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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 53 "Even and odd functions, adding si
ne and cosine graphs" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, N
anaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  26.3.20
07" }}{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 22 "Even and odd functions" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 6 "A set " }{TEXT 307 1 "S" }{TEXT -1 20 " of real numbers is " }
{TEXT 259 19 "symmetrical about 0" }{TEXT -1 14 " if whenever  " }
{XPPEDIT 18 0 "x in S" "6#-%#inG6$%\"xG%\"SG" }{TEXT -1 12 "  then als
o " }{XPPEDIT 18 0 "-x in S" "6#-%#inG6$,$%\"xG!\"\"%\"SG" }{TEXT -1 
94 ". In particular, the set of all real numbers |R is symmetrical abo
ut 0 as is any open interval" }{XPPEDIT 18 0 " ``(-r,r)" "6#-%!G6$,$%
\"rG!\"\"F'" }{TEXT -1 20 " or closed interval " }{XPPEDIT 18 0 "[-r,r
]" "6#7$,$%\"rG!\"\"F%" }{TEXT -1 8 ", where " }{TEXT 305 1 "r" }
{TEXT -1 29 " is a positive real number.  " }}{PARA 0 "" 0 "" {TEXT 
-1 11 "A function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 
47 " whose domain is symmetrical about 0 is called " }{TEXT 259 4 "eve
n" }{TEXT -1 15 " provided that " }{XPPEDIT 18 0 "f(-x) = f(x);" "6#/-
%\"fG6#,$%\"xG!\"\"-F%6#F(" }{TEXT -1 17 " for all numbers " }{TEXT 
289 1 "x" }{TEXT -1 29 " in the interval, and called " }{TEXT 259 3 "o
dd" }{TEXT -1 15 " provided that " }{XPPEDIT 18 0 "f(-x) = -f(x);" "6#
/-%\"fG6#,$%\"xG!\"\",$-F%6#F(F)" }{TEXT -1 17 " for all numbers " }
{TEXT 288 1 "x" }{TEXT -1 17 " in the interval." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 294 1 "a" }
{TEXT -1 26 " is a positive number, an " }{TEXT 259 13 "even function
" }{TEXT -1 41 " has the same value at a negative number " }{XPPEDIT 
18 0 "-a" "6#,$%\"aG!\"\"" }{TEXT -1 48 " as it has at the correspondi
ng positive number " }{TEXT 290 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 53 "A simple example of an even function is the function " 
}{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 33 "The graph of an even function is " }
{TEXT 259 22 "symmetrical about the " }{TEXT 291 1 "y" }{TEXT 259 5 " \+
axis" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 
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}}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "Given a point P" }
{XPPEDIT 18 0 "``(a,f(a))" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 21 " on
 the graph, where " }{TEXT 319 1 "a" }{TEXT -1 25 " is positive, the p
oint Q" }{XPPEDIT 18 0 "``(-a,f(a))" "6#-%!G6$,$%\"aG!\"\"-%\"fG6#F'" 
}{TEXT -1 31 " is also on the graph, and the " }{TEXT 295 1 "y" }
{TEXT -1 93 " axis is a right-angle bisector of the line QP, that is, \+
QM = MP and angle QMO is 90 degrees." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 293 1 "a" }{TEXT -1 39 " is \+
a positive number, the value of an " }{TEXT 259 12 "odd function" }
{TEXT -1 24 " at the negative number " }{XPPEDIT 18 0 "-a" "6#,$%\"aG!
\"\"" }{TEXT -1 67 " is the negative of its value at the corresponding
 positive number " }{TEXT 292 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 52 "A simple example of an odd function is the function " }
{XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 1 ".
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etrical about the the origin in the sense indicated by the following p
icture." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 393 475 475 
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{TEXT -1 21 " on the graph, where " }{TEXT 296 1 "a" }{TEXT -1 25 " is
 positive, the point Q" }{XPPEDIT 18 0 "``(-a,-f(a))" "6#-%!G6$,$%\"aG
!\"\",$-%\"fG6#F'F(" }{TEXT -1 91 " is also on the graph, and the line
 QP has the origin O as its mid-point, that is, QO = OP." }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 26 "Examples of even functions" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Example 1" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "plot(x^2,x=-2..2,labels=[x,y]);" }}{PARA 13 "" 1 "" 
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!1LLL))Qj^<F0$\"1_xvy7AoIF07$$!1LLL=Kvl;F0$\"1dyQ%yLZx#F07$$!1nm;C2G!e
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&Fbo7$$!1nmmmj^NmFbo$\"1@,BXx+.WFbo7$$!1ommm9'=(eFbo$\"1$[s$3d(yW$Fbo7
$$!1,++v#\\N)\\Fbo$\"1.`jPjd$[#Fbo7$$!1pmmmCC(>%Fbo$\"15!*RKWoh<Fbo7$$
!1*****\\FRXL$Fbo$\"1_<l<_\">6\"Fbo7$$!1+++D=/8DFbo$\"1H$*>9#z`J'!#<7$
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1*******R^bJ$Fbo$\"1&>/'3\")G*4\"Fbo7$$\"1'*****\\5a`TFbo$\"12NSD.>D<F
bo7$$\"1(****\\7RV'\\Fbo$\"1u0![HmWY#Fbo7$$\"1'*****\\@fkeFbo$\"1eTe3T
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0$\"1MgH-E<r6F07$$\"1LLL$)G[k6F0$\"18jd&Q?gN\"F07$$\"1++]7yh]7F0$\"1&G
UH\"\\/k:F07$$\"1nmm')fdL8F0$\"1uR@7\\Uy<F07$$\"1nmm,FT=9F0$\"1K$\\Af%
*=,#F07$$\"1LL$e#pa-:F0$\"1+7LkskdAF07$$\"1+++Sv&)z:F0$\"1^[pY)\\f\\#F
07$$\"1LLLGUYo;F0$\"1XzA\")Gx$y#F07$$\"1nmm1^rZ<F0$\"1y32%43X0$F07$$\"
1++]sI@K=F0$\"1\"*)RIu/qN$F07$$\"1++]2%)38>F0$\"1')3\"\\D2*fOF07$$\"\"
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" 0 "" {TEXT 263 9 "Example 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+
x^4-x^2;" "6#/-%\"fG6#%\"xG,&*$F'\"\"%\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "plot(x^4-x^2,x=-1.2..1.2,tickmarks=[3,2],labels=[x,y]
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F-7$$!1,+++j%zZ&F-$!16eGkdJ+@F-7$$!1+++!y[q(\\F-$!1bDf$H)\\j=F-7$$!1,+
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#eFen7$$\"1*****\\Z.'yHF-$!1-Bcj3%\\3)Fen7$$\"1+++!Hb(=NF-$!10>%***)e[
3\"F-7$$\"1(*****>d5/SF-$!1.,+yNBY8F-7$$\"1********3KAXF-$!1)RU(H%zoi
\"F-7$$\"1(****\\(3!>*\\F-$!1$yH$or%4(=F-7$$\"1+++?eF0bF-$!11]1svA7@F-
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\\#F-7$$\"1)****\\(oq.vF-$!1(>Sf>R-Y#F-7$$\"1****\\(R\"e_xF-$!11e\\B\"
RzR#F-7$$\"1+++?fX,!)F-$!1vbuXrM.BF-7$$\"1+++5iZ5&)F-$!1Y,31c(p*>F-7$$
\"1,++b:G:!*F-$!1!4)3:b&=_\"F-7$$\"1++](R8sC*F-$!1mA.J?(*Q7F-7$$\"1+++
S_9z%*F-$!1:YpA=V;\"*Fen7$$\"1+++0`'\\u*F-$!1Lz#QPt?y%Fen7$$\"1+++P&y5
+\"F*$\"1>M0I_*G;#F[o7$$\"1++]+Q&[-\"F*$\"1)*ztY\\z&G&Fen7$$\"1+++k!H'
[5F*$\"1;dK)Qwa4\"F-7$$\"1++v`%yR2\"F*$\"1(\\#GyRqp<F-7$$\"1++]VyK*4\"
F*$\"1Us!**4I+_#F-7$$\"1+++W/fB6F*$\"1!)))**yOQ8LF-7$$\"1++]WI&y9\"F*$
\"1[nxY;:%=%F-7$$\"1+]P$y*)3;\"F*$\"1hFaL5O&o%F-7$$\"1++DAl#R<\"F*$\"1
BNf4sl5_F-7$$\"1+]7hK'p=\"F*$\"1mxUf4mgdF-7$$\"1+++++++7F*F+-%'COLOURG
6&%$RGBG$\"#5!\"\"\"\"!F]cl-%+AXESLABELSG6$%\"xG%\"yG-%*AXESTICKSG6$\"
\"$\"\"#-%%VIEWG6$;$!#7F\\cl$\"#7F\\cl%(DEFAULTG" 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 299 9 "Example 3" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "f(x) = sqrt(1-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"
\"F,*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
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'>!*4tx*F.7$$!1++]P'psm\"F.$\"1Fh>K5.g)*F.7$$!1++]74_c7F.$\"1pBi\"oV2#
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g&*F.7$$\"1lmmmZvOLF.$\"1c!ez0!)oU*F.7$$\"1+++]2goPF.$\"1D!*\\-?qi#*F.
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)RQ\\fL6NF.7$$\"1++]P?Wl&*F.$\"1\"G)fhN(e\"HF.7$$\"1+]7G:3u'*F.$\"1H;B
q;AKDF.7$$\"1++v=5s#y*F.$\"1vO>U4Dt?F.7$$\"1+D1k2/P)*F.$\"1MW(eL^zz\"F
.7$$\"1+]P40O\"*)*F.$\"1H.f%oH+Z\"F.7$$\"1^7.#Q?&=**F.$\"1#>z^*=&RF\"F
.7$$\"1+voa-oX**F.$\"1P%\\8H')3/\"F.7$$\"1\\PMF,%G(**F.$\"1&=hJ*p=ltF1
7$$\"\"\"F*F*-%'COLOURG6&%$RGBG$\"#5F)F*F*-%(SCALINGG6#%,CONSTRAINEDG-
%+AXESLABELSG6$%\"xG%\"yG-%*AXESTICKSG6$\"\"#Fcal-%%VIEWG6$;F(F_`l%(DE
FAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 264 9 "Example 4" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
f(x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "pl
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F0$\"1LLL=Kvl;F07$$!1nm;C2G!e\"F0$\"1nm;C2G!e\"F07$$!1LL$3yO5]\"F0$\"1
LL$3yO5]\"F07$$!1++]nU)*=9F0$\"1++]nU)*=9F07$$!1LL$3WDTL\"F0$\"1LL$3WD
TL\"F07$$!1++]d(Q&\\7F0$\"1++]d(Q&\\7F07$$!1nmmc4`i6F0$\"1nmmc4`i6F07$
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mmm9'=(eFbo$\"1ommm9'=(eFbo7$$!1,++v#\\N)\\Fbo$\"1,++v#\\N)\\Fbo7$$!1p
mmmCC(>%Fbo$\"1pmmmCC(>%Fbo7$$!1*****\\FRXL$Fbo$\"1*****\\FRXL$Fbo7$$!
1+++D=/8DFbo$\"1+++D=/8DFbo7$$!1mmm;a*el\"Fbo$\"1mmm;a*el\"Fbo7$$!1pmm
;Wn(o)!#<$\"1pmm;Wn(o)Fjr7$$!1.++D^bUWFjr$\"1.++D^bUWFjr7$$!1qLLL$eV(>
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**\\nZ)H;FboF_t7$$\"1lmm;$y*eCFboFbt7$$\"1*******R^bJ$FboFet7$$\"1'***
**\\5a`TFboFht7$$\"1(****\\7RV'\\FboF[u7$$\"1'*****\\@fkeFboF^u7$$\"1J
LLL&4Nn'FboFau7$$\"1*******\\,s`(FboFdu7$$\"1lmm\"zM)>$)FboFgu7$$\"1**
*****pfa<*FboFju7$$\"1HLLeg`!)**FboF]v7$$\"1++]#G2A3\"F0F`v7$$\"1LLL$)
G[k6F0Fcv7$$\"1++]7yh]7F0Ffv7$$\"1nmm')fdL8F0Fiv7$$\"1nmm,FT=9F0F\\w7$
$\"1LL$e#pa-:F0F_w7$$\"1+++Sv&)z:F0Fbw7$$\"1LLLGUYo;F0Few7$$\"1nmm1^rZ
<F0Fhw7$$\"1++]sI@K=F0F[x7$$\"1++]2%)38>F0F^x7$F+F+-%'COLOURG6&%$RGBG$
\"#5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%\"yG-%*A
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT 
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+
cos*x;" "6#/-%\"fG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 44 "Note that the Maclaurin series expansion of " }
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volves even powers of " }{TEXT 297 1 "x" }{TEXT -1 1 ":" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x = 1-x^2/2!+x^4/4!-x^6/6!+x
^8/8!-x^10/10!+` . . . `;" "6#/*&%$cosG\"\"\"%\"xGF&,0F&F&*&F'\"\"#-%*
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-F,6#F8F.F&*&F'\"#5-F,6#F<F.F.%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 0 
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1+]P%[L'*)[F1$\"1O6B)>zJw\"F-7$$\"1mm;H!o-*\\F1$\"1@F1on;VFF-7$$\"1ML$
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7$$\"1.+]P$[/a)F1$!19pYP<3QjF-7$$\"1OLLLbdQ()F1$!1MT[SudOxF-7$$\"1om\"
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@E*F1$!1B[egt1o)*F-7$$\"1,+++PDj$*F1$!1-8$fzz5)**F-7$$\"1-]P%y+QT*F1$!
1`>/9vR****F-7$$\"1-+voyMk%*F1$!1BNQL@<#***F-7$$\"1,]7`\\*[^*F1$!1+#H4
6A%f**F-7$$\"1-+]P?Wl&*F1$!1H^U06B,**F-7$$\"1,+v=5s#y*F1$!1,!e%4X>m$*F
-7$$\"#5F*F+-%'COLOURG6&%$RGBG$F\\_n!\"\"F*F*-%+AXESLABELSG6$%\"xG%\"y
G-%*AXESTICKSG6$%(DEFAULTG\"\"$-%%VIEWG6$;F(F[_nF[`n" 1 2 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 9 "Example 6" }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "f(x) = PIECEWISE([x, 0 <= x and x < 1],[2-x, 1 <= x \+
and x < 2]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$F'31\"\"!F'2F'\"\"\"7$
,&\"\"#F0F'!\"\"31F0F'2F'F3" }{TEXT -1 10 " ,    and " }{XPPEDIT 18 0 
"f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " is periodic with period 2, " }
}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = 1/2+(x-floor(
x)-1/2)*(-1)^floor(x);" "6#/-%\"fG6#%\"xG,&*&\"\"\"F*\"\"#!\"\"F**&,(F
'F*-%&floorG6#F'F,*&F*F*F+F,F,F*),$F*F,-F06#F'F*F*" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
145 "f := x -> 1/2+(x-floor(x)-1/2)*(-1)^floor(x);\nplot(f(x),x=-2.5..
4.5,-0.2..1.2,numpoints=100,\n                         labels=[x,y],yt
ickmarks=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)
operatorG%&arrowGF(,&#\"\"\"\"\"#F.*&,(9$F.-%&floorG6#F2!\"\"#F.F/F6F.
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$$\"37ihh,*R\"RMF*$\"3!)y$QQ)4g3cF-7$$\"3zbbb&**H%3NF*$\"34UWWW+q:\\F-
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\"3lsss-!zJs$F*$\"3atsss*4#oFF-7$$\"3cSTTJ*\\Lz$F*$\"3O%feeo+l1#F-7$$
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{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Products and compositions of even
 and odd functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 160 "The following table shows how the even or odd nature of \+
a product or composition of two functions depends on the even or odd n
ature of the component functions. " }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "matrix([[f(x), g(x), f(x)*`.`*g(x), f(g(x))], [even,
 even, even, even], [even, odd, odd, even], [odd, even, odd, even], [o
dd, odd, even, odd]]);" "6#-%'matrixG6#7'7&-%\"fG6#%\"xG-%\"gG6#F+*(-F
)6#F+\"\"\"%\".GF2-F-6#F+F2-F)6#-F-6#F+7&%%evenGF;F;F;7&F;%$oddGF=F;7&
F=F;F=F;7&F=F=F;F=" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 
"" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"
xG" }{TEXT -1 10 " is even, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" 
}{TEXT -1 12 " is odd and " }{XPPEDIT 18 0 "h(x) = f(x)*`.`*g(x)" "6#/
-%\"hG6#%\"xG*(-%\"fG6#F'\"\"\"%\".GF,-%\"gG6#F'F," }{TEXT -1 6 ", the
n" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(-x) = f(-x)*`.
`*g(-x)" "6#/-%\"hG6#,$%\"xG!\"\"*(-%\"fG6#,$F(F)\"\"\"%\".GF/-%\"gG6#
,$F(F)F/" }{XPPEDIT 18 0 "`` = f(x)*`.`*(-g(x))" "6#/%!G*(-%\"fG6#%\"x
G\"\"\"%\".GF*,$-%\"gG6#F)!\"\"F*" }{XPPEDIT 18 0 "`` = - f(x)*`.`*g(x
)" "6#/%!G,$*(-%\"fG6#%\"xG\"\"\"%\".GF+-%\"gG6#F*F+!\"\"" }{XPPEDIT 
18 0 "`` = -h(x)" "6#/%!G,$-%\"hG6#%\"xG!\"\"" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 3 "so " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"x
G" }{TEXT -1 8 " is odd." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }
{XPPEDIT 18 0 "h(x) = x^2*sin*x;" "6#/-%\"hG6#%\"xG*(F'\"\"#%$sinG\"\"
\"F'F+" }{TEXT -1 20 " is an odd function." }}{PARA 0 "" 0 "" {TEXT 
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\"fG6#%\"xG" }{TEXT -1 10 " is even, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG
6#%\"xG" }{TEXT -1 12 " is odd and " }{XPPEDIT 18 0 "h(x) = f(g(x))" "
6#/-%\"hG6#%\"xG-%\"fG6#-%\"gG6#F'" }{TEXT -1 7 ", then " }}{PARA 256 
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%!G-%\"fG6#,$-%\"gG6#%\"xG!\"\"" }{XPPEDIT 18 0 "`` = f(g(x))" "6#/%!G
-%\"fG6#-%\"gG6#%\"xG" }{XPPEDIT 18 0 "`` = h(x)" "6#/%!G-%\"hG6#%\"xG
" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }{XPPEDIT 18 0 "
h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 9 " is even." }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Thus " }{XPPEDIT 18 0 "h(x) = sin^2*x;" "6#/-%\"hG6#%\"xG*
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\\93m#F1$\"19!Q`=y(Q@F-7$$!1mmT&)G*og#F1$\"1`hp_cu'f#F-7$$!1L$3-GrHb#F
1$\"1-!4'pij#3$F-7$$!1+++v'\\!*\\#F1$\"1_F6EX!3f$F-7$$!1++]im!\\W#F1$
\"1(4\"f*HDv6%F-7$$!1+++]Ow!R#F1$\"12*eKu$eaYF-7$$!1++]P1iOBF1$\"1F[ZM
&)o&>&F-7$$!1+++DwZ#G#F1$\"1RX-]4]MdF-7$$!1++++3IIAF1$\"1Hl(4*onXiF-7$
$!1+++vR7y@F1$\"1%)RK4,IVnF-7$$!1+++]r%f7#F1$\"18K6hl&>A(F-7$$!1+++D.x
t?F1$\"1Uk7d&Qkn(F-7$$!1n;a8s+z>F1$\"14'*3)>lTU)F-7$$!1LL3-TC%)=F1$\"1
M'z(3HE\\!*F-7$$!1+](ofh:x\"F1$\"1fH.v%>Bg*F-7$$!1mmm\"4z)e;F1$\"1t>T4
ZhA**F-7$$!1n;/w[,N;F1$\"1$\\A+W;)e**F-7$$!1nmTg1:6;F1$\"1g(*R:Ts$)**F
-7$$!1m;zWkG(e\"F1$\"186b15G(***F-7$$!1mm;HAUj:F1$\"1KzpOiX****F-7$$!1
m;a8!e&R:F1$\"1r'*f^[C!***F-7$$!1mm\"zz$p::F1$\"1@v$3$ymp**F-7$$!1m;H#
eH=\\\"F1$\"1+>.S?xP**F-7$$!1mmmm`'zY\"F1$\"15F5C,j%*)*F-7$$!1L$3FMEpN
\"F1$\"1WX'Q%y_\\&*F-7$$!1++v=t)eC\"F1$\"1&zsxfp4)*)F-7$$!1L$3x'*)fZ6F
1$\"1c3#=9PMJ)F-7$$!1nmm;1J\\5F1$\"1_;5/tF=vF-7$$!1**\\(=n;R&**F-$\"1&
>^*[HuQqF-7$$!1JL3xrs9%*F-$\"1z!fL*H_NlF-7$$!1i;H#oPb())F-$\"1&H='>PY9
gF-7$$!1$***\\(=[jL)F-$\"1WDri'=;[&F-7$$!1%**\\7G7H#yF-$\"1Nx@Z3$*o\\F
-7$$!1&****\\Pw%4tF-$\"1*R3vNqlX%F-7$$!1&**\\(o//'z'F-$\"1))z`8^$*\\RF
-7$$!1'***\\iXg#G'F-$\"1>n`<FOaMF-7$$!1FL3_:<6_F-$\"1*zotTR&yCF-7$$!1e
mmT&Q(RTF-$\"1>T@fg0=;F-7$$!1hm\"HdGe:$F-$\"1rf$*e2*Hj*F\\s7$$!1lm;/'=
><#F-$\"1@s.u6_VYF\\s7$$!1K$3Fpy7k\"F-$\"1@fu1SppEF\\s7$$!1++D\"yQ16\"
F-$\"16KyV\\XG7F\\s7$$!1L$3_D)=`%)F\\s$\"1#4uBKN'GrF\\t7$$!1pm\"zp))**
z&F\\s$\"171=Xk@gLF\\t7$$!11]iS\"*yYJF\\s$\"1tD%3#Q,**)*Fbu7$$!1EMLLe*
e$\\F\\t$\"1wQ^#)pGOCF\\u7$$\"1l;a)3RBE#F\\s$\"1eOa>]I<^Fbu7$$\"1tmTgx
E=]F\\s$\"1k.M%*y=;DF\\t7$$\"1#o\"HKk>uxF\\s$\"1AcYKpkJgF\\t7$$\"1pmT5
D,`5F-$\"1pXX:Iu/6F\\s7$$\"1r;zW#)>/;F-$\"1FI4*4_9b#F\\s7$$\"1sm;zRQb@
F-$\"1z8Z(4$=uXF\\s7$$\"1PLL$e,]6$F-$\"1f'z5nCMR*F\\s7$$\"1-+](=>Y2%F-
$\"1.%ec\"4Qq:F-7$$\"1QLe*[K56&F-$\"1)*)y\"zud#R#F-7$$\"1vmm\"zXu9'F-$
\"1ciypUSELF-7$$\"11+voR!Go'F-$\"1k;l(o-&RQF-7$$\"1QL$e9i\"=sF-$\"11d,
IF*eO%F-7$$\"1pm\"HK?Nv(F-$\"1**3;[aa**[F-7$$\"1,+++&y))G)F-$\"1la86([
VV&F-7$$\"1,+DJ?i7))F-$\"1RPUnzx_fF-7$$\"1,+]ibOO$*F-$\"1>!*\\@EwgkF-7
$$\"1++v$44,')*F-$\"1K9A7St_pF-7$$\"1++]i_QQ5F1$\"1oz/?!*HBuF-7$$\"1+]
(=-N(R6F1$\"19)4w(p8a#)F-7$$\"1,+D\"y%3T7F1$\"1HD%H1H<&*)F-7$$\"1+]P4k
h`8F1$\"1FedV\")pN&*F-7$$\"1++]P![hY\"F1$\"1&e:&Rl)3*)*F-7$$\"1n\"HdqE
9\\\"F1$\"1ba+-o8P**F-7$$\"1L$eRP0n^\"F1$\"1VJ/7-xq**F-7$$\"1+v=US)>a
\"F1$\"1*p`%>3q\"***F-7$$\"1nmT5FEn:F1$\"1=E2O^()****F-7$$\"1Leky8a#f
\"F1$\"1\\[VsAF&***F-7$$\"1+](o/?yh\"F1$\"1;y:\"*R!z(**F-7$$\"1mT5:()4
V;F1$\"1_?)on9y%**F-7$$\"1LLL$Qx$o;F1$\"1hi%GA\"30**F-7$$\"1nm;z)Qjx\"
F1$\"1c&z/vQMe*F-7$$\"1+++v.I%)=F1$\"1#)pQnF$*[!*F-7$$\"1L$ek`H@)>F1$
\"1gH,:iM,%)F-7$$\"1mm\"zpe*z?F1$\"1XZt$ejRi(F-7$$\"1+voa_VL@F1$\"1LTV
***GY:(F-7$$\"1M$e9\"=\"p=#F1$\"1e]On6ngmF-7$$\"1n\"H#o$)QSAF1$\"1O=Yr
]tZhF-7$$\"1,++D\\'QH#F1$\"1Hwb!e#o@cF-7$$\"1MeR(>#=WBF1$\"1_!fGP6,7&F
-7$$\"1n;zp%*\\%R#F1$\"1x**3'yCth%F-7$$\"1+v=Un\"[W#F1$\"1KW)3S5%=TF-7
$$\"1LLe9S8&\\#F1$\"1S'ofh;%GOF-7$$\"1nmT5hK+EF1$\"1c;I/h`aEF-7$$\"1,+
D1#=bq#F1$\"1pY_8q3%y\"F-7$$\"1n;H2FO3GF1$\"1Yj\"fYC*p5F-7$$\"1LLL3s?6
HF1$\"1?&*p.-]9_F\\s7$$\"1+D1R:/lHF1$\"1I%e\"ewx%3$F\\s7$$\"1n;zpe()=I
F1$\"1x$>&QlR)\\\"F\\s7$$\"1]i:NIzXIF1$\"1J[&RJD&\\\"*F\\t7$$\"1M3_+-r
sIF1$\"1v:GNFHPZF\\t7$$\"1<a)eOF'*4$F1$\"1c4g2=0g<F\\t7$$\"1++DJXaEJF1
$\"1<huQ*)GkAFbu7$$\"1$3x1'*oC:$F1$\"1+1O9R*G=\"Fbu7$$\"1nT5!R$RyJF1$
\"1dDXZHo`8F\\t7$$\"1]7`>yJ/KF1$\"1['HA>*GHRF\\t7$$\"1M$e*[ACIKF1$\"1i
))yjc=QyF\\t7$$\"1,D\"y5\"4#G$F1$\"1E_BCq-h>F\\s7$$\"1ommm*RRL$F1$\"1n
$)>IiMaOF\\s7$$\"1omTge)*RMF1$\"1(\\#>FhqU')F\\s7$$\"1om;a<.YNF1$\"1Y'
GF[Z%[:F-7$$\"1,]PM&*>^OF1$\"1Ida_7#)zBF-7$$\"1NLe9tOcPF1$\"1W_ERxoELF
-7$$\"1o;H#e0I&QF1$\"1b&>adWGE%F-7$$\"1,++]Qk\\RF1$\"1(=f0%GXE_F-7$$\"
1N$3-.B]+%F1$\"1uC<'Qerx&F-7$$\"1omT5ASgSF1$\"1P0+]*R$=jF-7$$\"1,]i!R
\"y:TF1$\"1><g<aOVoF-7$$\"1NL$3dg6<%F1$\"1gX`l1!eM(F-7$$\"1,+voTAqUF1$
\"1eMto3/p\")F-7$$\"1ommmxGpVF1$\"1o?C?()Go))F-7$$\"1M$eRA5\\Z%F1$\"1d
B8f2cY%*F-7$$\"1++D\"oK0e%F1$\"1n$)H<W9F)*F-7$$\"1+]il(z5j%F1$\"17CH!o
LS$**F-7$$\"1,++]oi\"o%F1$\"1^c'z*)R0***F-7$$\"1,v=#R+pq%F1$\"1v4-d()p
****F-7$$\"1,]PMR<KZF1$\"1i<pmg3'***F-7$$\"1,DcwuWdZF1$\"1,YYb5rz**F-7
$$\"1,+v=5s#y%F1$\"1lDr_bh]**F-7$$\"1,]P40O\"*[F1$\"1+!HZD(4$o*F-7$$\"
\"&F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%\"xG%\"yG-%*AX
ESTICKSG6$%(DEFAULTG\"\"#-%%VIEWG6$;F(F]_nF^`n" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 14 " is odd, then " }{XPPEDIT 18 0 "Int(f(x),x = -a ..
 a) = 0;" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%\"aG!\"\"F.\"\"!" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-
%\"fG6#%\"xG" }{TEXT -1 49 " is an odd function defined on the interva
l from " }{XPPEDIT 18 0 "x = -a" "6#/%\"xG,$%\"aG!\"\"" }{TEXT -1 4 " \+
to " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 7 ", then " }
{XPPEDIT 18 0 "Int(f(x),x = -a .. a) = 0;" "6#/-%$IntG6$-%\"fG6#%\"xG/
F*;,$%\"aG!\"\"F.\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 
"For example, " }{XPPEDIT 18 0 "Int(sin(x),x = -Pi .. Pi) = 0;" "6#/-%
$IntG6$-%$sinG6#%\"xG/F*;,$%#PiG!\"\"F.\"\"!" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 22 "Because the graph of  " }{XPPEDIT 18 0 "y = sin
*x" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 70 "  is symmetrical abo
ut the origin, the same area is cut off below the " }{TEXT 301 1 "x" }
{TEXT -1 38 " axis as above over the interval from " }{XPPEDIT 18 0 "x
 = -Pi;" "6#/%\"xG,$%#PiG!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x =
 Pi;" "6#/%\"xG%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 82 "A
 general abstract proof of this result can be obtained by splitting th
e integral " }{XPPEDIT 18 0 "Int(f(x),x = -a .. a);" "6#-%$IntG6$-%\"f
G6#%\"xG/F);,$%\"aG!\"\"F-" }{TEXT -1 31 " into the sum of two integra
ls." }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "Int(f(x),x =
 -a .. a) = Int(f(x),x = -a .. 0)+Int(f(x),x = 0 .. a);" "6#/-%$IntG6$
-%\"fG6#%\"xG/F*;,$%\"aG!\"\"F.,&-F%6$-F(6#F*/F*;,$F.F/\"\"!\"\"\"-F%6
$-F(6#F*/F*;F8F.F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "Mak
ing the substitution  " }{XPPEDIT 18 0 "t = -x" "6#/%\"tG,$%\"xG!\"\"
" }{TEXT -1 41 " in the first integral of the sum gives: " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Int(f(x),x = -a .. 0)" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$%\"aG!\"\"\"
\"!" }{TEXT -1 11 "   ...     " }{XPPEDIT 18 0 "PIECEWISE([t = -x, `x \+
=`*``-a*` implies  t =`*a],[dt = -dx, `x =`*0*` implies  t =`*0],[-dt \+
= dx, ``])" "6#-%*PIECEWISEG6%7$/%\"tG,$%\"xG!\"\",&*&%$x~=G\"\"\"%!GF
/F/*(%\"aGF/%.~implies~~t~=GF/F2F/F+7$/%#dtG,$%#dxGF+**F.F/\"\"!F/F3F/
F:F/7$/,$F6F+F8F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "`` = -Int(f(-t),t = \+
a .. 0);" "6#/%!G,$-%$IntG6$-%\"fG6#,$%\"tG!\"\"/F-;%\"aG\"\"!F." }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 Int(f(-t),t = 0 .. a);" "6#/%!G-%$IntG6$-%\"fG6#,$%\"tG!\"\"/F,;\"\"!
%\"aG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "( since switchin
g the limits changes the sign of the integral )" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=-Int(f(t),t = 0 .. a)" "6#/%!G,$-%$I
ntG6$-%\"fG6#%\"tG/F,;\"\"!%\"aG!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 
0 "" {TEXT -1 10 "( because " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" 
}{TEXT -1 22 " is an odd function ) " }}{PARA 258 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "`` = -Int(f(x),x = 0 .. a);" "6#/%!G,$-%$IntG6$-%\"
fG6#%\"xG/F,;\"\"!%\"aG!\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" 
{TEXT -1 75 "(since it does not matter what variable we use in the def
inite integral ). " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -a .. a) = Int(f(x
),x = -a .. 0)+Int(f(x),x = 0 .. a)" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%
\"aG!\"\"F.,&-F%6$-F(6#F*/F*;,$F.F/\"\"!\"\"\"-F%6$-F(6#F*/F*;F8F.F9" 
}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int
(f(x),x = -a .. a) = -Int(f(x),x =0..a)+Int(f(x),x = 0 .. a)" "6#/-%$I
ntG6$-%\"fG6#%\"xG/F*;,$%\"aG!\"\"F.,&-F%6$-F(6#F*/F*;\"\"!F.F/-F%6$-F
(6#F*/F*;F7F.\"\"\"" }{XPPEDIT 18 0 " ``=0" "6#/%!G\"\"!" }{TEXT -1 3 
".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }
}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)=x*cos*x" "6#
/-%\"fG6#%\"xG*(F'\"\"\"%$cosGF)F'F)" }{TEXT -1 22 "  is an odd functi
on, " }{XPPEDIT 18 0 "Int(x*cos*x,x=-Pi/4..Pi/4)=0" "6#/-%$IntG6$*(%\"
xG\"\"\"%$cosGF)F(F)/F(;,$*&%#PiGF)\"\"%!\"\"F1*&F/F)F0F1\"\"!" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 93 "This can also be verifi
ed by evaluating the integral using the integration by parts formula: \+
" }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v-Int(v*``(du/dx),x)" "6#/-%$In
tG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6
$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*cos*x,x)" "6#-%$IntG6$*(%\"xG\"\"
\"%$cosGF(F'F(F'" }{TEXT -1 11 "  ...      " }{XPPEDIT 18 0 "PIECEWISE
([u=x,v=sin*x],[du/dx=1,dv/dx=cos*x])" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG
/%\"vG*&%$sinG\"\"\"F)F.7$/*&%#duGF.%#dxG!\"\"F./*&%#dvGF.F3F4*&%$cosG
F.F)F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "``=x*sin*x-Int(sin*x,x)" "6#/%!G,&*(%\"xG\"\"\"%$sinGF(F'F(F(-%$
IntG6$*&F)F(F'F(F'!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "``=x*sin*x+cos*x+c" "6#/%!G,(*(%\"xG\"\"\"%$sinG
F(F'F(F(*&%$cosGF(F'F(F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"Int(x*cos*x,x = -Pi/4 .. Pi/4)=x*sin*x+cos*x" "6#/-%$IntG6$*(%\"xG\"
\"\"%$cosGF)F(F)/F(;,$*&%#PiGF)\"\"%!\"\"F1*&F/F)F0F1,&*(F(F)%$sinGF)F
(F)F)*&F*F)F(F)F)" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([Pi/4,``
],[-Pi/4,``])" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"\"\"\"%!\"\"%!G7$,$*&F(F
)F*F+F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(``(Pi/4)*sin(Pi/4)+co
s(Pi/4))-``(``(-Pi/4)*sin(-Pi/4)+cos(-Pi/4))" "6#/%!G,&-F$6#,&*&-F$6#*
&%#PiG\"\"\"\"\"%!\"\"F.-%$sinG6#*&F-F.F/F0F.F.-%$cosG6#*&F-F.F/F0F.F.
-F$6#,&*&-F$6#,$*&F-F.F/F0F0F.-F26#,$*&F-F.F/F0F0F.F.-F66#,$*&F-F.F/F0
F0F.F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "``=Pi/(4*sqrt(2))+1/sqrt(2)-``(Pi/(4*sqrt(2))+1/sqrt(2))" "6#/%!
G,(*&%#PiG\"\"\"*&\"\"%F(-%%sqrtG6#\"\"#F(!\"\"F(*&F(F(-F,6#F.F/F(-F$6
#,&*&F'F(*&F*F(-F,6#F.F(F/F(*&F(F(-F,6#F.F/F(F/" }{XPPEDIT 18 0 " ``=0
" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "Int(x*cos(x),x=-Pi/4..Pi/4);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%$cosG6#F'F(/F';,$*&\"\"%!\"\"%#
PiGF(F1,$*&F0F1F2F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "
" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x) = x^2*sin*x;" "6#/-%\"
fG6#%\"xG*(F'\"\"#%$sinG\"\"\"F'F+" }{TEXT -1 21 " is an odd function,
 " }{XPPEDIT 18 0 "Int(x^2*sin*x,x = -Pi/3 .. Pi/3) = 0;" "6#/-%$IntG6
$*(%\"xG\"\"#%$sinG\"\"\"F(F+/F(;,$*&%#PiGF+\"\"$!\"\"F2*&F0F+F1F2\"\"
!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 40 "Int(x^2*sin(x),x=-Pi/3..Pi/3);\nvalue(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#F
(F*/F(;,$%#PiG#!\"\"\"\"$,$F1#F*F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+
" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)=arctan*x
" "6#/-%\"fG6#%\"xG*&%'arctanG\"\"\"F'F*" }{TEXT -1 4 " is " }{TEXT 
259 3 "odd" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)=cos*x" "6#/-%\"gG
6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 4 " is " }{TEXT 259 4 "even" }
{TEXT -1 15 ", the function " }{XPPEDIT 18 0 "h(x)=f(x)*g(x)" "6#/-%\"
hG6#%\"xG*&-%\"fG6#F'\"\"\"-%\"gG6#F'F," }{XPPEDIT 18 0 "``=arctan*x*c
os*x" "6#/%!G**%'arctanG\"\"\"%\"xGF'%$cosGF'F(F'" }{TEXT -1 4 " is " 
}{TEXT 259 3 "odd" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Henc
e  " }{XPPEDIT 18 0 "Int(arctan*x*cos*x,x = -4 .. 4) = 0;" "6#/-%$IntG
6$**%'arctanG\"\"\"%\"xGF)%$cosGF)F*F)/F*;,$\"\"%!\"\"F/\"\"!" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "Maple appears to \"know\" the \+
result " }{XPPEDIT 18 0 "Int(phi(x),x=-a..a)=0" "6#/-%$IntG6$-%$phiG6#
%\"xG/F*;,$%\"aG!\"\"F.\"\"!" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "phi(
x)" "6#-%$phiG6#%\"xG" }{TEXT -1 6 " odd. " }}{PARA 0 "" 0 "" {TEXT 
-1 54 "Even though Maple cannot find the indefinite integral " }
{XPPEDIT 18 0 "Int(arctan*x*cos*x,x);" "6#-%$IntG6$**%'arctanG\"\"\"%
\"xGF(%$cosGF(F)F(F)" }{TEXT -1 43 ", Maple can nevertheless obtain th
e result " }{XPPEDIT 18 0 "Int(arctan*x*cos*x,x = -4 .. 4) = 0" "6#/-%
$IntG6$**%'arctanG\"\"\"%\"xGF)%$cosGF)F*F)/F*;,$\"\"%!\"\"F/\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "Int(arctan(x)*cos(x),x);\nvalue(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#%\"xG\"\"\"-%$cosGF)F+F*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&-%'arctanG6#%\"xG\"\"\"
-%$cosGF)F+F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "Int(arctan(x)*cos(x),x=-4..4);\nvalue(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%'arctanG6#%\"xG\"\"\"-%$c
osGF)F+/F*;!\"%\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x
G" }{TEXT -1 15 " is even, then " }{XPPEDIT 18 0 "Int(f(x),x = -a .. a
) = 2*Int(f(x),x = 0 .. a);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%\"aG!\"
\"F.*&\"\"#\"\"\"-F%6$-F(6#F*/F*;\"\"!F.F2" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 50 " is an even function defined on the interval from " }
{XPPEDIT 18 0 "x = -a" "6#/%\"xG,$%\"aG!\"\"" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 7 ", then " }{XPPEDIT 
18 0 "Int(f(x),x = -a .. a) = 2*Int(f(x),x = 0 .. a);" "6#/-%$IntG6$-%
\"fG6#%\"xG/F*;,$%\"aG!\"\"F.*&\"\"#\"\"\"-F%6$-F(6#F*/F*;\"\"!F.F2" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "For example, " }
{XPPEDIT 18 0 "Int(cos*x,x = -Pi .. Pi) = 2*Int(cos*x,x = 0 .. Pi);" "
6#/-%$IntG6$*&%$cosG\"\"\"%\"xGF)/F*;,$%#PiG!\"\"F.*&\"\"#F)-F%6$*&F(F
)F*F)/F*;\"\"!F.F)" }{TEXT -1 5 " = 2." }}{PARA 0 "" 0 "" {TEXT -1 21 
"Because the graph of " }{XPPEDIT 18 0 "y = cos*x" "6#/%\"yG*&%$cosG\"
\"\"%\"xGF'" }{TEXT -1 26 " is symmetrical about the " }{TEXT 302 1 "y
" }{TEXT -1 7 " axis, " }{XPPEDIT 18 0 "Int(cos*x,x = -Pi .. 0) = Int(
cos*x,x = 0 .. Pi);" "6#/-%$IntG6$*&%$cosG\"\"\"%\"xGF)/F*;,$%#PiG!\"
\"\"\"!-F%6$*&F(F)F*F)/F*;F0F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 128 "As with the result in the previous subsection, a general
 abstract proof of the result can be obtained by splitting the integra
l " }{XPPEDIT 18 0 "Int(f(x),x = -a .. a);" "6#-%$IntG6$-%\"fG6#%\"xG/
F);,$%\"aG!\"\"F-" }{TEXT -1 31 " into the sum of two integrals." }}
{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "Int(f(x),x = -a .. \+
a) = Int(f(x),x = -a .. 0)+Int(f(x),x = 0 .. a);" "6#/-%$IntG6$-%\"fG6
#%\"xG/F*;,$%\"aG!\"\"F.,&-F%6$-F(6#F*/F*;,$F.F/\"\"!\"\"\"-F%6$-F(6#F
*/F*;F8F.F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 25 "Making the substitution  " }{XPPEDIT 18 0 "t = \+
-x" "6#/%\"tG,$%\"xG!\"\"" }{TEXT -1 41 " in the first integral of the
 sum gives: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -a .. 0)" "6#-%$IntG6$-%\"
fG6#%\"xG/F);,$%\"aG!\"\"\"\"!" }{TEXT -1 11 "   ...     " }{XPPEDIT 
18 0 "PIECEWISE([t = -x, `x =`*``-a*` implies  t =`*a],[dt = -dx, `x =
`*0*` implies  t =`*0],[-dt = dx, ``])" "6#-%*PIECEWISEG6%7$/%\"tG,$%
\"xG!\"\",&*&%$x~=G\"\"\"%!GF/F/*(%\"aGF/%.~implies~~t~=GF/F2F/F+7$/%#
dtG,$%#dxGF+**F.F/\"\"!F/F3F/F:F/7$/,$F6F+F8F0" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "`` = -Int(f(-t),t = a .. 0);" "6#/%!G,$-%$IntG6$-%\"fG6
#,$%\"tG!\"\"/F-;%\"aG\"\"!F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(-t),t = 0 .. a);" "6#/%!G-%$
IntG6$-%\"fG6#,$%\"tG!\"\"/F,;\"\"!%\"aG" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 64 "( since switching the limits changes the sign of t
he integral ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+
= Int(f(t),t = 0 .. a);" "6#/%!G-%$IntG6$-%\"fG6#%\"tG/F+;\"\"!%\"aG" 
}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 10 "( because " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " is an even funct
ion ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(f(
x),x = 0 .. a);" "6#/%!G-%$IntG6$-%\"fG6#%\"xG/F+;\"\"!%\"aG" }{TEXT 
-1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 75 "(since it does not matter w
hat variable we use in the definite integral ). " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"Int(f(x),x = -a .. a) = Int(f(x),x = -a .. 0)+Int(f(x),x = 0 .. a)" "
6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%\"aG!\"\"F.,&-F%6$-F(6#F*/F*;,$F.F/\"
\"!\"\"\"-F%6$-F(6#F*/F*;F8F.F9" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = -a .. a) = Int(f(x),x = 0 \+
.. a)+Int(f(x),x = 0 .. a);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%\"aG!\"
\"F.,&-F%6$-F(6#F*/F*;\"\"!F.\"\"\"-F%6$-F(6#F*/F*;F7F.F8" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Int(f(
x),x = 0 .. a)" "6#/%!G*&\"\"#\"\"\"-%$IntG6$-%\"fG6#%\"xG/F.;\"\"!%\"
aGF'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x) = sin^2*x;" "6#/-%\"fG6#%\"xG
*&%$sinG\"\"#F'\"\"\"" }{TEXT -1 22 "  is an even function," }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin^2*x,x = -Pi/3 .. Pi
/3) = 2*Int(sin^2*x,x = 0 .. Pi/3);" "6#/-%$IntG6$*&%$sinG\"\"#%\"xG\"
\"\"/F*;,$*&%#PiGF+\"\"$!\"\"F2*&F0F+F1F2*&F)F+-F%6$*&F(F)F*F+/F*;\"\"
!*&F0F+F1F2F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Int(2*sin^2*x,x = 0 .. Pi/3);" "6#/%!G-%$IntG6$*(
\"\"#\"\"\"*$%$sinGF)F*%\"xGF*/F-;\"\"!*&%#PiGF*\"\"$!\"\"" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(``(1
-cos*2*x),x = 0 .. Pi/3);" "6#/%!G-%$IntG6$-F$6#,&\"\"\"F+*(%$cosGF+\"
\"#F+%\"xGF+!\"\"/F/;\"\"!*&%#PiGF+\"\"$F0" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x-sin*2*x/2" "6#/%!G,&%
\"xG\"\"\"**%$sinGF'\"\"#F'F&F'F*!\"\"F+" }{TEXT -1 2 "  " }{XPPEDIT 
18 0 "PIECEWISE([Pi/3, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"
\"\"\"$!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi/3-sqrt
(3)/4;" "6#/%!G,&*&%#PiG\"\"\"\"\"$!\"\"F(*&-%%sqrtG6#F)F(\"\"%F*F*" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(sin(x)^2
,x=-Pi/3..Pi/3);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG
6$*$)-%$sinG6#%\"xG\"\"#\"\"\"/F+;,$*&\"\"$!\"\"%#PiGF-F3,$*&F2F3F4F-F
-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%!\"\"\"\"$#\"\"\"\"\"#F&
*&F'F&%#PiGF)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam
ple 2" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x) = x*
sin*x;" "6#/-%\"fG6#%\"xG*(F'\"\"\"%$sinGF)F'F)" }{TEXT -1 22 "  is an
 even function," }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "I
nt(x*sin*x,x = -Pi/4 .. Pi/4) = 2*Int(x*sin*x,x = 0 .. Pi/4);" "6#/-%$
IntG6$*(%\"xG\"\"\"%$sinGF)F(F)/F(;,$*&%#PiGF)\"\"%!\"\"F1*&F/F)F0F1*&
\"\"#F)-F%6$*(F(F)F*F)F(F)/F(;\"\"!*&F/F)F0F1F)" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 13 "The integral " }{XPPEDIT 18 0 "Int(x*sin*
x,x)" "6#-%$IntG6$*(%\"xG\"\"\"%$sinGF(F'F(F'" }{TEXT -1 57 " can be f
ound by 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 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(x*sin*x,x);" "6#-%$IntG6$*(%\"xG\"\"\"%$sinGF(F'F(F
'" }{TEXT -1 11 "  ...      " }{XPPEDIT 18 0 "PIECEWISE([u = x, v = -c
os*x],[du/dx = 1, dv/dx = sin*x]);" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/%
\"vG,$*&%$cosG\"\"\"F)F/!\"\"7$/*&%#duGF/%#dxGF0F//*&%#dvGF/F5F0*&%$si
nGF/F)F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = -x*cos*x+Int(cos*x,x);" "6#/%!G,&*(%\"xG\"\"\"%$co
sGF(F'F(!\"\"-%$IntG6$*&F)F(F'F(F'F(" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -x*cos*x+sin*x+c;" "6#/%!G,(*
(%\"xG\"\"\"%$cosGF(F'F(!\"\"*&%$sinGF(F'F(F(%\"cGF(" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(x*sin*x,x = -Pi/4 .. Pi/4) = 2*(-x*cos*x+sin
*x);" "6#/-%$IntG6$*(%\"xG\"\"\"%$sinGF)F(F)/F(;,$*&%#PiGF)\"\"%!\"\"F
1*&F/F)F0F1*&\"\"#F),&*(F(F)%$cosGF)F(F)F1*&F*F)F(F)F)F)" }{TEXT -1 2 
"  " }{XPPEDIT 18 0 "PIECEWISE([Pi/4, ``],[0, ``]);" "6#-%*PIECEWISEG6
$7$*&%#PiG\"\"\"\"\"%!\"\"%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*(-Pi/(4*sqrt(2))+1/sqrt(2))
" "6#/%!G*&\"\"#\"\"\",&*&%#PiGF'*&\"\"%F'-%%sqrtG6#F&F'!\"\"F0*&F'F'-
F.6#F&F0F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=sqrt(2)-Pi/(2*sqrt(2)
)" "6#/%!G,&-%%sqrtG6#\"\"#\"\"\"*&%#PiGF**&F)F*-F'6#F)F*!\"\"F0" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(x*sin(x)
,x=-Pi/4..Pi/4);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG
6$*&%\"xG\"\"\"-%$sinG6#F'F(/F';,$*&\"\"%!\"\"%#PiGF(F1,$*&F0F1F2F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"##\"\"\"F%F'*(\"\"%!\"\"%#
PiGF'F%F&F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example
 3 " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x) = x^3*
sin(x);" "6#/-%\"fG6#%\"xG*&F'\"\"$-%$sinG6#F'\"\"\"" }{TEXT -1 22 " i
s an even function, " }{XPPEDIT 18 0 "Int(x^3*sin(x),x = -Pi/3 .. Pi/3
) = 2*Int(x^3*sin(x),x = 0 .. Pi/3);" "6#/-%$IntG6$*&%\"xG\"\"$-%$sinG
6#F(\"\"\"/F(;,$*&%#PiGF-F)!\"\"F3*&F2F-F)F3*&\"\"#F--F%6$*&F(F)-F+6#F
(F-/F(;\"\"!*&F2F-F)F3F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(x^3*sin(x),x=0..P
i/3);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG
\"\"$\"\"\"-%$sinG6#F(F*/F(;\"\"!,$%#PiG#F*F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**$)%#PiG\"\"$\"\"\"#!\"\"\"#a*&)F&\"\"#F(-%%sqrtG6#F'
F(#F(\"\"'*$F/F(!\"$F&F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(x^3*sin(x),x=-Pi/3..Pi/3);\nval
ue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"$\"\"\"
-%$sinG6#F(F*/F(;,$%#PiG#!\"\"F),$F1#F*F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**$)%#PiG\"\"$\"\"\"#!\"\"\"#F*&)F&\"\"#F(-%%sqrtG6#F'
F(#F(F'*$F/F(!\"'F&F." }}}{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 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "
Checking whether a function is even or odd: " }{TEXT 0 38 "type(..even
func(x)),type(..oddfunc(x))" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 286 8 "Question" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 82 "(a) Plot graphs of the follow
ing functions and determine whether each function is " }{TEXT 284 4 "e
ven" }{TEXT -1 2 ", " }{TEXT 284 3 "odd" }{TEXT -1 4 " or " }{TEXT 
284 20 "neither even nor odd" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 9 "   (i)   " }{XPPEDIT 18 0 "f(x) = x^3*sin*x;" "6#/-%\"fG6#%\"xG*
(F'\"\"$%$sinG\"\"\"F'F+" }{TEXT -1 11 "    (ii)   " }{XPPEDIT 18 0 "f
(x) = 1+x-x^3;" "6#/-%\"fG6#%\"xG,(\"\"\"F)F'F)*$F'\"\"$!\"\"" }{TEXT 
-1 10 "    (ii)  " }{XPPEDIT 18 0 "f(x) = exp(1+x)-exp(1-x);" "6#/-%\"
fG6#%\"xG,&-%$expG6#,&\"\"\"F-F'F-F--F*6#,&F-F-F'!\"\"F1" }{TEXT -1 2 
" ." }}{PARA 0 "" 0 "" {TEXT -1 49 "(b) Which of the functions in part
 (a) satisfies " }{XPPEDIT 18 0 "Int(f(x),x = -2 .. 2) = 0;" "6#/-%$In
tG6$-%\"fG6#%\"xG/F*;,$\"\"#!\"\"F.\"\"!" }{TEXT -1 62 "?  Verify your
 answer by using Maple to evaluate the integral." }}{PARA 0 "" 0 "" 
{TEXT 285 8 "Solution" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "(
a) (i)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(x^3*sin(x),x=
-3.5..3.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-
%'CURVESG6$7ap7$$!3++++++++N!#<$!38X#>()3$)R]\"!#;7$$!3QL$3-)\\&=Y$F*$
!3kE\\[k\">hI\"F-7$$!3vmmTg*4PU$F*$!3xOp**)\\Ms6\"F-7$$!37+]iS\\c&Q$F*
$!3#ea`'R!GQP*F*7$$!3]LL$3#*>uM$F*$!3s3P=x<*em(F*7$$!3'o\"H2V,B9LF*$!3
U'p$>2x]`iF*7$$!3?+DJl./\"G$F*$!3JnIWPY]4\\F*7$$!37$3_ve]yC$F*$!3<G<#e
jcNj$F*7$$!3Ym;z43m9KF*$!3!4vghV(>DCF*7$$!39$e*[LaLxJF*$!3clkAX]EY6F*7
$$!3#)*\\(=d+,SJF*$\"3/^e7D!G'**[!#>7$$!3\\;a)3o%o-JF*$\"3-6qW)GG=;\"F
*7$$!3;LLe/$f`1$F*$\"3Mn!HI^bO>#F*7$$!3/LLL=P@!*HF*$\"3[:nk\\D#>.%F*7$
$!3PLL3K\"o]\"HF*$\"3y_4h%[1Mc&F*7$$!3r*\\(o*pz-%GF*$\"3Xsw]O-,+oF*7$$
!3]m;Hn7\\lFF*$\"37iY>V6XoxF*7$$!3=+](o\"G:'p#F*$\"3:;;)[A;VW)F*7$$!3W
L$ekO9oi#F*$\"3t1=LD[!R#*)F*7$$!3++](=R;4f#F*$\"3u$pYeyC35*F*7$$!3cm;H
<%=]b#F*$\"3GgC([tpAB*F*7$$!38L$3FW?\">DF*$\"35Y5:JZr?$*F*7$$!3q**\\7o
CA$[#F*$\"3mD::73ko$*F*7$$!3%GLe9t'4YCF*$\"3?@!e\"=@Ay$*F*7$$!3Vm;z%*4
(*3CF*$\"39r$)fL%[)\\$*F*7$$!3-+]7e_%=P#F*$\"3GpN_b:B'G*F*7$$!3<L$e9_>
ZL#F*$\"3]\\7xs`0!>*F*7$$!3qm;athqgAF*$\"3HCrS4%R;\"*)F*7$$!3A+]iDGp'=
#F*$\"3>t>[IvsM&)F*7$$!3Cmm;u\"HW.#F*$\"3J::@WOSJvF*7$$!3?LL3n_J+>F*$
\"3%HO9E2/K\\'F*7$$!3q****\\sZL\\<F*$\"3Auz(*f;<o_F*7$$!3#*****\\PVt(f
\"F*$\"3u6JP*eWr2%F*7$$!3')****\\F#R;X\"F*$\"3.yf.)\\&GPIF*7$$!3DL$e9(
3(*=8F*$\"3?XR>-%>AA#F*7$$!3[mm;k`@h6F*$\"3BNDr:fHO9F*7$$!3MmmmcddF5F*
$\"3e)G;0+y$)G*!#=7$$!3(*)**\\7B67s)F]w$\"3]jx(4b$4z]F]w7$$!3?nmm;V<Xt
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jm\"H-V._\"F]w$\"3m$zo=e4AK&!#@7$$!3-)RLL3F^X$Fgx$\"3))=&HN[K^U\"!#F7$
$\"3fnmT&yo(3:F]w$\"3)f;aM*HGi^F]y7$$\"3'>+]7VLA&GF]w$\"3R%H$*4gN)GlFg
x7$$\"3jpm;a?@.VF]w$\"3:MbE%4uTK$Fjn7$$\"3w******\\\\@-eF]w$\"3UY!\\4#
y%32\"F]w7$$\"3%Q++v$oposF]w$\"3-y-v^a._DF]w7$$\"3g0+voMf(o)F]w$\"3s/$
4k&zP1]F]w7$$\"3#)***\\ii.j-\"F*$\"3?fjna\"3oC*F]w7$$\"3%GLL$oT'y;\"F*
$\"3_OT9&3%Hl9F*7$$\"3'3++DE5!>8F*$\"3Ws^%)H5WAAF*7$$\"3Mm;a)3rfX\"F*$
\"3%*eTft\"=h1$F*7$$\"3*4++vW0dg\"F*$\"3xe;\"f%>XPTF*7$$\"3;L$3-\"QfY<
F*$\"3/'H()*o3-Y_F*7$$\"3C+]PWF'Q*=F*$\"3$fW!)oVM8W'F*7$$\"3[LL$e/Xy.#
F*$\"3+rj+l?VcvF*7$$\"3m**\\(=<\"e)=#F*$\"3iq@6TkSX&)F*7$$\"3wL3Fu&p6E
#F*$\"3sj,<&))4P\"*)F*7$$\"3%ymmm(zvLBF*$\"3ijwcFd:(=*F*7$$\"3kn\"z%RS
(3P#F*$\"3;a8[z'=TG*F*7$$\"3Vn;H-,*zS#F*$\"3U&)Gf@!4'[$*F*7$$\"3AnT5lh
5XCF*$\"3!*R*4P!e%zP*F*7$$\"3-nm\"zAAA[#F*$\"3m7mJ#)*>%p$*F*7$$\"3&Qe*
)4&4.>DF*$\"3\"Q?Lqb$)3K*F*7$$\"3o+D1u'Reb#F*$\"3&38,Jm_(H#*F*7$$\"3]<
a8(R[Ef#F*$\"3a7$=Mn_L4*F*7$$\"3LM$3-7d%HEF*$\"3ddBWvd,4*)F*7$$\"3inTg
2R5(p#F*$\"3'p![')eETO%)F*7$$\"3#4++]p]Zw#F*$\"3ENkiflwwxF*7$$\"3Pn;HZ
:GUGF*$\"3/JH<W,\\qnF*7$$\"3$QL$e*R7)>HF*$\"3o&4f\"eGZvaF*7$$\"3S+]7=p
:*)HF*$\"3o+9$=Z\\b0%F*7$$\"3'pmmmV,&eIF*$\"3GsLur+buBF*7$$\"3dL3xcrVK
JF*$\"3.W\\n*e#*R\"GF]w7$$\"3<+](o(GP1KF*$!3&3@kKtKR8#F*7$$\"30+v$f$ev
TKF*$!3kP&*)RL6mS$F*7$$\"3Q+++&zQrF$F*$!3%pmJ-d!*fv%F*7$$\"3r+D1a<_7LF
*$!3ad;4Y-m#='F*7$$\"3g+]78Z!zM$F*$!3+F%>VzHqo(F*7$$\"3W]P%[`GfQ$F*$!3
s5U3**Qa!R*F*7$$\"3I+DccB&RU$F*$!3C%)p-3jS=6F-7$$\"39]7Gyh(>Y$F*$!3U/(
[OlLnI\"F-7$$\"3++++++++NF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FgflFff
l-%+AXESLABELSG6$Q\"x6\"Q!F\\gl-%%VIEWG6$;$!#NFefl$\"#NFefl%(DEFAULTG
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) =
 x^3*sin*x;" "6#/-%\"fG6#%\"xG*(F'\"\"$%$sinG\"\"\"F'F+" }{TEXT -1 7 "
 is an " }{TEXT 284 13 "even function" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 9 "In fact  " }{XPPEDIT 18 0 "f(-x)=(-x)^3*sin(-x)" "6#/
-%\"fG6#,$%\"xG!\"\"*&,$F(F)\"\"$-%$sinG6#,$F(F)\"\"\"" }{XPPEDIT 18 
0 "``=(-x^3)*(-sin*x)" "6#/%!G*&,$*$%\"xG\"\"$!\"\"\"\"\",$*&%$sinGF+F
(F+F*F+" }{XPPEDIT 18 0 "``=x^3*sin*x" "6#/%!G*(%\"xG\"\"$%$sinG\"\"\"
F&F)" }{XPPEDIT 18 0 "``=f(x)" "6#/%!G-%\"fG6#%\"xG" }{TEXT -1 22 ", f
or any real number " }{TEXT 317 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "y=x^3*sin*x" "6#/%\"yG*(%
\"xG\"\"$%$sinG\"\"\"F&F)" }{TEXT -1 26 " is symmetrical about the " }
{TEXT 314 1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT 259 4 "No
te" }{TEXT -1 10 ": Maple's " }{TEXT 0 4 "type" }{TEXT -1 39 " procedu
re can be used to check that   " }{XPPEDIT 18 0 "f(x) = x^3*sin*x;" "6
#/-%\"fG6#%\"xG*(F'\"\"$%$sinG\"\"\"F'F+" }{TEXT -1 22 "  is an even f
unction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "type(x^3*sin(x),evenfunc(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 183 "There are other ways of using Maple to help in dete
rmining whether a function is even or odd or neither even nor odd. In \+
this example, Maple immediately simplifies the expression for " }
{XPPEDIT 18 0 "f(-x)" "6#-%\"fG6#,$%\"xG!\"\"" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x^3*sin(x);" "6#*&%\"xG\"\"$-%$sinG6#F$\"\"\"" }{TEXT 
-1 41 " (or in the usual mathematical notation: " }{XPPEDIT 18 0 "x^3*
sin*x" "6#*(%\"xG\"\"$%$sinG\"\"\"F$F'" }{TEXT -1 32 " ), which is the
 expression 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 34 "f := x -> x^3*sin(x);\nf(-x);\nf(x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&)9$\"\"
$\"\"\"-%$sinG6#F.F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG
\"\"$\"\"\"-%$sinG6#F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG\"
\"$\"\"\"-%$sinG6#F%F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 5 "(ii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "pl
ot(1+x-x^3,x=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"#\"\"!$\"\"(F*7$$!3MLLL$Q6G\">!#<$
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\"3#3=W/[mFi%F07$$!3ALLL=Kvl;F0$\"3?p*GI^oi&RF07$$!3wmm;C2G!e\"F0$\"3I
q^#)[S8mLF07$$!3OLL$3yO5]\"F0$\"3K^8-Gj'4)GF07$$!3&*****\\nU)*=9F0$\"3
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7$$\"3)Qjmm\"f`@')Fjr$\"3!)e:!G^ub3\"F07$$\"3%z****\\nZ)H;Fbo$\"3o=M_M
_le6F07$$\"3ckmm;$y*eCFbo$\"3@(z?kVH5B\"F07$$\"3f)******R^bJ$Fbo$\"3-S
^sxw5&H\"F07$$\"3'e*****\\5a`TFbo$\"3VOPk;wpV8F07$$\"3'o****\\7RV'\\Fb
o$\"3%px/![%*3u8F07$$\"3Y'*****\\@fkeFbo$\"3#e['R5dv%Q\"F07$$\"3_ILLL&
4Nn'Fbo$\"3g9z*zLT,P\"F07$$\"3A*******\\,s`(Fbo$\"3$Q()yk?ObK\"F07$$\"
3%[mm;zM)>$)Fbo$\"3E=u$)=u3c7F07$$\"3M*******pfa<*Fbo$\"3%*p#3jjs]9\"F
07$$\"39HLLeg`!)**Fbo$\"3l&Rc.V\")Q+\"F07$$\"3w****\\#G2A3\"F0$\"3-cvI
KwbZ\")Fbo7$$\"3;LLL$)G[k6F0$\"3_)\\'y[N?aeFbo7$$\"3#)****\\7yh]7F0$\"
3!)pOC[Q&f%HFbo7$$\"3xmmm')fdL8F0$!3,Zj;Nq()3QFjr7$$\"3bmmm,FT=9F0$!3]
]E%RUTGN%Fbo7$$\"3FLL$e#pa-:F0$!3yma.3.u'*))Fbo7$$\"3!*******Rv&)z:F0$
!3Y/FwWwQj8F07$$\"3ILLLGUYo;F0$!3w5**zV7;w>F07$$\"3_mmm1^rZ<F0$!3]8mxK
[p!f#F07$$\"34++]sI@K=F0$!3#3>U@!\\`=LF07$$\"34++]2%)38>F0$!3ak^RMxj)3
%F07$$\"\"#F*$!\"&F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXESLABELS
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = 1+x-x
^3;" "6#/-%\"fG6#%\"xG,(\"\"\"F)F'F)*$F'\"\"$!\"\"" }{TEXT -1 4 " is \+
" }{TEXT 284 20 "neither even nor odd" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT 259 4 "Note" }{TEXT -1 10 ": Maple's " }{TEXT 0 4 "type" }
{TEXT -1 38 " procedure can be used to check that  " }{XPPEDIT 18 0 "f
(x) = 1+x-x^3;" "6#/-%\"fG6#%\"xG,(\"\"\"F)F'F)*$F'\"\"$!\"\"" }{TEXT 
-1 26 " is neither even nor odd. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "type(1+x-x^3,evenfunc(x));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "type(1+x-x^3,oddfunc(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 58 "Alternatively, one could simply demonstrate the fa
ct that " }{XPPEDIT 18 0 "f(x) = 1+x-x^3;" "6#/-%\"fG6#%\"xG,(\"\"\"F)
F'F)*$F'\"\"$!\"\"" }{TEXT -1 50 " is neither even nor odd with a nume
rical example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "f := x -> 1+x-x^3;\nf(2);\nf(-2);" }}{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#!\"
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(ii) " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "plot(exp(1+x)-exp(1-x),x=-3..3);" }}{PARA 13 "" 1 "
" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"$\"\"!$!3Ai
2*\\Z\"GYa!#;7$$!3!******\\2<#pG!#<$!3yP!p,6i]x%F-7$$!3#)***\\7bBav#F1
$!3!e'ePv*[zD%F-7$$!36++]K3XFEF1$!3r!>&=4t.UPF-7$$!3%)****\\F)H')\\#F1
$!3;q?90tm%G$F-7$$!3#****\\i3@/P#F1$!3Mf^HAun$)GF-7$$!3;++Dr^b^AF1$!37
YBr$*>WaDF-7$$!3$****\\7Sw%G@F1$!3!R2+2&Hc^AF-7$$!3*****\\7;)=,?F1$!3)
GjYiL(>u>F-7$$!3/++DO\"3V(=F1$!3**e]KD9gH<F-7$$!3#******\\V'zV<F1$!3&*
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\\\"F1$!3]V/X%Hno:\"F-7$$!3/+++vl[p8F1$!3GN8KK73+5F-7$$!3\"******\\>iU
C\"F1$!3]=A&[qq,l)F17$$!3-++DhkaI6F1$!3B^/&)\\2&=a(F17$$!3s******\\XF`
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Q[#Fgp$!3wS`Z:]Gk8F17$$!32)***\\i6:.8Fgp$!3k>[#)yKt/rFgp7$$!3Wb+++v`hH
!#?$!3l5#pw1h+h\"!#>7$$\"3]****\\(QIKH\"Fgp$\"31IxoIKM]qFgp7$$\"38****
\\7:xWCFgp$\"3]Q+4@_RU8F17$$\"3E,++vuY)o$Fgp$\"3]Wg)4nP50#F17$$\"3!z**
****4FL(\\Fgp$\"3^61avii;GF17$$\"3A)****\\d6.B'Fgp$\"3N_k!4Qr0h$F17$$
\"3s****\\(o3lW(Fgp$\"3a.T?$\\!*HV%F17$$\"35*****\\A))oz)Fgp$\"3W)[m$=
\"=OU&F17$$\"3e******Hk-,5F1$\"3u%)o\"\\CqwR'F17$$\"36+++D-eI6F1$\"3%=
\"z]eY;UvF17$$\"3u***\\(=_(zC\"F1$\"3S&p?/Sl\")o)F17$$\"3M+++b*=jP\"F1
$\"3=GG:F>)y+\"F-7$$\"3g***\\(3/3(\\\"F1$\"3]JL/'QnQ:\"F-7$$\"33++vB4J
B;F1$\"3,f-))f5^C8F-7$$\"3u*****\\KCnu\"F1$\"3(32WG\"fv6:F-7$$\"3s***
\\(=n#f(=F1$\"3Lr(GLBQDt\"F-7$$\"3P+++!)RO+?F1$\"3)e2yAM5D(>F-7$$\"30+
+]_!>w7#F1$\"3%y$))QGzd\\AF-7$$\"3O++v)Q?QD#F1$\"3-#)Qg;XOgDF-7$$\"3G+
++5jypBF1$\"3Y:Wdm_\"=)GF-7$$\"3<++]Ujp-DF1$\"3wb48rLB)H$F-7$$\"3++++g
Ed@EF1$\"3p*36-cu)>PF-7$$\"39++v3'>$[FF1$\"3(RS()[PivA%F-7$$\"37++D6Ej
pGF1$\"3=@6r_&fqx%F-7$$\"\"$F*$\"3Ai2*\\Z\"GYaF--%'COLOURG6&%$RGBG$\"#
5!\"\"$F*F*Fc[l-%+AXESLABELSG6$Q\"x6\"Q!Fh[l-%%VIEWG6$;F(Fhz%(DEFAULTG
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) =
 exp(1+x)-exp(1-x);" "6#/-%\"fG6#%\"xG,&-%$expG6#,&\"\"\"F-F'F-F--F*6#
,&F-F-F'!\"\"F1" }{TEXT -1 7 " is an " }{TEXT 284 12 "odd function" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "In fact " }{XPPEDIT 18 
0 "f(-x) = exp(1-x)-exp(1+x)" "6#/-%\"fG6#,$%\"xG!\"\",&-%$expG6#,&\"
\"\"F/F(F)F/-F,6#,&F/F/F(F/F)" }{XPPEDIT 18 0 "``=-(exp(1+x)-exp(1-x))
" "6#/%!G,$,&-%$expG6#,&\"\"\"F+%\"xGF+F+-F(6#,&F+F+F,!\"\"F0F0" }
{XPPEDIT 18 0 "``=-f(x)" "6#/%!G,$-%\"fG6#%\"xG!\"\"" }{TEXT -1 21 " f
or any real number " }{TEXT 316 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 33 " is symmetrical about the origin." }}{PARA 0 "" 0 "" 
{TEXT 259 4 "Note" }{TEXT -1 10 ": Maple's " }{TEXT 0 4 "type" }{TEXT 
-1 39 " procedure can be used to check that   " }{XPPEDIT 18 0 "f(x) =
 exp(1+x)-exp(1-x);" "6#/-%\"fG6#%\"xG,&-%$expG6#,&\"\"\"F-F'F-F--F*6#
,&F-F-F'!\"\"F1" }{TEXT -1 20 " is an odd function." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "type(exp(1
+x)-exp(1-x),oddfunc(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "(b)  T
he function " }{XPPEDIT 18 0 "f(x) = exp(1+x)-exp(-x);" "6#/-%\"fG6#%
\"xG,&-%$expG6#,&\"\"\"F-F'F-F--F*6#,$F'!\"\"F1" }{TEXT -1 11 " satisf
ies " }{XPPEDIT 18 0 "Int(f(x),x=-2..2)=0" "6#/-%$IntG6$-%\"fG6#%\"xG/
F*;,$\"\"#!\"\"F.\"\"!" }{TEXT -1 32 ", because it is an odd function.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 41 "Int(exp(1+x)-exp(1-x),x=-2..2);\nvalue(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$IntG6$,&-%$expG6#,&%\"xG\"\"\"F,F,F,-F(6#,&F,F,F+!
\"\"F0/F+;!\"#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 7 "Warning" }{TEXT -1 8 ": Using \+
" }{TEXT 0 19 "type(..evenfunc(x))" }{TEXT -1 6 " and  " }{TEXT 0 18 "
type(..oddfunc(x))" }{TEXT -1 4 " is " }{TEXT 259 22 "not guaranteed t
o work" }{TEXT -1 14 " in all cases." }}{PARA 0 "" 0 "" {TEXT -1 22 "C
onsider the function " }{XPPEDIT 18 0 "f(x)=ln((1+x)/(1-x))" "6#/-%\"f
G6#%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F-,&F-F-F'!\"\"F/" }{TEXT -1 38 " whic
h is defined on the open interval" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6
$,$\"\"\"!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+
" }{XPPEDIT 18 0 "f(-x)=ln((1-x)/(1+x))" "6#/-%\"fG6#,$%\"xG!\"\"-%#ln
G6#*&,&\"\"\"F/F(F)F/,&F/F/F(F/F)" }{XPPEDIT 18 0 "``=-ln((1+x)/(1-x))
" "6#/%!G,$-%#lnG6#*&,&\"\"\"F+%\"xGF+F+,&F+F+F,!\"\"F.F." }{XPPEDIT 
18 0 "``=-f(x)" "6#/%!G,$-%\"fG6#%\"xG!\"\"" }{TEXT -1 21 " for any re
al number " }{TEXT 318 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "-1<x
" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT 
-1 10 ", so that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 
7 " is an " }{TEXT 284 12 "odd function" }{TEXT -1 12 ". The graph " }
{XPPEDIT 18 0 "y=ln((1+x)/(1-x))" "6#/%\"yG-%#lnG6#*&,&\"\"\"F*%\"xGF*
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Qhx**F47$Fc]l$!3=F!fZC6A&**F47$Fh]l$!333l*[G?t\"**F47$F]^l$!3SY*yU]:#>
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%)fin:!*fF47$Fgdl$\"3_38kydNzpF47$F\\el$\"3k[()*3Hlp(yF47$Fael$\"391w$
==(=K')F47$Ffel$\"33O,o#Q2,A*F47$F[fl$\"3QrC,TO]['*F47$F`fl$\"3o]1fk\\
='z*F47$Fefl$\"3`8*)fq[:/**F47$Fjfl$\"3'**f(HC\\5V**F47$F_gl$\"3#[WnXn
v>(**F47$Fdgl$\"3#Q(G>byt!***F47$Figl$\"3MiH*pWs$****F47$F^hl$\"3w\")y
H:T3(***F47$Fchl$\"3]\\zo'[-B)**F47$Fhhl$\"3)=0xO.Y]&**F47$F]il$\"3CqL
\\=)[`\"**F47$Fbil$\"3-M&eU-Y))z*F47$Fgil$\"3b7'\\MHwLj*F47$F\\jl$\"3<
mtJhks:#*F47$Fajl$\"3%=9Z\"p`[\\')F47$F[[m$\"3=`![s%f%z)pF47$F`[m$\"3#
)3#H!fP.egF47$Fe[m$\"3%G>WYb!oO]F47$Fj[m$\"3*z@EM?iO$QF47$F_\\m$\"3A*p
`w#z\\hDF47$Fi\\m$\"3C/84*oy=K\"F47$Fc]m$\"3A@e2m$eZ6'F77$F]^m$!3[[))z
]xRd7F47$Fg^m$!3;Wv<:O+aDF47$F\\_m$!3;#G@[W\\(yPF47$Fa_m$!3`A?.`%o/%\\
F47$Ff_m$!3[,[Fe13ogF47$F\\`m$!3;#fVJoyZ3(F47$Fa`m$!3rz\"o\"[TvTzF47$F
f`m$!3YiB11I1k')F47$F[am$!3Aea&e]/2D*F47$F`am$!3S4>;+*4Ln*F47$Feam$!3d
\"\\*ydyf>)*F47$Fjam$!3];MQh\\-B**F47$F_bm$!3g<?7]s^e**F47$Fdbm$!3'**G
V%p(RJ)**F47$Fibm$!3C8\"\\Pklo***F47$F^cm$!3;W<=\"*)z'****F47$Fccm$!3O
*)*>$)*Hk#***F47$Fhcm$!3'y2x]T'Rw**F47$F]dm$!3OW#p]6b4&**F47$Fbdm$!3H5
n>YDM;**F47$Fgdm$!3KTQUF'R(>)*F47$F\\em$!3;x\")*=rVpo*F47$Faem$!3c`eXY
9&)[#*F47$Ffem$!3eYlK!=!)=j)F47$F[fm$!3&GK.7G\">QzF47$F`fm$!3:fY<OHk@r
F47$Fefm$!3&yejj5i*HhF47$Fjfm$!388Su0\"[/.&F47$F_gm$!3=p%[a(e\"\\*QF47
$Fdgm$!3+Y[/q(3mp#F47$F^hm$!3%y7T1cn4O\"F47$Fhhm$!3BXiH,t\"f^#!#D-F^im
6&F`imF(F(Faim-%+AXESLABELSG6%Q\"x6\"Q\"yFhao-%%FONTG6$%*HELVETICAG\"
\"*-%*AXESTICKSG6$7+/F)%\"0G/$\"+Fjzq:!\"*%$p/2G/$\"+aEfTJFhbo%\"pG/$
\"+\")*)Q7ZFhbo%%3p/2G/$\"+3`=$G'Fhbo%#2pG/$\"+N;)R&yFhbo%%5p/2G/$\"+i
zxC%*Fhbo%#3pG/$\"+Hub*4\"Fcim%%7p/2G/$\"+iqjc7Fcim%#4pG\"\"%-F[bo6$%'
SYMBOLGF^bo-%%VIEWG6$;F(Fcdo%(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 17 "Notice also that " }{XPPEDIT 18 0 "sin^3*
x;" "6#*&%$sinG\"\"$%\"xG\"\"\"" }{TEXT -1 29 " always has the same si
gn as " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "sin*x;" "6#*&
%$sinG\"\"\"%\"xGF%" }{TEXT -1 50 " is positive and less than one, the
 red graph of  " }{XPPEDIT 18 0 "y = sin^3*x;" "6#/%\"yG*&%$sinG\"\"$%
\"xG\"\"\"" }{TEXT -1 28 " is below the blue graph of " }{XPPEDIT 18 
0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 127 ", simply
 because, when we cube a positive number less than 1, we obtain a valu
e which is less than the number we started with. " }}{PARA 0 "" 0 "" 
{TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "x = Pi/3,sin*x = 1/2;
" "6$/%\"xG*&%#PiG\"\"\"\"\"$!\"\"/*&%$sinGF'F$F'*&F'F'\"\"#F)" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin^3*x = 1/8;" "6#/*&%$sinG\"\"$%
\"xG\"\"\"*&F(F(\"\")!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Now we investigate the graph of
  " }{XPPEDIT 18 0 "y = sin^3*x;" "6#/%\"yG*&%$sinG\"\"$%\"xG\"\"\"" }
{TEXT -1 20 " in a different way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 45 "The standard expansion for the sine of a \+
sum:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(alpha+bet
a) = sin*alpha*cos*beta+cos*alpha*sin*beta;" "6#/-%$sinG6#,&%&alphaG\"
\"\"%%betaGF),&**F%F)F(F)%$cosGF)F*F)F)**F-F)F(F)F%F)F*F)F)" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "and the double angle formulas:
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([sin*2
*x = 2*sin*x*cos*x, ``],[cos*2*x = 1-2*sin^2*x, ``]);" "6#-%*PIECEWISE
G6$7$/*(%$sinG\"\"\"\"\"#F*%\"xGF**,F+F*F)F*F,F*%$cosGF*F,F*%!G7$/*(F.
F*F+F*F,F*,&F*F**(F+F**$F)F+F*F,F*!\"\"F/" }{TEXT -1 2 "  " }}{PARA 0 
"" 0 "" {TEXT -1 50 "can be used to establish the triple angle formula
:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x = 3*sin*
x-4*sin^3*x;" "6#/*(%$sinG\"\"\"\"\"$F&%\"xGF&,&*(F'F&F%F&F(F&F&*(\"\"
%F&*$F%F'F&F(F&!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{TEXT 271 14 "______________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 
18 0 "alpha=x,beta=2*x" "6$/%&alphaG%\"xG/%%betaG*&\"\"#\"\"\"F%F*" }
{TEXT -1 27 " in the sum formula gives: " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "sin*3*x = sin(x+2*x);" "6#/*(%$sinG\"\"\"\"
\"$F&%\"xGF&-F%6#,&F(F&*&\"\"#F&F(F&F&" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 
18 0 "`` = sin*x*cos*2*x+sin*2*x*cos*x;" "6#/%!G,&*,%$sinG\"\"\"%\"xGF
(%$cosGF(\"\"#F(F)F(F(*,F'F(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 "`` = sin*x*(1-2*sin^2*x)+2*sin*x*cos*x*cos*x;" "6#/%!G,
&*(%$sinG\"\"\"%\"xGF(,&F(F(*(\"\"#F(*$F'F,F(F)F(!\"\"F(F(*0F,F(F'F(F)
F(%$cosGF(F)F(F0F(F)F(F(" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin*x-2*s
in^3*x+2*sin*x*cos^2*x;" "6#/%!G,(*&%$sinG\"\"\"%\"xGF(F(*(\"\"#F(*$F'
\"\"$F(F)F(!\"\"*,F+F(F'F(F)F(%$cosGF+F)F(F(" }{TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = sin*x-2*sin^3*x+2*sin*x*(1-sin^2*x);" "6#/%!G,(*&%
$sinG\"\"\"%\"xGF(F(*(\"\"#F(*$F'\"\"$F(F)F(!\"\"**F+F(F'F(F)F(,&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 "``= 3*sin*x-4*sin^3*x
" "6#/%!G,&*(\"\"$\"\"\"%$sinGF(%\"xGF(F(*(\"\"%F(*$F)F'F(F*F(!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Thus we have: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x= 3*sin*x-4*si
n^3*x" "6#/*(%$sinG\"\"\"\"\"$F&%\"xGF&,&*(F'F&F%F&F(F&F&*(\"\"%F&*$F%
F'F&F(F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "We can \+
make " }{XPPEDIT 18 0 "sin^3*x;" "6#*&%$sinG\"\"$%\"xG\"\"\"" }{TEXT 
-1 41 " the subject of this equation to obtain: " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "sin^3*x = 3/4" "6#/*&%$sinG\"\"$%\"xG\"
\"\"*&F&F(\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-1/4;" "6#,
&*&%$sinG\"\"\"%\"xGF&F&*&F&F&\"\"%!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 45 " These steps can be carried out using Map
le. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "expand(sin(alpha+beta));\nsubs(\{alpha=x,beta=2*x\},
%);\ne1 := subs(\{cos(2*x)=1-2*sin(x)^2,sin(2*x)=2*sin(x)*cos(x)\},%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%&alphaG\"\"\"-%$cosG
6#%%betaGF)F)*&-F+F'F)-F&F,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*
&-%$sinG6#%\"xG\"\"\"-%$cosG6#,$*&\"\"#F)F(F)F)F)F)*&-F+F'F)-F&F,F)F)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,&*&-%$sinG6#%\"xG\"\"\",&F+
F+*&\"\"#F+)F'F.F+!\"\"F+F+*(F.F+)-%$cosGF)F.F+F'F+F+" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Then convert the exp
ression into one which involves only " }{XPPEDIT 18 0 " sin(x" "6#-%$s
inG6#%\"xG" }{TEXT -1 23 " by using the identity " }{XPPEDIT 18 0 "cos
(x)^2=1-sin(x)^2" "6#/*$-%$cosG6#%\"xG\"\"#,&\"\"\"F+*$-%$sinG6#F(F)!
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 106 "#subs(\{cos(2*x)=1-2*sin(x)^2,sin(2*x)=2
*sin(x)*cos(x)\},e1);\nsubs(cos(x)^2=1-sin(x)^2,e1);\nsx := normal(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\",&F)F)*&\"
\"#F))F%F,F)!\"\"F)F)*(F,F),&F)F)*$F-F)F.F)F%F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#sxG,&*&\"\"$\"\"\"-%$sinG6#%\"xGF(F(*&\"\"%F()F)F'F(
!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 
"Thus we have established the identity . . ." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sin(3*x)=sx;\neq :
= %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#,$*&\"\"$\"\"\"%\"xG
F*F*,&*&F)F*-F%6#F+F*F**&\"\"%F*)F.F)F*!\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We can make " }{XPPEDIT 
18 0 "sin(x)^3" "6#*$-%$sinG6#%\"xG\"\"$" }{TEXT -1 30 " the subject o
f this equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "isolate(eq,sin(x)^3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*$)-%$sinG6#%\"xG\"\"$\"\"\",&*&#F+\"\"%F+-F'6#,$*&F*F
+F)F+F+F+!\"\"*&#F*F/F+F&F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 18 "The Maple command " }{TEXT 0 7 "combine" 
}{TEXT -1 67 " uses formulas like the one above in manipulating trig e
xpressions." }}{PARA 0 "" 0 "" {TEXT -1 43 "The following equation is \+
the same formula." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "sin(x)^3=combine(sin(x)^3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*$)-%$sinG6#%\"xG\"\"$\"\"\",&-F'6#,$F)F*#!\"\"\"\"
%F&#F*F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Now we investigate
 the formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin^
3*x = 3/4" "6#/*&%$sinG\"\"$%\"xG\"\"\"*&F&F(\"\"%!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "sin*x-1/4;" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&F&F&\"\"%
!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"
\"$F%%\"xGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 " graphic
ally." }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y
 = sin^3*x;" "6#/%\"yG*&%$sinG\"\"$%\"xG\"\"\"" }{TEXT -1 23 " can be \+
constructed by " }{TEXT 259 20 "adding the ordinates" }{TEXT -1 17 " o
n the graphs of" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "y
 = 3/4;" "6#/%\"yG*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "y = -1/4;" "6#/%\"yG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "sin*3*x;" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 315 "plot([3/4*sin(x),-1/4*sin(3*x)],x=0..4*Pi,y,col
or=[blue,green],\n   thickness=2,labelfont=[HELVETICA,9],ytickmarks=6,
\n    font=[SYMBOL,9],xtickmarks=[0=`0`,evalf(Pi/2)=`p/2`,evalf(Pi)=`p
`,\n    evalf(3*Pi/2)=`3p/2`,evalf(2*Pi)=`2p`,evalf(5*Pi/2)=`5p/2`,eva
lf(3*Pi)=`3p`,\n      evalf(7*Pi/2)=`7p/2`,evalf(4*Pi)=`4p`]);" }}
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EWG6$;F(Fjfq%(DEFAULTG" 1 2 0 1 10 2 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 18 "For example, when " }{XPPEDIT 18 0 "x=Pi/2" "6#/%\"xG*&%#
PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"
$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG
\"\"\"%\"xGF%" }{TEXT -1 24 "  has its maximum value " }{XPPEDIT 18 0 
"3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 9 " , while " }{XPPEDIT 
18 0 "-1/4;" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 23 " has its
 maximum value " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=Pi/2" 
"6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 23 ", the value of the sum
 " }{XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "sin*x+1/4;" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&F&F&\"\"%!\"
\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"\"$F
%%\"xGF%" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "3/4+1/4=1" "6#/,&*&\"\"$
\"\"\"\"\"%!\"\"F'*&F'F'F(F)F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x=P
i/3" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "-1/4;" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"sin*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 14 " is zero, so \+
 " }{XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "sin*x-1/4;" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&F&F&\"\"%!\"
\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"\"$F
%%\"xGF%" }{TEXT -1 24 "  has the same value as " }{XPPEDIT 18 0 "3/4;
" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x" "
6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 12 ", which is  " }{XPPEDIT 18 0 "3
/8 = 0;" "6#/*&\"\"$\"\"\"\"\")!\"\"\"\"!" }{TEXT -1 6 ".375. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In order \+
to form the graph of the sum  " }{XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"
\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-1/4;" "6#,&*&%$sinG
\"\"\"%\"xGF&F&*&F&F&\"\"%!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin
*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 50 ", we add the amou
nts given by the green graph of  " }{XPPEDIT 18 0 "y = -1/4;" "6#/%\"y
G,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x" "6
#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 44 " to the amounts given by \+
the blue graph of  " }{XPPEDIT 18 0 "y = 3*sin*x/4;" "6#/%\"yG**\"\"$
\"\"\"%$sinGF'%\"xGF'\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 24 "The resulting graph of  " }{XPPEDIT 18 0 "y = sin^3*x;" "
6#/%\"yG*&%$sinG\"\"$%\"xG\"\"\"" }{TEXT -1 11 ", that is, " }
{XPPEDIT 18 0 "y = 3/4;" "6#/%\"yG*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "sin*x-1/4;" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&F&F&\"
\"%!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*3*x" "6#*(%$sinG\"\"\"
\"\"$F%%\"xGF%" }{TEXT -1 46 " \"wiggles\" above and below the blue gr
aph of  " }{XPPEDIT 18 0 "y = 3/4;" "6#/%\"yG*&\"\"$\"\"\"\"\"%!\"\"" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }
{TEXT -1 45 ", to correspond to the way that the graph of " }{XPPEDIT 
18 0 "y = -1/4;" "6#/%\"yG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 31 
" \"wiggles\" above and below the " }{TEXT 304 1 "x" }{TEXT -1 7 " axi
s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 311 "plot([3/4*sin(x),sin(x)^3],x=0..4*Pi,y,color=[blue,g
reen],\n    thickness=2,labelfont=[HELVETICA,9],ytickmarks=6,\n    fon
t=[SYMBOL,9],xtickmarks=[0=`0`,evalf(Pi/2)=`p/2`,evalf(Pi)=`p`,\n    e
valf(3*Pi/2)=`3p/2`,evalf(2*Pi)=`2p`,evalf(5*Pi/2)=`5p/2`,evalf(3*Pi)=
`3p`,\n      evalf(7*Pi/2)=`7p/2`,evalf(4*Pi)=`4p`]);" }}{PARA 13 "" 
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{XPPEDIT 18 0 "y=1/4" "6#/%\"yG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 1 " " }
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. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 327 "plot([3/4*sin(x),1/4*sin(3*x),sin(x)^3],x=0..4*Pi,y,
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The sum of sine waves: \+
 " }{XPPEDIT 18 0 "sin^3*x = 3/4;" "6#/*&%$sinG\"\"$%\"xG\"\"\"*&F&F(
\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-1/4;" "6#,&*&%$sinG
\"\"\"%\"xGF&F&*&F&F&\"\"%!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin
*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT -1 17 " is the (finite) \+
" }{TEXT 259 24 "Fourier series expansion" }{TEXT -1 4 " of " }
{XPPEDIT 18 0 "f(x)=sin^3*x" "6#/-%\"fG6#%\"xG*&%$sinG\"\"$F'\"\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "The term " }{XPPEDIT 18 
0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "si
n*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 8 " is the " }{TEXT 259 16 "f
undamental wave" }{TEXT -1 28 ". which has the same period " }
{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 26 " as the pe
riodic function " }{XPPEDIT 18 0 "f(x)=sin^3*x" "6#/-%\"fG6#%\"xG*&%$s
inG\"\"$F'\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "The t
erm " }{XPPEDIT 18 0 "-1/4" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "sin*3*x" "6#*(%$sinG\"\"\"\"\"$F%%\"xGF%" }{TEXT 
-1 8 " is the " }{TEXT 259 5 "first" }{TEXT -1 29 " (and in this case \+
the only) " }{TEXT 259 8 "harmonic" }{TEXT -1 16 ". Its period is " }
{XPPEDIT 18 0 "2*Pi/3" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT -1 
19 " and its frequency " }{XPPEDIT 18 0 "3/(2*Pi)" "6#*&\"\"$\"\"\"*&
\"\"#F%%#PiGF%!\"\"" }{TEXT -1 30 " is three times the frequency " }
{XPPEDIT 18 0 "1/(2*Pi)" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT 
-1 26 " of the fundamental wave. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 66 "Extending a function defined on an interval to a period
ic function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 
"" 0 "" {TEXT -1 17 "Given a function " }{XPPEDIT 18 0 "f(x);" "6#-%\"
fG6#%\"xG" }{TEXT -1 13 " defined for " }{XPPEDIT 18 0 "a <= x;" "6#1%
\"aG%\"xG" }{XPPEDIT 18 0 "``<b" "6#2%!G%\"bG" }{TEXT -1 16 ", we can \+
extend " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " to a \+
periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT 
-1 13 " with period " }{XPPEDIT 18 0 "T = b -a" "6#/%\"TG,&%\"bG\"\"\"
%\"aG!\"\"" }{TEXT -1 55 " defined for all real numbers as follows. Fo
r a number " }{TEXT 272 1 "x" }{TEXT -1 14 " greater than " }{TEXT 
273 1 "b" }{TEXT -1 19 ", keep subtracting " }{TEXT 274 1 "T" }{TEXT 
-1 7 "  from " }{TEXT 276 1 "x" }{TEXT -1 28 " until the resulting num
ber " }{XPPEDIT 18 0 "u = x-k*T;" "6#/%\"uG,&%\"xG\"\"\"*&%\"kGF'%\"TG
F'!\"\"" }{TEXT -1 6 ", for " }{TEXT 275 1 "k" }{TEXT -1 43 " a positi
ve integer, lies in the interval  " }{XPPEDIT 18 0 "a <= x;" "6#1%\"aG
%\"xG" }{XPPEDIT 18 0 "``<b" "6#2%!G%\"bG" }{TEXT -1 26 ". Similarly, \+
for a number " }{TEXT 278 1 "x" }{TEXT -1 11 " less than " }{TEXT 279 
1 "a" }{TEXT -1 34 ", add a suitable integer multiple " }{XPPEDIT 18 
0 "k*`.`*T;" "6#*(%\"kG\"\"\"%\".GF%%\"TGF%" }{TEXT -1 4 " of " }
{TEXT 277 1 "T" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "u = x+k*T;" "6
#/%\"uG,&%\"xG\"\"\"*&%\"kGF'%\"TGF'F'" }{TEXT -1 23 "  lies in the in
terval " }{XPPEDIT 18 0 "a <= x;" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<b
" "6#2%!G%\"bG" }{TEXT -1 36 ". In each of these two cases define " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 27 " for the extended
 function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 7 " to \+
be " }{XPPEDIT 18 0 "f(u)" "6#-%\"fG6#%\"uG" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 39 "More succinctly, given any real number " }
{TEXT 283 1 "x" }{TEXT -1 51 ", there is a unique (positive or negativ
e) integer " }{TEXT 313 1 "k" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "
u=x+k*T" "6#/%\"uG,&%\"xG\"\"\"*&%\"kGF'%\"TGF'F'" }{TEXT -1 23 "  lie
s in the interval " }{XPPEDIT 18 0 "a <= x;" "6#1%\"aG%\"xG" }
{XPPEDIT 18 0 "``<b" "6#2%!G%\"bG" }{TEXT -1 16 ", and we define " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " to be equal to \+
" }{XPPEDIT 18 0 "f(u)" "6#-%\"fG6#%\"uG" }{TEXT -1 2 ". " }}{PARA 
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goQ!Fefo-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fc\\lFc[oFcho" 1 2 0 1 10 0 2 
9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1
0" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "
Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curv
e 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29
" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 19 "The Maple function " }{TEXT 0 5 "floor" }{TEXT -1 64 " \+
can be used to define and such an extension. For a real number " }
{TEXT 308 1 "x" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "floor(x)" "6#-%&floo
rG6#%\"xG" }{TEXT -1 58 " is the largest integer which is greater than
 or equal to " }{TEXT 309 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 
7 "Define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for \+
" }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6
#2%!G\"\"\"" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = x^2+1;" "6#/-%
\"fG6#%\"xG,&*$F'\"\"#\"\"\"F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 20 "For any real number " }{TEXT 310 1 "x" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "u=x-floor(x)" "6#/%\"uG,&%\"xG\"\"\"-%&floorG6#F&!\"\"
" }{TEXT -1 20 " is in the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\"
\"!%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" }{TEXT -1 20 ", and \+
we can extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 "
 to a periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }
{TEXT -1 31 ", with period 1, by setting f_(" }{TEXT 280 1 "x" }{TEXT 
-1 4 ") = " }{XPPEDIT 18 0 "f(u) = f(x-floor(x))" "6#/-%\"fG6#%\"uG-F%
6#,&%\"xG\"\"\"-%&floorG6#F+!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "f := x -> x
^2+1;\nf_ := x -> f(x-floor(x));\np1 := plot(f_(x),x=-2..5,0..2.1,disc
ont=true,thickness=2,ytickmarks=2):\nplots[display](p1,symbol=circle);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a
rrowGF(,&*$)9$\"\"#\"\"\"F1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 
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{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" 
{TEXT -1 7 "Define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 6 " for  " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 
"`` < 2;" "6#2%!G\"\"#" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = 1+2*
x-x^2;" "6#/-%\"fG6#%\"xG,(\"\"\"F)*&\"\"#F)F'F)F)*$F'F+!\"\"" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "For any real number " }{TEXT 
312 1 "x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "u = x-2*floor(x/2);" "6#/%
\"uG,&%\"xG\"\"\"*&\"\"#F'-%&floorG6#*&F&F'F)!\"\"F'F." }{TEXT -1 21 "
 is in the interval  " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }
{XPPEDIT 18 0 "`` < 2;" "6#2%!G\"\"#" }{TEXT -1 20 ", and we can exten
d " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " to a period
ic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 31 ",
 with period 2, by setting f_(" }{TEXT 282 1 "x" }{TEXT -1 4 ") = " }
{XPPEDIT 18 0 "f(u) = f(x-2*floor(x/2));" "6#/-%\"fG6#%\"uG-F%6#,&%\"x
G\"\"\"*&\"\"#F,-%&floorG6#*&F+F,F.!\"\"F,F3" }{TEXT -1 1 "." }}{PARA 
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\"\"$F*F*F\\\\o-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$%(DEFAULTGF`\\o-%+AX
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"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" 
{TEXT -1 7 "Define " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 6 " for  " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }
{XPPEDIT 18 0 "`` < 2;" "6#2%!G\"\"#" }{TEXT -1 4 " by " }{XPPEDIT 18 
0 "f(x) = x^2+1;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F+F+" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 20 "For any real number " }{TEXT 311 1 
"x" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "u = x-3*floor((x+1)/3);" "6#/%\"u
G,&%\"xG\"\"\"*&\"\"$F'-%&floorG6#*&,&F&F'F'F'F'F)!\"\"F'F/" }{TEXT 
-1 21 " is in the interval  " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!
\"\"%\"xG" }{XPPEDIT 18 0 "`` < 2;" "6#2%!G\"\"#" }{TEXT -1 20 ", and \+
we can extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 24 "
 to a periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }
{TEXT -1 27 ", with period 3, by setting" }}{PARA 256 "" 0 "" {TEXT 
-1 4 " f_(" }{TEXT 281 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "f(u) =
 f(x-3*floor((x+1)/3));" "6#/-%\"fG6#%\"uG-F%6#,&%\"xG\"\"\"*&\"\"$F,-
%&floorG6#*&,&F+F,F,F,F,F.!\"\"F,F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "f := x ->
 x^2+1;\nf_ := x -> f(x-3*floor((x+1)/3));\np1 := plot(f_(x),x=-4..8,0
..5.1,thickness=2,discont=true):\nplots[display](p1,symbol=circle);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro
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>%#f_Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%\"fG6#,&9$\"\"\"*&\"\"$F1-%
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cin-%*THICKNESSG6#\"\"#-%'POINTSG6(7$$!\"%Fdin$FhinFdin7$$!\"\"FdinF_j
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\"Q!F^[o-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F]jnFhjn;Fcin$\"#^Fbjn" 1 2 4 
1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" 
}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The examples above
 suggest that we can extend a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"
fG6#%\"xG" }{TEXT -1 14 " defined  for " }{XPPEDIT 18 0 "a <= x;" "6#1
%\"aG%\"xG" }{XPPEDIT 18 0 "``<b" "6#2%!G%\"bG" }{TEXT -1 24 " to a pe
riodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 
13 " with period " }{XPPEDIT 18 0 "T=b-a" "6#/%\"TG,&%\"bG\"\"\"%\"aG!
\"\"" }{TEXT -1 13 " by defining " }{XPPEDIT 18 0 "f_(x) = f(x-T*floor
((x-a)/T)" "6#/-%#f_G6#%\"xG-%\"fG6#,&F'\"\"\"*&%\"TGF,-%&floorG6#*&,&
F'F,%\"aG!\"\"F,F.F5F,F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
50 "The third example above of extending the function " }{XPPEDIT 18 
0 "f(x) = x^2+1;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F+F+" }{TEXT -1 
32 ", defined on the interval  for  " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$
\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 2;" "6#2%!G\"\"#" }{TEXT -1 24 
" to a periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }
{TEXT -1 49 " with period 3, fits into the following template." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
160 "f := x -> x^2+1;\na := -1;\nT := 3;\nf_ := x -> f(x-T*floor((x-a)
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ts[display](p1,symbol=circle);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F1F1F(F(F(" }
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2 9 1 4 2 1.000000 45.000000 42.000000 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 10 "Example 4 " }}{PARA 0 "
" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 16 " is defined for " }{XPPEDIT 18 0 "1 <= x;" "6#1\"
\"\"%\"xG" }{XPPEDIT 18 0 "`` < 5;" "6#2%!G\"\"&" }{TEXT -1 4 " by " }
{XPPEDIT 18 0 "f(x) = (5-x)*(x-1);" "6#/-%\"fG6#%\"xG*&,&\"\"&\"\"\"F'
!\"\"F+,&F'F+F+F,F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "T
he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 40 " c
an be extended to a periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f
_G6#%\"xG" }{TEXT -1 25 " with period 4, that is, " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = (5-x)*(x-1);" "6#/-%#f_G6#%\"
xG*&,&\"\"&\"\"\"F'!\"\"F+,&F'F+F+F,F+" }{TEXT -1 7 "  when " }
{XPPEDIT 18 0 "1 <= x" "6#1\"\"\"%\"xG" }{XPPEDIT 18 0 " ``< 5" "6#2%!
G\"\"&" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 4 "and " }
{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 28 " is periodic wit
h period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 93 "f := x -> (5-x)*(x-1);\na := 1: T := 4:\nf_ := x -
> f(x-T*floor((x-a)/T)):\nplot(f_(x),x=-3..7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&,&9$\"\"\"
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PoF/$\"3KmEU(GWM(RF/7$$\"3a+]P40O\"*oF/$\"3ySn2hu>))RF/7$$\"3s+voa-oXp
F/$\"3?&=p_O\\q*RF/7$$\"\"(F*$\"\"%F*-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-
%+AXESLABELSG6$Q\"x6\"Q!F\\]m-%%VIEWG6$;F(F]\\m%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 16 " is defined for " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!
\"\"%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%!G\"\"\"" }{TEXT -1 4 " by " 
}{XPPEDIT 18 0 "f(x) = x-x^3;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"$!\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 43 " can be extended \+
to a periodic function f_(" }{TEXT 287 1 "x" }{TEXT -1 26 ") with peri
od 2, that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_
(x) = x-x^3;" "6#/-%#f_G6#%\"xG,&F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 7 "
  when " }{XPPEDIT 18 0 "-1 <= x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 
18 0 " `` < 1" "6#2%!G\"\"\"" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" 
{TEXT -1 4 "and " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 
28 " is periodic with period 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
88 "f := x -> x-x^3;\na := -1: T := 2:\nf_ := x -> f(x-T*floor((x-a)/T
)):\nplot(f_(x),x=-3..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6
#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*$)F-\"\"$F.!\"\"F(F(F(" }}
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gH:1O[W*H\"F87$$\"33;a)eOF'*4&F/$!32uq_#enYq\"F87$$\"3a***\\7`Wl7&F/$!
3AxQ50$\\22#F87$$\"3amT5!R$Ry^F/$!3u]SEDS\"*pEF87$$\"3cL$e*[ACI_F/$!3d
?u;P]bOJF87$$\"3e+D\"y5\"4#G&F/$!3md[+ra.zMF87$$\"3enmmm*RRL&F/$!3zIs'
pX=dq$F87$$\"3inT5S9Xg`F/$!3QVN&H)*)ezPF87$$\"3on;a8H'pQ&F/$!3)eF[5z)[
EQF87$$\"3sn\"zpQuMT&F/$!392$z'4f`ZQF87$$\"3wnmTge)*RaF/$!3<<XFn$[Q%QF
87$$\"3%ym\"H2)3I\\&F/$!3O\"**oITTnw$F87$$\"3%zmmTvJga&F/$!3VLZ&e:7Tg$
F87$$\"3eM3FWch)f&F/$!3!Q)=(fivrO$F87$$\"3A,]PM&*>^cF/$!3O$[pS&ykjIF87
$$\"3'y;zWU$y.dF/$!3wGd;9FD-FF87$$\"3]MLe9tOcdF/$!3'\\\")y-39<H#F87$$
\"3q+vV[ko/eF/$!3M)\\Ji]G'y=F87$$\"3yn;H#e0I&eF/$!3:>\"ydc#=Q9F87$$\"3
([$e9;ZK,fF/$!3Y[)Q,7]9x*F27$$\"31,++]Qk\\fF/$!3^\\+M!)f%G-&F27$$\"3%
\\L3-.B]+'F/$\"3*pC,JZ.H-&Fbr7$$\"3%zm;/@-/1'F/$\"3Od@*ft$==gF27$$\"3$
4+D1R\"y:hF/$\"3qxvFGIHU6F87$$\"3![LL3dg6<'F/$\"3?qIZ3EYh;F87$$\"31o;z
pBp?iF/$\"3)p**y(4_V*4#F87$$\"3K,+voTAqiF/$\"3V@HHB5#\\]#F87$$\"3fM$3x
'fv>jF/$\"3_x<ts(G1(GF87$$\"3%ymmmw(GpjF/$\"3o+D\"==n#*=$F87$$\"3/M$eR
A5\\Z'F/$\"35T8KY5*zn$F87$$\"3C++D\"oK0e'F/$\"3!\\XvX0E)[QF87$$\"3m+++
]oi\"o'F/$\"3k+$oA!oK\\OF87$$\"35,+v=5s#y'F/$\"3iE_a*p_=.$F87$$\"3Q+D1
k2/PoF/$\"3DkPuF$)y0DF87$$\"3a+]P40O\"*oF/$\"3ka#pH'e`J=F87$$\"3>]7.#Q
?&=pF/$\"3)e@M$))Q$eV\"F87$$\"3s+voa-oXpF/$\"3oHU'fT'y%***F27$$\"3O]PM
F,%G(pF/$\"3+Du-)R!o7_F27$$\"\"(F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%
+AXESLABELSG6$Q\"x6\"Q!Fiao-%%VIEWG6$;F(F\\ao%(DEFAULTG" 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 16 " is defined for " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!
\"\"%\"xG" }{XPPEDIT 18 0 "`` < 3;" "6#2%!G\"\"$" }{TEXT -1 4 " by " }
{XPPEDIT 18 0 "f(x) = PIECEWISE([x+1, -1 <= x and x < 0],[1, 0 <= x an
d x < 2],[3-x, 1 <= x and x < 3]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$
,&F'\"\"\"F-F-31,$F-!\"\"F'2F'\"\"!7$F-31F3F'2F'\"\"#7$,&\"\"$F-F'F131
F-F'2F'F;" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 14 "We can extend " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#
%\"xG" }{TEXT -1 24 " to a periodic function " }{XPPEDIT 18 0 "f_(x)" 
"6#-%#f_G6#%\"xG" }{TEXT -1 25 " with period 4, that is, " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "f_(x) = PIECEWISE([x+1, -1 <= x and x < 0],[1, 0 <= x and x < 2],[3
-x, 1 <= x and x < 3]);" "6#/-%#f_G6#%\"xG-%*PIECEWISEG6%7$,&F'\"\"\"F
-F-31,$F-!\"\"F'2F'\"\"!7$F-31F3F'2F'\"\"#7$,&\"\"$F-F'F131F-F'2F'F;" 
}{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 4 "and " }{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 
28 " is periodic with period 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "f := x -> piecewise(x<0,x+1
,x<2,1,3-x);\na := -1: T := 4:f_ := x -> f(x-T*floor((x-a)/T)):\nplot(
f_(x),x=-1..11,y=0..1.2,ytickmarks=2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6'29$\"\"!
,&F0\"\"\"F3F32F0\"\"#F3,&\"\"$F3F0!\"\"F(F(F(" }}{PARA 13 "" 1 "" 
{GLPLOT2D 503 135 135 {PLOTDATA 2 "6&-%'CURVESG6$7[t7$$!\"\"\"\"!$F*F*
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HyI\"F/7$$!3k+++DhDQ!)F/$\"3O*****\\(Quh>F/7$$!37++++:M%Q(F/$\"3*)****
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**\\d'*)o\\F/Fgp7$$\"3B*****\\(>ZIuF/Fgp7$$\"3G+++vnBw**F/Fgp7$$\"3%**
***\\FPQ^7!#<Fgp7$$\"3;+++IrS7:F[rFgp7$$\"3.+++&o;Bu\"F[rFgp7$$\"37++]
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$\"31+++!4X$4IF[r$\"3'e0+++!4X$*Ffp7$$\"31++]2#>m1$F[r$\"3]0++]2#>m'F2
7$$\"31+++DL*Q7$F[r$\"3a+++]K$*Q7F/7$$\"3/++]Uu;\"=$F[r$\"3`+++DWn6=F/
7$$\"30+++g:WQKF[r$\"3_++++cT%Q#F/7$$\"3-+]Pu\\10LF[r$\"3?++vV(\\10$F/
7$$\"3)****\\()Q)orLF[r$\"3()****\\()Q)or$F/7$$\"3&***\\7.=JQMF[r$\"3a
***\\7.=JQ%F/7$$\"3#*****\\<_$\\]$F[r$\"3A*****\\<_$\\]F/7$$\"3')**\\7
G#3Rc$F[r$\"3b)**\\7G#3RcF/7$$\"3z***\\(Q7)Gi$F[r$\"3)y***\\(Q7)GiF/7$
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F/7$$\"3()**\\(=)=\\GkF[r$\"3J,+D\"=\"3:dF/7$$\"3%*****\\P<I*['F[r$\"3
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(F[r$\"3#R++]_NQ'>F/7$$\"3A+++]/;hsF[r$\"36-+++Xg6EF/7$$\"3Z+](o%z&)>t
F[r$\"3k/+vo%z&)>$F/7$$\"3#)***\\PWb&ytF[r$\"3I)***\\PWb&y$F/7$$\"3?**
\\iSHDPuF[r$\"3)>**\\iSHDP%F/7$$\"3Y****\\P/&f\\(F[r$\"3_%****\\P/&f\\
F/7$$\"3w**\\i0B7gvF[r$\"3h(**\\i0B7g&F/7$$\"32++vtTHCwF[r$\"3q++]P<%H
C'F/7$$\"3Q+](=/m%)o(F[r$\"3y.+v=/m%)oF/7$$\"3q+++5zj_xF[r$\"3(o+++5zj
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**\\PO*Rt)F/7$$\"3c**\\i!4!yLzF[r$\"3f&**\\i!4!yL*F/7$$\"3=****\\<3;%*
zF[r$\"3$=****\\<3;%**F/7$$\"3Q*\\(oC-0-!)F[rFgp7$$\"3e**\\(=jR*4!)F[r
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[rFgp7$$\"3)*)****\\2ws0)F[rFgp7$$\"3+)**\\PqL))3)F[rFgp7$$\"3y)***\\K
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\\'[M\\)F[rFgp7$$\"3W****\\PM&=v)F[rFgp7$$\"3v+++gzs+!*F[rFgp7$$\"35++
+0\"Q_D*F[rFgp7$$\"3q++]x2k2&*F[rFgp7$$\"3d+++?EdR(*F[rFgp7$$\"3^++DOw
-1)*F[rFgp7$$\"3Y++]_E[s)*F[rFgp7$$\"3V+]ig,r0**F[rFgp7$$\"3S++vow$*Q*
*F[rFgp7$$\"3G,D\"GU^b&**F[rFgp7$$\"3Q+](o<l@(**F[rFgp7$$\"3#3D1R0s/)*
*F[rFgp7$$\"3[*\\P4$*y())**F[rFgp7$$\"3!*\\(oz!e3(***F[rFgp7$$\"3.++]o
#R0+\"!#;$\"3g'*****\\J2Y**F/7$$\"3.+]P%3$[15Ff]m$\"3W(***\\i:p^$*F/7$
$\"3-++D+pU75Ff]m$\"3I)****\\(*4tv)F/7$$\"3,+]7;2P=5Ff]m$\"3;****\\(QG
H;)F/7$$\"3++++KXJC5Ff]m$\"3-++++oaovF/7$$\"3,+vVz=lI5Ff]m$\"3I***\\i0
7[$pF/7$$\"3-+](oA*)p.\"Ff]m$\"3e)***\\7t2,jF/7$$\"3-+DJulKV5Ff]m$\"3'
y**\\(oDMncF/7$$\"3/++v@Rm\\5Ff]m$\"3:(****\\#ygL]F/7$$\"3)***\\(=dHd0
\"Ff]m$\"3l,+]7G/FWF/7$$\"3%******>A&zh5Ff]m$\"391+++yZ?QF/7$$\"33+]7s
3'y1\"Ff]m$\"3(G***\\(y7R@$F/7$$\"3/++DAl#R2\"Ff]m$\"3P(****\\xZtg#F/7
$$\"3)**\\(o\"*[W!3\"Ff]m$\"3[-+DJ3^b>F/7$$\"35+]7hK'p3\"Ff]m$\"3!)*)*
*\\()Qn.8F/7$$\"31+DcI;[$4\"Ff]m$\"3-\\**\\P%p$=lF27$$\"#6F*F+-%'COLOU
RG6&%$RGBG$\"#5F)F+F+-%*AXESTICKSG6$%(DEFAULTG\"\"#-%+AXESLABELSG6$Q\"
x6\"Q\"yFfcm-%%VIEWG6$;F(Febm;F+$\"#7F)" 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 "" }}{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 3 "Q1 " }}
{PARA 0 "" 0 "" {TEXT -1 78 "Plot graphs of the following functions an
d determine whether each function is " }{TEXT 284 4 "even" }{TEXT -1 
2 ", " }{TEXT 284 3 "odd" }{TEXT -1 4 " or " }{TEXT 284 20 "neither ev
en nor odd" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 8 "   (a)  " }{XPPEDIT 18 0 "f(x) = 1+x^2+x^4;" "6#/-
%\"fG6#%\"xG,(\"\"\"F)*$F'\"\"#F)*$F'\"\"%F)" }{TEXT -1 9 "    (b)  " 
}{XPPEDIT 18 0 "f(x)=1+x^3" "6#/-%\"fG6#%\"xG,&\"\"\"F)*$F'\"\"$F)" }
{TEXT -1 9 "    (c)  " }{XPPEDIT 18 0 "f(x)=1/(1+x^2)" "6#/-%\"fG6#%\"
xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 10 "     (d)  " }
{XPPEDIT 18 0 "f(x)=x/(1+x^2)" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F)F)*$F'\"
\"#F)!\"\"" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "   (e)   " }{XPPEDIT 18 0 "f(x) = x*sin^2*
x;" "6#/-%\"fG6#%\"xG*(F'\"\"\"*$%$sinG\"\"#F)F'F)" }{TEXT -1 11 "    \+
  (f)  " }{XPPEDIT 18 0 "f(x) = x^2*sin*x;" "6#/-%\"fG6#%\"xG*(F'\"\"#
%$sinG\"\"\"F'F+" }{TEXT -1 10 "     (g)  " }{XPPEDIT 18 0 "f(x)=x^2*s
in^2*x " "6#/-%\"fG6#%\"xG*(F'\"\"#%$sinGF)F'\"\"\"" }{TEXT -1 9 "    \+
(h)  " }{XPPEDIT 18 0 "f(x) = x+sin*2*x;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*
(%$sinGF)\"\"#F)F'F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 8 "   (i)  " }{XPPEDIT 18 0 "f(x) = x+cos
*3*x;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*(%$cosGF)\"\"$F)F'F)F)" }{TEXT -1 
10 "     (j)  " }{XPPEDIT 18 0 "f(x) = exp(x)+exp(-x);" "6#/-%\"fG6#%
\"xG,&-%$expG6#F'\"\"\"-F*6#,$F'!\"\"F," }{TEXT -1 10 "     (k)  " }
{XPPEDIT 18 0 "f(x)=exp(x)-exp(-x)" "6#/-%\"fG6#%\"xG,&-%$expG6#F'\"\"
\"-F*6#,$F'!\"\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 45 "_________________________________________
____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "____________________
_________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" 
{TEXT -1 75 "Find the following integrals, making use of the possible \+
simplification of " }{XPPEDIT 18 0 "Int(f(x),x=-a..a)" "6#-%$IntG6$-%
\"fG6#%\"xG/F);,$%\"aG!\"\"F-" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "f
(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " is even or odd. " }}{PARA 0 "" 
0 "" {TEXT -1 8 "   (a)  " }{XPPEDIT 18 0 "Int(``(1+x^4/4),x = -1 .. 1
);" "6#-%$IntG6$-%!G6#,&\"\"\"F**&%\"xG\"\"%F-!\"\"F*/F,;,$F*F.F*" }
{TEXT -1 9 "    (b)  " }{XPPEDIT 18 0 "Int(sin*2*x,x = -Pi .. Pi);" "6
#-%$IntG6$*(%$sinG\"\"\"\"\"#F(%\"xGF(/F*;,$%#PiG!\"\"F." }{TEXT -1 
11 "      (c)  " }{XPPEDIT 18 0 "Int(cos*2*x,x = -Pi/4 .. Pi/4);" "6#-
%$IntG6$*(%$cosG\"\"\"\"\"#F(%\"xGF(/F*;,$*&%#PiGF(\"\"%!\"\"F1*&F/F(F
0F1" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 9 "    (d)  " }
{XPPEDIT 18 0 "Int(x^3*cos*x,x=-Pi..Pi)" "6#-%$IntG6$*(%\"xG\"\"$%$cos
G\"\"\"F'F*/F';,$%#PiG!\"\"F." }{TEXT -1 11 "     (e)   " }{XPPEDIT 
18 0 "Int(x*sin*x,x = -Pi/3 .. Pi/3);" "6#-%$IntG6$*(%\"xG\"\"\"%$sinG
F(F'F(/F';,$*&%#PiGF(\"\"$!\"\"F0*&F.F(F/F0" }{TEXT -1 9 "    (f)  " }
{XPPEDIT 18 0 "Int(x^2*sin*x,x=-Pi..Pi)" "6#-%$IntG6$*(%\"xG\"\"#%$sin
G\"\"\"F'F*/F';,$%#PiG!\"\"F." }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 46 "_____________________________________________\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________________
______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 
20 "Plot the graphs of  " }{XPPEDIT 18 0 "y=sin*x" "6#/%\"yG*&%$sinG\"
\"\"%\"xGF'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=1/2" "6#/%\"yG*&\"\"\"
F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*x" "6#*(%$sinG\"\"
\"\"\"#F%%\"xGF%" }{TEXT -1 65 " in the same diagram with Maple and th
en construct the graph of  " }{XPPEDIT 18 0 "y=sin*x+1/2" "6#/%\"yG,&*
&%$sinG\"\"\"%\"xGF(F(*&F(F(\"\"#!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "sin*2*x" "6#*(%$sinG\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 10 " on pape
r." }}{PARA 0 "" 0 "" {TEXT -1 78 "Check your picture by drawing all t
hree graphs in the same diagram with Maple." }}{PARA 0 "" 0 "" {TEXT 
-1 46 "_____________________________________________\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 45 "_______________________________________
______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 8 "Express
 " }{XPPEDIT 18 0 "cos^3*x;" "6#*&%$cosG\"\"$%\"xG\"\"\"" }{TEXT -1 
14 " in terms of  " }{XPPEDIT 18 0 "cos*x;" "6#*&%$cosG\"\"\"%\"xGF%" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*3*x;" "6#*(%$cosG\"\"\"\"\"$F%
%\"xGF%" }{TEXT -1 36 ", and so indicate how the graph of  " }
{XPPEDIT 18 0 "y = cos^3*x;" "6#/%\"yG*&%$cosG\"\"$%\"xG\"\"\"" }
{TEXT -1 47 " can be built up by adding two \"cosine\" graphs." }}
{PARA 0 "" 0 "" {TEXT -1 46 "_________________________________________
____\n" }}{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 45 "____________________
_________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" 
{TEXT -1 19 "(a) Check that the " }{XPPEDIT 18 0 "f(x)=sin(x*sqrt(1-x^
2))" "6#/-%\"fG6#%\"xG-%$sinG6#*&F'\"\"\"-%%sqrtG6#,&F,F,*$F'\"\"#!\"
\"F," }{TEXT -1 20 " is an odd function." }}{PARA 0 "" 0 "" {TEXT -1 
83 "(b) Check that Maple cannot find an analytical formula for the ind
efinite integral " }{XPPEDIT 18 0 "Int(sin(x*sqrt(1-x^2)),x)" "6#-%$In
tG6$-%$sinG6#*&%\"xG\"\"\"-%%sqrtG6#,&F+F+*$F*\"\"#!\"\"F+F*" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "(c) Check using Maple that " 
}{XPPEDIT 18 0 "Int(sin(x*sqrt(1-x^2)),x=-1..1)=0" "6#/-%$IntG6$-%$sin
G6#*&%\"xG\"\"\"-%%sqrtG6#,&F,F,*$F+\"\"#!\"\"F,/F+;,$F,F3F,\"\"!" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 58 ": T
his illustrates the fact that Maple \"knows\" the result " }{XPPEDIT 
18 0 "Int(f(x),x=-a..a)=0" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;,$%\"aG!\"\"F
.\"\"!" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" 
}{TEXT -1 56 " is a (continuous) odd function defined on the interval \+
" }{XPPEDIT 18 0 "[-a,a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 46 "_________________________________________
____\n" }}{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 45 "____________________
_________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 16 " is defined for " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!
\"\"%\"xG" }{XPPEDIT 18 0 "`` < 4;" "6#2%!G\"\"%" }{TEXT -1 4 " by " }
{XPPEDIT 18 0 "f(x) = (4-x)*(x+1);" "6#/-%\"fG6#%\"xG*&,&\"\"%\"\"\"F'
!\"\"F+,&F'F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "G
ive a formula which extends " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" 
}{TEXT -1 24 " to a periodic function " }{XPPEDIT 18 0 "f_(x)" "6#-%#f
_G6#%\"xG" }{TEXT -1 25 " with period 5, that is, " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "f_(x) = (4-x)*(x+1);" "6#/-%#f_G6#%\"
xG*&,&\"\"%\"\"\"F'!\"\"F+,&F'F+F+F+F+" }{TEXT -1 6 " when " }
{XPPEDIT 18 0 "-1 <= x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` < 4
" "6#2%!G\"\"%" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 4 "and " 
}{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 28 " is periodic wi
th period 5. " }}{PARA 0 "" 0 "" {TEXT -1 22 "and plot the graph of " 
}{XPPEDIT 18 0 "f_(x)" "6#-%#f_G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "__________________
___________________________\n" }}{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 45 "_____________________________________________" }}{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 34 "Code for picture of even function " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
492 "p1 := plot(x^2,x=-1.4..1.4):\np2 := plot([[-1,0],[-1,1],[1,1],[1,
0]],\n       linestyle=2,color=COLOR(RGB,.1,.1,.1)):\np3 := plot([[[-1
,1],[1,1]]$3],style=point,symbol=[circle,diamond,cross],color=black):
\nt1 := plots[textplot]([[1,-0.05,`a`],[-1,-0.05,`-a`],\n   [-0.05,-0.
05,`O`],[1.4,-0.05,`x`],[-0.05,2,`y`],[-0.05,0.95,`M`],\n   [-1.25,1,`
Q(-a,f(a))`],[1.23,1,`P(a,f(a))`]],color=black):\nt2 := plots[textplot
]([1.15,1.8,`y = f(x)`],color=red):\nplots[display]([p1,p2,p3,t1,t2],t
ickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "Co
de for picture of odd function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 515 "p1 := plot(x^3,x=-1.3..1.3)
:\np2 := plot([[-1,0],[-1,-1],[1,1],[1,0]],\n       linestyle=2,color=
COLOR(RGB,.1,.1,.1)):\np3 := plot([[[-1,-1],[1,1]]$3],style=point,symb
ol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[1,-0.
1,`a`],[-1,0.2,`-a`],\n   [-0.05,0.15,`O`],[1.3,-0.1,`x`],[-0.05,2.2,`
y`],\n   [-1.35,-0.9,`Q(-a,-f(a))`],[1.35,1,`P(a,f(a))`]]):\nt2 := plo
ts[textplot]([.9,1.5,`y = f(x)`],color=red):\nplots[display]([p1,p2,p3
,t1,t2],tickmarks=[0,0],\n                   view=[-1.35..1.35,-2.36..
2.36]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 80 "Code for p
icture for extending a function on an interval to a periodic function \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1425 "f := x -> 0.2+0.7*sin(x)+1/10*cos(5*x):\nf_ := x -> f(x-2*fl
oor(x/2)):\nfa := .3: fb := f(2.):\np1 := plot(f(x),x=0..2,color=red):
\np2 := plot(f(x+2),x=-2..0,color=brown):\np3 := plot(f(x-2),x=2..4,co
lor=brown):\np4 := plot(f(x-4),x=4..6,color=brown):\np5 := plot(f(x-6)
,x=6..8,color=brown):\np6 := plot([[-2,0],[8,0]],color=black):\np7 := \+
plot([[[-2,0],[-2,f(0)]],seq([[2*i,0],[2*i,f(2)]],i=0..4)],\n         \+
       color=black,linestyle=2):\np8 := plot([[[.7,0],[.7,f(.7)]],[[4.
7,0],[4.7,f(.7)]]],\n                color=blue,linestyle=3):\np9 := p
lot([[[-2,fa],[2,fa],[4,fa],[6,fa]]$3],style=point,\n     symbol=[circ
le,diamond,cross],color=brown):\np10 := plot([[0,fb],[4,fb],[6,fb],[8,
fb]],style=point,symbol=circle,color=brown):\np11 := plot([[[0,fa]]$3]
,style=point,symbol=[circle,diamond,cross],color=red):\np12 := plot([[
2,fb]],style=point,symbol=circle,color=red):\np13 := [plottools[arrow]
([.7,-.4],[.7,-.15],.02,.15,.3,\n                       color=COLOR(RG
B,.6,0,.9))][1]:\np14 := [plottools[arrow]([4.7,-.4],[4.7,-.15],.02,.1
5,.3,\n                       color=COLOR(RGB,.6,0,.9))][1]:\np15 := p
lot([[0.7,-.4],[4.7,-.4]],color=COLOR(RGB,.6,0,.9)):\nt1 := plots[text
plot]([[0,-.05,a],[2,-.05,b]],\n                color=COLOR(RGB,0,0,.0
1)):\nt2 := plots[textplot]([[.7,-.07,`u=x-2T`],[4.7,-.07,x]],\n      \+
          color=blue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10
,p11,p12,p13,p14,p15,t1,t2],\n                      axes=none);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }