIT 18 0 "sin*alpha*cos*beta=1/2" "6#/**%$sinG\" \"\"%&alphaGF&%$cosGF&%%betaGF&*&F&F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(si n(alpha+beta)+sin(alpha-beta)" "6#-%!G6#,&-%$sinG6#,&%&alphaG\"\"\"%%b etaGF,F,-F(6#,&F+F,F-!\"\"F," }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\" #F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(-cos((k+2)*x)/(k+2)-cos((k-2)*x)/( k-2));" "6#-%!G6#,&*&-%$cosG6#*&,&%\"kG\"\"\"\"\"#F.F.%\"xGF.F.,&F-F.F /F.!\"\"F2*&-F)6#*&,&F-F.F/F2F.F0F.F.,&F-F.F/F2F2F2" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG 6%7$%#PiG%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*P i);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(-cos(( k+2)*Pi)/(k+2)-cos((k-2)*Pi)/(k-2)+1/(k+2)+1/(k-2));" "6#-%!G6#,**&-%$ cosG6#*&,&%\"kG\"\"\"\"\"#F.F.%#PiGF.F.,&F-F.F/F.!\"\"F2*&-F)6#*&,&F-F .F/F2F.F0F.F.,&F-F.F/F2F2F2*&F.F.,&F-F.F/F.F2F.*&F.F.,&F-F.F/F2F2F." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\" #F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(-(-1)^(k+2)/(k+2)-(-1)^(k-2)/(k-2) +1/(k+2)+1/(k-2));" "6#-%!G6#,**&),$\"\"\"!\"\",&%\"kGF*\"\"#F*F*,&F-F *F.F*F+F+*&),$F*F+,&F-F*F.F+F*,&F-F*F.F+F+F+*&F*F*,&F-F*F.F*F+F**&F*F* ,&F-F*F.F+F+F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/% !G*&\"\"\"F&*&\"\"#F&%#PiGF&!\"\"" }{XPPEDIT 18 0 "``(1-(-1)^k);" "6#- %!G6#,&\"\"\"F'),$F'!\"\"%\"kGF*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1 /(k+2)+1/(k-2));" "6#-%!G6#,&*&\"\"\"F(,&%\"kGF(\"\"#F(!\"\"F(*&F(F(,& F*F(F+F,F,F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = (1-(-1)^k)/(2*Pi);" " 6#/%!G*&,&\"\"\"F'),$F'!\"\"%\"kGF*F'*&\"\"#F'%#PiGF'F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(2*k/(k^2-4));" "6#-%!G6#*(\"\"#\"\"\"%\"kGF(,&*$ F)F'F(\"\"%!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = k*(1-(-1)^k)/((k ^2-4)*Pi);" "6#/%!G*(%\"kG\"\"\",&F'F'),$F'!\"\"F&F+F'*&,&*$F&\"\"#F' \"\"%F+F'%#PiGF'F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/Pi" "6#*&\"\"\"F$% #PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*2*x*cos*2*x,x = 0 . . Pi) = 1/(2*Pi);" "6#/-%$IntG6$*.%$sinG\"\"\"\"\"#F)%\"xGF)%$cosGF)F* F)F+F)/F+;\"\"!%#PiG*&F)F)*&F*F)F0F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin*4*x,x = 0 .. Pi);" "6#-%$IntG6$*(%$sinG\"\"\"\"\"%F(%\"x GF(/F*;\"\"!%#PiG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 1/(2*Pi);" "6#/%!G*&\"\"\"F&*&\"\"#F&%#PiGF&!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-cos(4*x)/4);" "6#-%!G6#,$*&-%$ cosG6#*&\"\"%\"\"\"%\"xGF-F-F,!\"\"F/" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$%#PiG%!G 7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 0" "6#/%!G\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 " b[2] = 0;" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 10 " and, if " } {XPPEDIT 18 0 "k<>2" "6#0%\"kG\"\"#" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "b[k]= (1-(-1)^k)/(k*Pi) - k*(1-(- 1)^k)/((k^2-4)*Pi)" "6#/&%\"bG6#%\"kG,&*&,&\"\"\"F+),$F+!\"\"F'F.F+*&F 'F+%#PiGF+F.F+*(F'F+,&F+F+),$F+F.F'F.F+*&,&*$F'\"\"#F+\"\"%F.F+F0F+F.F ." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -4*(1-(-1)^k)/(k*(k^2-4)*Pi);" "6#/%!G,$*(\"\"%\"\"\",&F(F(),$F( !\"\"%\"kGF,F(*(F-F(,&*$F-\"\"#F(F'F,F(%#PiGF(F,F," }{TEXT -1 2 " " } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 4*((-1)^k-1)/( k*(k^2-4)*Pi);" "6#/%!G*(\"\"%\"\"\",&),$F'!\"\"%\"kGF'F'F+F'*(F,F',&* $F,\"\"#F'F&F+F'%#PiGF'F+" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[k] = PIECEWISE([0, `if k is even`],[- 8/(k*(k^2-4)*Pi), `if k is odd`]);" "6#/&%\"bG6#%\"kG-%*PIECEWISEG6$7$ \"\"!%-if~k~is~evenG7$,$*&\"\")\"\"\"*(F'F2,&*$F'\"\"#F2\"\"%!\"\"F2%# PiGF2F8F8%,if~k~is~oddG" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "k=0,1,2,3, ` . . . `" "6'/%\"kG\"\"!\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[k, `|`, 1, 2, 3, 4, 5, 6, 7, 8], [b[k], `|`, 8 /3/Pi, 0, -8/15/Pi, 0, -8/105/Pi, 0, -8/315/Pi, 0]])" "6#-%'matrixG6#7 $7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")7,&%\"bG6#F(F) *(F1F*F,!\"\"%#PiGF7\"\"!,$*(F1F*\"#:F7F8F7F7F9,$*(F1F*\"$0\"F7F8F7F7F 9,$*(F1F*\"$:$F7F8F7F7F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "interface(showassume d=0): k := 'k': assume(k,integer):\nb[k]=2/Pi*Int(sin(x)^2*sin(k*x),x= 0..Pi);\n``=value(rhs(%));\nbb := unapply(rhs(%),k): k := 'k':\nmatrix ([[k,`|`,seq(k,k=1..8)],[b[k],`|`,bb(1),0,seq(bb(k),k=3..8)]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#%#k|irG,$,$-%$IntG6$*&)-%$sin G6#%\"xG\"\"#\"\"\"-F06#*&F'F4F2F4F4/F2;\"\"!%#PiG*&F3F4F;!\"\"F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*,\"\"%\"\"\"%#PiG!\"\",&F(F*)F* %#k|irGF(F(F-F*,&*$)F-\"\"#F(F(F'F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7,%\"kG%\"|grG\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"( \"\")7,&%\"bG6#F(F),$*(F1F*F,!\"\"%#PiGF8F*\"\"!,$*(F1F*\"#:F8F9F8F8F: ,$*(F1F*\"$0\"F8F9F8F8F:,$*(F1F*\"$:$F8F9F8F8F:Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "b[ 2]=2/Pi*Int(sin(x)^2*sin(2*x),x=0..Pi);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"#,$,$-%$IntG6$*&)-%$sinG6#%\"xG F'\"\"\"-F06#,$*&F'F3F2F3F3F3/F2;\"\"!%#PiG*&F'F3F;!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The Fourier half-range si ne series of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " i s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = 8*sin(x )/(3*Pi)" "6#/-%\"FG6#%\"xG*(\"\")\"\"\"-%$sinG6#F'F**&\"\"$F*%#PiGF*! \"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(``(4*((-1)^k-1)/(k*(k^2-4) *Pi))*sin*k*x,k = 3 .. infinity);" "6#-%$SumG6$**-%!G6#*(\"\"%\"\"\",& ),$F,!\"\"%\"kGF,F,F0F,*(F1F,,&*$F1\"\"#F,F+F0F,%#PiGF,F0F,%$sinGF,F1F ,%\"xGF,/F1;\"\"$%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x) = S um(``(-8/((2*k-1)*(2*k+1)*(2*k-3)*Pi))*sin((2*k-1)*x),k = 1 .. infinit y);" "6#/-%\"FG6#%\"xG-%$SumG6$*&-%!G6#,$*&\"\")\"\"\"**,&*&\"\"#F2%\" kGF2F2F2!\"\"F2,&*&F6F2F7F2F2F2F2F2,&*&F6F2F7F2F2\"\"$F8F2%#PiGF2F8F8F 2-%$sinG6#*&,&*&F6F2F7F2F2F2F8F2F'F2F2/F7;F2%)infinityG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 8*sin*x/(3*Pi)-8*sin*3*x/(15*Pi)-8*sin*5*x/(105*P i)-8*sin*7*x/(315*Pi)-8*sin*9*x/(693*Pi)-` . . . `;" "6#/%!G,.**\"\") \"\"\"%$sinGF(%\"xGF(*&\"\"$F(%#PiGF(!\"\"F(*,F'F(F)F(F,F(F*F(*&\"#:F( F-F(F.F.*,F'F(F)F(\"\"&F(F*F(*&\"$0\"F(F-F(F.F.*,F'F(F)F(\"\"(F(F*F(*& \"$:$F(F-F(F.F.*,F'F(F)F(\"\"*F(F*F(*&\"$$pF(F-F(F.F.%(~.~.~.~GF." } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The following picture shows the graphs of some truncated \+ series." }}{PARA 0 "" 0 "" {TEXT -1 40 "The Fourier series converges r apidly to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in \+ the interval " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 35 " , and to appropriate extensions of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 24 " outside this interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "g := x -> p iecewise(x<0,-sin(x)^2,sin(x)^2):\n'g(x)'=g(x);\ng_ := x -> g(x-2*Pi*f loor((x+Pi)/(2*Pi)));\nFS := (x,n)->sum(-8/((2*k-1)*(2*k+1)*(2*k-3)*Pi )*sin((2*k-1)*x),k=1..n);\nplot([g_(x),FS(x,1),FS(x,2),FS(x,3)],x=-Pi. .3*Pi,\n color=[black,red,blue,green],linestyle=[3,1$3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,$*$)-%$sinGF &\"\"#\"\"\"!\"\"2F'\"\"!7$F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g_Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%\"gG6#,&9$\" \"\"*(\"\"#F1%#PiGF1-%&floorG6#,$*&#F1F3F1*&,&F0F1F4F1F1F4!\"\"F1F1F1F =F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FSGf*6$%\"xG%\"nG6\"6$%) operatorG%&arrowGF)-%$sumG6$,$*.\"\")\"\"\",&*&\"\"#F3%\"kGF3F3F3!\"\" F8,&*&F6F3F7F3F3F3F3F8,&*&F6F3F7F3F3\"\"$F8F8%#PiGF8-%$sinG6#*&F4F39$F 3F3F8/F7;F39%F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 550 248 248 {PLOTDATA 2 "6(-%'CURVESG6%7ct7$$!3*****4tk#fTJ!#<$!3s[aIH.eVR!#M7$$!3 -q3EI:onGF*$!3?e9wG3'pJ(!#>7$$!3'z.'3Js^[FF*$!3W45mE#GrY\"!#=7$$!3*e?6 >$HNHEF*$!3*G!4&=G\"G-CF97$$!3Y:T[9-M&\\#F*$!333U9LQVEOF97$$!3]Dq0(\\F 8O#F*$!3cy*pH$3n[\\F97$$!3=(*zMckUEAF*$!31P'HN#4;$G'F97$$!3))o*QcTD:4# F*$!3M)>jW!4\"[_(F97$$!3_X$Gg`ls&>F*$!3cJ]$H=5$z&)F97$$!3jcu7;***F97$$!3@XDbt&4'4:F*$!3Q/xb4 &3E'**F97$$!3-#))zpi(Qx9F*$!3RE:\"eh,I\"**F97$$!3#)=sS!ol^W\"F*$!3>qmK 6!)*H%)*F97$$!3U#*=E(y@2Q\"F*$!3$H$***QkYIk*F97$$!3/ml6%*yF;8F*$!3S:>+ ;92m$*F97$$!3?*))e6'>)H=\"F*$!3'p7-vz/*p&)F97$$!3Q77?Ggo\\5F*$!3n&)H` \"z>:_(F97$$!3;F/\"*pdqW\"F97$$!39@#G\"fLX)p#F9$!3)HOvi9;m5(F37$ $!375OJ&[a3-#F9$!3A>l/L=cGSF37$$!33***)\\6cDV8F9$!3m_0k@'4Nz\"F37$$!3c $p\"fuhX/5F9$!3AK#p1iVb+\"F37$$!3Y!)Q%oPnll'F3$!3qbMDN#[WU%!#?7$$!3GD3 x2IdoKF3$!3=6KrUl(z1\"F_v7$$\"37*HAIh8U>\"F_v$\"3z=Ocva9E9!#B7$$\"3W' \\m2;F8_$F3$\"3Qk**yp?YR7F_v7$$\"3'HwI-'HBBpF3$\"37*\\!HpBY&y%F_v7$$\" 3&H]pf(Q^K5F9$\"3ahsJG=Ii5F37$$\"3gHf\"fX/FP\"F9$\"3_'f\"o5<^s=F37$$\" 3!Hy3eh&3`?F9$\"37^&eF9ni:%F37$$\"3@O;qvnYLFF9$\"3yx^!y.!f(G(F37$$\"3` \"[TRT8[/%F9$\"3wk6&H6`([:F97$$\"3'oK\"=_+;c`F9$\"3%=+hiF%y/EF97$$\"3[ (>c*G%))pa'F9$\"3>\")zuO.%yq$F97$$\"36o5t0o\"yt(F9$\"3m=&p=iXQ)[F97$$ \"3[_H5Dp#Q:*F9$\"350Du\\KD&G'F97$$\"3o$[ZWq$)p0\"F*$\"3#pqWOLmUe(F97$ $\"3E'f$RN$Qp<\"F*$\"3v*eGsoHt_)F97$$\"3%)3(Rj'H*oH\"F*$\"3SN$QzMW$o#* F97$$\"3k/heU3mm8F*$\"3%**[1&)HV!*e*F97$$\"3X+D$)=(GkV\"F*$\"3zz!QaTP0 #)*F97$$\"3Y)pbpl78Z\"F*$\"3D2G(e1c8!**F97$$\"3D'*)y]f'>1:F*$\"3MR)[Uw E$e**F97$$\"3/%4-K`!3T:F*$\"3[bf$\\Js6***F97$$\"3/#HD8Zkfd\"F*$\"3E^\" *>0Ht****F97$$\"3Nk,UnE%og\"F*$\"39$p.*RA,()**F97$$\"3(o.:N'3sP;F*$\"3 K58A#*yFb**F97$$\"3S4*4'f!*fo;F*$\"3&zg&z`3l/**F97$$\"3o\"y/dDx%*p\"F* $\"3,%y)HZTKN)*F97$$\"3_EX*ykL7w\"F*$\"3d`^a)G*pT'*F97$$\"3NrU3S+*H#=F *$\"3oa$=*)Q`tP*F97$$\"3%\\\"=FSJ]e>F*$\"3/1hgdxlq&)F97$$\"3_e$f/C;S4# F*$\"3lTlfA.G.vF97$$\"3PaX*yvcIA#F*$\"37(=p/)**p:jF97$$\"3A](H`F(4_BF* $\"3W!))*3!pr4/&F97$$\"3))eT.=vt'[#F*$\"3F(=1IB]$4PF97$$\"3an&Q2wx8i#F *$\"3'3?zD>_2Z#F97$$\"3DohnL&>]u#F*$\"3Gtz\\\"F97$$\"3%*oPh18moGF *$\"3s-4GwJ,msF37$$\"3WXoo<+RNJF*$\"3%>wQk()>s%Q!#A7$$\"3)*Q^pQ?-qJF*$ !3]0?@d@7!3)!#@7$$\"34KMqfSl/KF*$!3akX%eCs9(RF_v7$$\"3>D`d=YCT$F*$!3)3$Hbc:^erF37$$\"3tgB8!HNI`$F*$!3s; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "g := x -> piecewise(x<0,-sin(x)^2,sin(x)^2): \n'g(x)'=g(x);\nFourierSeries(g(x),x=-Pi..Pi,numterms=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,$*$)-%$sinGF &\"\"#\"\"\"!\"\"2F'\"\"!7$F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"\")\"\"$\"\"\"*&%#PiG!\"\"-%$sinG6#%\"xGF(F(F(*& #F&\"#:F(*&F*F+-F-6#,$*&F'F(F/F(F(F(F(F+*&#F&\"$0\"F(*&F*F+-F-6#,$*&\" \"&F(F/F(F(F(F(F+*&#F&\"$:$F(*&F*F+-F-6#,$*&\"\"(F(F/F(F(F(F(F+*&#F&\" $$pF(*&F*F+-F-6#,$*&\"\"*F(F/F(F(F(F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "If we use the procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 18 " with the option \"" }{TEXT 268 9 " type=sine" }{TEXT -1 34 "\", we only need the definition of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval \+ " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "f := x -> sin(x)^2:\n'f(x)'=f(x);\nFourierSeries(f(x),x=0..Pi,type=sine,nu mterms=10):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot(F(x),x=-Pi..3*Pi,c olor=COLOR(RGB,.8,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"fG6#%\"xG*$)-%$sinGF&\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,,*&#\"\")\"\"$\"\"\"*&%#PiG!\"\"-%$sinGF&F-F-F-*& #F+\"#:F-*&F/F0-F26#,$*&F,F-F'F-F-F-F-F0*&#F+\"$0\"F-*&F/F0-F26#,$*&\" \"&F-F'F-F-F-F-F0*&#F+\"$:$F-*&F/F0-F26#,$*&\"\"(F-F'F-F-F-F-F0*&#F+\" $$pF-*&F/F0-F26#,$*&\"\"*F-F'F-F-F-F-F0" }}{PARA 13 "" 1 "" {GLPLOT2D 635 252 252 {PLOTDATA 2 "6'-%'CURVESG6#7]t7$$!3*****4tk#fTJ!#<$!3WY[%) p\\Ww!)!#F7$$!3g)43'\\X*!#?7$$!3yMay)3PY+$F*$!3'z#) 3Lq@hE#!#>7$$!3S_J_4$fh$HF*$!3u4>lomLsUF97$$!3-q3EI:onGF*$!31oRZrFkhrF 97$$!3'z.'3Js^[FF*$!3'eMdSLh;W\"!#=7$$!3*e?6>$HNHEF*$!3=K8L3'*)>S#FI7$ $!3Y:T[9-M&\\#F*$!3!H'Hb([q_k$FI7$$!3]Dq0(\\F8O#F*$!3+;8ZHD\"R&\\FI7$$ !3=(*zMckUEAF*$!3Y!Q$3?JhpiFI7$$!3))o*QcTD:4#F*$!3UoEsEdA;vFI7$$!3_X$G g`ls&>F*$!35sosP*FI7$$!3?*))e6'>)H=\"F*$!3sW #H^:T(y&)FI7$$!3Q77?Ggo\\5F*$!3Nf&*>)o))G^(FI7$$!3;F/\"*pd-g6!*\\FI7$$!37kc-@fxskFI$!3I%z? 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" }}{PARA 0 "" 0 "" {TEXT -1 14 "We can exte nd " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " to an " } {TEXT 267 4 "even" }{TEXT -1 41 " function f_even defined on the inter val " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "``(-infinity,infinity);" "6#-%!G6$,$%)infinityG!\"\"F' " }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_even(x) = f(abs(x))" "6#/-%'f_evenG6#%\"xG-%\"fG6#-%$absG6#F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 85 "When using Maple to do t his there appears to be an advantage in using the expression " }{TEXT 268 19 "piecewise(x<0,-x,x)" }{TEXT -1 13 " in place of " }{TEXT 268 6 "abs(x)" }{TEXT -1 20 " because the use of " }{TEXT 268 9 "piecewise " }{TEXT -1 14 " implies that " }{TEXT 286 1 "x" }{TEXT -1 65 " is rea l, and this enables Maple to perform some simplifications." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Examples" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "f := x -> sqrt(x);\nf_even := unapply(simplify(f(pie cewise(x<0,-x,x))),x):\n'f_even(x)'=f_even(x);\nplot(f_even(x),x=-3..3 ,ytickmarks=2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%%sqrtG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_evenG6#%\"xG-%*PIECEWISEG6$7$*$- %%sqrtG6#,$F'!\"\"\"\"\"2F'\"\"!7$*$-F.6#F'F21F4F'" }}{PARA 13 "" 1 " " {GLPLOT2D 462 206 206 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$!\"$\"\"!$\"3 ?x)ov!30K&o%*f;F-7$$!36++]K3XFEF-$\"3qWTZT8%4i\"F-7$$!3%)****\\F)H') \\#F-$\"3iX96%[02e\"F-7$$!3#****\\i3@/P#F-$\"3&R&f@)=<'R:F-7$$!3;++Dr^ b^AF-$\"3E-9(3I=0]\"F-7$$!3$****\\7Sw%G@F-$\"3sC`A#))H*e9F-7$$!3***** \\7;)=,?F-$\"3O9+WyNj99F-7$$!3/++DO\"3V(=F-$\"3aw=`PP0p8F-7$$!3#****** \\V'zViUC\"F-$\"3@3#)*G,la6\"F-7$$!3-++DhkaI6F-$\"3+*zcRerK1\"F -7$$!3s******\\XF`**!#=$\"3!pO3-#*4m(**Ffp7$$!3u*******>#z2))Ffp$\"3a# 30s&Q)\\Q*Ffp7$$!3S++]7RKvuFfp$\"3UC9Iab*fk)Ffp7$$!3s,+++P'eH'Ffp$\"3c FiG\")ykMzFfp7$$!3q)***\\7*3=+&Ffp$\"3!pB$\\!zYB2(Ffp7$$!3[)***\\PFcpP Ffp$\"3#3p7y[w'RhFfp7$$!3;)****\\7VQ[#Ffp$\"3k[M%e$p\"Q)\\Ffp7$$!35)** \\P9(\\$*=Ffp$\"3CMNWiKV^VFfp7$$!32)***\\i6:.8Ffp$\"3#y*yp1&=*4OFfp7$$ !3%p)*\\i:sw%)*!#>$\"3aX^.(3+\"QJFfp7$$!39$***\\(oKQm'Fds$\"3)GOI9/S9e #Ffp7$$!3O***\\(=K**zMFds$\"3y&4^I*RZl=Ffp7$$!3Wb+++v`hH!#?$\"3f.yl55+ UaFds7$$\"3l%**\\ilg4,$Fds$\"3aW=t\"*>@NoHO%>BVFfp7$$ \"38****\\7:xWCFfp$\"3X!>Cv2jW%\\Ffp7$$\"3E,++vuY)o$Ffp$\"3[(>5[XvK2'F fp7$$\"3!z******4FL(\\Ffp$\"3q3jv-?=_qFfp7$$\"3A)****\\d6.B'Ffp$\"3#[f $=vAB$*yFfp7$$\"3s****\\(o3lW(Ffp$\"3[^Qy]bJH')Ffp7$$\"35*****\\A))oz) Ffp$\"3;\"**\\8zs\"z$*Ffp7$$\"3e******Hk-,5F-$\"3/EFP$380+\"F-7$$\"36+ ++D-eI6F-$\"3m[<8huGj5F-7$$\"3u***\\(=_(zC\"F-$\"3Vyy07\"Gr6\"F-7$$\"3 M+++b*=jP\"F-$\"3A6?X4i;t6F-7$$\"3g***\\(3/3(\\\"F-$\"3#4u7F-7$$\"3u*****\\KCnu\"F-$\"3wOi%o)pj@8F- 7$$\"3s***\\(=n#f(=F-$\"3q3N!=uW'p8F-7$$\"3P+++!)RO+?F-$\"3>G=)GCUVT\" F-7$$\"30++]_!>w7#F-$\"3'H'>oCgje9F-7$$\"3O++v)Q?QD#F-$\"3oVe!G#HF,:F- 7$$\"3G+++5jypBF-$\"3x@S)p-6%R:F-7$$\"3<++]Ujp-DF-$\"3mztme7*>e\"F-7$$ \"3++++gEd@EF-$\"3-l?/Hr7>;F-7$$\"39++v3'>$[FF-$\"3)HZI2m0yl\"F-7$$\"3 7++D6EjpGF-$\"37N$y(***)*Rp\"F-7$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"#5!\" \"$F*F*Fh]l-%+AXESLABELSG6$Q\"x6\"Q!F]^l-%*AXESTICKSG6$%(DEFAULTG\"\"# -%%VIEWG6$;F(F_]lFb^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 80 "When f is already even, the even extension defined in this way coincides with f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "f := x -> x*sin(x);\nf_even := unapply(simplify(f(piecewise(x<0,-x,x))),x):\n'f_even(x)'=f_even(x );\nplot(f_even(x),x=-4..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG f*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%$sinG6#F-F.F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_evenG6#%\"xG*&F'\"\"\"-%$sinGF&F )" }}{PARA 13 "" 1 "" {GLPLOT2D 475 198 198 {PLOTDATA 2 "6%-%'CURVESG6 $7jo7$$!\"%\"\"!$!3\"GrJ7)*4s-$!#<7$$!3cLLL$Q6G\"RF-$!3tqE%RLnss#F-7$$ !3ommmmFiDQF-$!3X)e)o7$$!3M+++0vJ\"3$F-$\"3yR%zPIWh&=FQ7$$!3smmmhN2-IF-$\"3+Lo^20* [<%FQ7$$!34LLL[5-?HF-$\"3+H@w[\"GrT'FQ7$$!3!******\\`oz$GF-$\"33,$\\2$ G(\\[)FQ7$$!3!omm;)3DoEF-$\"3V=tQ-tN;7F-7$$!3?+++:v2*\\#F-$\"3oRFrnXZ( \\\"F-7$$!3BLLL8>1DBF-$\"3EJA =F-7$$!3;+++S(R#**>F-$\"3&z\"f\"[aN&==F-7$$!35+++?4h7>F-$\"3..YH'\\i>! 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NF1$\"3=UGn_$>7W%F-7$$!3YmmmOk]JLF1$\"3#)3q;V7L1DBF1$\"3pCRv#fD\\,'F-7$$!3kmmmw))yr@F1$\"3[V#\\( H**e0jF-7$$!3;+++S(R#**>F1$\"3q.lehlNomF-7$$!30++++@)f#=F1$\"3A.rGic=x qF-7$$!3-+++gi,f;F1$\"38Ur'eny:_(F-7$$!3qmmm\"G&R2:F1$\"3!RtVc*[SwzF-7 $$!3XLLLtK5F8F1$\"35_![dxvVf)F-7$$!3eLLL$HsV<\"F1$\"3_j,6lo0)>*F-7$$!3 +-++]&)4n**F-$\"3&e'GA$yZ;+\"F17$$!37PLLL\\[%R)F-$\"33le%\\T#G(3\"F17$ $!3G)*****\\&y!pmF-$\"3q'eE&pj#)*>\"F17$$!3'))******4\"eZeF-$\"3E`4s_A -i7F17$$!3Y******\\O3E]F-$\"3(4)>T6)=5L\"F17$$!3!fmm;CP*oTF-$\"3=\\&3. 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" }}{PARA 0 " " 0 "" {TEXT -1 14 "We can extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 7 " to an " }{TEXT 267 3 "odd" }{TEXT -1 10 " function " }{XPPEDIT 18 0 "f_odd(x)" "6#-%&f_oddG6#%\"xG" }{TEXT -1 25 " defin ed on the interval " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "``(-infinity,infinity);" "6#-%!G6$,$ %)infinityG!\"\"F'" }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_odd(x) = f(abs(x))*signum(x);" "6#/-%&f_oddG6#% \"xG*&-%\"fG6#-%$absG6#F'\"\"\"-%'signumG6#F'F/" }{TEXT -1 2 ".," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "signum(x)=PIECEWIS E([1,x>0],[0, x=0],[-1 , x<0])" "6#/-%'signumG6#%\"xG-%*PIECEWISEG6%7$ \"\"\"2\"\"!F'7$F./F'F.7$,$F,!\"\"2F'F." }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "When using Maple t o do this there appears to be an advantage in using the expression " } {TEXT 268 19 "piecewise(x<0,-x,x)" }{TEXT -1 13 " in place of " } {TEXT 268 6 "abs(x)" }{TEXT -1 20 " because the use of " }{TEXT 268 9 "piecewise" }{TEXT -1 14 " implies that " }{TEXT 285 1 "x" }{TEXT -1 65 " is real, and this enables Maple to perform some simplifications. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 8 "Examp les" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "f := x -> sqrt(x);\nf_odd := unapply(sim plify(f(piecewise(x<0,-x,x))*signum(x)),x):\n'f_odd(x)'=f_odd(x);\nplo t(f_odd(x),x=-3..3,ytickmarks=2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fG%%sqrtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6#%\"xG-%* PIECEWISEG6$7$,$*$-%%sqrtG6#,$F'!\"\"\"\"\"F21F'\"\"!7$*$-F/6#F'F32F5F '" }}{PARA 13 "" 1 "" {GLPLOT2D 352 250 250 {PLOTDATA 2 "6&-%'CURVESG6 $7V7$$!\"$\"\"!$!3?x)ov!30K&o%*f;F-7$$!36++]K3XFEF-$!3qWTZT8%4i\"F-7$$! 3%)****\\F)H')\\#F-$!3iX96%[02e\"F-7$$!3#****\\i3@/P#F-$!3&R&f@)=<'R:F -7$$!3;++Dr^b^AF-$!3E-9(3I=0]\"F-7$$!3$****\\7Sw%G@F-$!3sC`A#))H*e9F-7 $$!3*****\\7;)=,?F-$!3O9+WyNj99F-7$$!3/++DO\"3V(=F-$!3aw=`PP0p8F-7$$!3 #******\\V'zViUC\"F-$!3@3#)*G,la6\"F-7$$!3-++DhkaI6F-$!3+*zcRerK1\" F-7$$!3s******\\XF`**!#=$!3!pO3-#*4m(**Ffp7$$!3u*******>#z2))Ffp$!3a#3 0s&Q)\\Q*Ffp7$$!3S++]7RKvuFfp$!3UC9Iab*fk)Ffp7$$!3s,+++P'eH'Ffp$!3cFiG \")ykMzFfp7$$!3q)***\\7*3=+&Ffp$!3!pB$\\!zYB2(Ffp7$$!3[)***\\PFcpPFfp$ !3#3p7y[w'RhFfp7$$!3;)****\\7VQ[#Ffp$!3k[M%e$p\"Q)\\Ffp7$$!32)***\\i6: .8Ffp$!3#y*yp1&=*4OFfp7$$!39$***\\(oKQm'!#>$!3)GOI9/S9e#Ffp7$$!3Wb+++v `hH!#?$!3f.yl55+UaF_s7$$\"3%[***\\i]2=jF_s$\"3+&[E$R\"yN^#Ffp7$$\"3]** **\\(QIKH\"Ffp$\"3%))[P'e7:'f$Ffp7$$\"3K******\\4+p=Ffp$\"3#o>oHO%>BVF fp7$$\"38****\\7:xWCFfp$\"3X!>Cv2jW%\\Ffp7$$\"3E,++vuY)o$Ffp$\"3[(>5[X vK2'Ffp7$$\"3!z******4FL(\\Ffp$\"3q3jv-?=_qFfp7$$\"3A)****\\d6.B'Ffp$ \"3#[f$=vAB$*yFfp7$$\"3s****\\(o3lW(Ffp$\"3[^Qy]bJH')Ffp7$$\"35*****\\ A))oz)Ffp$\"3;\"**\\8zs\"z$*Ffp7$$\"3e******Hk-,5F-$\"3/EFP$380+\"F-7$ $\"36+++D-eI6F-$\"3m[<8huGj5F-7$$\"3u***\\(=_(zC\"F-$\"3Vyy07\"Gr6\"F- 7$$\"3M+++b*=jP\"F-$\"3A6?X4i;t6F-7$$\"3g***\\(3/3(\\\"F-$\"3#4u7F-7$$\"3u*****\\KCnu\"F-$\"3wOi%o)p j@8F-7$$\"3s***\\(=n#f(=F-$\"3q3N!=uW'p8F-7$$\"3P+++!)RO+?F-$\"3>G=)GC UVT\"F-7$$\"30++]_!>w7#F-$\"3'H'>oCgje9F-7$$\"3O++v)Q?QD#F-$\"3oVe!G#H F,:F-7$$\"3G+++5jypBF-$\"3x@S)p-6%R:F-7$$\"3<++]Ujp-DF-$\"3mztme7*>e\" F-7$$\"3++++gEd@EF-$\"3-l?/Hr7>;F-7$$\"39++v3'>$[FF-$\"3)HZI2m0yl\"F-7 $$\"37++D6EjpGF-$\"37N$y(***)*Rp\"F-7$$\"\"$F*$\"3?x)ov!30K " 0 "" {MPLTEXT 1 0 143 "f := x -> x^2*sin(x);\nf_odd := unapply(simplify(f(p iecewise(x<0,-x,x))*signum(x)),x):\n'f_odd(x)'=f_odd(x);\nplot(f_odd(x ),x=-4..4,ytickmarks=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6# %\"xG6\"6$%)operatorG%&arrowGF(*&)9$\"\"#\"\"\"-%$sinG6#F.F0F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6#%\"xG*&)F'\"\"#\"\"\"-%$si nGF&F+" }}{PARA 13 "" 1 "" {GLPLOT2D 406 272 272 {PLOTDATA 2 "6&-%'CUR VESG6$7]o7$$!\"%\"\"!$\"37&o#\\#*R)3@\"!#;7$$!3cLLL$Q6G\"R!#<$\"3U.W'p EGr1\"F-7$$!3ommmmFiDQF1$\"33-Jm/!>%[#*F17$$!37++]Q]1tG lc&F17$$!3nmmmwxE.NF1$\"3BtB4TelUVF17$$!3Mmmm1rQ!=$F17 $$!3YmmmOk]JLF1$\"3[5G5AH>&4#F17$$!3_LLL[9cgJF1$\"3g!fP(yuq%*=!#=7$$!3 smmmhN2-IF1$!3eTpaTGL`7F17$$!3!******\\`oz$GF1$!3jR'*R\"f3!3CF17$$!3!o mm;)3DoEF1$!35^m^WkaXKF17$$!3]LLL)>kOe#F1$!3Jb(HEe9T`$F17$$!3?+++:v2* \\#F1$!3o__X@]IUPF17$$!3$pmmTrp?T#F1$!3qO@\"*fB$y(QF17$$!3BLLL8>1DBF1$ !3IH\"4Z=#pRRF17$$!3%******\\RD%[AF1$!3q\"o7+%)z%QRF17$$!3kmmmw))yr@F1 $!3/]jb!*y=!*QF17$$!3SLLL3V^&3#F1$!31.())3**Gey$F17$$!3;+++S(R#**>F1$! 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\"\"F*7$$!3ommmmFiDQ!#1DBF0F+7$$!3kmmmw))y r@F0F+7$$!3;+++S(R#**>F0F+7$$!30++++@)f#=F0F+7$$!3-+++gi,f;F0F+7$$!3qm mm\"G&R2:F0F+7$$!3XLLLtK5F8F0F+7$$!3eLLL$HsV<\"F0F+7$$!3!pmmT2Tb3\"F0F +7$$!3+-++]&)4n**!#=F_o7$$!3cpmmT(******z-6j'FaoFbq7$$\" 3q\"******4#32$)FaoFeq7$$\"3q#****\\qM8F0F`r7$$\"3%)*******HSu]\"F0F`r7$$\"3'HLL$ep'Rm\"F0F`r7$$\"3')** ****R>4N=F0F`r7$$\"3#emm;@2h*>F0F`r7$$\"3]*****\\c9W;#F0F`r7$$\"3Lmmmm d'*GBF0F`r7$$\"3j*****\\iN7]#F0F`r7$$\"3aLLLt>:nEF0F`r7$$\"35LLL.a#o$G F0F`r7$$\"3ammm^Q40IF0F`r7$$\"3y******z]rfJF0F`r7$$\"3gmmmc%GpL$F0F`r7 $$\"3/LLL8-V&\\$F0F`r7$$\"3=+++XhUkOF0F`r7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 127 "f := x -> 2/(x+1);\nf_odd := unapp ly(simplify(f(piecewise(x<0,-x,x))*signum(x)),x):\n'f_odd(x)'=f_odd(x) ;\nplot(f_odd(x),x=-4..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.,&9$F.F.F.!\"\"\"\"#F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6#%\"xG-%*PIECEWISEG6%7 $,$*&\"\"\"F.,&F'F.F.!\"\"F0\"\"#2F'\"\"!7$F3/F'F37$,$*&F.F.,&F.F.F'F. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(i) \+ Find the first few terms of the Fourier half-range " }{TEXT 268 11 "si ne series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" "6 #-%\"fG6#%\"xG" }{TEXT -1 64 ", that is, the Fourier series for the od d periodic extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 16 " with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated series." }}{PARA 0 "" 0 "" {TEXT -1 56 "(ii) Find the first few terms of the Fourier half-range " }{TEXT 268 13 "c osine series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 64 ", that is, the Fourier series for the \+ odd periodic extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 16 " with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "(i) The periodic odd extension of " }{XPPEDIT 18 0 " f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 27 " can be graphed as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "f := x -> piecewise(x<1,x,1):\n'f(x)'=f(x);\nf_odd := unapply(sim plify(f(piecewise(x<0,-x,x))*signum(x)),x):\n'f_odd(x)'=f_odd(x);\nf_ \+ := x -> f_odd(x-4*floor((x+2)/4)):\n'f_(x)'='f_odd(x-4*floor((x+2)/4)) ';\nplot(f_(x),x=-4.3..8.3,color=COLOR(RGB,.4,0,1),thickness=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$F'2F'\" \"\"7$F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6#% \"xG-%*PIECEWISEG6%7$!\"\"1F'F,7$F'2F'\"\"\"7$F01F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%&f_oddG6#,&F'\"\"\"*&\"\"%F,-%&flo orG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 775 159 159 {PLOTDATA 2 "6'-%'CURVESG6#7\\s7$$!3#)*************H%!#<$! 3?)*************H!#=7$$!3)****\\(GzniTF*$!3y****\\(Gzni\"F-7$$!38++]de NDSF*$!3T8++]deND!#>7$$!3-+DcdE(e!RF*$\"3M)**\\PCMFT*F87$$!3!***\\id%* Q'y$F*$\"3,,+vBa5O@F-7$$!3%)*\\PH5=?l$F*$\"3c,]iq*=)zMF-7$$!3z***\\#[n kF*$!\"\"F[o7$$!3y\\(of41g)>F*Fjp7$$!3oC\"GVfzw(>F*Fjp7$ $!3#)*\\(o#4`$p>F*Fjp7$$!3%)\\iS*3+F&>F*Fjp7$$!3()**\\7'3Zg$>F*Fjp7$$! 3k*\\7)*z4!*z\"F*Fjp7$$!3U****\\8D(>m\"F*Fjp7$$!3]***\\2[n0U\"F*Fjp7$$ !3[**\\iN]o%G\"F*Fjp7$$!3W****\\!f-)[6F*Fjp7$$!3]***\\ZR#e!3\"F*Fjp7$$ !3d******)>iB,\"F*Fjp7$$!3S'***\\K+UT%*F-Fas7$$!31(****\\2=#f()F-Fds7$ $!3'e****\\3iVW(F-Fgs7$$!3m%****\\41&HhF-Fjs7$$!38'*\\i!*3\\N\\F-F]t7$ $!3g(**\\iov9u$F-F`t7$$!3%e*\\i?hn@BF-Fct7$$!3&3%****\\bw=!*F8Fft7$$\" 3Hs++D\"*z3IF8Fit7$$\"3a3++!QOO]\"F-F\\u7$$\"3'y+v=3GF!HF-F_u7$$\"3=2+ v$y>=I%F-Fbu7$$\"3W/](o+`-a&F-Feu7$$\"3q,++IioynF-Fhu7$$\"3l/](ovVu8)F -F[v7$$\"3g2+v$G,i\\*F-F^v7$$\"3)y](o/fm>)*F-Fav7$$\"3#3]iD08V,\"F*Fin 7$$\"3'4DcY^fm/\"F*Fin7$$\"3)3+]n(f+z5F*Fin7$$\"3$4]P4!*)pV6F*Fin7$$\" 3)4+D^#=R37F*Fin7$$\"33,vV%Q#RV8F*Fin7$$\"3?,+vVHRy9F*Fin7$$\"3X+](eDQ js\"F*Fin7$$\"3?+Dc8&f+'=F*Fin7$$\"3)****\\7x!y$*>F*Fin7$$\"3(fSml%>Y- ?F*Fjp7$$\"3T7G)=7V6,#F*Fjp7$$\"3&)=#*>(HC)>?F*Fjp7$$\"3&[i:DZ0&G?F*Fj p7$$\"3uP%[J#y'e/#F*Fjp7$$\"3=]7yt,Bj?F*Fjp7$$\"3]vo/v[&z4#F*Fjp7$$\"3 Q+DJw&zE8#F*Fjp7$$\"3e]P%)y*G@?#F*Fjp7$$\"3!3+v8Qy:F#F*Fjp7$$\"3_++]*> !\\#R#F*Fjp7$$\"3E+]iLpq,&F*Fin7$$\"3!*\\i:&4p/3&F*Fin7$$\"3&) **\\Pe)oQ9&F*Fin7$$\"3]+]7**3Tw_F*Fin7$$\"39,]()RH&*3aF*Fin7$$\"3!**** *\\#3@\"ocF*Fin7$$\"3P+v$ff$y.eF*Fin7$$\"3%3+v$4hWRfF*Fin7$$\"3giS\"** \\yd&fF*Fin7$$\"3GDJX!*36sfF*Fin7$$\"3gcEs&3x-)fF*Fin7$$\"3#z=#*4GV%)) fF*Fin7$$\"3E>BmbsF*$!3(G***\\2oPVuF-7$$\"3O+]i_C[!Q(F*$!3U'**\\PZv^>'F-7$$\"3++++' e-`](F*$!3-+++ST(p%\\F-7$$\"3^+v=#)oQQwF*$!3'\\*\\7y68;OF-7$$\"3-,]Py6 ZrxF*$!3!**)*\\i@)G&G#F-7$$\"3[,++J)\\))*yF*$!38&)*****o,:,\"F-7$$\"33 ,]i$[Gi-)F*$\"3c2,]i$[Gi#F87$$\"3)3]7=C9J;)F*$\"3$*3]7=C9J;F-7$$\"3q++ +++++$)F*$\"332++++++IF--%+AXESLABELSG6$Q\"x6\"Q!Fghl-%*THICKNESSG6#\" \"#-%&COLORG6&%$RGBG$\"\"%F[q$F[oF[oFin-%%VIEWG6$;$!#VF[q$\"#$)F[q%(DE FAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "T he following code determines the first few terms of the half-range sin e series of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 97 " \+ and then plots the graph of the truncated series along with the wavefo rm which it approximates. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "f := x -> piecewise(x<1,x,1):\n'f( x)'=f(x);\nFourierSeries(f(x),x=0..2,type=sine,numterms=9,info=1):\nF \+ := unapply(%,x):\n'F(x)'=F(x);\nf_odd := unapply(simplify(f(piecewise( x<0,-x,x))*signum(x)),x):\nf_ := x -> f_odd(x-4*floor((x+2)/4)):\nplot ([f_(x),F(x)],x=-4.2..4.2,color=[black,red],linestyle=[3,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$F'2F'\" \"\"7$F-%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~sine~co efficient~..~G/,$**\"\"#\"\"\",&*(-%$cosG6#*&%\"kGF(%#PiGF(F(F/F(F0F(F (*&F'F(-%$sinG6#,$*(F'!\"\"F/F(F0F(F(F(F7F(F/!\"#F0F8F7,$**F'F(,&*()F7 F/F(F/F(F0F(F(*&F'F(F2F(F7F(F/F8F0F8F7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,4**\"\"#\"\"\",&%#PiGF+F*F+F+F-!\"#-%$sinG6#,$*(F *!\"\"F-F+F'F+F+F+F+*&F-F4-F06#*&F-F+F'F+F+F4*&#F*\"\"*F+*(,&*&\"\"$F+ F-F+F+F*F4F+F-F.-F06#,$**F?F+F*F4F-F+F'F+F+F+F+F+*&#F+F*F+*&F-F4-F06#, $*(F*F+F-F+F'F+F+F+F+F4*&#F*\"#DF+*(,&*&\"\"&F+F-F+F+F*F+F+F-F.-F06#,$ **FQF+F*F4F-F+F'F+F+F+F+F+*&#F+F?F+*&F-F4-F06#,$*(F?F+F-F+F'F+F+F+F+F4 *&#F*\"#\\F+*(,&*&\"\"(F+F-F+F+F*F4F+F-F.-F06#,$**F]oF+F*F4F-F+F'F+F+F +F+F+*&#F+\"\"%F+*&F-F4-F06#,$*(FdoF+F-F+F'F+F+F+F+F4*&#F*\"#\")F+*(,& *&F;F+F-F+F+F*F+F+F-F.-F06#,$**F;F+F*F4F-F+F'F+F+F+F+F+" }}{PARA 13 " " 1 "" {GLPLOT2D 577 214 214 {PLOTDATA 2 "6&-%'CURVESG6%7[p7$$!3;+++++ ++U!#<$!3w,++++++?!#=7$$!3R+++0R!p,%F*$!3aQ+++0R!p\"!#>7$$!3%****\\<(H fdQF*$\"3n++]#GqSU\"F-7$$!39++]l6VyOF*$\"3X)****\\M)o:KF-7$$!39++]e<3) \\$F*$\"3i)****\\T#=>]F-7$$!3C++v?&*e=LF*$\"3c(***\\#z/T\"oF-7$$!3I++v Rs<_JF*$\"3'p***\\-wAy%)F-7$$!39++v+@-mIF*$\"3a)***\\#**y(R$*F-7$$!3++ +vhp')zHF*$\"\"\"\"\"!7$$!31++v$>l2*GF*FU7$$!3;++vDMm,GF*FU7$$!3#**** \\2RJSi#F*FU7$$!3?+++4]JTCF*FU7$$!3m****\\?$y.G#F*FU7$$!3k***\\P-!z*=# F*FU7$$!3i*****ps,#*4#F*FU7$$!3'*****\\'f@P0#F*FU7$$!3')*****fYT#3?F*F U7$$!3\"**\\(o\\kb-?F*FU7$$!3%***\\PL9(o*>F*$!\"\"FW7$$!3++D1F* Fcp7$$!3/++v+9]&)>F*Fcp7$$!37+]7o88u>F*Fcp7$$!3)*****\\N8wi>F*Fcp7$$!3 $****\\-F@+%>F*Fcp7$$!35+++07G<>F*Fcp7$$!3()******QTiH=F*Fcp7$$!3l**** *H2n>u\"F*Fcp7$$!3)****\\d/lFe\"F*Fcp7$$!3/+++P%eMR\"F*Fcp7$$!3))***** z!44L7F*Fcp7$$!3#***\\(y7=)R6F*Fcp7$$!3'****\\xMXl/\"F*Fcp7$$!3E***\\( G8$)R'*F-F]s7$$!3())*****z\"4U\"))F-F`s7$$!3T'***\\xC`-qF-Fcs7$$!3)y** *\\KyQx_F-Ffs7$$!3_'****\\P!QxMF-Fis7$$!34\"***\\F;TC=F-F\\t7$$!3OA(** **\\_h9%!#?F_t7$$\"3L****\\UD_5=F-Fct7$$\"3z)***\\<,oAMF-Fft7$$\"3y,++ lW&Q;&F-Fit7$$\"3E.++SzlipF-F\\u7$$\"3G*****\\?OCs)F-F_u7$$\"3A,+v$=uP d*F-Fbu7$$\"3K++D;7^U5F*FU7$$\"3K+](QyPq8\"F*FU7$$\"3K++]^VcJ7F*FU7$$ \"3x*****>+P9S\"F*FU7$$\"3'3++]J7Ge\"F*FU7$$\"3M++D1`;Z,++Pl%o#>F*FU7$$\"3)3D19zzz%>F*FU7$$\"3!3]7e/8\"p>F*FU 7$$\"3'ei:Inz'z>F*FU7$$\"3q](=-IY-*>F*FU7$$\"378.#QhHb*>F*FU7$$\"3bv=U FH\"3+#F*Fcp7$$\"3)zVB5C'41?F*Fcp7$$\"3S+]ia&z8,#F*Fcp7$$\"3A+vVjgk`?F *Fcp7$$\"3g***\\Ad7f4#F*Fcp7$$\"3-++vKRF%=#F*Fcp7$$\"3Y++D$HNEF#F*Fcp7 $$\"3a+++bSTXCF*Fcp7$$\"3(****\\iS(HEEF*Fcp7$$\"31,++s&40!GF*Fcp7$$\"3 k++vA\")e*)GF*Fcp7$$\"3E++]tmmyHF*Fcp7$$\"3Y+]()3w+nIF*$!3V&**\\7\"R#* H$*F-7$$\"3m++DW&[`:$F*$!3O$***\\dX^Y%)F-7$$\"3$4++S$3q#F-7$F:$\"3!**G!oX*QA&GF-7$$! 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}{TEXT -1 27 " can be graphed as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "f := x -> p iecewise(x<1,x,1):\n'f(x)'=f(x);\nf_even := unapply(simplify(f(piecewi se(x<0,-x,x))),x):\n'f_even(x)'=f_even(x);\nf_ := x -> f_even(x-4*floo r((x+2)/4)):\n'f_(x)'='f_even(x-4*floor((x+2)/4))';\nplot(f_(x),x=-4.3 ..8.3,color=COLOR(RGB,.4,0,1),thickness=2,numpoints=60);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$F'2F'\"\"\"7$F-%* otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_evenG6#%\"xG-%*PIE CEWISEG6&7$\"\"\"1F'!\"\"7$,$F'F.2F'\"\"!7$F'2F'F,7$F,1F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_evenG6#,&F'\"\"\"*&\"\"%F ,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 778 149 149 {PLOTDATA 2 "6'-%'CURVESG6#7`x7$$!3#)*********** **H%!#<$\"3?)*************H!#=7$$!3\\znSu,9WUF*$\"3([znSu,9W#F-7$$!39f N\")[.G)=%F*$\"3]\"fN\")[.G)=F-7$$!3#)Q.AB0UKTF*$\"3:)Q.AB0UK\"F-7$$!3 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6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The first few ter ms of the half-range cosine series of " }{XPPEDIT 18 0 "f(x);" "6#-%\" fG6#%\"xG" }{TEXT -1 25 " can be found as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "f := x -> p iecewise(x<1,x,1):\nFourierSeries(f(x),x=0..2,type=cos,numterms=3,info =true):\nF := unapply(%,x):\n'F(x)'=F(x);\nf_even := unapply(simplify( f(piecewise(x<0,-x,x))),x):\nf_ := x -> f_even(x-4*floor((x+2)/4)):\np lot([f_(x),F(x)],x=-4.2..4.2,color=[red,green],thickness=[1,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G#\"\"$\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%G,$*&\"\"%F,F+!\"#F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%/coefficient~ofG-%$cosG6#*&%#PiG\"\"\"%\"xGF)%% ~-->G,$*&\"\"#F)F(!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%/coeff icient~ofG-%$cosG6#,$**\"\"$\"\"\"\"\"#!\"\"%#PiGF*%\"xGF*F*%%~-->G,$* (\"\"%F*\"\"*F,F-!\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#% \"xG,*#\"\"$\"\"%\"\"\"*(F+F,%#PiG!\"#-%$cosG6#,$*(\"\"#!\"\"F.F,F'F,F ,F,F6*(F5F,F.F/-F16#*&F.F,F'F,F,F6*&#F+\"\"*F,*&F.F/-F16#,$**F*F,F5F6F .F,F'F,F,F,F,F6" }}{PARA 13 "" 1 "" {GLPLOT2D 687 206 206 {PLOTDATA 2 "6&-%'CURVESG6%7[p7$$!3;+++++++U!#<$\"3w,++++++?!#=7$$!3G++]_>X3TF*$\" 3#G++]_>X3\"F-7$$!3R+++0R!p,%F*$\"3aQ+++0R!p\"!#>7$$!3;+]PQ%[s$RF*$\"3 5%)**\\ih:viF87$$!3%****\\<(HfdQF*$\"3n++]#GqSU\"F-7$$!3/+]io?,oPF*$\" 3c***\\PJz)>BF-7$$!39++]l6VyOF*$\"3X)****\\M)o:KF-7$$!3O+++ikD)e$F*$\" 3J'*****z`V]F-7$$!3>+]iRcL3MF*$ \"34)**\\PgVm\"fF-7$$!3C++v?&*e=LF*$\"3c(***\\#z/T\"oF-7$$!30++D!Q$QNK F*$\"3[****\\(>mhk(F-7$$!3I++vRs<_JF*$\"3'p***\\-wAy%)F-7$$!39++v+@-mI F*$\"3a)***\\#**y(R$*F-7$$!3+++vhp')zHF*$\"\"\"\"\"!7$$!31++v$>l2*GF*F ]p7$$!3;++vDMm,GF*F]p7$$!3#****\\2RJSi#F*F]p7$$!3?+++4]JTCF*F]p7$$!3m* ***\\?$y.G#F*F]p7$$!3i*****ps,#*4#F*F]p7$$!35+++07G<>F*F]p7$$!3l*****H 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\"3o5@^%[G(*e'F-7$Fd\\l$\"3tf=5=`OWgF-7$$\"3#3+D\"ojDdMF*$\"3I\"HWEB<* zaF-7$Fi\\l$\"3_iu,:!Qf!\\F-7$$\"3!3+D1f\"QXNF*$\"3c;\"[PrDIR%F-7$F^]l $\"3#GQi/io())QF-7$$\"3'3+vG,&fGOF*$\"3 " 0 "" {MPLTEXT 1 0 270 "f := x -> piecewise(x<1,x,1):\nFourierSeries(f(x),x= 0..2,type=cos,numterms=15):\nF := unapply(%,x):\n'F(x)'=F(x);\nf_even \+ := unapply(simplify(f(piecewise(x<0,-x,x))),x):\nf_ := x -> f_even(x-4 *floor((x+2)/4)):\nplot([f_(x),F(x)],x=-4.2..4.2,color=[red,green],thi ckness=[1,2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,<#\" \"$\"\"%\"\"\"*(F+F,%#PiG!\"#-%$cosG6#,$*(\"\"#!\"\"F.F,F'F,F,F,F6*(F5 F,F.F/-F16#*&F.F,F'F,F,F6*&#F+\"\"*F,*&F.F/-F16#,$**F*F,F5F6F.F,F'F,F, F,F,F6*&#F+\"#DF,*&F.F/-F16#,$**\"\"&F,F5F6F.F,F'F,F,F,F,F6*&#F5F=F,*& F.F/-F16#,$*(F*F,F.F,F'F,F,F,F,F6*&#F+\"#\\F,*&F.F/-F16#,$**\"\"(F,F5F 6F.F,F'F,F,F,F,F6*&#F+\"#\")F,*&F.F/-F16#,$**F=F,F5F6F.F,F'F,F,F,F,F6* &#F5FEF,*&F.F/-F16#,$*(FKF,F.F,F'F,F,F,F,F6*&#F+\"$@\"F,*&F.F/-F16#,$* 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" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 55 "(i) Find the first few terms of the F ourier half-range " }{TEXT 268 11 "sine series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 64 ", t hat is, the Fourier series for the odd periodic extension of " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " with period 6. \+ " }}{PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated ser ies." }}{PARA 0 "" 0 "" {TEXT -1 56 "(ii) Find the first few terms of \+ the Fourier half-range " }{TEXT 268 13 "cosine series" }{TEXT -1 18 " \+ for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 64 ", that is, the Fourier series for the odd periodic extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " with period 6. " }}{PARA 0 "" 0 "" {TEXT -1 41 "Draw the graphs of some truncated se ries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 " Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "(i) The per iodic odd extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 27 " can be graphed as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "f := x -> 3-x:\n'f(x )'=f(x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x)), x):\n'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-6*floor((x+3)/6)):\n'f_( x)'='f_odd(x-6*floor((x+3)/6))';\nplot(f_(x),x=-6.3..12.3,color=COLOR( RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6# %\"xG,&\"\"$\"\"\"F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_odd G6#%\"xG-%*PIECEWISEG6%7$,&F'!\"\"\"\"$F-2F'\"\"!7$F0/F'F07$,&F.\"\"\" F'F-2F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%&f_oddG6# ,&F'\"\"\"*&\"\"'F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }} {PARA 13 "" 1 "" {GLPLOT2D 574 185 185 {PLOTDATA 2 "6'-%'CURVESG6#7gp7 $$!3#)*************H'!#<$!3;+++++++FF*7$$!3')**\\7LKk)>'F*$!38+](owc8! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 298 "f := x -> 3-x:\n'f(x)'=f(x) ;\nFourierSeries(f(x),x=0..3,type=sin,numterms=9,info=1):\nF := unappl y(%,x):\n'F(x)'=F(x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x) )*signum(x)),x):\nf_ := x -> f_odd(x-6*floor((x+3)/6)):\nplot([f_(x),F (x)],x=-6.3..12.3,color=[black,red],linestyle=[3,1],numpoints=100);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"F'!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~sine~coefficient~..~G/,$** \"\"'\"\"\",&*&%\"kGF(%#PiGF(F(-%$sinG6#F*!\"\"F(F+!\"#F,F1F(,$*(F'F(F +F0F,F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,4*(\"\"'\" \"\"%#PiG!\"\"-%$sinG6#,$*(\"\"$F-F,F+F'F+F+F+F+*(F3F+F,F--F/6#,$**\" \"#F+F3F-F,F+F'F+F+F+F+*(F9F+F,F--F/6#*&F,F+F'F+F+F+*&#F3F9F+*&F,F--F/ 6#,$**\"\"%F+F3F-F,F+F'F+F+F+F+F+*&#F*\"\"&F+*&F,F--F/6#,$**FHF+F3F-F, F+F'F+F+F+F+F+*&F,F--F/6#,$*(F9F+F,F+F'F+F+F+F+*&#F*\"\"(F+*&F,F--F/6# ,$**FUF+F3F-F,F+F'F+F+F+F+F+*&#F3FEF+*&F,F--F/6#,$**\"\")F+F3F-F,F+F'F +F+F+F+F+*&#F9F3F+*&F,F--F/6#,$*(F3F+F,F+F'F+F+F+F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 623 231 231 {PLOTDATA 2 "6&-%'CURVESG6%7\\t7$$!3#)***** ********H'!#<$!3;+++++++FF*7$$!3/III]Yr,iF*$!3'*ppp\\`G)z#F*7$$!3Eggg+ $HM5'F*$!3uRRR*pql*GF*7$$!3C[[)>_q11'F*$!3w^^,y%H$RHF*7$$!3@OOOV<\"z,' F*$!3!QOOmD)3#)HF*7$$!3g%)f.'*oc7gF*$!3S:S'R5Lu)HF*7$$!3(QL3([?A2gF*$! 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This is because the odd perio dic extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " is discontinuous at 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "f := x -> 3-x:\n'f(x)'=f(x);\nFour ierSeries(f(x),x=0..3,type=sin,numterms=40):\nF := unapply(%,x):\nf_od d := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x)),x):\nf_ := x - > f_odd(x-6*floor((x+3)/6)):\nplot([f_(x),F(x)],x=-3.3..9.3,color=[bla ck,red],linestyle=[3,1],numpoints=150);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"F'!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 574 286 286 {PLOTDATA 2 "6&-%'CURVESG6%7hu7$$!3#)*********** **H$!#<$\"3?)*************H!#=7$$!3@s3jMS_6KF*$\"3:A(3jMS_6#F-7$$!3S5; Cq:aMJF*$\"3U3hT-dTX8F-7$$!3!\\F\"pPn'z/$F*$\"3y!\\F\"pPn'z%!#>7$$!3)) >I&4.=3'HF*$!3_7!)p/p>=RF=7$$!3#)o`!o_$3uGF*$!3u6j%>tk\"f7F-7$$!3)49r' Q&pOz#F*$!3;!f)G8YIj?F-7$$!3]?\"=<^0/r#F*$!3-&z=G)[%f*GF-7$$!3&f)G$ft$ HCEF*$!3[S6nSE1dPF-7$$!3w(p/<7e%QDF*$!3WAI&Hy=ah%F-7$$!3m&)G$>kl,X#F*$ !3OV6n!eV$)\\&F-7$$!3ou7pxvRsBF*$!3;`s3BU-wiF-7$$!3US6nc\"\\[G#F*$!3'e f)GL%3::(F-7$$!3_=GU`7%p>#F*$!3f9=xlueI!)F-7$$!3w(p/(*pDA6#F*$!3WAI&H+ Vx())F-7$$!3(pbL5'eHN?F*$!3EIWm*QTqk*F-7$$!3Ao-/U)=Q%>F*$!3yJ(fz:\"=c5 F*7$$!3EuO8\"F*7$$!3a9A$o)H=wRo3OI\"F*7$$!3dUk'p?Z)3;F*$!3UdN.$z_6R\"F*7$$! 3&*39r7X[D:F*$!31\"f)G([:XZ\"F*7$$!3)3=xlG/&Q9F*$!37>GU8d\\h:F*7$$!35' )zp'GH'e8F*$!3!R,-Lrq8k\"F*7$$!3#Rg!f3LZs7F*$!33'R49pEvs\"F*7$$!3%pbL5 Y\")H=\"F*$!30Vk'*Q&=q\"=F*7$$!3#yrdYay]5\"F*$!3;#GU`X@\\*=F*7$$!3Cx;v g7%4-\"F*$!3wA$[#R(e!z>F*7$$!3yPePO@>S$*F-$!3A;CO'y!)f1#F*7$$!3+7A$[[H )*[)F-$!3!)yn^^q,^@F*7$$!3!*p3j%*)fqm(F-$!3,8p`5SHLAF*7$$!3_Ws3VX^`nF- $!3bv7pX&[YK#F*7$$!3ogV:8MlKfF-$!3$Rc%oeYt1CF*7$$!3#>+^w%*3i0&F-$!3!)* *[B0\"zV\\#F*7$$!3)GHWm**>?E%F-$!3sqbL+!)ztDF*7$$!3mxES+HJwwb()* HF*7$$\"3Of)\\C*H&\\(QF_x$\"3+]v+Z]7'*HF*7$$\"3@!\\\">;On0lF_x$\"3_3QQ EV\\$*HF*7$$\"31@J$*RURO\"*F_x$\"3/n+w0O'3*HF*7$$\"3GQ;u[NyR9F=$\"3k$e 7X;-c)HF*7$$\"3Yk**[tw#f'>F=$\"3C+^EB2M!)HF*7$$\"3z;m)H#f@=IF=$\"3!R8q 2%y\")pHF*7$$\"39pK[sT]qSF=$\"3an^Fe\\HfHF*7$$\"3#Qdw9n!3vhF=$\"3PM_G$ >\\#QHF*7$$\"3[y)p/ZHEUF-$\"33Xo-G0PxD F*7$$\"3'GcL](\\]I]F-$\"3sVm\\-&\\p\\#F*7$$\"35_s3V3'z)eF-$\"3zu7p:R?6 CF*7$$\"3HF\"=x^b'3nF-$\"3F(=G#[W8HBF*7$$\"3^8hT#[J*>wF-$\"3l)Qe<&o+QA F*7$$\"37>v74&H!4%)F-$\"35[s3\\q4f@F*7$$\"3(3y;v#Gm/$*F-$\"3#>K[srL&p? 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)'=f(x);\nf_even := unapply(simplify(f(piecewise(x<0,-x,x))),x):\n'f_e ven(x)'=f_even(x);\nf_ := x -> f_even(x-6*floor((x+3)/6)):\n'f_(x)'='f _even(x-6*floor((x+3)/6))';\nplot(f_(x),x=-3.3..12.3,color=COLOR(RGB,. 4,0,1),thickness=2,numpoints=75);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"fG6#%\"xG,&\"\"$\"\"\"F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%'f_evenG6#%\"xG-%*PIECEWISEG6$7$,&F'\"\"\"\"\"$F-2F'\"\"!7$,&F.F-F'! \"\"1F0F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_evenG6 #,&F'\"\"\"*&\"\"'F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }} {PARA 13 "" 1 "" {GLPLOT2D 740 189 189 {PLOTDATA 2 "6'-%'CURVESG6#7[r7 $$!3#)*************H$!#<$\"3?)*************H!#=7$$!3%=;i@==(*=$F*$\"3_ =;i@==(*=F-7$$!3KCVKkjVzIF*$\"33xCVKkjVz!#>7$$!3cnvJfuWbIF*$\"3U+ovJfu WbF87$$!3z53Ja&e9.$F*$\"3wB63Ja&e9$F87$$!3TKu!=5k%>IF*$\"3U&GV2=5k%>F8 7$$!3.aSI\\'pu+$F*$\"3#4ZaSI\\'pu!#?7$$!3kv1!o>va*HF*$\"3\\cVK*>.[_%FM 7$$!3E(H(HW2[$)HF*$\"3)RFq-dD>l\"F87$$!3t$y$GMH]NHF*$\"3IF;irlq\\kF87$ 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7$$\"3mS0MbB,y6Fdil$\"3_1aS`N7!y#F*7$$\"3uH(H.\">u)=\"Fdil$\"3M(H(H.\" >u)GF*7$$\"33++![=tO>\"Fdil$\"3'3++![=tOHF*7$$\"3Wq-FfWg)>\"Fdil$\"3S/ Fq#fWg)HF*7$$\"3zS0uLd`.7Fdil$\"33#f%fiEkkHF*7$$\"3:63@3qY37Fdil$\"3c) )=*y\"*H`\"HF*7$$\"3h0a5/NB>7Fdil$\"3#R%f%*e\\m2GF*7$$\"32++++++I7Fdil $\"3G*************p#F*-%+AXESLABELSG6$Q\"x6\"Q!Ff^m-%*THICKNESSG6#\"\" #-%&COLORG6&%$RGBG$\"\"%!\"\"$\"\"!Fd_m$\"\"\"Fd_m-%%VIEWG6$;$!#LFb_m$ \"$B\"Fb_m%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 55 "The first few terms of the half-range cosine serie s of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " can be \+ found as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "f := x -> 3-x:\n'f(x)'=f(x);\nFourierSeries( f(x),x=0..3,type=cos,numterms=5,info=1):\nF := unapply(%,x):\n'F(x)'=F (x);\nplot(F(x),x=-3.3..12.3,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#)*HJF*$\"3ng#3r;jgL\"F-7$$!3 U**\\(ouxu3$F*$\"3!zw5Vmsf:\"F-7$$!3N***\\iHt\\/$F*$\"3?WNr7?`W5F-7$$! 3s**\\iX)oC+$F*$\"3^!H(Qp!*G/5F-7$$!3k*****\\Rk*fHF*$\"395))GU%ph.\"F- 7$$!3t**\\79G.7GF*$\"3akkcCb'[o\"F-7$$!3!)***\\KB,Tm#F*$\"3CJFmtax5IF- 7$$!3#)**\\())pOx\\#F*$\"3lg2s2RZ=\\F-7$$!3%)****\\k@PJBF*$\"3ZR^x%GO+ %oF-7$$!3a*****z&[!R;#F*$\"3WiFS=c4Z&)F-7$$!3o****\\^vV'*>F*$\"3S>bF\" H$*f+\"F*7$$!3y**\\(y=m(H=F*$\"39yH?J3hc6F*7$$!3))***\\U#[4j;F*$\"3(*4 QD7\\\"HK\"F*7$$!3()***\\Z8p&3:F*$\"3)\\u(\\>Ya!\\\"F*7$$!3')***\\_WVS N\"F*$\"3!Q*H^#*3(*e;F*7$$!3y***\\U/TS>\"F*$\"3V#*o(o>x2#=F*7$$!3s*** \\KkQS.\"F*$\"3'3vg2\\\"4n>F*7$$!3a(***\\7$Rco)F-$\"3)[6tp%)R\\6#F*7$$ !3'y***\\#>#*3.(F-$\"3qM5['yN(yAF*7$$!3#y***\\n=X\"Q&F-$\"396!3Q$oelCF *7$$!3y(***\\U:,KPF-$\"3aw,CIHReEF*7$$!3L'**\\iUf`.#F-$\"3\"=)f$\"3))QgdM;H(*GF*7$$\"35q/](or&*[$!#?$\"3'4bu%H *e&**GF*7$$\"3!G/+vV()\\3%F^s$\"3yF2E*y^i*GF*7$$\"3cQ+D1x,@yF^s$\"3We \"4E/Hu)GF*7$$\"3W.+](z/d:\"F-$\"3WYgq#)oCtGF*7$$\"3f-+D^3\"H!>F-$\"3V PRExU))HGF*7$$\"3u,++0p6]EF-$\"3u!>)Qy*4\"pFF*7$$\"3P.+]()4ZKVF-$\"3_B x8T'H(*e#F*7$$\"3*\\+++2D[,'F-$\"3aR!35we:R#F*7$$\"3@-++gq3/xF-$\"3`t \"HF=$>4AF*7$$\"3U******\\!\\LR*F-$\"3)oubvnX30#F*7$$\"3V+++*fE@5\"F*$ \"3cEup.JC1>F*7$$\"3#4++IH=\\E\"F*$\"3C'Roxa17v\"F*7$$\"3j+](ou[FT\"F* $\"3gU:WZ@#ef\"F*7$$\"3M++v+#z0c\"F*$\"3=F*$\"3Q@mE+>(33\"F*7$$\"3_++]A81h? F*$\"3scY.7?,)[*F-7$$\"3U+++GS(*4AF*$\"30h::a/x-\")F-7$$\"3M+]P0\\>$Q# F*$\"3sBm@\\G'*fiF-7$$\"3E++v#y:kb#F*$\"3L[<-SbyBUF-7$$\"3<+]P5cu4FF*$ \"3EfZ0`(Qoa#F-7$$\"33+++Qa2jGF*$\"3Oq,Z\\;ur8F-7$$\"3E]Pf=A80HF*$\"35 a.Kv$\\C=\"F-7$$\"3W+v=***)=ZHF*$\"3I&*3;!o\"yf5F-7$$\"3i]7yzdC*)HF*$ \"3WZs1Y*zk+\"F-7$$\"3M+]PgDIJIF*$\"34)*z;9PuB5F-7$$\"3D+Dc@hT:JF*$\"3 Gf'RV1QoE\"F-7$$\"3g++v#oH&*>$F*$\"3'fG#=>venNF*$\"3SDKQHa6G^F-7$$\"3%3+vyBdqo$F*$\"3 rdHN1hnSqF-7$$\"3%3++vy+U&QF*$\"3Bvs*\\gFsr)F-7$$\"3!3+DEu!p2SF*$\"3dP WA(HN'45F*7$$\"3w++v(p!=hTF*$\"3oA8L:f5[6F*7$$\"39+]7].uEVF*$\"3?'pqOo ]AJ\"F*7$$\"3Q++]-+I#\\%F*$\"3wW\"\\&pN]\"\\\"F*7$$\"37+]i^*pUm%F*$\"3 [%>&\\[5Ky;F*7$$\"3w++v+*Ri$[F*$\"3Ia'zu(fZ\\=F*7$$\"3G+++Z-%f)\\F*$\" 3!frcscWZ)>F*7$$\"3y***\\KfSc8&F*$\"3J52`3.$)=@F*7$$\"3/+]Po5K(H&F*$\" 3u@8')\\GR@iCF*7$$\"3.++v%HLgi&F*$\"3i= 7,x4adEF*7$$\"3s*****f/lIz&F*$\"3EYknozshF*$\"3)Q$fUMhFrGF*7$$\"3#***\\7Mm) zF'F*$\"3cEX%p%>1dFF*7$$\"3Y++veA4OkF*$\"3I%o>\"[JS'e#F*7$$\"3y+]i)fT; h'F*$\"3U\"fUB$3!*zBF*7$$\"35,+]Q4>(y'F*$\"3)zHTc\\:D>#F*7$$\"3F++DG)H \\%pF*$\"3wi*fiGke/#F*7$$\"3W*****zroE5(F*$\"3`$3yC4_d!>F*7$$\"3S++]^' )3rsF*$\"3g\"G:Bn')\\u\"F*7$$\"3M,++&e3&RuF*$\"3c!R.%3sfm:F*7$$\"3S+]( o2A@f(F*$\"3I@+&[7$)))R\"F*7$$\"3Z***\\(obtWxF*$\"3,4paKW6R7F*7$$\"3F+ ](eU#e6zF*$\"3Uu3](Q\"Q\"3\"F*7$$\"3'>++IGH%y!)F*$\"3F$Gn]VAFL*F-7$$\" 3#***\\(G<>aB)F*$\"3)RxhW\\`&\\yF-7$$\"3'y**\\F14CR)F*$\"3%o$\\?7`fahF -7$$\"3?++DK(3lb)F*$\"3.\"3(**pTpAUF-7$$\"3^-+v,%31s)F*$\"37:Nf%fhEW#F -7$$\"3],Dc7r#3!))F*$\"3Mi'e%\\J(\\w\"F-7$$\"3[+]PBe/\")))F*$\"390-XmR %HG\"F-7$$\"3(*\\7yy^:@*)F*$\"3wz&*4?qnF6F-7$$\"3Y*\\(=MXEh*)F*$\"3)3B I/%z7M5F-7$$\"3'*[Pf*)QP,!*F*$\"3]:9e<\\?/5F-7$$\"3X)****\\C$[T!*F*$\" 3\"*p?,$H@&Q5F-7$$\"3o)*\\(oNY%4#*F*$\"3QY7ck5Ff[m$\"3ap-0nj(4=\"F*7$$\"3;+ ]2,L*f.\"Ff[m$\"3EXj\">,\"HZ8F*7$$\"39+v$3()o50\"Ff[m$\"3;OMLiQz6:F*7$ $\"37++gSW9m5Ff[m$\"3%R1dV)yMv;F*7$$\"35+]#[uAM3\"Ff[m$\"3E/[LdleZ=F*7 $$\"33++0\\5q+6Ff[m$\"3lRIH!pLL+#F*7$$\"38+]Kq\\:;6Ff[m$\"3Y(*fOMS1V@F *7$$\"3=++g\"*)3;8\"Ff[m$\"3[5'3pSN$*H#F*7$$\"31+v$\\*f3[6Ff[m$\"3?.`p 3m5)[#F*7$$\"3%***\\F)4jX;\"Ff[m$\"3')f(yR[2\"zEF*7$$\"3'*****f)yL.=\" Ff[m$\"3r)zoORJ`#GF*7$$\"3)***\\#*yW5'>\"Ff[m$\"3S$*3PrJb'*GF*7$$\"3* \\Pf!>9M+7Ff[m$\"3q0WFq*f&**GF*7$$\"3+]P>f$yX?\"Ff[m$\"3#p!*)Gj.S&*GF* 7$$\"3-D\"G$*H:)37Ff[m$\"3I2Ih7%pT)GF*7$$\"3.+DYRA087Ff[m$\"3%))y6!GA7 mGF*7$$\"30]7t>h_@7Ff[m$\"3bD<\"4`L8\"GF*7$$\"32++++++I7Ff[m$\"3^UW&*f GvNFF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F[bmFjam-%+AXESLABELSG6$Q\"x6 \"Q!F`bm-%*THICKNESSG6#\"\"#-%%VIEWG6$;$!#LFiam$\"$B\"Fiam%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Taking a \+ few more terms gives a better approximation for " }{XPPEDIT 18 0 "f(x) ;" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "f := x -> 3-x:\n'f(x)'=f(x );\nFourierSeries(f(x),x=0..3,type=cos,numterms=15):\nF := unapply(%,x ):\n'F(x)'=F(x);\nplot(F(x),x=-3.3..12.3,thickness=2,numpoints=60);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"F'!\"\"" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,4#\"\"$\"\"#\"\"\"*(\" #7F,%#PiG!\"#-%$cosG6#,$*(F*!\"\"F/F,F'F,F,F,F,*&#\"\"%F*F,*&F/F0-F26# *&F/F,F'F,F,F,F,*&#F.\"#DF,*&F/F0-F26#,$**\"\"&F,F*F6F/F,F'F,F,F,F,F,* &#F.\"#\\F,*&F/F0-F26#,$**\"\"(F,F*F6F/F,F'F,F,F,F,F,*&#F9\"#FF,*&F/F0 -F26#,$*(F*F,F/F,F'F,F,F,F,F,*&#F.\"$@\"F,*&F/F0-F26#,$**\"#6F,F*F6F/F ,F'F,F,F,F,F,*&#F.\"$p\"F,*&F/F0-F26#,$**\"#8F,F*F6F/F,F'F,F,F,F,F,*&# F9\"#vF,*&F/F0-F26#,$*(FFF,F/F,F'F,F,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 633 200 200 {PLOTDATA 2 "6&-%'CURVESG6$7ip7$$!3#)*********** **H$!#<$\"3#HZ)eDG$*oI!#=7$$!3]1W'=V!ohJF*$\"3g5P#)fX8G:F-7$$!3>8)GP'3 OBIF*$\"3gWu^$p\\W3%!#>7$$!38XZ3Z(4I!HF*$\"3qeuoT9qj%)F87$$!3_x1WI'eEy #F*$\"3)4,Ezv=5=#F-7$$!3Ndu%[zj>^#F*$\"3e[%pb6YX$[F-7$$!37\"\\p\"=YZRA F*$\"3QIZ))pnJ.wF-7$$!3c]I)\\`!Go>F*$\"3.WsuveOM5F*7$$!3:p,h/*[or\"F*$ 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32=\"F*7$$\"3s$fNlT'=Y5Fbel$\"3w&*GBNXFj9F*7$$\"3g.A8.YFq5Fbel$\"3\"G1 /W<8Mq\"F*7$$\"31&30fB3x4\"Fbel$\"3!4p]X/vW(>F*7$$\"3HR.#3x2U7\"Fbel$ \"3_+-?\"HG@C#F*7$$\"3)*30B%***o]6Fbel$\"3u&o#H&*yP6DF*7$$\"3GW'=Vzuq< \"Fbel$\"391$RS2?#oFF*7$$\"3y8)G2c[(*=\"Fbel$\"31V\"oWHA.\"HF*7$$\"3D$ )*QrKAC?\"Fbel$\"33H/tb&Q*eHF*7$$\"3m\"\\pN;6i@\"Fbel$\"3%yt+pO'oYGF*7 $$\"32++++++I7Fbel$\"3e^6W " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 275 8 "Question" } {TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f (x);" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be defined on the interval " } {XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G \"\"\"" }{TEXT -1 4 " by " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = x-x^3;" "6#/-%\"fG6# %\"xG,&F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(i) Find the first few te rms of the Fourier half-range " }{TEXT 268 11 "sine series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 64 ", that is, the Fourier series for the odd periodic extens ion of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " with \+ period 2. " }}{PARA 0 "" 0 "" {TEXT -1 40 "Draw the graph of your trun cated series." }}{PARA 0 "" 0 "" {TEXT -1 56 "(ii) Find the first few \+ terms of the Fourier half-range " }{TEXT 268 13 "cosine series" } {TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\" xG" }{TEXT -1 65 ", that is, the Fourier series for the even periodic \+ extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 40 "Draw the graph of you r truncated series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 274 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(i) " }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }{XPPEDIT 18 0 " f(x)=x-x^3" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 41 " \+ defines on odd function on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7 $,$\"\"\"!\"\"F%" }{TEXT -1 52 " so it can be used \"as is\" for the o dd extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " to this interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := x -> x-x^3:\n'f(x)'=f(x);\nplo t(f(x),x=-1..1,thickness=2,tickmarks=[4,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*$)F'\"\"$F)!\"\"" }}{PARA 13 " " 1 "" {GLPLOT2D 484 323 323 {PLOTDATA 2 "6'-%'CURVESG6$7hn7$$!\"\"\" \"!$F*F*7$$!3!RLL$e%G?y*!#=$!3600#>nJz@%!#>7$$!3ommm;p0k&*F/$!3MdE3\" \\2q:)F27$$!3E++vV5Su$*F/$!3N7*y#GTBO6F/7$$!3wKL$3s%HaF/$!3V8]cZ*3*GQF/7$$!3Q+++]$*4 )*\\F/$!3K(4)p&HC&\\PF/7$$!39+++]_&\\c%F/$!3T7`80En8OF/7$$!31+++]1aZTF /$!3))**QyJo2MMF/7$$!3umm;/#)[oPF/$!3vV\")zDiILKF/7$$!3hLLL$=exJ$F/$!3 _,H))3`b_HF/7$$!3*RLLLtIf$HF/$!3[G-\"piiGo#F/7$$!3]++]PYx\"\\#F/$!3sI' *GC=1PBF/7$$!3EMLLL7i)4#F/$!3o+nUCN>1?F/7$$!3c****\\P'psm\"F/$!3WvLXvI #4i\"F/7$$!3')****\\74_c7F/$!3iwuO(R#oO7F/7$$!3)3LLL3x%z#)F2$!3.:Jb#[@ FA)F27$$!3KMLL3s$QM%F2$!3yK/#f&3kNVF27$$!3]^omm;zr)*!#@$!3C7 ^4)F27$$\"3GKLLe\"*[H7F/$\"3e&*fFkO!4@\"F/7$$\"3I*******pvxl\"F/$\"3\\ Ck:sj@7;F/7$$\"3#z****\\_qn2#F/$\"3)*oYb9**>()>F/7$$\"3U)***\\i&p@[#F/ $\"3,q4)=)*Q#HBF/7$$\"3B)****\\2'HKHF/$\"3%fgXUpm,o#F/7$$\"3ElmmmZvOLF /$\"3#>R(\\ZFClHF/7$$\"3i******\\2goPF/$\"3E#f)f?3PLKF/7$$\"3UKL$eR<*f TF/$\"3'=x@/KZ+W$F/7$$\"3m******\\)Hxe%F/$\"3J)4cie[QF/7$$\"3WKLeRm/>7BqHF/7$$\"3]mmmT6KU $)F/$\"3U6,],$Rl`#F/7$$\"3fKLLLbdQ()F/$\"3eK#H!4Jcl?F/7$$\"3[++]i`1h\" *F/$\"3#=E(>%RIEZ\"F/7$$\"3Y++++PDj$*F/$\"3+gGG10Wa6F/7$$\"3W++]P?Wl&* F/$\"3fY/J,O%G8)F27$$\"3A++v=5s#y*F/$\"3\\]UPN\\(\\?%F27$$\"\"\"F*F+-% 'COLOURG6&%$RGBG$\"#5F)F+F+-%*AXESTICKSG6$\"\"%Fg^l-%+AXESLABELSG6$Q\" x6\"Q!F\\_l-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(F\\^l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "The next picture shows the graph of a truncated sine series along with the graph of th e periodic odd extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 247 "f := x -> x-x^3:\n'f(x)'=f(x);\nf_ := x -> f( x-2*floor((x+1)/2)):\n'f_(x)'='f(x-2*floor((x+1)/2))';\nFourierSeries( f(x),x=0..1,type=sin,info=1):\nF := unapply(%,x):\n'F(x)'=F(x);\nplot( [f_(x),F(x)],x=-1..3,color=[red,green],thickness=[1,3],ytickmarks=3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*$)F'\"\"$ F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%\"fG6#,&F' \"\"\"*&\"\"#F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%:k~th~sine~coefficient~..~G/,$**\"\"%\"\"\",(*()% \"kG\"\"#F()%#PiGF-F(-%$sinG6#*&F,F(F/F(F(F(**\"\"$F(F,F(F/F(-%$cosGF2 F(F(*&F5F(F0F(!\"\"F(F,!\"%F/F:F9,$**\"#7F()F9,&F(F(F,F(F(F,!\"$F/F@F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,**(\"#7\"\"\"%#PiG! \"$-%$sinG6#*&F,F+F'F+F+F+*&#\"\"$\"\"#F+*&F,F--F/6#,$*(F5F+F,F+F'F+F+ F+F+!\"\"*&#\"\"%\"\"*F+*&F,F--F/6#,$*(F4F+F,F+F'F+F+F+F+F+*&#F4\"#;F+ *&F,F--F/6#,$*(F>F+F,F+F'F+F+F+F+F;" }}{PARA 13 "" 1 "" {GLPLOT2D 598 262 262 {PLOTDATA 2 "6'-%'CURVESG6%7jr7$$!\"\"\"\"!$F*F*7$$!3!RLL$e%G? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "f := x -> x-x^3:\n'f(x)'=f(x);\nFourierSeries(f(x),x =0..1,type=cos,numterms=6,info=1):\nF := unapply(%,x):\n'F(x)'=F(x);\n g := x -> abs(x-x^3):\ng_ := x -> g(x-2*floor((x+1)/2)):\nplot([g_(x), F(x)],x=-1..3,color=[black,red],linestyle=[3,1],thickness=[1,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*$)F'\"\"$F)! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%9constant~coefficient~-->G#\" \"\"\"\"%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%7$$! 3!RLL$e%G?y*F/$\"3600#>nJz@%F27$$!3u****\\(oUIn*F/$\"3c)eFl?Q>A'F27$$! 3ommm;p0k&*F/$\"3MdE3\"\\2q:)F27$$!3[*****\\P&3Y$*F/$\"39J(ei6WB=\"F/7 $$!3PLLLLQ6G\"*F/$\"3CKN:RYMA:F/7$$!3'pm;/'zXQ*)F/$\"3'Qc\\L%[)pz\"F/7 $$!3a++](3-)[()F/$\"3a2Y5;WL_?F/7$$!3-LLe9i9f&)F/$\"3_;&G`m-))G#F/7$$! 3immmT.\\p$)F/$\"3\"QF/7$$!3Vn;z%4HL7'F/$\"3&QVo@5xt#QF/7$$!3zLL 3x%)[;gF/$\"3Uc$G/%>jQQF/7$$!39+]Pfyk4fF/$\"3CB'ya6md%QF/7$$!3hnmmTs!G !eF/$\"3)Q.I>!G&)[QF/7$$!39,+]7Bv.dF/$\"3[MPKDD;[QF/7$$!3nMLL$Q(p/cF/$ \"3ZEz1xV6WQF/7$$!3?om;aCk0bF/$\"3gB81smwOQF/7$$!3i+++Dve1aF/$\"3@SE?D xGFVymi$F/7$$!3i+++vE%)*=%F/$\"3K%z\"HU\\KaMF/ 7$$!3InmmT&[bw$F/$\"3^&=#Q:!=;B$F/7$$!3)RLL$3WDTLF/$\"3agnJ+uBoHF/7$$! 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    T$*F27$Fifm$\"3IQH,0jim6F/7$F^gm$\"3wec6x%eAS\"F/ 7$Fcgm$\"3$)el`81**H;F/7$Fhgm$\"3!og&es'*H^=F/7$F]hm$\"3?c9%op..0#F/7$ Fbhm$\"3tyPcgFSEAF/7$Fghm$\"3)3eT'e:o#Q#F/7$F\\im$\"3ShV]STR\\EF/7$Fai m$\"3<`04[`n2HF/7$Ffim$\"37JRac#He=$F/7$F[jm$\"3#z&zKDqYgMF/7$F`jm$\"3 amri?q5$o$F/7$Fejm$\"3)H)*[*zqi8QF/7$$\"3gmTg_2@7DFegl$\"3pV0x***3(GQF /7$Fjjm$\"3%)>*e^?L%QQF/7$$\"3DLe91%Q:`#Fegl$\"3WvmB7Z6VQF/7$F_[n$\"3l V-M#3oJ%QF/7$Fd[n$\"3V^*QK#)*3RQF/7$Fi[n$\"3GJ[V!=P9$QF/7$F^\\n$\"3'>) \\OpS!3#QF/7$Fc\\n$\"3*of4e,&z2QF/7$Fg]n$\"3Ml7)p3KGt$F/7$Fa^n$\"3)G(R Sc>[_OF/7$Ff^n$\"3=r(Q>a!**fNF/7$F[_n$\"37GTsQke%R$F/7$F`_n$\"3)\\[W[U 8\\D$F/7$Fe_n$\"3CQhmNh2qIF/7$Fj_n$\"3#4o!*Gy(ePGF/7$F_`n$\"3Y&\\i*38M gDF/7$Fd`n$\"3i:k\"*R^8hAF/7$Fi`n$\"3md5IXYcT>F/7$F^an$\"3\"p+D^\"f<=; F/7$Fcan$\"3_>#HV/(358F/7$Fhan$\"3Lg`m4o!)=5F/7$F]bn$\"3_O$RlLZM\"zF27 $Fgbn$\"3#>iyO'y'fY'F27$FacnFbdn-Fdcn6&Ffcn$\"*++++\"!\")F+F+-Fhcn6#\" \"#-F\\dnFicn-%+AXESLABELSG6$Q\"x6\"Q!Fefp-%%VIEWG6$;F(Facn%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{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 277 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 " " 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 28 " be defined on the interval " }{XPPEDIT 18 0 "0<=x" "6#1 \"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 4 " by \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([sin(Pi/2*x), 0 <= x and x < 1],[2-x, 1 <= x and x <= 2]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$sinG6#*(%# PiG\"\"\"\"\"#!\"\"F'F131\"\"!F'2F'F17$,&F2F1F'F331F1F'1F'F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(i) Find the first few terms of the Fourier half-range " }{TEXT 268 11 "sine series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 64 ", that is, the Fourier series \+ for the odd periodic extension of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6# %\"xG" }{TEXT -1 16 " with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 40 " Draw the graph of your truncated series." }}{PARA 0 "" 0 "" {TEXT -1 87 "(ii) Find the first few terms of the Fourier half-range cosine ser ies for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 65 ", that is, the Fourier series for the even periodic exten sion of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 40 "Draw the graph of your tru ncated series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(i) " }}{PARA 0 "" 0 "" {TEXT -1 109 "The picture shows the graph of \+ a truncated sine series along with the graph of the periodic odd exten sion of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 305 "f := x -> piecewise(x<1,sin(Pi/2*x),2-x):\n'f(x)'=f(x);\nFourierS eries(f(x),x=0..2,type=sin,numterms=8,info=1):\nF := unapply(%,x):\n'F (x)'=F(x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x) ),x):\nf_ := x -> f_odd(x-4*floor((x+2)/4)):\nplot([f_(x),F(x)],x=-2.. 6,color=[red,green],thickness=[1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$sinG6#,$*(\"\"#!\"\"%#PiG\"\"\"F' F4F42F'F47$,&F'F2F1F4%*otherwiseG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$% :k~th~sine~coefficient~..~G/,$*,\"\"#\"\"\",,*(F'F(-%$sinG6#*&%\"kGF(% #PiGF(F()F/F'F(!\"\"*&F'F(F+F(F(*(F'F(-F,6#,$*(F'F2F/F(F0F(F(F(F1F(F(* &F'F(F5F(F2*(F/F(F0F(-%$cosGF6F(F2F(F/!\"#F0F=,&*$F1F(F(F(F2F2F(,$*,F' F(,(*(F'F(F5F(F1F(F(*&F'F(F5F(F2F:F2F(F/F=F0F=F>F2F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"xG,2*&#\"\"\"\"\"#F+*(,&*$)%#PiGF,F+F+\" \")F+F+F1!\"#-%$sinG6#,$*(F,!\"\"F1F+F'F+F+F+F+F+*&#F+\"\"$F+*&F1F9-F5 6#*&F1F+F'F+F+F+F+*&#\"\"%\"\"*F+*&F1F3-F56#,$**F5E$F27$$!3 nmmmwxE.:F/$!3MLLLLAKn\\F27$$!3ommmOk]J8F/$!3=LLLLc$\\o'F27$$!3_LLL[9c g6F/$!3wkmm;bQ%R)F27$$!37+++0vJ\"3\"F/$!3w)*****\\\\#o=*F27$$!3smmmhN2 -5F/$!3wKLL$Qk#z**F27$$!3EMLLLVg:)*F2$!3qu(HS\\0e***F27$$!3H,++]IZ5'*F 2$!3iP$Htu'G\")**F27$$!3Kommm)*F27$$!3E,++]`oz$)F2$!3 +/?)GdYyn*F27$$!3hMLL$3(4JvF2$!33\"y:\"Q!ytD*F27$$!3'zmmm\")3Do'F2$!3e wp41'pEn)F27$$!3&\\LLL)>kOeF2$!3'pL(\\&>'pOzF27$$!3%>+++:v2*\\F2$!3%[a /(fT\"31(F27$$!37nmmTrp?TF2$!3ZJ`b>!p,.'F27$$!3IKLLL\">1D$F2$!3S\\nYA4 1()[F27$$!3Q******\\RD%[#F2$!3)z(\\:b:(R!QF27$$!3Ymmmm())yr\"F2$!3M88t KP#em#F27$$!3)*RLLL3V^&)!#>$!3c\\Kf$y>#R8F27$$\"3fg%)******f-w!#@$\"3G 1]bJL@%>\"!#?7$$\"3p*)******z!*Q()Fdr$\"3#*[3]d#F2$\"3Q^jqx+UNRF27$$\"3!)*** ****RP)4MF2$\"3omwZN(3P5&F27$$\"3Ummm\"HUz;%F2$\"3ocUHZQ?*3'F27$$\"3.L LL$=Zg#\\F2$\"3qe[[V;X))pF27$$\"3clmmms'*GnF2$\"3#>9TdMp(3()F27$$\"3![ mmm;AE\\(F2$\"31CW_vTNM#*F27$$\"31kmmmqFc#)F2$\"3+74k'3Csi*F27$$\"3a(* **\\iJV+()F2$\"3=)>n?1mBz*F27$$\"3.JLLe#*eW\"*F2$\"3$evSGZi)4**F27$$\" 3K)**\\iInmO*F2$\"3c5VR*zb0&**F27$$\"3]km;a`u)e*F2$\"3-?][I;9z**F27$$ \"3qIL3-M#3\")*F2$\"3=`F&Q=&e&***F27$$\"3!)*****\\9!H.5F/$\"3+-++]&)4n **F27$$\"3/LL$e#3#>3\"F/$\"3cpmmTF/$\"3gSnmmmr[RF]s7$$\"3oKLL=2Vs@F/$!3yELL$=2Vs \"F27$$\"3f*****\\`pfK#F/$!3)e*****\\`pfKF27$$\"3!HLLLm&z\"\\#F/$!36HL LLm&z\"\\F27$$\"3s******z-6jEF/$!3>(******z-6j'F27$$\"3<******4#32$GF/ $!3q\"******4#32$)F27$$\"3F****\\qMLF/$!3#=c\\\\G&[\\')F27$$\"3%)*******HSu]$F/ $!3kd=NBe%z)pF27$$\"3Smm;HOq&e$F/$!3sIzE>O.egF27$$\"3'HLL$ep'Rm$F/$!3U 44NQF/$!3sd %o-v(\\hDF27$$\"3&GLLed*f:RF/$!3$GR%o/&y=K\"F27$$\"3#emm;@2h*RF/$!3H.= H4&RZ6'F]s7$$\"37LLL))3E!3%F/$\"3e<\"*\\SzRd7F27$$\"3]*****\\c9W;%F/$ \"3AdDh-Q+aDF27$$\"3#HLLe;!pYUF/$\"3g-wqE'\\(yPF27$$\"3Lmmmmd'*GVF/$\" 3-7R3E'o/%\\F27$$\"3)HLLep+^T%F/$\"3m!4M'=33ogF27$$\"3j*****\\iN7]%F/$ \"3QSdVF)yZ3(F27$$\"39mm;*z$>%e%F/$\"3Bf)**RFa<%zF27$$\"3aLLLt>:nYF/$ \"3)=Uh26jSm)F27$$\"3KLLL)o))>v%F/$\"3!HS-le/2D*F27$$\"35LLL.a#o$[F/$ \"3KA3na*4Ln*F27$$\"3CmmT:D*)y[F/$\"3.jgp)*yf>)*F27$$\"3E++]F'f4#\\F/$ \"3*4aP$))\\-B**F27$$\"3Gn;a$=$*>%\\F/$\"3g5q)*ps^e**F27$$\"3SLLeRn-j \\F/$\"3#p%R:#yRJ)**F27$$\"3a**\\i&HgS)\\F/$\"3?IzC\\c'o***F27$$\"3amm m^Q40]F/$\"3iMLL$[h!\\**F27$$\"3;LL$eY/C3&F/$\"3QommT`&f<*F27$$\"3y*** ***z]rf^F/$\"38-+++#\\GS)F27$$\"3gmmmc%GpL&F/$\"3-MLLLarImF27$$\"3/LLL 8-V&\\&F/$\"3kpmmmypX]F27$$\"3=+++XhUkcF/$\"35)*****\\&QdN$F27$$\"3=++ +:onnKzmG$F27$F9$!3S0$[;R!)\\'\\F27$F>$!3y\"Q2+yArk'F27$FC$ !3/DgSAW2$[)F27$FH$!3![m[&=8!pA*F27$FM$!3glA\"4gg'Q(*F27$FR$!3nT+8@.%4 #)*F27$FW$!3YZ`e,B`!))*F27$Ffn$!35C&*4W?.<**F27$F[o$!3twxHG0VI**F27$$! 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$F)F*F+F1*&-F&6#,$F(\"\"&F)F*F+#F,F<*&-F&6#,$F(F,F)F*F+#F2F7*&-F&6#,$F (\"\"(F)F*F+#F,FG*&-F&6#,$F(\"\"*F)F*F+#F1F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be defined on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG " }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 4 " by " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([0, 0 <= x and x < 1],[1, 1 <= x and x <= 2]); " "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$\"\"!31F,F'2F'\"\"\"7$F031F0F'1F' \"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(a) Find the first few terms of the Fourier half-ran ge " }{TEXT 268 13 "cosine series" }{TEXT -1 18 " for the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 65 ", that is, the Fo urier series for the even periodic extension of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 57 " with period 4. Draw the graphs of s ome truncated series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "f_even := x -> piecewise(x<-1,1,x<1,0,1) :\n'f_even(x)'=f_even(x);\nf_ := x -> f_even(x-4*floor((x+2)/4)):\n'f_ (x)'='f_even(x-4*floor((x+2)/4))';\nplot(f_(x),x=-4.3..8.3,color=COLOR (RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_ev enG6#%\"xG-%*PIECEWISEG6%7$\"\"\"2F'!\"\"7$\"\"!2F'F,7$F,%*otherwiseG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_evenG6#,&F'\"\" \"*&\"\"%F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 775 130 130 {PLOTDATA 2 "6'-%'CURVESG6#7_r7$$!3#)****** *******H%!#<$\"\"!F,7$$!38++]deNDSF*F+7$$!3!***\\id%*Q'y$F*F+7$$!3z*** \\#[nkXIF*F+7$$!3 v(=#*HZm$GIF*F+7$$!32DJqv!R:,$F*F+7$$!3&RfeqPDJ+$F*F+7$$!3QiSTy;r%*HF* $\"\"\"F,7$$!3EJ&p(zzH')HF*FT7$$!39+]7\"G%)y(HF*FT7$$!33+]Pq]2`GF*FT7$ $!3,+]ifeEGFF*FT7$$!3)***\\iU/!)pCF*FT7$$!3z**\\iQ^\\-AF*FT7$$!3()**\\ 7'3Zg$>F*FT7$$!3U****\\8D(>m\"F*FT7$$!3]***\\2[n0U\"F*FT7$$!3[**\\iN]o %G\"F*FT7$$!3W****\\!f-)[6F*FT7$$!3]***\\ZR#e!3\"F*FT7$$!3d******)>iB, \"F*FT7$$!3k\\7`uY$Q+\"F*FT7$$!3!f*\\i+:2`**!#=F+7$$!3W&\\PfD'zn)*F]qF +7$$!33'**\\7,@Dy*F]qF+7$$!3B'*\\(=_q>h*F]qF+7$$!3S'***\\K+UT%*F]qF+7$ $!3s'**\\P0>.5*F]qF+7$$!31(****\\2=#f()F]qF+7$$!3'e****\\3iVW(F]qF+7$$ !3m%****\\41&HhF]qF+7$$!3g(**\\iov9u$F]qF+7$$!3&3%****\\bw=!*!#>F+7$$ \"3a3++!QOO]\"F]qF+7$$\"3=2+v$y>=I%F]qF+7$$\"3q,++IioynF]qF+7$$\"3l/]( ovVu8)F]qF+7$$\"3g2+v$G,i\\*F]qF+7$$\"3?d(=UfLzl*F]qF+7$$\"3)y](o/fm>) *F]qF+7$$\"3o#)=#*f?`+**F]qF+7$$\"3eei::#)R\")**F]qF+7$$\"3Xj!RqVEi+\" F*FT7$$\"3#3]iD08V,\"F*FT7$$\"3'4DcY^fm/\"F*FT7$$\"3)3+]n(f+z5F*FT7$$ \"3$4]P4!*)pV6F*FT7$$\"3)4+D^#=R37F*FT7$$\"33,vV%Q#RV8F*FT7$$\"3?,+vVH Ry9F*FT7$$\"3X+](eDQjs\"F*FT7$$\"3)****\\7x!y$*>F*FT7$$\"3!3+v8Qy:F#F* FT7$$\"3E+]iXv`)HF*FT7$$\"39DcEVt!Q*HF*FT7$$\"3b(oaXBRA+$F*F+7$$ \"3'*\\P%e7r1,$F*F+7$$\"3yu=U3\\`FIF*F+7$$\"3g*****4p)RWIF*F+7$$\"3'** *\\(31#QwJF*F+7$$\"3M++vIaO3LF*F+7$$\"3[+]PCowjNF*F+7$$\"3#4+]s_Yt%QF* F+7$$\"3l*****H]b@5%F*F+7$$\"3'*****\\s%=UP%F*F+7$$\"3g**\\Pfzu?YF*F+7 $$\"3q+vV#))3bv%F*F+7$$\"3y,+]0)p-*[F*F+7$$\"3Kw$4rop>#\\F*F+7$$\"3u^( =(o&pO&\\F*F+7$$\"3eQM_4&>&p\\F*F+7$$\"3GE\"G.Xp`)\\F*F+7$$\"3qp/t?WH$ *\\F*F+7$$\"3+9G8\"R>7+&F*FT7$$\"3Td^`hV94]F*FT7$$\"3#3]P>Lpq,&F*FT7$$ \"3!*\\i:&4p/3&F*FT7$$\"3&)**\\Pe)oQ9&F*FT7$$\"3]+]7**3Tw_F*FT7$$\"39, ]()RH&*3aF*FT7$$\"3!*****\\#3@\"ocF*FT7$$\"3%3+v$4hWRfF*FT7$$\"3f,++eV w+iF*FT7$$\"3#3+]-,+!okF*FT7$$\"3c+]P;G-LnF*FT7$$\"3w+voLqyaoF*FT7$$\" 3'4++5D^l(pF*FT7$$\"378G)=Zt_)pF*FT7$$\"3;Ecw#p&*R*pF*FT7$$\"3KQ%[O\"z r-qF*F+7$$\"3[]7`M,W6qF*F+7$$\"3pvoHwX))GqF*F+7$$\"3!4]i!=!Hj/(F*F+7$$ \"3I^Pf,z@\"3(F*F+7$$\"3$3+D^y1h6(F*F+7$$\"3w+v=_X)e=(F*F+7$$\"3s++D>B mbsF*F+7$$\"3O+]i_C[!Q(F*F+7$$\"3++++'e-`](F*F+7$$\"3-,]Py6ZrxF*F+7$$ \"33,]i$[Gi-)F*F+7$$\"3q+++++++$)F*F+-%+AXESLABELSG6$Q\"x6\"Q!Feal-%*T HICKNESSG6#\"\"#-%&COLORG6&%$RGBG$\"\"%!\"\"F+FT-%%VIEWG6$;$!#VFabl$\" #$)Fabl%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 55 "(b) Find the first few terms of the Fourier half-r ange " }{TEXT 268 11 "sine series" }{TEXT -1 18 " for the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 64 ", that is, the Fo urier series for the odd periodic extension of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 57 " with period 4. Draw the graphs of s ome truncated series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "f_odd := x -> piecewise(x<-1,-1,x<1,0,1) :\n'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-4*floor((x+2)/4)):\n'f_(x) '='f_odd(x-4*floor((x+2)/4))';\nplot(f_(x),x=-3.3..8.3,color=COLOR(RGB ,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6# %\"xG-%*PIECEWISEG6%7$!\"\"2F'F,7$\"\"!2F'\"\"\"7$F1%*otherwiseG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%&f_oddG6#,&F'\"\"\"*& \"\"%F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 657 172 172 {PLOTDATA 2 "6'-%'CURVESG6#7`t7$$!3#)*********** **H$!#<$\"\"!F,7$$!3;LL$e]wN<$F*F+7$$!3_mmm6I:ZIF*F+7$$!3+3-)3'HSLIF*F +7$$!3#*\\P45Hl>IF*F+7$$!3m?0q%)yx7IF*F+7$$!3%=H2$fG!f+$F*F+7$$!3eiS\" R$y-**HF*$\"\"\"F,7$$!3KL3_3G:#*HF*FB7$$!3s;z%pq_Y'HF*FB7$$!36+]P0E:PH F*FB7$$!3!p;HAS_@)GF*FB7$$!3ELL3*>_r#GF*FB7$$!3++]iP_W.FF*FB7$$!3Gmm;w #Q(zDF*FB7$$!3Ymm;LVoIBF*FB7$$!3,+]i;#\\n?#F*FB7$$!37LL3+T\"G3#F*FB7$$ !3g**\\iq\")3a?F*FB7$$!3_mm;TAOD?F*FB7$$!3Z$3-)e2==?F*FB7$$!3)**\\PkF* *4,#F*FB7$$!3\\;H2%zF*$!\"\"F,7$$!3Ym\"zpMtA)>F *F^p7$$!3#****\\AQ5z'>F*F^p7$$!3LLLLB&e/\">F*F^p7$$!3]mmTkm+`=F*F^p7$$ !3.LL3?0.M&*\\P* H/UM*FcsF+7$$!3tGLLVxR8()FcsF+7$$!3m%***\\F#o@g(FcsF+7$$!3egmm6(Q4\\'F csF+7$$!3C%*****Hi(*))RFcsF+7$$!3)e*****\\/uw9FcsF+7$$\"3Kq+++BkU%*!#> F+7$$\"3)oLLeToF9$FcsF+7$$\"3]vmmOD+ddFcsF+7$$\"31wmmT@IkoFcsF+7$$\"3k wmmYX,\"F*FB7$$\"32%3-LRpD-\"F*FB7$$\"3KUgx^+nQ5F*FB7$$\"3 e++D52xa5F*FB7$$\"3'QLeu>&yo6F*FB7$$\"3;nmm%o*z#G\"F*FB7$$\"3T,+D5O)H` \"F*FB7$$\"3A,+]S25_;F*FB7$$\"3/,+vqy@rF*FB7$$\"3-,Dc9skd>F*FB7$$\"3_UN@$*H=t>F *FB7$$\"3C%ek=x=())>F*FB7$$\"350,>hm['*>F*FB7$$\"3&fi:0baU+#F*F^p7$$\" 3fY6%)RC-7?F*F^p7$$\"3nnm;H.z>?F*F^p7$$\"3zn;a&)Q#R8#F*F^p7$$\"3!zm;>W d![AF*F^p7$$\"3wLL$3OuU\\#F*F^p7$$\"3UMLeTX-]FF*F^p7$$\"3Unmm$=S8'GF*F ^p7$$\"3U++vDelsHF*F^p7$$\"3_/,>A)p,)HF*F^p7$$\"3g3-j=Qo()HF*F^p7$$\"3 o7.2:y>&*HF*F^p7$$\"3AZ!)*\\F*F+7$$\"3a%ek$)H,e+&F*FB7$$\"3-E\"G)*\\48-&F*F B7$$\"3Ro;H,x\"o.&F*FB7$$\"3?Ne920&))4&F*FB7$$\"3--++8L)3;&F*FB7$$\"3; M$e\\Q>wF&F*FB7$$\"3Gmm\"pXbVR&F*FB7$$\"3)4+]#>6SQcF*FB7$$\"3YM$e/u+xv &F*FB7$$\"3%zmm;O+q(eF*FB7$$\"3Y%e*[tUA3fF*FB7$$\"3'4]7`=[%RfF*FB7$$\" 3AfRAT,1bfF*FB7$$\"3[+v)f9RNnF*F^p7$$\"3Mom\"\\3'QdoF*F^p7$$\"3c,v=!GOM\"p F*F^p7$$\"3yM$eaZ'[ppF*F^p7$$\"3kUgFC!*\\$)pF*F^p7$$\"3Q^P4t:^(*pF*F^p 7$$\"3v0E]Zy^/qF*F+7$$\"38g9\">7C:,(F*F+7$$\"3h8.K'RI&=qF*F+7$$\"3*z;H 2nOb-(F*F+7$$\"3g%ek$o " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 28 " be defined on the interval " }{XPPEDIT 18 0 " 0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 3;" "6#1%!G\"\"$" }{TEXT -1 4 " by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+ PIECEWISE([x, 0 <= x and x < 2],[6-2*x, 2 <= x and x <= 3]);" "6#/-%\" fG6#%\"xG-%*PIECEWISEG6$7$F'31\"\"!F'2F'\"\"#7$,&\"\"'\"\"\"*&F0F4F'F4 !\"\"31F0F'1F'\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) (i) Find the first few terms of the Fourier half-range " }{TEXT 268 13 "cosine series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 65 " , that is, the Fourier series for the even periodic extension of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " with period 6. \+ " }}{PARA 0 "" 0 "" {TEXT -1 52 " (ii) Draw the graphs of some tru ncated series. " }}{PARA 0 "" 0 "" {TEXT -1 60 " (iii) Comment on \+ the convergence of the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "Here is the graph of the even periodi c extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " for comparison purposes." }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" } {TEXT -1 50 ": Part (b) of this question appears further down. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 275 "f := x -> piecewise(x<2,x,6-2*x):\n'f(x)'=f(x);\nf_even := unappl y(simplify(f(piecewise(x<0,-x,x))),x):\n'f_even(x)'=f_even(x);\nf_ := \+ x -> f_even(x-6*floor((x+3)/6)):\n'f_(x)'='f_even(x-6*floor((x+3)/6))' ;\nplot(f_(x),x=-4.3..8.3,color=COLOR(RGB,.4,0,1),thickness=2,numpoint s=60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6 $7$F'2F'\"\"#7$,&\"\"'\"\"\"*&F-F1F'F1!\"\"%*otherwiseG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%'f_evenG6#%\"xG-%*PIECEWISEG6&7$,&*&\"\"#\"\" \"F'F/F/\"\"'F/1F'!\"#7$,$F'!\"\"2F'\"\"!7$F'2F'F.7$,&F0F/*&F.F/F'F/F5 1F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_evenG6#,&F '\"\"\"*&\"\"'F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 675 145 145 {PLOTDATA 2 "6'-%'CURVESG6#7gv7$$!3#) *************H%!#<$\"3;+++++++<\"=F* 7$$!3[=ri(pgl2%F*$\"3_\")GP-$RM#>F*7$$!3i?K%)H!fA0%F*$\"3Qzn:q4uZ>F*7$ $!3(=Kf?Odz-%F*$\"39y1%zjU?(>F*7$$!3\\st;Gl!e,%F*$\"3_FE$=Z$>%)>F*7$$! 37BaF%pbO+%F*$\"3)odCdIWj*>F*7$$!3=uMQg[]\"*RF*$\"3[Zpw?(4I)>F*7$$!3ED :\\ESNzRF*$\"3]]I)H03(e>F*7$$!3kGP#4p]2$RF*$\"3Gdu%=Q,:'=F*7$$!3-KfNbt 9#)QF*$\"31k=r5ZHkt)o**!#=7$$!3X]I)p!= UVMF*$\"33+5mRhVo))F\\q7$$!3m6FEj9m)Q$F*$\"3SLUDl#HKx(F\\q7$$!3XsBa>6! 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" }}{PARA 0 "" 0 "" {TEXT -1 60 " (iii) Comment on \+ the convergence of the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "Here is the graph of the odd periodi c extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " for comparison purposes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "f := x -> piecewise(x<2,x,6-2*x): \n'f(x)'=f(x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signu m(x)),x):\n'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-6*floor((x+3)/6)); \n'f_(x)'='f_odd(x-6*floor((x+3)/6))':\nplot(f_(x),x=-5..10,color=COLO R(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG 6#%\"xG-%*PIECEWISEG6$7$F'2F'\"\"#7$,&\"\"'\"\"\"*&F-F1F'F1!\"\"%*othe rwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6#%\"xG-%*PIECEWIS EG6%7$,&*&\"\"#\"\"\"F'F/!\"\"\"\"'F01F'!\"#7$F'2F'F.7$,&F1F/*&F.F/F'F /F01F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f_Gf*6#%\"xG6\"6$%)oper atorG%&arrowGF(-%&f_oddG6#,&9$\"\"\"*&\"\"'F1-%&floorG6#,&*&#F1F3F1F0F 1F1#F1\"\"#F1F1!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 606 189 189 {PLOTDATA 2 "6'-%'CURVESG6#7ip7$$!\"&\"\"!$\"\"\"F*7$$!3u****\\(oUIn%! 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" }} {PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be defined on the interval " } {XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G \"\"\"" }{TEXT -1 5 " by " }{XPPEDIT 18 0 "f(x) = x/2+sin(Pi/2*x);" " 6#/-%\"fG6#%\"xG,&*&F'\"\"\"\"\"#!\"\"F*-%$sinG6#*(%#PiGF*F+F,F'F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) (i) Find the first \+ few terms of the Fourier half-range " }{TEXT 268 13 "cosine series" } {TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 72 " that is, t he Fourier series for the even periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " with " }{TEXT 268 8 "period 2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 " (ii) Draw the gra phs of some truncated series. " }}{PARA 0 "" 0 "" {TEXT -1 60 " (i ii) Comment on the convergence of the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Here is the graph of t he even periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 25 " for comparison purposes." }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 50 ": Part (b) of this question appears further down. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "f := x -> x/2+sin(Pi*x/2):\n'f(x)'=f(x);\nf_even := \+ unapply(simplify(f(piecewise(x<0,-x,x))),x):\n'f_even(x)'=f_even(x);\n f_ := x -> f_even(x-2*floor((x+1)/2)):\n'f_(x)'='f_even(x-2*floor((x+1 )/2))';\nplot(f_(x),x=-2.2..6.2,y=0..1.6,color=COLOR(RGB,.4,0,1),thick ness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"#!\" \"F'\"\"\"F,-%$sinG6#,$*(F*F+%#PiGF,F'F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'f_evenG6#%\"xG-%*PIECEWISEG6$7$,&*&\"\"#!\"\"F'\"\" \"F/-%$sinG6#,$*(F.F/%#PiGF0F'F0F0F/2F'\"\"!7$,&*&F.F/F'F0F0F1F01F8F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%'f_evenG6#,&F'\"\" \"*&\"\"#F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" }}{PARA 13 "" 1 " " {GLPLOT2D 656 169 169 {PLOTDATA 2 "6'-%'CURVESG6#7cu7$$!3;+++++++A!# <$\"3ix%\\P%*p,4%!#=7$$!3C++DwfAa@F*$\"3!o[Ii'*y+<$F-7$$!3G++]_>X3@F*$ \"39Y%)>C6fPAF-7$$!3M++vGzni?F*$\"3a#z'z[CM'H\"F-7$$!3R+++0R!p,#F*$\"3 1(\\J&QC9+N!#>7$$!3_v=nrp%p+#F*$\"3Y/>M&oa&Q9FB7$$!3A]PMQ+*p*>F*$\"3[z aAE***HB'!#?7$$!3ODc,0J.()>F*$\"3AG-T;!R]o#FB7$$!3G+vorh2x>F*$\"3-40ON )ziu%FB7$$!3K]7.0B;d>F*$\"3!>t$**ettl))FB7$$!3;+]PQ%[s$>F*$\"3iKj!zihy H\"F-7$$!3E+D102U(*=F*$\"34i,rL[C<@F-7$$!3;++vrHfd=F*$\"3kJ)e2\"3NIHF- 7$$!3@+v=?D!G\"=F*$\"3cbh_'['GMQF-7$$!3/+]io?,o8)Qs%F-7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 52 " (ii) Draw the g raphs of some truncated series. " }}{PARA 0 "" 0 "" {TEXT -1 51 "Here \+ is the graph of the odd periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " for comparison purposes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "f : = x -> x/2+sin(Pi*x/2);\nf_odd := unapply(simplify(f(piecewise(x<0,-x, x))*signum(x)),x):\n'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-2*floor(( x+1)/2)):\n'f_(x)'='f_odd(x-2*floor((x+1)/2))';\nplot(f_(x),x=-2.2..5. 2,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/9$F/F /-%$sinG6#,$*&F.F/*&%#PiGF/F1F/F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&f_oddG6#%\"xG,&*&\"\"#!\"\"F'\"\"\"F,-%$sinG6#,$*(F *F+%#PiGF,F'F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-% &f_oddG6#,&F'\"\"\"*&\"\"#F,-%&floorG6#,&*&F.!\"\"F'F,F,#F,F.F,F,F4" } }{PARA 13 "" 1 "" {GLPLOT2D 534 202 202 {PLOTDATA 2 "6'-%'CURVESG6#7_r 7$$!3;+++++++A!#<$!3ix%\\P%*p,4%!#=7$$!3[LLeH0N>@F*$!3%z7%*emV0Y#F-7$$ !3xmm;f5qQ?F*$!3XdX3auX5!)!#>7$$!3E+v='QG&o>F*$\"3bvBB5p6:lF87$$!3]L$3 Krb$)*=F*$\"3S7a=W72)4#F-7$$!3?+DJ.#R%>=F*$\"3E9Z;iJ<,PF-7$$!3)om;MpA0 u\"F*$\"3'z'f&R#*G8E&F-7$$!3kmmmtI3h;F*$\"3kj'QusZ.x'F-7$$!3gmm\"RXV;e \"F*$\"3I())eqwM/?)F-7$$!39+D\"oR\"e-:F*$\"3G(G$Gm1VH&*F-7$$!3mL$3(R$> NU\"F*$\"3?f;8M]*\\2\"F*7$$!3-n;aW!=pF\"F*$\"3kPBmp'=%o7F*7$$!3iL$3(p% >5?\"F*$\"3AS3z320]8F*7$$!3A+]([*37D6F*$\"3CJ**Qnx==9F*7$$!3!omTvote3 \"F*$\"31Z8cw#zzW\"F*7$$!3QL$3-[Em/\"F*$\"3c2bFee+u9F*7$$!3ym;awG+F5F* $\"3OTHCP\"*f&[\"F*7$$!3=+](GFzt+\"F*$\"3S2Vz%=Vi\\\"F*7$$!3kL$e>PtC+ \"F*$\"3=yAymdv)\\\"F*7$$!3womT5Znv**F-$!3yc.[Nkx)\\\"F*7$$!3:,+D,dhE* *F-$!3ajQ(yMki\\\"F*7$$!3kML3#pcv()*F-$!3-yHVzGp$\\\"F*7$$!3a++vt'Q%z( *F-$!3](eE<$=P)[\"F*7$$!3anmTb1K\"o*F-$!3wCL(4R8G[\"F*7$$!3gMLeMQ*))*) )F-$!3B&o!HHT-I9F*7$$!3o,+v8qY;\")F-$!3/?fWTPPi8F*7$$!3QMLL)pAo]'F-$!3 #yQ`[xS&y6F*7$$!3qmm;4r/*3&F-$!39lOf'*G!Qr*F-7$$!3gLL3_t+\"H%F-$!3o>)z J%***oQ)F-7$$!3_+++&fnH\\$F-$!3aKui*eZ?'pF-7$$!3;+++55l\"p#F-$!3m:T4+H ,\\aF-7$$!3!)*****\\UM.*=F-$!3)Qlf3hb5(QF-7$$!3]*****\\T<\"=6F-$!3a+4' )3xP1BF-7$$!3!>*****\\S+fMF8$!3)Q$3-;(F87$$\"3'[l;z>mMb$F8$\"3!>o 1tv1cN(F87$$\"3;ILeWOfc5F-$\"3E&z>B7\"Q!=#F-7$$\"3S(*\\7$=W/*=F-$\"3?4 AI_`FrQF-7$$\"3kkmm@ZHCFF-$\"3S//7bF-7$$\"3ykm;/bnIMF-$\"3QB8$H[nr %oF-7$$\"3#\\mmmGcq8%F-$\"34Qg%fOw\">\")F-7$$\"3#*)**\\7%QV!y&F-$\"3'f pT$R#*Gx5F*7$$\"3A95@'[x9F*7$$\"31-D\"ywe 4y*F-$\"3]w$Q)ygX)[\"F*7$$\"3C-]7`+$4(**F-$\"33]@9xg`)\\\"F*7$$\"3y7`W (GU=+\"F*$!3+&=s!pY2*\\\"F*7$$\"3MD\"y&p:f15F*$!3M*H%*Gh]m\\\"F*7$$\"3 \"z$4r^3M65F*$!3MCh/14<%\\\"F*7$$\"3C]P%Q8!4;5F*$!3)[&\\Pabj\"\\\"F*7$ $\"39v$4\")p)eD5F*$!3y+lVbzR'[\"F*7$$\"3E+]Pis3N5F*$!3'=)QyCz$4[\"F*7$ $\"3bLe90LP96F*$!3f1O`f$=nU\"F*7$$\"3#om;zMfO>\"F*$!3sh+!**QdsN\"F*7$$ \"3+n;HB!y#R8F*$!3iYWda(z;>\"F*7$$\"3=LL3PuM'\\\"F*$!3.)eyYz$yH'*F-7$$ \"37L$eWT)\\f;F*$!3y!4?R>-(*z'F-7$$\"3emm;<,^I=_,=F*$!31G\"H]K<)fSF-7$$\"3om\"HUS;#y=F*$!3#)f)e%RTC5DF-7$$\"3 MLLe))4\"\\&>F*$!3**35!*fg4J$*F87$$\"3_m;z)*R9M?F*$\"3Asl,0e&z1(F87$$ \"3q******3qP8@F*$\"3KY75dwSQBF-7$$\"3'***\\i^5*3>#F*$\"3oZYJ%[L#3RF-7 $$\"3E++D%40%oAF*$\"3)*\\G7NIKMaF-7$$\"3***\\(=;RSVBF*$\"3![BnvT$Q`oF- 7$$\"3t**\\7QFS=CF*$\"3x<2G!=S7?)F-7$$\"3'****\\xa\\\\e#F*$\"3y(4j]/rt 3\"F*7$$\"3]mmmj#*fMFF*$\"3'3g(>'3_;G\"F*7$$\"3ILLeq2\\9GF*$\"3[&==&)y )3l8F*7$$\"35++]xAQ%*GF*$\"3+A1U=1YL9F*7$$\"3]$eRs1z0$HF*$\"3K\")G%=*) \\$f9F*7$$\"3Ym\"zp&exmHF*$\"3u.e*\\UE?[\"F*7$$\"35iSTa]#e(HF*$\"339uq /;>([\"F*7$$\"3;e*[=Du[)HF*$\"3SQO+$)[:#\\\"F*7$$\"3?1kc]))R*)HF*$\"3S L)G)z2c%\\\"F*7$$\"3y`QG\\M#R*HF*$\"3K]0Drh\"p\\\"F*7$$\"3Q,8+[![%)*HF *$\"3='[mD0@#*\\\"F*7$$\"3U\\(=nksH+$F*$!3qtp!\\x-&)\\\"F*7$$\"37T&)eT 52@IF*$!3U?)*)*zn\"*)[\"F*7$$\"3#GLekVp\"RIF*$!3a`Z@!3B&y9F*7$$\"3om\" z/u9$=JF*$!3^-v&QZAOU\"F*7$$\"3_++]W+Y(>$F*$!3IoTnA=b`8F*7$$\"3/m;zq\" *RYLF*$!3mt`jkAQ#=\"F*7$$\"3I+]isM3-NF*$!3!*)Q:)f5ZP&*F-7$$\"3aLeR.%)= yNF*$!3)R*R)Rr)eg#)F-7$$\"3!ommTL$HaOF*$!3Ah6cP*)*e*oF-7$$\"3TLekV\"oR t$F*$!3e65'H?9%)Q&F-7$$\"3,+]7`Hk8QF*$!3?AJKUpWc_hAF-7$$\"3gLLLvb6nRF*$!3V'=(R\"z(R2oF87$$\"3WL$3n`*eXSF*$\"3n B0/X`aM%*F87$$\"3GLL3)\\jS7%F*$\"3!>Gbgs!zcDF-7$$\"3(3]Pal()=?%F*$\"3S GeZ$e%zFTF-7$$\"3en;z7=rzUF*$\"3a^0bTbD_cF-7$$\"3#R$e*38C7N%F*$\"3)yVw 9r\"\\(*pF-7$$\"3H+++\\ktAWF*$\"3u[7y5,mw#)F-7$$\"37nmTA)eme%F*$\"3c5; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 59 "(a) (i) Find the first few terms of the F ourier half-range " }{TEXT 281 13 "cosine series" }{TEXT -1 18 " for t he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 72 " that is, the Fourier series \+ for the even periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 6 " with " }{TEXT 268 8 "period 6" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 52 " (ii) Draw the graphs of some trunca ted series. " }}{PARA 0 "" 0 "" {TEXT -1 60 " (iii) Comment on the convergence of the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 52 "Here is the graph of the even periodic e xtension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " f or comparison purposes." }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 50 ": Part (b) of this question appears further down. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "f := x -> piecewise(x<2,3-x,1):\n'f(x)'=f(x);\nf_even := unapply(simplify( f(piecewise(x<0,-x,x))),x):\n'f_even(x)'=f_even(x);\nf_ := x -> f_even (x-6*floor((x+3)/6)):\n'f_(x)'='f_even(x-6*floor((x+3)/6))';\nplot(f_( x),x=-4.3..10.3,y=0..3,color=COLOR(RGB,.4,0,1),thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&\"\"$\"\"\" F'!\"\"2F'\"\"#7$F.%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% 'f_evenG6#%\"xG-%*PIECEWISEG6&7$\"\"\"1F'!\"#7$,&F'F,\"\"$F,2F'\"\"!7$ ,&F1F,F'!\"\"2F'\"\"#7$F,1F8F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%# f_G6#%\"xG-%'f_evenG6#,&F'\"\"\"*&\"\"'F,-%&floorG6#,&*&F.!\"\"F'F,F,# F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 632 203 203 {PLOTDATA 2 "6 '-%'CURVESG6#7ep7$$!3#)*************H%!#<$\"3#)*************H\"F*7$$!3 fLLeu2)39%F*$\"3fLLeu2)39\"F*7$$!3[mm;\\:w\")RF*$\"\"\"\"\"!7$$!3#)*\\ P>w7L%QF*F57$$!3F*$\"3S +]ih%3I0\"F*7$$!3;m;/fybp=F*$\"3%QLe49U/8\"F*7$$!3)HLe>;(=:+]7NN=R9F*7$$!3;mTNGs--9F*$\"3%Q$e krF(zf\"F*7$$!3]KL$=*zBV7F*$\"3]nm;3?wc#F*7$$!3Q!****\\DDh['Ffp$\"3'4++XZ(Q^BF*7$$!3n#******G\\^!\\Ffp$\" 3t+++r][4DF*7$$!3'\\****\\KtTK$Ffp$\"3]++]nEenEF*7$$!31#****\\.51!=Ffp $\"3z++]'**Q*>GF*7$$!3e\"*)***\\uYqF!#>$\"33,+]D`HsHF*7$$\"3Ou;H#o%\\1 6Ffp$\"3cK3xJ0N*)GF*7$$\"3)yL$e4h.!\\#Ffp$\"3Am;/*Q'*4v#F*7$$\"3w1]i&o +_8%Ffp$\"3K*\\P9$*zke#F*7$$\"3kvmmh_O!y&Ffp$\"3VKL$QZj>U#F*7$$\"3)Rnm TXNS<(Ffp$\"3gKLeakf#G#F*7$$\"3MsmmYcqn&)Ffp$\"3xKLLN%HK9#F*7$$\"37Mek Rv))=5F*$\"3)e;a.Y76)>F*7$$\"3+,]i9X+\"=\"F*$\"3+**\\P&[&**==F*7$$\"3/ Me92b]C8F*$\"3'f;aG\\%\\v;F*7$$\"33nmm*\\1!o9F*$\"3$HLL.]$*>`\"F*7$$\" 3;Me9K)\\ai\"F*$\"3%e;ay;]XP\"F*7$$\"3C,]ikJ*Gy\"F*$\"3v)*\\PNo5<7F*7$ $\"32,+D\\k\"G$>F*$\"3%*)**\\2b$=n5F*7$$\"3)3+vQtRF3#F*F57$$\"3bMeR`*o \"RAF*F57$$\"3?om\"H<)f&R#F*F57$$\"3?o;z$))**Go#F*F57$$\"3=ML3#f$z#*HF *F57$$\"3SM$e41'o9LF*F57$$\"3#3+v8S%*[f$F*F57$$\"31o\"zMu5iu$F*F57$$\" 3ENLe&3Fv*QF*F57$$\"3in;Ht;&Q0%F*$\"3in;Ht;&Q0\"F*7$$\"3(******4Ew,@%F *$\"3(******4Ew,@\"F*7$$\"3s+]7s$4JO%F*$\"3s+]7s$4JO\"F*7$$\"3Y,+D$[Ug ^%F*$\"3Y,+D$[Ug^\"F*7$$\"3a,v$>88Sm%F*$\"3a,v$>88Sm\"F*7$$\"3l,]i!y$) >\"[F*$\"3l,]i!y$)>\"=F*7$$\"3k,vod*zi(\\F*$\"3k,vod*zi(>F*7$$\"3k,+vM hdS^F*$\"3k,+vMhdS@F*7$$\"3!RL3sb.#)G&F*$\"3!RL3sb.#)G#F*7$$\"3'F*$\"3.K3Fb(3r!GF*7$$\"3I,+]!zU !\\jF*$\"3q)***\\4s&4l#F*7$$\"39Me*eApf\\'F*$\"3'e;/TxIS]#F*7$$\"3)pm \"Hhc*Gk'F*$\"3-L$3(QV5dBF*7$$\"3AM$377wkz'F*$\"3yl;zyQ_.AF*7$$\"3Y,]7 \"ec+&pF*$\"3a)*\\()=M%*\\?F*7$$\"3UMek_&4-5(F*$\"3clTNZ/z**=F*7$$\"3Q nm;CDO]sF*$\"3iKL$eZP'\\N3x`e'\\%zF*$\"3![;Hi9M]0\"F*7$$\"3v M$3iNzB-)F*F57$$\"3IMekeGz*4)F*F57$$\"3kNL3hj?x\")F*F57$$\"3`-v=?8vI$) F*F57$$\"3Up;HziH%[)F*F57$$\"37.++@+[m()F*F57$$\"39nmTLW*)*3*F*F57$$\" 3eOLLR,;z$*F*F57$$\"3;+]i9xd(o*F*F57$$\"3?+++,_ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(b) (i) Find the first few terms of the Fourier half-range " }{TEXT 282 11 "s ine series" }{TEXT -1 19 " for the function " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 73 " \+ that is, the Fourier series for the odd periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " with " }{TEXT 268 8 "period 6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 " \+ (ii) Draw the graphs of some truncated series. " }}{PARA 0 "" 0 "" {TEXT -1 51 "Here is the graph of the odd periodic extension of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " for comparison p urposes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "f := x -> piecewise(x<2,3-x,1):\n'f(x)'=f(x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x)),x):\n'f_odd(x)' =f_odd(x);\nf_ := x -> f_odd(x-6*floor((x+3)/6)):\n'f_(x)'='f_odd(x-6* floor((x+3)/6))';\nplot(f_(x),x=-4.3..10.3,color=COLOR(RGB,.4,0,1),thi ckness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWI SEG6$7$,&\"\"$\"\"\"F'!\"\"2F'\"\"#7$F.%*otherwiseG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%&f_oddG6#%\"xG-%*PIECEWISEG6'7$!\"\"1F'!\"#7$,&F'F ,\"\"$F,2F'\"\"!7$F3/F'F37$,&F1\"\"\"F'F,2F'\"\"#7$F81F:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f_G6#%\"xG-%&f_oddG6#,&F'\"\"\"*&\"\"'F, -%&floorG6#,&*&F.!\"\"F'F,F,#F,\"\"#F,F,F4" }}{PARA 13 "" 1 "" {GLPLOT2D 664 201 201 {PLOTDATA 2 "6'-%'CURVESG6#7`s7$$!3#)*********** **H%!#<$\"3#)*************H\"F*7$$!3fLLeu2)39%F*$\"3fLLeu2)39\"F*7$$!3 [mm;\\:w\")RF*$\"\"\"\"\"!7$$!3#)*\\P>w7L%QF*F57$$!3*o(zvsHF*FP7$$!31Tg_q(3I'HF*FP7$$!3]qUzd.^VHF*FP7$$!3^*\\i]%>,CH F*FP7$$!3S;a8%H=g%GF*FP7$$!3IL$3KkC!oFF*FP7$$!3%***\\iYWSBEF*FP7$$!3em ;/]UyyCF*FP7$$!30L$3KTQ!HBF*FP7$$!3]**\\PwDHz@F*FP7$$!3Hm;/(*)e=5#F*FP 7$$!31L$3x@DW-#F*FP7$$!3g**\\PQ:*p%>F*$!3S+]ih%3I0\"F*7$$!3;m;/fybp=F* $!3%QLe49U/8\"F*7$$!3)HLe>;(=:+]7NN=R9F*7$$!3]KL$=*zBV7F*$!3]nm;3?wcGF*7$$!3e\"*)***\\uYqF!#> $!33,+]D`HsHF*7$$!3*=gUz0ad!>Fds$!3!Qd?%fC%4)HF*7$$!3>7`)emS5/\"Fds$!3 %p9TLf*e*)HF*7$$!3#\\A!GQFFjU#F*7$$\"3MsmmYcqn&)F_r$\"3xKLLN%HK9#F*7$$\"3+,]i9X +\"=\"F*$\"3+**\\P&[&**==F*7$$\"33nmm*\\1!o9F*$\"3$HLL.]$*>`\"F*7$$\"3 C,]ikJ*Gy\"F*$\"3v)*\\PNo5<7F*7$$\"3;,v$p![&y&=F*$\"3%))\\iI>X@9\"F*7$ $\"32,+D\\k\"G$>F*$\"3%*)**\\2b$=n5F*7$$\"3#4D1/F(Hq>F*$\"34\\PfHFqH5F *7$$\"3w+Dc\"4yx+#F*F57$$\"3#3v=F\"*e_/#F*F57$$\"3)3+vQtRF3#F*F57$$\"3 bMeR`*o\"RAF*F57$$\"3?om\"H<)f&R#F*F57$$\"3?o;z$))**Go#F*F57$$\"3>,v$z tYy$GF*F57$$\"3=ML3#f$z#*HF*F57$$\"38yctJF&G+$F*FP7$$\"3j@!)Qr=\"H,$F* FP7$$\"39l./65(H-$F*FP7$$\"344Fp],.LIF*FP7$$\"3c'R(**H%[J0$F*FP7$$\"3+ %3-$4nEtIF*FP7$$\"3$*e9\"zE.N6$F*FP7$$\"3IM3_E)RP:$F*FP7$$\"3d%eRP%H@M KF*FP7$$\"3SM$e41'o9LF*FP7$$\"3inm;J-zaMF*FP7$$\"3#3+v8S%*[f$F*FP7$$\" 31o\"zMu5iu$F*FP7$$\"3ENLe&3Fv*QF*FP7$$\"3e=/^K#3m$RF*FP7$$\"3Y,vVz$*o vRF*FP7$$\"3mU5!H&*H_*RF*FP7$$\"3J%ekj_qZ,%F*$!3J%ekj_qZ,\"F*7$$\"3(f7 G)*46V.%F*$!3(f7G)*46V.\"F*7$$\"3in;Ht;&Q0%F*$!3in;Ht;&Q0\"F*7$$\"3DMe 9nR,KTF*$!3DMe9nR,K6F*7$$\"3(******4Ew,@%F*$!3(******4Ew,@\"F*7$$\"3Y, +D$[Ug^%F*$!3Y,+D$[Ug^\"F*7$$\"3l,]i!y$)>\"[F*$!3l,]i!y$)>\"=F*7$$\"3k ,+vMhdS^F*$!3k,+vMhdS@F*7$$\"3'F*$\" 3.K3Fb(3r!GF*7$$\"3I,+]!zU!\\jF*$\"3q)***\\4s&4l#F*7$$\"3)pm\"Hhc*Gk'F *$\"3-L$3(QV5dBF*7$$\"3Y,]7\"ec+&pF*$\"3a)*\\()=M%*\\?F*7$$\"3Qnm;CDO] sF*$\"3iKL$eZP'\\N3x`e'\\% zF*$\"3![;Hi9M]0\"F*7$$\"3vM$3iNzB-)F*F57$$\"3IMekeGz*4)F*F57$$\"3kNL3 hj?x\")F*F57$$\"3`-v=?8vI$)F*F57$$\"3Up;HziH%[)F*F57$$\"3FOe9]\")QD')F *F57$$\"37.++@+[m()F*F57$$\"3-qT5COLZ))F*F57$$\"38N$3sA(=G*)F*F57$$\"3 'oP%)zi+%[*)F*F57$$\"3!oTg(GSho*)F*F57$$\"3cQ%[\"H2sy*)F*F57$$\"3_ek`H u#)))*)F*F57$$\"3]yW#*HT$*)**)F*F57$$\"3D+DJI3/4!*F*FP7$$\"3q$ek=jn%\\ !*F*FP7$$\"39nmTLW*)*3*F*FP7$$\"3(=+vjGFXB*F*FP7$$\"3eOLLR,;z$*F*FP7$$ \"3;+]i9xd(o*F*FP7$$\"3?+++,_ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 29 "___________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be defined on the interva l " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 3;" "6 #1%!G\"\"$" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = PIECEWISE([x^2, \+ 0 <= x and x < 2],[4, 2 <= x and x <= 3]);" "6#/-%\"fG6#%\"xG-%*PIECEW ISEG6$7$*$F'\"\"#31\"\"!F'2F'F-7$\"\"%31F-F'1F'\"\"$" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(a) ( i) Find the first few terms of the Fourier half-range " }{TEXT 283 13 "cosine series" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 73 " \+ that is, the Fourier series for the even periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " with " } {TEXT 268 8 "period 6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 " (ii) Draw the graphs of some truncated series. " }}{PARA 0 "" 0 "" {TEXT -1 60 " (iii) Comment on the convergence of the Fourier s eries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Here is the graph of the even periodic extension of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " for comparison purposes." }} {PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 50 ": Part (b) of this qu estion appears further down. 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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "(b) (i) Find th e first few terms of the Fourier half-range " }{TEXT 284 11 "sine seri es" }{TEXT -1 19 " for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 73 " tha t is, the Fourier series for the odd periodic extension of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " with " }{TEXT 268 8 "period 6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 " \+ (ii) Draw the graphs of some truncated series. " }}{PARA 0 "" 0 "" {TEXT -1 51 "Here is the graph of the odd periodic extension of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " for comparison p urposes. 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Fgfm-%*THICKNESSG6#\"\"#-%&COLORG6&%$RGBG$Fip!\"\"F+$\"\"\"F*-%%VIEWG6 $;F(Fafm%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 29 "_____________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{PARA 0 "" 0 "" {TEXT -1 68 "(a) Check, using Maple, that the Fourier half-range sine series of " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=sqrt(2*x-x^2),0<=x" "6$/-%\"fG6#% \"xG-%%sqrtG6#,&*&\"\"#\"\"\"F'F.F.*$F'F-!\"\"1\"\"!F'" }{XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[k]*sin(k *Pi*x/2),k = 1 .. infinity);" "6#-%$SumG6$*&&%\"bG6#%\"kG\"\"\"-%$sinG 6#**F*F+%#PiGF+%\"xGF+\"\"#!\"\"F+/F*;F+%)infinityG" }{TEXT -1 1 "," } }{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "b[k] = 2/k;" "6#/ &%\"bG6#%\"kG*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin (k*Pi/2)*J[1](k*Pi/2);" "6#*&-%$sinG6#*(%\"kG\"\"\"%#PiGF)\"\"#!\"\"F) -&%\"JG6#F)6#*(F(F)F*F)F+F,F)" }{TEXT -1 7 ", with " }{XPPEDIT 18 0 "J [1](x)" "6#-&%\"JG6#\"\"\"6#%\"xG" }{TEXT -1 55 " is the Bessel functi on given in Maple by the function " }{TEXT 268 12 "BesselJ(1,x)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 74 "(b) Compare the graphs \+ of some truncated Fourier series with the graph of " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 " __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 35 "Code for picture of even extension " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 412 "p1 := plot([[t,1- cos(t),t=0..4],[-t,1-cos(t),t=0..4]],\n color=[red,brown],linesty le=[1,1],\n tickmarks=[0,0],thickness=[2, 1]):\np2 := plot([[[3.8,-0.05],[3.8,0.05]],[[-3.8,-0.05],[-3.8,0.05]]] ,\n color=black):\nt1 := plots[textplot]([[3.75,-0.2,`L`], \n [-3.85,-0.2,`-L`],[-.2,2.1,`y`],[4.7,-0.1,`x`]],color=black):\np lots[display]([p1,p2,t1],view=[-4.8..4.8,-0.2..2.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Code for picture of odd extension " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 419 "p1 := plot([[t,1-cos(t),t=0..3.8],[-t,cos(t)-1,t=0.. 3.8]],\n color=[red,brown],linestyle=[1,1],\n \+ tickmarks=[0,0],thickness=[2,1]):\np2 := plot([[[3.8,-0.0 5],[3.8,0.05]],[[-3.8,-0.05],[-3.8,0.05]]],\n color=black): \nt1 := plots[textplot]([[3.75,-0.2,`L`],\n [-3.85,-0.2,`-L`],[-.2, 2.1,`y`],[4.7,-0.1,`x`]],color=black):\nplots[display]([p1,p2,t1],view =[-4.8..4.8,-2.2..2.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "f := t -> 1.3+cos(t)/2:\np1 := plot([[t,f(t),t=0..3.8],[-t,-f(t),t=0..3.8] ],\n color=[red,brown],linestyle=[1,1],\n \+ tickmarks=[0,0],thickness=[2,1]):\np2 := plot([[0,1.8]],style =point,symbol=circle,color=red):\np3 := plot([[0,-1.8]],style=point,sy mbol=circle,color=brown):\np4 := plot([[[0,0]]$3],style=point,symbol=[ circle,diamond,cross],\n color=red):\np5 := plot([[[3.8,-0.05],[3 .8,0.05]],[[-3.8,-0.05],[-3.8,0.05]]],\n color=black):\nt1 \+ := plots[textplot]([[3.75,-0.2,`L`],\n [-3.85,-0.2,`-L`],[-.2,2.1,` y`],[4.7,-0.1,`x`]],\n font=[HELVETICA,10],color=COLOR(RGB,.01,0,0) ):\nplots[display]([p1,p2,p3,p4,p5,t1],view=[-4.8..4.8,-2.2..2.2]);" } }{PARA 13 "" 1 "" {GLPLOT2D 358 269 269 {PLOTDATA 2 "62-%'CURVESG6&7S7 $$\"\"!F)$\"3/+++++++=!#<7$$\"3ELLL$e=HG)!#>$\"3KB.(=\"eG)z\"F,7$$\"3m mmTvT)*[:!#=$\"3e'G$*Rg8Sz\"F,7$$\"3ALL$31y%fBF6$\"3%HassgYhy\"F,7$$\" 3aLL$eUW`<$F6$\"39HJ,dS+v(4Pr[u\"F,7$$\"3,++ve%\\'>bF6$\"3$oN,C\"zuD6\"F,$\"3.2>A&G\\9_\"F,7$$\"3?L$37C()R=\"F,$\"3 -Y!y%*e<')[\"F,7$$\"3smm;Xfip7F,$\"3%f[Sh0>$[9F,7$$\"3pmmmgJE,:F,$ \"3KpZiq\"RZL\"F,7$$\"3!***\\()o(=Ke\"F,$\"3/&G3(Q*)y$H\"F,7$$\"33+]iE 5Eh;F,$\"3'*Q'G2KH[D\"F,7$$\"3tmmTN**oU)>F,$\"3<8WK_s=+6F,7$$\"3++]7Hb$[0#F,$\"3!*4\"=Y#4Kn5F,7$$\"3=LL3SH gL@F,$\"3gc>#**)*=K.\"F,7$$\"3\"******HQx\\@#F,$\"3![*\\0/4G(***F67$$ \"3c***\\(*R'e%H#F,$\"3Y)f&*4O`))o*F67$$\"3++](o@7;P#F,$\"3x46>ReAQ$)F67$$\"3Qm;aD4:[GF,$\"3_nm$)Q)H P@)F67$$\"33+]P=p4GHF,$\"3)>))Q7!*=N6)F67$$\"3Umm;R(ei+$F,$\"3%>D9%G$= d/)F67$$\"3c**\\(=#p3)3$F,$\"3U.;)4Xbr+)F67$$\"3OLLL(=(*o;$F,$\"32MK:8 2g,!)F67$$\"3MLLem?\\ZKF,$\"3\"4Xi?_5!G!)F67$$\"3cm;az&>uK$F,$\"3E$H>z (43'3)F67$$\"37+++jY'3S$F,$\"3M9?pGc6n\")F67$$\"3emm\"p,T][$F,$\"3%o$4 Y5Y+#H)F67$$\"3%HLL8NH.c$F,$\"3EI=p=L)>V)F67$$\"3\"***\\()=CgSOF,$\"3Q s[>Bcr4')F67$$\"3t**\\7()RV(4Pr[u\"F,7$$!3,++ve%\\'>bF6$!3$oN,C\" zuD6\"F,$!3.2>A&G\\9_\"F,7$$!3?L$37C()R=\"F,$!3-Y!y% *e<')[\"F,7$$!3smm;Xfip7F,$!3%f[Sh0>$[9F,7$$!3pmmmgJE,:F,$!3KpZiq\"RZ L\"F,7$$!3!***\\()o(=Ke\"F,$!3/&G3(Q*)y$H\"F,7$$!33+]iE5Eh;F,$!3'*Q'G2 KH[D\"F,7$$!3tmmTN**oU)>F,$!3<8WK_s=+6F,7$ $!3++]7Hb$[0#F,$!3!*4\"=Y#4Kn5F,7$$!3=LL3SHgL@F,$!3gc>#**)*=K.\"F,7$$! 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