Latex Equation Compiler Code

111 lines (111 with data), 60.4 kB

÷
ƒ’À;è TeX output 2012.12.29:1852‹
ÿÿÿÿïheader=pstricks.proïheader=pst-algparser.proïheader=pst-dots.pro ¹ðü ý;9‘ègiïcolor push gray 0’þñï	color popŽŽ ¦Æs‘ègi ýr9ïcolor push gray 0Ÿ<~ˆ’’ßNïcolor push gray 0ï	color popŽŽ’˜¿JóÆîê½q½qecrm2074»T‘þfKour–d—of“EQC“functionalit‘ÿwmyŽŸ(žæ’Ö±¸ó& Šffffecrm1440¼Jan‘‘Rheinl€änderŽŽŽŽŽŸ!\’Ï]Decemžnba‘er–‘29,“2012ŽŸ:4ós©^dG®G®ecbx1728ÆCon–ÿ{kten“tsŽŸaeó¥!¢Necbx1200Ç1Ž‘žæNumeric–¸ev‘ÿ@alution“of“equations’
?a2ŽŽ¤?D2Ž‘žæW‘þàorking–¸with“phš ysical“quan˜tities’
4È3ŽŽ¡3Ž‘žæUser–¸dened“functions’
sD"4ŽŽ¡4Ž‘žæAsking–¸for“v›ÿ@alues“of“v˜ariables’
J—5ŽŽ¡5Ž‘žæSym b_úolic‘¸computations’
n%Ü7ŽŽ¤€‘žæóÓ·åecrm1200º5.1Ž‘,£žExample:–êlSolving“a“quadratic“equation‘™l‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘ëÍïcolor push gray 07Ž‘Ëoï	color popŽ¡‘žæ5.2Ž‘,£žExample:–êlFinding“the“extrema“of“a“function‘Ë̍‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘ëÍïcolor push gray 09Ž‘Ëoï	color popŽ¡‘žæ5.3Ž‘,£žLibrary–êlof“substititions‘+‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 010Ž‘Ëoï	color popŽ¤?DÇ6Ž‘žæMatrices’
½†>10ŽŽ¡7Ž‘žæGraphs’
Åàa11ŽŽ¡8Ž‘žæUtilit y‘¸functions’
ÓÁ14ŽŽ¡9Ž‘žæPitfalls’
Ç'i14ŽŽ¤€‘žæº9.1Ž‘,£žUnexpSˆected‘êlnewlines‘j1‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 014Ž‘Ëoï	color popŽ¡‘žæ9.2Ž‘,£žMultiple–êlpSˆossible“v‘ÿXíalues“for“equations‘э‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 015Ž‘Ëoï	color popŽ¡‘žæ9.3Ž‘,£žMultiple‘êlsubstitutions‘¬~‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 016Ž‘Ëoï	color popŽ¡‘žæ9.4Ž‘,£žOutput–êlof“n•¬wum“bSˆers‘õA‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 016Ž‘Ëoï	color popŽ¡‘žæ9.5Ž‘,£žLibrary‘êlequations‘]Œ‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 016Ž‘Ëoï	color popŽ¡‘žæ9.6Ž‘,£žPstric•¬wks‘êlheadac“hes‘šo‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 017Ž‘Ëoï	color popŽŽŸ‘ègiïcolor push gray 0’
¨1Ž’þñï	color popŽŽŒ‹* ¹ðü ý;9‘ègiïcolor push gray 0ó-]Kdïecsl1200Ø1‘¿DNUMERIC–êlEV‘þ±ÜALUTION“OF“EQUA‘ÿeTIONS’
&æº2Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÆ1Ž‘^…Numeric–6êev‘þöØalution“of“equationsŽŸÏ‘ègiºEQC‘Jjadds–J‚the“pSˆo•¬ww“er–Jƒof›J‚a“spreadsheet˜to“Latex.‘KIt“is˜pSˆossible“to˜write˜do¬wwn“equations˜and“obtain˜theŽ¤€‘ègin¬wumerical–êlv›ÿXíalue“of“v˜ariables:ŽŸÈò‘â‡mïcolor push gray 0ï	color popŽŽ‘ègió+D7`±ectt1200Ö$$\eq{x–,j=“3}$$Ž¡‘ègi$$\eq{y–,j=“4}$$Ž¡‘ègi$$\eq{z–,j=“\sqrt{x^2“+“y^2}}“=“\val{z}$$Ž©Èñ’â9%ó·ág£cmmi12Êx–URóX«Qcmr12¹=“3Ž¤m¦’â{^Êy‘˹=‘UR4Ž¡’ÂBÊz‘Þ5¹=‘URŸõxó%ú±u
cmex10ÐpŽ‘UTŸõx‰zà$ÏŸ
ýˆÊxŸüˆ‰ó|{Ycmr8È2Ž‘j¬¹+‘ª¨Êyn9Ÿüˆ‰È2ŽŽŽŽ‘7y·¹=‘UR5ŽŸÛL‘ègiºIf–êlyš¬wou“only“w˜an˜t“to“do“a“n˜umerical“calculation“in“y˜our“text,“using“the“ó"!",š
cmsy10ÍnÇv‘ÿ@al“ºk˜eyw˜ord“is“sucien˜t.Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖThe–,jsquare“root“of“two“is“\val{\sqrt{2}}.ŽŸÈò‘ègiºThe–êlsquare“roSˆot“of“t•¬ww“o–êlis“1.414.ŽŸí¦‘ègiEquations–êlcan“bšSˆe“assigned“a“lab˜el“to“reuse“them“later“on:Ž©g`‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{equation}Ž¤€‘ègi\eq[eq:important]{a–,j=“x“+“2}Ž¡‘ègi\end{equation}Ž¡‘ègi\centerline{\emph{Equation–,j\ref{eq:important}“defines“the“value“of“aŽ¡‘ègito–,jbe“$\printeq{"eq:important"}“=“\val{a}$.}}Ž¦¡’×ç8Êa–UR¹=“Êx–ª¨¹+“2Ž’
éc¼º(1)ŽŽŽŸm¦‘fŽºó,둇¡ecti1200×Equation–2ð1“denes“the“value“of“a“to“b‘ÿfpe“Êa–UR¹=“Êx–ª¨¹+“2–UR=“5×.ŽŽŸí¦‘ègiºIf›ÑÃy•¬wou‘ÑÂw“an“t˜to˜use‘ÑÂmathematical˜or˜ph“ysical‘ÑÂconstan“ts˜in˜y“our–ÑÂequations,‘Òdene˜them˜with“the˜ÍnÇconstan tŽ¡‘ègiºk•¬weyw“ord.–êlThe“le“mathconstanš¬wts.tex“con˜tains“denitions“for“the“most“common“constan˜ts“Ê‘X¥ºand“Êeº.Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\constant{\Pi–,j=“3.141}$$Ž¡‘ègi$$\eq{r–,j=“3}$$Ž¡‘ègi$$\eq{d–,j=“2\Pi“r}“=“\val{d}$$Ž¦¡’Ö¶™¹–UR=“3Ê:¹141Ž¤m¦’âÀAÊr‘¨à¹=‘UR3Ž¡’ÆxÊd–UR¹=“2Êr‘¨à¹=“18Ê:¹85ŽŸÛL‘ègiºEquations–¤mcan›¤ntak¬we“an˜optional“list˜of“options,–¤Áfor˜example,“Ö\eq[label=eq:x;–,jeqraw=false]{x“=“y^2}º.ŽŸ€‘ègiThe–êloptions“will“apply“only“to“this“spSˆecic“equation.“Some“of“the“a¬wv‘ÿXíailable“options“are:ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹t ¹ðü ý;9‘ègiïcolor push gray 0Ø2‘¿DW¬wORKING–êlWITH“PHYSICAL“QUANTITIES’
¸àº3Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0‘ßüÇunits–¸=“{unit;“unit;“...}ï	color popŽŽ‘xQ÷ºT‘ÿeell–êlEQC“what“units“it“should“use“in“the“output.Ž©zˆ‘â‡mïcolor push gray 0‘ßüÇprecision–¸=“in tegerï	color popŽŽ‘^µSºThe–?¤precision“(total›?¥n•¬wum“bSˆer–?¤of“digits)“with“whic¬wh“oating˜pSˆoinš¬wt“n˜um˜bSˆers“areŽ¤€‘ÇUprin¬wted.ŽŸz‡‘â‡mïcolor push gray 0‘ßüÇxeddigits–¸=“true|falseï	color popŽŽ‘tsºY‘ÿeou–wScan“decide“whether‘wTyš¬wou“w˜an˜t“to“round“all“n˜um˜bSˆers“to‘wTa“xed“n˜um˜bSˆer“ofŽ¡‘ÇUdigits–@;(e.g.,‘@Q12345“rounded›@<to“a“precision“of“three˜w¬would“bSˆe“12300)“or“whether“to˜displa¬wy“a“xedŽ¡‘ÇUn•¬wum“bSˆer–êlof“digits“after“the“decimal“mark¬wer.Ž¦‘â‡mïcolor push gray 0‘ßüÇlo wsclimit–¸=“realï	color popŽŽ‘R°ÁºIt–â>is“pSˆossible“to“conš¬wtrol“whic˜h‘â?n˜um˜bSˆers“are“prin˜ted“in“scien˜tic“notation“(e.g.,Ž¡‘ÇU¹1Ê:¹23‘
ÿþÍ
Ž‘UR¹10Ÿû¥2È5Ž‘Àº).–êlThis“options“givš¬we“the“smallest“n˜um˜bSˆer“that“is“still“prin˜ted“normally‘ÿe.ŽŸz‡‘â‡mïcolor push gray 0‘ßüÇhighsclimit–¸=“realï	color popŽŽ‘XPgºThis–êloptions“givš¬we“the“largest“n˜um˜bSˆer“that“is“still“prin˜ted“normally‘ÿe.Ž¦‘â‡mïcolor push gray 0‘ßüÇeqalign–¸=“none|eqnarra y|amsï	color popŽŽ’™‘¥ºThis–Õ8option“conš¬wtrols“equation‘Õ9formatting.‘ÖàÖnone“ºprin˜ts“no“ampSˆersands,Ž¡‘ÇUÖeqnarray–ž¨ºprin¬wts“ampšSˆersands“on“b˜oth“sides“of“the“equals“sign“(Ö&=&º),‘ž»and“Öams“ºprin¬wts“an“amp˜ersandŽ¡‘ÇUbšSˆefore–‹>the‘‹=equals“sign“(Ö&=º).‘Œ4By“default,‘‹gEQC‘‹automatically“adds“amp˜ersands“to‘‹=y¬wour“equationsŽ¡‘ÇUdepšSˆending–êlon“the“en•¬wvironmen“t–êlthey“app˜ear“in.ŽŸz‡‘â‡mïcolor push gray 0‘ßüÇeqc hain–¸=“true|falseï	color popŽŽ‘dpºOmits–Ó\the“left“hand“side›Ó[of“an“equation“if“it“is“iden¬wtical“to˜the“left“hand“side“ofŽ¡‘ÇUthe–êlequation“directly“preceding“it“in“an“Öeqnarray“ºor“AMS“en•¬wvironmen“t.Ž¦‘â‡mïcolor push gray 0‘ßüÇeqra w–¸=“true|falseï	color popŽŽ‘Z••ºPrinš¬wt–êlequations“as“y˜ou“t˜ypšSˆed“them“(if“p˜ossible),“or“with“EQC“formatting.ŽŸgb‘ègiThe›êlk•¬weyw“ord˜ÍnÇeqcoptions˜ºcan˜bSˆe˜used˜to˜set˜these˜options˜globally‘ÿe.ŽŸ-	,‘ègiÆ2Ž‘^…W‘þrDorking–6êwith“ph›ÿ{kysical“quan˜titiesŽŸY0‘ègiºEQC‘üÓadds›üÙthe–üØclass“Unit“to“the˜functionalitš¬wy“of“the“GiNaC‘üÓlibrary‘ÿe.‘üõThis“means“that“y˜ou‘üÙcan“use“unitsŽ¡‘ègiin–êla“natural“w•¬wa“y–êlinside“equations:ŽŸ a鍍‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\clearequationsŽ¡‘ègi\eqcoptions{units–,j=“{\mm}}Ž¡‘ègi$$\eq{x–,j=“\unit{3}{\mm}}$$Ž¡‘ègi$$\eq{y–,j=“\unit{4}{\mm}}$$Ž¡‘ègi$$\eq{z–,j=“x“+“y}“=“\quantity{z}$$ŽŸ aꡍ’ÖÃØÊx–UR¹=“3“mmŽŽ¤÷ˍ’×Êy‘˹=–UR4“mmŽŽ¡’ÀŒÍÊz‘Þ5¹=›URÊx–ª¨¹+“Êy‘˹=˜7‘UPmmŽŽŸ‘ègiºUpSˆon–³	startup,‘³}EQC›²“kno¬wws“only“the‘³base“SI˜units.‘µÆT‘ÿeo›³get“full“suppSˆort˜for“all“units“dened˜in“theŽ¤€‘ègiSIunits.st•¬wy›¼Cpac“k‘ÿXíage,‘¼ºinclude˜units.tex‘¼Dat˜the˜bSˆeginning˜of˜y“our˜doSˆcumen“t.‘¿New‘¼Dunits˜can˜easily˜bSˆeŽ¡‘ègidened–êlusing“the“ÍnÇdefunit“ºk•¬weyw“ord.–êlF‘ÿeor“example:ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹» ¹ðü ý;9‘ègiïcolor push gray 0Ø3‘¿DUSER–êlDEFINED“FUNCTIONS’
S׺4Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\defunit['']{\inch}{2.54\mm}Ž¤€‘ègi\eqcoptions{units–,j=“{\inch}}Ž¡‘ègi$$z–,j=“x“+“y“=“\quantity{z}$$Ž© €¡’½WxÊz‘Þ5¹=›URÊx–ª¨¹+“Êy‘˹=˜2Ê:¹756‘UPŸû™ó#¾KÈcmsy8Î00ŽŽŽŸ‘ègiºThe–­option“Öunits›­ºtells“EQC‘­what“units“it“should“use˜in“the“output.‘­pIf“no˜units“are“giv¬wen,‘­"EQC‘­will“useŽ¡‘ègithe–DäSI‘DÍbase›Dåunits.‘EoThe“units“can˜also“bSˆe“giv¬wen“as˜an“optional“argumen¬wt˜to“the“ÍnÇv‘ÿ@al“ºk•¬weyw“ord˜(and‘DäwillŽ¡‘ègionly–êlapply“to“that“spSˆecic“v‘ÿXíalue):Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\defunit['']{\inch}{2.54\mm}Ž¡‘ègi\eqcoptions{units–,j=“{\inch}}Ž¡‘ègi$$z–,j=“x“+“y“=“\quantity{z}“=“\quantity[units“=“{\mm}]{z}$$Ž¦¡’¦Êz‘Þ5¹=›URÊx–ª¨¹+“Êy‘˹=˜2Ê:¹756‘UPŸû™Î00ŽŽ‘+ßâ¹=˜7‘UPmmŽŽŸ‘ègiºA–êlshort“form“also“exists,“with“the“syn¬wtax“Ö\quantity[\mm]{z}º.Ž©€‘ègiNote–êlthat“there“are“four“dierenš¬wt“k˜eyw˜ords“that“can“bSˆe“used“for“nding“the“v›ÿXíalue“of“a“v˜ariable:ŽŸ€‘â‡mïcolor push gray 0‘ßüÇn• um“v‘ÿ@alï	color popŽŽ‘tºThis–êlprinš¬wts“a“n˜umeric“v‘ÿXíalue,“or“prin˜ts“an“error“if“the“v‘ÿXíariable“doSˆes“not“ha˜v˜e“one.Ž¤€‘â‡mïcolor push gray 0‘ßüÇunitsï	color popŽŽ‘˜ÃºPrin¬wts–êlonly“the“units“of“the“v‘ÿXíariable.Ž¡‘â‡mïcolor push gray 0‘ßüÇquan• tit“yï	color popŽŽ‘”PºPrinš¬wts–êla“n˜umeric“v‘ÿXíalue“plus“units“(if“there“are“an˜y).Ž¡‘â‡mïcolor push gray 0‘ßüÇv‘ÿ@alï	color popŽŽ‘þöZºPrin•¬wts›êlan“y˜kind˜of˜sym“bSˆolic˜expressionŽŸ-4‘ègiÆ3Ž‘^…User–6êdened“functionsŽŸae‘ègiºEQC‘qnextends–qGiNaC‘qmb¬wy›q‘oering“user-dened“functions˜whic¬wh“can“bSˆe“created˜at“run¬wtime“in˜the“LatexŽ¤€‘ègile.›ÎA‘Vfunction–Œis“declared‘‹using“the“ÍnÇfunction“ºk•¬weyw“ord.˜A‘Vdenition–Œfor›‹the“function“ma¬wy“bSˆe˜giv¬wen“withŽ¡‘ègiÍnÇdeuncº,››Cbut–›/this“is“not“strictly“necessary‘ÿe.‘›©Of“course,˜only“functions“with“denitions“can“bSˆe“ev‘ÿXíaluated!Ž¦‘ègiF‘ÿeunctions–L*can“bSˆe‘L+givš¬wen“hin˜ts‘FÅwhen“they›L+are“dened“using“the˜synš¬wtax“Ö\function[hintlist]...º.‘MHin˜tsŽ¡‘ègienable––öEQC‘–áto“handle‘–÷and“prinš¬wt“functions“bSˆetter.‘—vF‘ÿeor“example,‘—the“hin˜t“Ötrig“ºwill“result‘–÷in“the“functionŽ¡‘ègibSˆeing–êlprin¬wted“as“¹sinŽ‘WŠŸû3ÎÈ2Ž‘ŒÊx“ºinstead“of“¹(sinŽ‘m(Êx¹))Ÿû¥2È2Ž‘Àº.Ž¦‘ègiConsider–êlthe“functionŽ©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\function{f}{x}Ž¡‘ègi$$\deffunc{f}{ax^2–,j+“bx“+“c}$$Ž¦’ÅTÊf‘Q¹=‘URÊaxŸû™È2Ž‘j¬¹+–ª¨Êbx“¹+“ÊcŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹)’ ¹ðü ý;9‘ègiïcolor push gray 0Ø4‘¿DASKING–êlF¬wOR“V›þ±ÜALUES“OF“V˜ARIABLES’

÷º5Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖWith–,j$\eq{a“=“4}$,“$\eq{b“=“-2}$“and“$\eq{c“=“7}$“we“can“compute“$f(x)$“for“$x“=“3$:Ž¤€‘ègi$$f(x)–,j=“\val{f(x)}$$Ž¡‘ègi$$f(3)–,j=“\val{f(3)}“=“\numval{\val{f(3)}}$$ŽŸ‘ègiºWith›êlÊa–UR¹=“4º,˜Êb“¹=“͹2˜ºand˜Êc“¹=“7˜ºw¬we˜can˜compute˜Êf‘
Gÿ¹(Êx¹)˜ºfor˜Êx“¹=“3º:Ž¤íލ’ºŸÊf‘
Gÿ¹(Êx¹)–UR=“7–ª¨+“4›
ÿþÊxŸû™È2Ž‘j¬Í“¹2˜ÊxŽ¡’±8•f‘
Gÿ¹(3)–UR=“Êc–ª¨¹+“3›
ÿþÊb“¹+“9˜Êa–UR¹=“37ŽŸS¨‘ègiºNote–Ú©what›Ú¨eect“the“ÍnÇv‘ÿ@al˜ºstatemen¬wt“has“on˜the“dieren¬wt“parameters.‘ÚÁT‘ÿeo˜bSˆoth“expand“the˜function“andŽ¤€‘ègisubstitute–êlthe“v›ÿXíalues“of“kno¬wwn“v˜ariables,“a“double“ÍnÇv‘ÿ@al“ºstatemen¬wt“is“necessary‘ÿe.ŽŸ,,„‘ègiÆ4Ž‘^…Asking–6êfor“v›þöØalues“of“v˜ariablesŽŸµ
‘ègiºMost–a‚došSˆcumen¬wts‘awill“b˜e“of‘athe“t¬wyp˜e“that‘adene“a“n•¬wum“b˜er›aof–a‚equations“and˜then“try“to˜calculate“n¬wumericŽ¡‘ègiv›ÿXíalues–êlfor“the“v˜ariables“used“in“these“equations.“There“are“sev¬weral“methoSˆds“of“doing“this“ecienctly:ŽŸzù‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºThe›¹òmost‘¹ñstraigh•¬wt-forw“ard˜w“a“y–¹ñis˜to“dene˜an“equation˜and“then˜other“equations˜that“giv¬we˜theŽ¡‘ÇUkno¬wwn–&v›ÿXíalues‘'of“v˜ariables.‘jThe“ÍnÇv‘ÿ@al“ºstatemen¬wt›'can“then“bSˆe˜used“to˜calculated“the“v‘ÿXíalues˜of“unk•¬wo“wnŽ¡‘ÇUv‘ÿXíariables.Ž©íލçYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\begin{eqnarray*}Ž¡‘xý\eq{x–,j=“3y“+“4}\\Ž¡‘xý\eq{y–,j=“5}\\Ž¡‘xýx–,j&=&“\numval{x}Ž¡‘ )\end{eqnarray*}ŽŸ&[¼’Üý“ÊxŽŽ’íªå¹=ŽŽ’
Îm3Êy‘á¹+‘ª¨4ŽŽŽ¤€’Ý‚ÊyŽŽ’íªå¹=ŽŽ’
Îm5ŽŽŽ¡’Üý“ÊxŽ’íªå¹=ŽŽ’
Îm19ŽŽŽŸíލ‘ÇUºThis–±ÏmethošSˆd“b˜ecomes“a•¬wwkw“ard–±Ïif“yš¬wou“w˜an˜t‘±Ðto“ask“for“the“v‘ÿXíalue“of“Êx“ºagain,‘±Þusing“a“dieren˜t“v‘ÿXíalueŽ¤€‘ÇUfor–êlÊyn9º.“Y‘ÿeou“wš¬would“ha˜v˜e“to“delete“the“equation“dening“Êy‘X¥ºand“create“a“new“one.ŽŸ
‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºIf–?ey¬wou›?fneed“the“v‘ÿXíalue“of˜a“v‘ÿXíariable“for“man•¬wy˜dieren“t–?eparameters,‘?¼the˜bSˆest“w•¬wa“y–?eis“to˜dene“aŽ¡‘ÇUfunction.Ž¦çYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\clearequationsŽ¡‘ÇU\begin{eqnarray*}Ž¡‘ )\function{x}{y}%Ž¡‘ )\deffunc{x}{3y+4}\\Ž¡‘ )x–,j&=&“\val{x(5)}\\Ž¡‘ )x–,j&=&“\val{x(7)}\\Ž¡‘ÇU\end{eqnarray*}ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹4T ¹ðü ý;9‘ègiïcolor push gray 0Ø4‘¿DASKING–êlF¬wOR“V›þ±ÜALUES“OF“V˜ARIABLES’

÷º6Ž’þñï	color popŽŽ ¦Æs ý~9©€’Üý“ÊxŽŽ’íªå¹=ŽŽ’
Îm3Êy‘á¹+‘ª¨4ŽŽŽ¤€’Üý“ÊxŽ’íªå¹=ŽŽ’
Îm19ŽŽŽ¡’Üý“ÊxŽ’íªå¹=ŽŽ’
Îm25ŽŽŽ¡¦‘ÇUºThe›{only‘zdra•¬wwbac“k˜is˜that˜y“ou–zcannot˜easily˜substitute“the˜denition˜Êx‘ƹ=‘Å3Êy‘6C¹+‘È
4“ºin¬wto˜otherŽ¤€‘ÇUequations.–êlY‘ÿeou“wš¬would“ha˜v˜e“to“do“something“lik˜e“Ö\eqsubst{z–,j=“3x}{x“=“\val{x}}º.ŽŸ€‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºThe–êllast“w•¬wa“y–êlis“to“use“the“Öwithº-form“of“the“ÍnÇv‘ÿ@al“ºstatemen¬wt.ŽŸ€çYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\clearequations%Ž¡‘ÇU\begin{eqnarray*}Ž¡‘ )\eq{x–,j=“3y“+“4}\\Ž¡‘ )x–,j&=&“\numvalwith{x}{y“=“5}\\Ž¡‘ )x–,j&=&“\numvalwith{x}{y“=“7}\\Ž¡‘ÇU\end{eqnarray*}ŽŸ)€’Üý“ÊxŽŽ’íªå¹=ŽŽ’
Îm3Êy‘á¹+‘ª¨4ŽŽŽ¤€’Üý“ÊxŽ’íªå¹=ŽŽ’
Îm19ŽŽŽ¡’Üý“ÊxŽ’íªå¹=ŽŽ’
Îm25ŽŽŽ¡¦‘ÇUºAll–Èthe“assignmenš¬wts“giv˜en“as‘Éthe“second“argumen˜t“are“dened“as“equations,‘the“searc˜h“for“theŽ¤€‘ÇUv‘ÿXíalue–is“pšSˆerformed,‘$and“then“the“temp˜orary“equations‘are“deleted“again.‘eTherefore,‘$y¬wou“can“ndŽ¡‘ÇUev¬wen–êlv›ÿXíalues“of“v˜ariables“that“are“indirectly“dened:Ž©€çYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\clearequationsŽ¡‘ÇU\begin{eqnarray*}Ž¡‘ )\eq{x–,j=“3y“+“4}\\Ž¡‘ )\eq{z–,j=“4x}\\Ž¡‘ )z–,j&=&“\numvalwith{z}{y“=“5}\\Ž¡‘ÇU\end{eqnarray*}Ž¦Ÿ€’Üý“ÊxŽŽ’íªå¹=ŽŽ’
Îm3Êy‘á¹+‘ª¨4ŽŽŽ¤€’ݬ°ÊzŽŽ’íªå¹=ŽŽ’
Îm4ÊxŽŽŽ¡’ݬ°zŽ’íªå¹=ŽŽ’
Îm76ŽŽŽ¡ŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹>. ¹ðü ý;9‘ègiïcolor push gray 0Ø5‘¿DSYMBOLIC‘êlCOMPUT–ÿeA“TIONS’
SD
º7Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÆ5Ž‘^…Sym‘ÿ{kb‘„”olic‘6êcomputationsŽŸae‘ègiºEQC‘‡oers–‡the“pSˆossibilitš¬wy“of‘‡‚sym˜bSˆolic“manipulation“of“equations.‘‰ûY‘ÿeou“can‘‡‚use“it“to“add,‘‡ësubtract,Ž¤€‘ègim¬wultiply–
?or›
@divide“an˜equation“with˜an“expression.‘
pAlso,‘
Hit“is˜pSˆossible“to˜substitute“an˜expression“withŽ¡‘ègianother–êlexpression,“and“to“dieren¬wtiate“an“expression.ŽŸ'ɞ‘ègió.&Lt$ffffecbx1440Ù5.1Ž‘y¯Example:–G\Solving“a“quadratic“equationŽŸ?„‘ègiºConsider–êlthe“equationŽ©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\clearequationsŽ¡‘ègi\function{f}{x}Ž¡‘ègi$$\deffunc{f}{ax^2–,j+“bx“+“c}“=“0$$Ž¦’º|ûÊf‘Q¹=›URÊaxŸû™È2Ž‘j¬¹+–ª¨Êbx“¹+“Êc˜¹=˜0ŽŸ‘ègiºWhat–êlv‘ÿXíalues“of“x“will“solv¬we“this“equation?ŽŸ€‘ègiW‘ÿee–·need“to›·bring“the“equation“in¬wto“the“form˜ÊxŸû¥2È2Ž‘¤_¹+–ä[2ÊAx“¹+“ÊAŸû¥2È2Ž‘%–¹=‘e’ÊB‘›º,‘·†whic¬wh–·can“then˜bSˆe“written“asŽ¡‘ègi¹(Êx–ª¨¹+“ÊA¹)Ÿû¥2È2Ž‘V¹=‘URÊB‘…rºand‘êlsolv¬wed.ŽŸ €‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqdiv{\val{f(x)}–,j=“0}{a},“a“\ne“0$$Ž¡‘ègi$$\eqadd{"prev"}{(1/2–,jb/a)^2“-“c/a}$$Ž¡‘ègi$$\eqsimpf{"prev"}{expand}$$ŽŸ%„vŸ÷áō’¨ï¹1Ž’¨ÉŸ[«‰zà+
Ÿ
ý΍ÊaŽŽŽŽ’´'½ŸöG©ÐŽ’¹§¿Êc–ª¨¹+“Êa›
ÿþxŸû™È2Ž‘j¬¹+“Êb˜xŸöG©Ð
ŽŽ’
Áá¹=–UR0Ê;‘
ÿþa“Í6¹=“0Ž¤!\Ÿ÷áō’ˆ] 1Ž’ˆ7Ÿ[«‰zà+
Ÿ
ý΍ÊaŽŽŽŽ’“•ÍŸöG©ÐŽ’™ÏÊc–ª¨¹+“Êa›
ÿþxŸû™È2Ž‘j¬¹+“Êb˜xŸöG©Ð
ŽŽ’ä…G¹+Ÿ÷áō‘ÝÛ1Ž‘Ý۟[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘ºIÊbŸû¥2È2ŽŽ‘$=Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘í͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽ‘‘a¹=Ÿ÷áō‘ˆ…1Ž‘ˆ…Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
dóÊbŸû¥2È2ŽŽ‘ÎçŸ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘—Ç͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽ¡’«%xŸû™È2Ž‘j¬¹+Ÿ÷áō‘ÝÛ1Ž‘Ý۟[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘ºIÊbŸû¥2È2ŽŽ‘$=Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘í¹+Ÿ÷áō‘ÝÛÊb‘
ÿþxŽ‘Ý۟[«‰zà
¬:Ÿ
ý΍‘ÀaŽŽŽŽ‘š¹=Ÿ÷áō‘ˆ…1Ž‘ˆ…Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
dóÊbŸû¥2È2ŽŽ‘ÎçŸ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘—Ç͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŸþþ‘ègiºThe–³`spšSˆecial“equation“lab˜el“Ö"prev"“ºis“a“shortcut“to“access“the“last“equation“pro˜cessed“b¬wy“EQC.“WithŽŸ€‘ègiÍnÇeqsimpf–êlºit“is“pSˆossible“to“do“a“v‘ÿXíariet¬wy“of“simplications“on“the“equation:Ž¦‘â‡mïcolor push gray 0‘ßüÇexpandï	color popŽŽ‘¢ºF‘ÿeully–êlexpands“all“expressions,“including“function“argumen¬wts.Ž¤€‘â‡mïcolor push gray 0‘ßüÇexpandfï	color popŽŽ‘½`ºOnly–êlexpand“function“denition,“not“argumen¬wts.Ž¡‘â‡mïcolor push gray 0‘ßüÇev‘ÿ@alï	color popŽŽ‘ÚºNumerically–êlev‘ÿXíaluate“the“equation“as“far“as“pSˆossible.Ž¡‘â‡mïcolor push gray 0‘ßüÇnormalï	color popŽŽ‘˜]ºNormalize–#the“equation,‘#Ãthat“is,‘#Âforce“all›#terms“to“ha•¬wv“e–#a“common˜denominator“(see“descriptionŽ©€‘ÇUof–êlGiNaC's“Önormal()“ºmethoSˆd“for“details).Ž¡‘â‡mïcolor push gray 0‘ßüÇcollect-commonï	color popŽŽ‘IꍺCollect–uÄcommon“factors“from“all“terms‘uÅof“the“equation“(see“description“of“GiNaC'sŽ¦‘ÇUÖcollect_common_factors()–êlºmethoSˆd“for“details).ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹DË ¹ðü ý;9‘ègiïcolor push gray 0Ø5‘¿DSYMBOLIC‘êlCOMPUT–ÿeA“TIONS’
SD
º8Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0‘ßüÇunsafeï	color popŽŽ‘q(ºDo‘kunsafe–jsimplications,›qfor“example,˜Ÿõ ÿÍpŽ‘sŸõ ÿ‰zàmVŸ
_
ÊxŸüˆ‰È2ŽŽŽŽ‘荹=›zÄÊx–kºand“¹arctanŽ‘&£átanŽ‘9œ·Êx˜¹=˜Êxº.‘Note“that‘jthe“oppSˆosite,Ž©€‘ÇU¹tanŽ‘À+arctanŽ‘;c¡Êxº,–êlis“not“unsafe“and“thš¬wus“is“done“automatically“(b˜y“GiNaC).ŽŸ€‘ègiAs–êlcan“bSˆe“seen,“ÊA›UR¹=ŸûFu‘ˆ…È1Ž‘ˆ…Ÿ
öû‰zà@Ÿåê2ŽŽŽŽŸûFu‘Ÿió ×
2cmmi8ËbŽ‘.ïŸ
öû‰zàƒÏŸåêaŽŽŽŽ‘Ð]ºand“ÊB‘ðX¹=˜(ŸûFu‘
33È1Ž‘
33Ÿ
öû‰zà@Ÿåê2ŽŽŽŽŸûFu‘JËbŽ‘ٝŸ
öû‰zàƒÏŸåêaŽŽŽŽ‘
Ÿ¹)Ÿû¥2È2Ž‘j¬ÍŸûFu‘H†ËcŽ‘Ý۟
öû‰zàƒÏŸåêaŽŽŽŽ‘	”ݺ.“The“equation“can“no¬ww“bSˆe“written“as:Ž¤ €‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eq{\left(x–,j+“\frac{1}{2}\frac{b}{a}\right)^2“=“\rhs{"prev"}}$$ŽŸ)Ÿ÷áō’°$¹1Ž’°$Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽ’¹7ïŸïqÐŽŸ÷áō’ÃֆÊbŽ’Ã@zŸ[«‰zà+
Ÿ
ý΍aŽŽŽŽ’ÍIV¹+‘ª¨2‘
ÿþÊxŸïqÐŽŽŽ’ðz*ŸñÄ<È2Ž’ø€¹=Ÿ÷áō‘ˆ…1Ž‘ˆ…Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
dóÊbŸû¥2È2ŽŽ‘ÎçŸ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘—Ç͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŸ *0‘ègiºApplying–êlthe“square“rošSˆot“to“b˜oth“sides“yields“the“t•¬ww“o‘êlsolutions:Ž¡‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqpow{"prev"}{1/2}$$Ž¦‘ègi$$\eqsimpf[eq:poss1]{"prev"}{unsafe}$$Ž¦‘ègi$$\mbox{or}\quad\eq[eq:poss2]{\lhs{"prev"}–,j=“-\rhs{"prev"}}$$Ž¡Ÿ™´Ÿ÷áō’¤$À¹1Ž’¤$ÀŸ[«‰zàßüŸ
ý΍2ŽŽŽŽ’­7íŸé\ÐsŽ’¹7ïŸé\‰zà<?Ÿ£ôŸïqŽŸ÷áō‘
ž—ÊbŽ‘
‹Ÿ[«‰zà+
Ÿ
ý΍aŽŽŽŽ‘g¹+‘ª¨2‘
ÿþÊxŸïqÐŽŽŽ‘7B;ŸñÄ<È2ŽŽŽŽ’ø€¹=‘URŸì³ÐrŽ‘UTŸì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽŽŸ(£ë’¹Jx‘ª¨¹+Ÿ÷áō‘cäÊbŽ‘Ý۟[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽ‘q[¹=‘URŸì³ÐrŽ‘UTŸì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽŽŸ$Ç¿’©‡:ºorŽ’¿·žÊx‘ª¨¹+Ÿ÷áō‘cäÊbŽ‘Ý۟[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽ‘q[¹=‘UR͍Ÿì³ÐrŽ‘Ÿì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽŽŸÈ׍‘ègiºThe›êlt•¬ww“o˜pSˆossible˜v‘ÿXíalues˜for˜Êx˜ºare˜therefore:ŽŸ€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eqsubst*[eq:temp1]{"eq:poss1"}{x–,j=“x_1}Ž¦‘ègi\eqsubst*[eq:temp2]{"eq:poss2"}{x–,j=“x_2}Ž¡‘â‡mïcolor push gray 0ï	color popŽŽ‘ègi$$\eqsub[eq:sol1]{"eq:temp1"}{\frac{b}{2a}}$$Ž¦‘ègi$$\eqsub[eq:sol2]{"eq:temp2"}{\frac{b}{2a}}$$ŽŸ)í(’¶Ñ.ÊxŸ
ÌÌÈ1Ž‘V¹=‘URŸì³ÐrŽ‘UTŸì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽ‘>Ó×͍Ÿ÷áō‘cäÊbŽ‘Ý۟[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽŽŸ$Ç¿’²&ƒxŸ
ÌÌÈ2Ž‘V¹=‘UR͍Ÿ÷áō‘¹<ÊbŽ‘
33Ÿ[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽ‘	͍‘ª¨Ÿì³ÐrŽ‘ªªŸì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽŽŸ…‘ègiºA–êlfaster“w•¬wa“y–êlto“ac•¬whiev“e–êlthe“same“result“is“b¬wy“using“ÍnÇeqsolv eº.ŽŸ€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqsubst{\eqsolve{f(x)–,j=“0}{x}{1}}{x“=“x_1}$$Ž¦‘ègi$$\eqsubst{\eqsolve{f(x)–,j=“0}{x}{2}}{x“=“x_2}$$ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹	Q• ¹ðü ý;9‘ègiïcolor push gray 0Ø5‘¿DSYMBOLIC‘êlCOMPUT–ÿeA“TIONS’
SD
º9Ž’þñï	color popŽŽ ¦Æs ý†|5’¶Ñ.ÊxŸ
ÌÌÈ1Ž‘V¹=‘URŸì³ÐrŽ‘UTŸì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽ‘>Ó×͍Ÿ÷áō‘cäÊbŽ‘Ý۟[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽŽŸ"’²&ƒxŸ
ÌÌÈ2Ž‘V¹=‘UR͍Ÿ÷áō‘¹<ÊbŽ‘
33Ÿ[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽ‘	͍‘ª¨Ÿì³ÐrŽ‘ªªŸì³‰zà,Ó۟L荍Ÿ÷áō‘
33¹1Ž‘
33Ÿ[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘
¡ÊbŸüˆ‰È2ŽŽ‘	y•Ÿ[«‰zà
ëŸ
ý΍ÊaŸüˆ‰È2ŽŽŽŽŽ‘Bu͍Ÿ÷áō‘lÊcŽ‘Ý۟[«‰zà+
Ÿ
ý΍aŽŽŽŽŽŽŽŽŸ`÷‘ègiºWith,–êlfor“example,“Êa–UR¹=“3º,›êlÊb“¹=“4˜ºand˜Êc“¹=“͹5˜ºthe˜pSˆossible˜v‘ÿXíalues˜for˜Êx˜ºare:ŽŸv«‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$x_1–,j=“\numval[3]{x_1}$$Ž©€‘ègi$$x_2–,j=“\numval[3]{x_2}$$ŽŸv¬’ÕgcÊxŸ
ÌÌÈ1Ž‘V¹=‘UR0Ê:¹786Ž¤CF’Ó¬µÊxŸ
ÌÌÈ2Ž‘V¹=‘UR͹2Ê:¹12Ž¡‘ègiºThe–	ÒpÖ3›	Òoºoption“to“ÍnÇn• um“v‘ÿ@al–	Òpºsets˜the“precision“to“three˜digits,‘	Óòit“is“equiv‘ÿXíalen¬wt˜to“writingŽ¦‘ègiÖ\numval[precision=3]{x_1}º.ŽŸÃF‘ègiWhen–êlapplying“Êf‘
Gÿ¹(Êx¹)“ºon“these“v‘ÿXíalues“of“Êxº,“the“result“is“(almost)“zero:Ž¤Iҍ‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$f(x_1)–,j=“\numval{f(x_1)}$$Ž¦‘ègi$$f(x_2)–,j=“\numval{f(x_2)}$$Ž¡’·H»Êf‘
Gÿ¹(ÊxŸ
ÌÌÈ1Ž‘À¹)–UR=“͹1Ê:¹776‘
ÿþÍ
Ž‘UR¹10Ÿû™ÎÈ15ŽŽŸCF’×¾Êf‘
Gÿ¹(ÊxŸ
ÌÌÈ2Ž‘À¹)–UR=“0ŽŸ$™‘ègiÙ5.2Ž‘y¯Example:–G\Finding“the“extrema“of“a“functionŽŸ‚ʍ‘ègiºT‘ÿeo–êlnd“the“extrema“of“Êf‘
Gÿ¹(Êx¹)º,“wš¬we“set“the“rst“deriv‘ÿXíativ˜e“to“zero“and“solv˜e“for“Êxº:Ž¡‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\deleq{"eq:a";"eq:b";"eq:c"}Ž¦‘ègi$$\eq[eq:f]{f(x)–,j=“\val{f(x)}}$$Ž¦‘ègi$$\eqdiff{"eq:f"}{x}–,j=“0$$Ž¦‘ègi\eqsub*{\rhs{"prev"}–,j=“0}{b}%Ž¦‘ègi\eqsubst*{"prev"}{x–,j=“x_{extr}}%Ž¦‘ègi$$\eqdiv[eq:xextr]{"prev"}{2a}$$Ž¡’»k¤Êf‘
Gÿ¹(Ž‘ÙÃÊx¹)ŽŽŽ‘n+=‘URÊc–ª¨¹+“Êa›
ÿþxŸû™È2Ž‘j¬¹+“Êb˜xŽŸCF’»·Ôf‘
GÿŸû™Î0Ž‘8¹(Ž‘§üÊx¹)ŽŽŽ‘<d=›URÊb–ª¨¹+“2–
ÿþÊa“x˜¹=˜0ŽŸ6Ս’Ð…ÊxŸ
ÌÌËextrŽ‘³Õ¹=‘UR͍Ÿ÷áō‘¹<ÊbŽ‘
33Ÿ[«‰zà
ûŸ
ý΍¹2‘
ÿþÊaŽŽŽŽŽŸAʍ‘ègiºDepSˆending–ùœon›ùthe“sign˜of“the˜second“deriv‘ÿXíate˜Êf‘
GÿŸû¥2Î00Ž‘dq¹(Ž‘
ö5Êx¹)ŽŽŽ‘¤y=‘o.2‘
ÿþÊaº,‘ù¡this“will˜bSˆe“a˜minim¬wum“or˜a“maxim¬wum.‘ù´TheŽ¦‘ègiv‘ÿXíalue–êlof“the“extrem¬wum“isŽ¡‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqsubstc[eq:yextr]{y_{extr}–,j=“\rhs{"eq:f"}}{x“=“x_{extr};“"eq:xextr"}$$ŽŸ.’É`ÁÊyŸ
ÌÌËextrŽ‘³Õ¹=‘URÊc‘ª¨ÍŸ÷áō‘ÝÛ¹1Ž‘Ý۟[«‰zàßüŸ
ý΍4ŽŽŽŽŸ÷áō‘$=ÊbŸû¥2È2ŽŽ‘$=Ÿ[«‰zà	¾îŸ
ý΍‘
É÷ÊaŽŽŽŽŽŸS؍‘ègiºor–êlwith“the“v‘ÿXíalues“used“in“the“previous“section“Êa–UR¹=“3º,›êlÊb“¹=“4˜ºand˜Êc“¹=“͹5º:Ž¡‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$x_{extr}–,j=“\numval{x_{extr}}\quad“y_{extr}“=“\numval{y_{extr}}$$Ž¡’ÄÊxŸ
ÌÌËextrŽ–³Õ¹=›UR͹0Ê:¹6667‘¿DÊyŸ
ÌÌËextrŽ“¹=˜Í¹6Ê:¹333ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹
_Þ ¹ðü ý;9‘ègiïcolor push gray 0Ø6‘¿DMA‘ÿeTRICES’
³ªÄº10Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÙ5.3Ž‘y¯Library–G\of“substititionsŽŸ0‘ègiºT‘ÿeo›O+mak•¬we‘O*w“orking˜with˜equations˜easier,‘ODthere˜is˜a–O*library˜of˜mainly˜trigonometric“substitutions˜whic¬whŽ¤€‘ègican–êlbSˆe“included“with“Ö\input‘,jsubstitutions.texº.“F‘ÿeor“example,“consider“this“equation:Ž©Q²‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\clearequationsŽ¡‘ègi\input‘,j../examples/substitutions.texŽ¡‘ègi$$\eq[eq:r1]{\frac{r_1}{\sin(\pi–,j-“\frac{\theta}{2})}“=“\frac{r_2}{\sin(\frac{\pi}{2}“-“\frac{\theta}{2})}}$$Ž¡‘ègi$$\eqsubst{"prev"}{"lib:trig:sina+pi/2";‘,j"lib:trig:sina+pi"}$$Ž¡‘ègi$$\eqsubst{"prev"}{"lib:trig:sin-a";‘,j"lib:trig:cos-a"}$$Ž¦¡Ÿ÷áō’È
>ÊrŸ
ÌÌÈ1ŽŽ’³çŸ[«‰zà3ú<Ÿ¸d¹sinŽ‘m(Ê‘á͍ŸûFu‘äËŽ‘Ý۟
öû‰zà@ŸåêÈ2ŽŽŽŽ‘	Q¹)ŽŽŽŽ’똨=Ÿ÷áō‘¼QÊrŸ
ÌÌÈ2ŽŽ‘ˆ…Ÿ[«‰zà4s'Ÿ¸d¹sinŽ‘m(ŸûFu‘
33ËŽ‘
33Ÿ
öû‰zà+
ŸåꍐuƒÈ2ŽŽŽŽ‘
<͍ŸûFu‘äËŽ‘Ý۟
öû‰zà@ŸåêÈ2ŽŽŽŽ‘	Q¹)ŽŽŽŽŽŸ!Ò¥’µ^½ÍŸ÷áō‘Œ„ÊrŸ
ÌÌÈ1ŽŽ‘Ý۟[«‰zà+h៸d¹sinŽ‘mŸöG©ÐŽ‘í͍ŸûFu‘
9mËŽ‘
33Ÿ
öû‰zà@ŸåêÈ2ŽŽŽŽ‘¦jŸöG©Ð
ŽŽŽŽŽŽŽ‘3ÏA¹=Ÿ÷áō‘ÞJÊrŸ
ÌÌÈ2ŽŽ‘ˆ…Ÿ[«‰zà,·Ÿ¸d¹cosŽ‘»TŸöG©ÐŽ‘;V͍ŸûFu‘
9mËŽ‘
33Ÿ
öû‰zà@ŸåêÈ2ŽŽŽŽ‘¦jŸöG©Ð
ŽŽŽŽŽŽŽŽŸ!⃍Ÿ÷áō’ЕñÊrŸ
ÌÌÈ1ŽŽ’Ä‘óŸ[«‰zà"ŠŸ¸d¹sinŽ‘mŸöG©ÐŽŸûFu‘&‹ËŽ‘ QŸ
öû‰zà@ŸåêÈ2ŽŽŽŽ‘“ˆŸöG©Ð
ŽŽŽŽŽŽŽ’ë.¹=Ÿ÷áō‘3ŸÊrŸ
ÌÌÈ2ŽŽ‘ˆ…Ÿ[«‰zà#aŸ¸d¹cosŽ‘»TŸöG©ÐŽŸûFu‘tÃËŽ‘n‰Ÿ
öû‰zà@ŸåêÈ2ŽŽŽŽ‘áÀŸöG©Ð
ŽŽŽŽŽŽŽŽŸ#Ñò‘ègiºThis–êlequation“can“noš¬ww“bSˆe“solv˜ed“to“giv˜e“an“expression“forŸûFu‘#ÙËŽ‘ŸŸ
öû‰zà@ŸåêÈ2ŽŽŽŽŽ‘
Öº:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqsubst{"prev"–,j*“\sin{\theta/2}“/“r_2}{\eqrev{"lib:trig:tansincos"}}$$Ž¡‘ègi$$\eqrev{\eqsimpf{\eqfunc{"prev"}{\arctan}}{unsafe}–,j*“2}$$ŽŸ#⡍Ÿ÷áō’ÍÊrŸ
ÌÌÈ1ŽŽ’ÍŸ[«‰zà
Ÿ
ý΍ÊrŸ
ÌÌÈ2ŽŽŽŽŽ’Û™$¹=‘URtanŽ‘N(ŸïqÐŽŸ÷áō‘ `	ÊŽ‘ V³Ÿ[«‰zàßüŸ
ý΍¹2ŽŽŽŽ‘'iâŸïqÐŽŽŽŽŸ$=p’ÀIÊ‘¨à¹=‘UR2‘ÿüarctanŽ‘&£rŸïqÐŽŸ÷áō‘0«ýÊrŸ
ÌÌÈ1ŽŽ‘0«ýŸ[«‰zà
Ÿ
ý΍ÊrŸ
ÌÌÈ2ŽŽŽŽŽ‘;꿟ïqÐŽŽŽŽŸ×
‘ègiºThe›^|k•¬weyw“ord‘^{ÍnÇeqrev˜ºsw“aps‘^{left˜and˜righ“t–^{hand˜side“of˜the“equation.‘_RÍnÇeqfunc˜ºapplies“a˜function“to˜bSˆothŽ¡‘ègisides.ŽŸ,ÿt‘ègiÆ6Ž‘^…MatricesŽŸQö‘ègiºV‘ÿeectors–êland“matrices“can“bSˆe“created“with“the“follo¬wwing“input:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{align*}Ž¡‘ôÀ=\matrix{v_1}%Ž¡‘ôÀ=\matrix{v_2}%Ž¡‘ôÀ=\matrix{M_1}%Ž¡‘ôÀ=\eq[eqraw=false]{v_1–,j=“{x;“y;“z}}\\Ž¡‘ôÀ=\eq[eqraw=false]{v_2–,j=“\transpose{x;“y;“z}}\\Ž¡‘ôÀ=\eq[eqraw=false]{M_1–,j=“{{x_1;“y_1;“z_1};{x_2;“y_2;“z_2};{x_3;“y_3;“z_3}}}Ž¡‘ègi\end{align*}ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹k£ ¹ðü ý;9‘ègiïcolor push gray 0Ø7‘¿DGRAPHS’
À¢|º11Ž’þñï	color popŽŽ ¦Æs ý~9’¼‘ëÊvŸ
ÌÌÈ1ŽŽŽŽ’ÊUu¹=‘URŸöG©ÐŽžæc‘
ÕTÊxŽ‘‚¦yŽ‘.«…zŽŽŽ‘9©ºŸöG©Ð
ŽŽŽŽŽŽŸ!›’¼‘ëÊvŸ
ÌÌÈ2ŽŽŽŽ’ÊUu¹=Ÿçá9‘URÐ0ŽŸ™§‘UR@ŽŽŸñfd‘ÕTÊxŽŽ¤ÿ‘ŽyŽŽ¡‘,ãzŽŽŽŸçá9‘‚¦Ð1ŽŸ™§‘‚¦AŽŽŽŽŽŽŽŸ/ý’¶âÊMŸ
ÌÌÈ1ŽŽŽŽ’ÊUu¹=Ÿçá9‘URÐ0ŽŸ™§‘UR@ŽŽŸñfd‘ÕTÊxŸ
ÌÌÈ1ŽŽ‘(BªÊyŸ
ÌÌÈ1ŽŽ‘<½TÊzŸ
ÌÌÈ1ŽŽŽ¤ÿ‘ÕTÊxŸ
ÌÌÈ2ŽŽ‘(BªÊyŸ
ÌÌÈ2ŽŽ‘<½TÊzŸ
ÌÌÈ2ŽŽŽ¡‘ÕTÊxŸ
ÌÌÈ3ŽŽ‘(BªÊyŸ
ÌÌÈ3ŽŽ‘<½TÊzŸ
ÌÌÈ3ŽŽŽŽŸçá9‘KòªÐ1ŽŸ™§‘KòªAŽŽŽŽŽŽŽŸ#©\‘ègiºIt–_tis‘_svš¬wery“impSˆortan˜t›_sto“indicate˜to“EQC‘^Òthat˜a“v‘ÿXíariable“con¬wtains˜a“matrix˜(bSˆecause“of˜the“non-Ž¤€‘ègicomm•¬wutativit“y–_vof“proSˆducts).›a²This“is“done“with“the“ÍnÇmatrix“ºk•¬weyw“ord.˜Note–_vthat“curren¬wtly“it“is“notŽ¡‘ègipSˆossible–ϕto›ϖcreate“a“standing‘Ê0v¬wector˜with“Ö\eq{v_2–,j=“{{x};{y};{z}}º.‘ÐõInstead,‘ÏÏuse˜the‘ϕÍnÇtransp_úoseŽ¡‘ègiºfunction–êlas“sho¬wwn“in“the“example.ŽŸQø‘ègiThe–êlv‘ÿXíalue“of“matrix“elemen¬wts“can“bSˆe“found“with“the“function“Ö\mindexº:Ž©õ鍍‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{align*}Ž¡‘ôÀ=\eq{m–,j=“\mindex{M_1;“b;“2}}\\Ž¡‘ôÀ=\eq{b–,j=“3}\\Ž¡‘ôÀ=m_{32}–,j&=“\val{m}Ž¡‘ègi\end{align*}ŽŸß׍’ËxtÊmŸ
ÌÌÈ32ŽŽŽŽ’âã¹=‘URÊMŸ
ÌÌÈ1Ž‘À¹[Êb;‘
ÿþ¹2]ŽŽŽŽ¤€’ÙÀ§ÊbŽŽŽ’âã¹=‘UR3ŽŽŽŽ¡’ËxtÊmŸ
ÌÌÈ32ŽŽŽŽ’âã¹=‘URÊyŸ
ÌÌÈ3ŽŽŽŽŽŸiègiºY‘ÿeou–êlcan“use“the“Ö\wild“ºfunction“as“the“index“to“access“a“whole“ro¬ww“or“column.ŽŸQù‘ègiEQC–êlsuppšSˆorts“the“follo¬wwing“op˜erations“with“matrices“and“v¬wectors:Ž¦‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºA¬wddition–êland“subtractionŽ¤û‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºMultiplicationŽŸú‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºExpSˆonen¬wtiationŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºT‘ÿeranspSˆosition–êlwith“the“ÍnÇtransp_úoseŽ¦‘ègiºEQC–êlwill“terminate“with“an“error“if“the“matrices“or“v¬wectors“are“not“compatible“with“one“another.ŽŸ*&‘ègiÆ7Ž‘^…GraphsŽŸ3^‘ègiºEQC‘û!has–û%some›û&suppSˆort“for˜creating“graphs“with˜the“pac•¬wk‘ÿXíage˜pstric“ks.‘û?As‘û%an˜example,‘û)w“e˜will‘û%compareŽŸ€‘ègithe–êlgraph“of“a“function“with“its“T‘ÿea¬wylor“series“expansion.ŽŸQø‘ègiThe–êldenition“of“the“T‘ÿea¬wylor“series“expansion“isŽŸä’–n(ÊTŸ
ÌÌËnŽ‘¨PÊf›
Gÿ¹(Êx;‘
ÿþxŸ
ÌÌÈ0Ž‘À¹)–UR=Ÿðÿü‘	kÖËnŽŸ“Ÿô™–ÐXŽŽŸ
㇍‘SËiÈ=0ŽŽ‘ª¨Êf˜Ÿû™È(ËiÈ)Ž‘IQ¹(ÊxŸ
ÌÌÈ0Ž›À¹)‘
ÿþÍ
ŽŸ÷áō‘ˆ…¹(Êx–ª¨Í“ÊxŸ
ÌÌÈ0Ž˜¹)Ÿû¥2ËiŽŽ‘ˆ…Ÿ[«‰zà-M±Ÿ
ý΍‘ÿÊi¹!ŽŽŽŽŽŸ ¨‘ègiºApplying–êlthis“to“¹arctanŽ‘$ä(Êx¹)“ºaround“the“pSˆoinš¬wt“ÊxŸ
ÌÌÈ0Ž‘V¹=‘UR0Ê:¹7º,“w˜e“ha˜v˜eŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹vÓ ¹ðü ý;9‘ègiïcolor push gray 0Ø7‘¿DGRAPHS’
À¢|º12Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\function{f}{x}%Ž¤€‘ègi\begin{align}Ž¡‘ôÀ=\eq[eq:f0]{f(x)–,j=“\arctan“x}\tag*{}\\Ž¡‘ôÀ=\eqdiff[eq:f1]{"prev"}{x}\tag*{}\\Ž¡‘ôÀ=\eqdiff[eq:f2]{"prev"}{x}\tag*{}Ž¡‘ègi\end{align}Ž©&Ô͍’¿ùŒÊf‘
Gÿ¹(Êx¹)ŽŽŽ’Ú2_=‘URarctanŽ‘%øÈÊxŽŽŽŽŸ3’’½+Sf‘
GÿŸû™Î0Ž‘8¹(Ž‘§üÊx¹)ŽŽŽŽŽŽ’Ú2_=Ÿ÷áō‘{œ1Ž‘ˆ…Ÿ[«‰zàÆ*Ÿ
ý΍1–ª¨+“ÊxŸüˆ‰È2ŽŽŽŽŽŽŽŽŽŸn̍’ºÝÊf‘
GÿŸû™Î00Ž‘dq¹(Ž‘
ö5Êx¹)ŽŽŽŽŽŽ’Ú2_=‘UR͹2Ÿ÷áō‘±eÊxŽ‘
33Ÿ[«‰zà-©¶Ÿmȍ¹(Ž‘‘Ä1–ª¨+“ÊxŸüˆ‰È2Ž‘À¹)ŽŽŽ‘(鲟ú*«È2ŽŽŽŽŽŽŽŽŽŸ č‘ègiºThe–êlv‘ÿXíalues“of“the“function“at“ÊxŸ
ÌÌÈ0Ž‘ªpºareŽŸÿ®‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{align*}Ž¡‘ôÀ=\eqsubstc{"eq:f0"}{x–,j=“x_0;“"eq:x_0"}\\Ž¡‘ôÀ=\eqsubstc{"eq:f1"}{x–,j=“x_0;“"eq:x_0"}\\Ž¡‘ôÀ=\eqsubstc{"eq:f2"}{x–,j=“x_0;“"eq:x_0"}Ž¡‘ègi\end{align*}Ž¦’ÆtÃÊf‘
Gÿ¹(Ž‘ÙÃ0Ê:¹7)ŽŽŽŽŽŽ’éÈ=‘UR0Ê:¹6107ŽŽŽŽ¤€’æŠÊf‘
GÿŸû™Î0Ž‘8¹(Ž‘§ü0Ê:¹7)ŽŽŽŽŽŽ’éÈ=‘UR0Ê:¹6711ŽŽŽŽ¡’ÁXQÊf‘
GÿŸû™Î00Ž‘dq¹(Ž‘
ö50Ê:¹7)ŽŽŽŽŽŽ’éÈ=‘UR͹0Ê:¹6306ŽŽŽŽ©U‘ègiºThe–êlrst“parts“of“the“expansion“areŽŸÿ®‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{align}Ž¤€‘ôÀ=\eq{\arctan(x)‘,j=Ž¡‘
\rhs{\eqsubst{"eq:f0"}{x–,j=“x_0}}“+Ž¡‘
\rhs{\eqsubst{"eq:f1"}{x–,j=“x_0}}“\frac{x“-“x_0}{1!}“+Ž¡‘
\rhs{\eqsubst{"eq:f2"}{x–,j=“x_0}}“\frac{(x“-“x_0)^2}{2!}Ž¡‘
}\notag\\Ž¡‘ôÀ=\eqsubst[eq:series]{"prev"}{"eq:x_0"}\tag*{}\\Ž¡‘ôÀ=\eqsimpf{"prev"}{expand}Ž¡‘ègi\end{align}ŽŸ/§‘vÚG¹arctanŽ’™}½ÊxŽŽŽ’£€a¹=‘URarctanŽ‘%øÈÊxŸ
ÌÌÈ0ŽŽ‘4Æ͍Ÿ÷áō‘ÝÛÊxŸ
ÌÌÈ0Ž‘À¹(Ž‘QÆÊxŸ
ÌÌÈ0Ž‘j¬Í‘ª¨Êx¹)ŽŽŽ‘0¨ÙŸú*«È2ŽŽ‘Ý۟[«‰zà</Ÿ·¼‘6=¹(Ž‘È
1–ª¨+“ÊxŸûÝߍÈ2ŽŸS0ŽŽ‘À¹)ŽŽŽ‘0ïŸùà·È2ŽŽŽŽŽ‘CÑå͍Ÿ÷áō‘ÝÛÊxŸ
ÌÌÈ0Ž‘j¬Í‘ª¨ÊxŽ‘Ý۟[«‰zà ÅOŸ
ý΍“¹1–ª¨+“ÊxŸûÝߍÈ2ŽŸS0ŽŽŽŽŽŽŽŽŽŽŸòT’£€a¹=‘UR0Ê:¹1409–ª¨+“0Ê:¹6711›
ÿþÊx“Í“¹0Ê:¹3153˜(Ž‘‘ÂÊx“Í“¹0Ê:¹7)ŽŽŽ‘/Ÿú*«È2ŽŽŽŽŽŸ€’£€a¹=‘UR1Ê:¹113›
ÿþÊx–ª¨Í“¹0Ê:¹3153˜ÊxŸû™È2Ž‘j¬Í“¹0Ê:¹01357ŽŽŽ’
éc¼º(2)ŽŽŽŽŽ¦‘ègiThe–êlsame“result“can“bSˆe“obtained“more“easily“with“the“ÍnÇtseries“ºk•¬weyw“ord:ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹
ß ¹ðü ý;9‘ègiïcolor push gray 0Ø7‘¿DGRAPHS’
À¢|º13Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ôÀ=Ö\arctan–,jx“=“\val{\tseries{\arctan“x}{x“=“0.7}{3}}Ž©à’‰*;¹arctanŽ’«Í±Êx–UR¹=“1Ê:¹113›
ÿþÊx–ª¨Í“¹0Ê:¹3153˜ÊxŸû™È2Ž‘j¬Í“¹0Ê:¹01357ŽŸ•c‘ègiºThe›êlt•¬ww“o˜functions˜are˜compared˜in˜gure˜1.ŽŸà‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{figure}[!htb]Ž¤€‘ôÀ=\begin{center}Ž¡‘
\psset{xunit=40mm,‘,jyunit=25mm}Ž¡‘
\mmakepspicture{-1}{-1.6}{2}{1.2}{0.5}{0.2}{$x$}{$\arctan(x)$}{%IMPORTANT‘,jCOMMENT!Ž¡‘
qå\printvector{\val{\eqeval{y–,j=“\arctan“x}{x“=“-1:2:0.1}}}%Ž¡‘
\psset{linecolor=red}%Ž¡‘
\pscurve{-}\printvector{\val{\eqeval{"eq:series"}{x–,j=“-1:2:0.1}}}}Ž¡‘ôÀ=\end{center}Ž¡‘ôÀ=\caption{Comparison–,jof“$\arctan“x$“with“its“Taylor“series“expansion\label{fig:func}}Ž¡‘ègi\end{figure}Ž 
Œ‘ègi ÿèDïcolor push gray 0 ú¼ŸØ*Ѝ‘M?…ïcolor push gray 0ï	color popŽŽŸŽ0F’Äï ò
C"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { BeginArrow 1.  1.  scale  false 0.4 1.4 1.5 2.   1. .setopacityalpha  Arrow  EndArrow  } def  239.00383  0  -113.81102  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 2.0 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
J"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { BeginArrow 1.  1.  scale  false 0.4 1.4 1.5 2.   1. .setopacityalpha  Arrow  EndArrow  } def  89.62578  0 exch -113.81143  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 2.0 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïñ"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  0 rotate /n 3 def /dx 56.9055  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ïò"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  0 rotate /n -2 def /dx 56.9055  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ïó"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  90 rotate /n 5 def /dx 14.22615  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ïô"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  90 rotate /n -8 def /dx 14.22615  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ŽŽŽ‘Sò
;"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { BeginArrow 1.  1.  scale 2 setlinecap 0 0 moveto 0 0.1 L stroke 0 0 moveto  EndArrow  moveto } def /ArrowB { } def  341.66055  0  0.0  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
B"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { BeginArrow 1.  1.  scale 2 setlinecap 0 0 moveto 0 0.1 L stroke 0 0 moveto  EndArrow  moveto } def /ArrowB { } def  199.2552  0 exch 0.0  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò

"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  0 rotate /n 6 def /dx 56.9055  def n 0 lt { /dx dx neg def /n n neg def } if /y2 199.16946 CLW 2 div add def /y1 y2 neg def /y1 0 def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ŽŽŽ’Äï ïì"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  227.84953  0  -113.81102  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
B"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { BeginArrow 1.  1.  scale 2 setlinecap 0 0 moveto 0 0.1 L stroke 0 0 moveto  EndArrow  moveto } def /ArrowB { } def  199.2552  0 exch 0.0  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end Ÿ
"!‘ý¹0Ž‘1f
0Ê:¹5Ž‘jMÜ1Ê:¹0Ž’£5«1Ê:¹5Ž’Üz2Ê:¹0ŽŽŽŸ
"!‘ºëÃ͹0Ê:¹5Ž‘‚ô͹1Ê:¹0ŽŽŽŽŽŽŸŽ0FŸ™’
¸}Vó·ág£ffcmmi12¾xŽŽŽŽŽ‘Sò
;"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { BeginArrow 1.  1.  scale 2 setlinecap 0 0 moveto 0 0.1 L stroke 0 0 moveto  EndArrow  moveto } def /ArrowB { } def  341.66055  0  0.0  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
B"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { BeginArrow 1.  1.  scale 2 setlinecap 0 0 moveto 0 0.1 L stroke 0 0 moveto  EndArrow  moveto } def /ArrowB { } def  199.2552  0 exch 0.0  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  90 rotate /n 14 def /dx 14.22615  def n 0 lt { /dx dx neg def /n n neg def } if /y2 341.43306 CLW 2 div add def /y1 y2 neg def /y2 0 def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ŽŽŽŸŽ0F‘Sò
;"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { BeginArrow 1.  1.  scale 2 setlinecap 0 0 moveto 0 0.1 L stroke 0 0 moveto  EndArrow  moveto } def /ArrowB { } def  341.66055  0  0.0  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïó"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  85.44377  0 exch -113.81143  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ŸÝݍ‘ô¹ž¹0ŽŽ¤ñƍŸÝݍ‘ë–0Ê:¹2ŽŽ¡ŸÝݍ‘ë–0Ê:¹4ŽŽ¡ŸÝݍ‘ë–0Ê:¹6ŽŽ¡ŸÝݍ‘ë–0Ê:¹8ŽŽ¡ŸÝݍ‘ë–1Ê:¹0ŽŽ¡ŸÝݍ‘ë–1Ê:¹2ŽŽŽ¤9卟]ݍ‘â@¿Í¹0Ê:¹2ŽŽ¡Ÿ]ݍ‘â@¿Í¹0Ê:¹4ŽŽ¡Ÿ]ݍ‘â@¿Í¹0Ê:¹6ŽŽ¡Ÿ]ݍ‘â@¿Í¹0Ê:¹8ŽŽ¡Ÿ]ݍ‘â@¿Í¹1Ê:¹0ŽŽ¡Ÿ]ݍ‘â@¿Í¹1Ê:¹2ŽŽ¡Ÿ]ݍ‘â@¿Í¹1Ê:¹4ŽŽ¡Ÿ]ݍ‘â@¿Í¹1Ê:¹6ŽŽŽŽŽŽ ÿ2Û
Ÿüfg’§ÜóX«Qffcmr12½arctanŽ’Ï(¾x½)ŽŽŽŽŽ’Äï ŸŽ0Fò"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  [ 227.62204 78.75346 216.24023 77.27191 204.86017 75.66336 193.47838 73.91154 182.09831 71.9991 170.71652 69.90756 159.33473 67.61414 147.95467 65.09279 136.57288 62.31528 125.19281 59.25124 113.81102 55.867 102.42921 52.12675 91.04915 47.99577 79.66736 43.44258 68.2873 38.44112 56.9055 32.98055 45.52371 27.06627 34.14365 20.73196 22.76186 14.04164 11.38179 7.08974 0.0 0.0 -11.38179 -7.08974 -22.76186 -14.04164 -34.14365 -20.73196 -45.52371 -27.06627 -56.9055 -32.98055 -68.2873 -38.44112 -79.66736 -43.44258 -91.04915 -47.99577 -102.42921 -52.12675 -113.81102 -55.867   1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò."  tx@Dict begin STP newpath 0.8 SLW 1 0 0  setrgbcolor  /ArrowA { moveto } def /ArrowB { } def  [ 227.62204 67.60002 216.24023 68.43361 204.86017 68.81783 193.47838 68.7538 182.09831 68.24149 170.71652 67.27983 159.33473 65.86992 147.95467 64.01173 136.57288 61.70529 125.19281 58.9495 113.81102 55.74544 102.42921 52.09311 91.04915 47.99144 79.66736 43.44258 68.2873 38.44438 56.9055 32.99683 45.52371 27.1021 34.14365 20.75801 22.76186 13.96565 11.38179 6.72397 0.0 -0.9649 -11.38179 -9.10313 -22.76186 -17.68962 -34.14365 -26.72546 -45.52371 -36.20956 -56.9055 -46.14194 -68.2873 -56.52257 -79.66736 -67.35255 -91.04915 -78.62973 -102.42921 -90.35623 -113.81102 -102.5321   1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 0.8 SLW 1 0 0  setrgbcolor  1. .setopacityalpha  0  setlinecap stroke  grestore end ŽŽŽŸ%€‘_î]ïcolor push gray 0ºFigure–êl1:“Comparison“of“¹arctanŽ‘&âÊx“ºwith“its“T‘ÿea¬wylor“series“expansionï	color popŽŽŽï	color popŽŸŽm‘ègiThe–êlargumen¬wts“to“ÍnÇmmak epspicture“ºare:Ž¦‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºNegativš¬we–êlÊxº-axis“v‘ÿXíalue,“then“negativ˜e“Êyn9º-axis“v‘ÿXíalueŽ¤\v‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºP•¬wositiv“e–êlÊxº-axis“v›ÿXíalue,“then“pSˆositiv¬we“Êyn9º-axis“v˜alueŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊxº-axis–êlstep“v›ÿXíalue,“then“Êyn9º-axis“step“v˜alue“(for“the“axis“labSˆels)ŽŸ\w‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊx–êlºand“Êy‘X¥ºaxis“namesŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºAn–êlarra¬wy“of“¹(Êx¹;‘
ÿþÊyn9¹)“ºv‘ÿXíalues“to“plotŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹Šù ¹ðü ý;9‘ègiïcolor push gray 0Ø8‘¿DUTILITY‘êlFUNCTIONS’
u\åº14Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiThe–9›last“is“generated‘9œbš¬wy“the“k˜eyw˜ord“ÍnÇprin• tv“ectorº,‘9Éwhic˜h–9›con˜v˜erts“the“output“of‘9œÍnÇeqev‘ÿ@al“ºin˜to“a“formatŽ¤€‘ègirecognized–!»bš¬wy“pstric˜ks.‘#˜ÍnÇeqev‘ÿ@al“ºev›ÿXíaluates“an“equation‘!ºfor“all“the“v˜alues“of‘!ºthe“indepSˆenden¬wt“v˜ariableŽ¡‘ègigiv¬wen–ªZas›ªYthe“second˜parameter.‘ª¼In˜the“example,‘ªj͹1–UR:“2“:“0Ê:¹1˜ºcreates–ªZa˜v¬wector“of˜n•¬wum“bSˆers–ªZfrom˜Í¹1“ºto˜¹2“ºinŽ¡‘ègiincremen¬wts›¥"of–¥!¹0Ê:¹1º.‘¥ŒThe“result˜is˜a“matrix˜with“the˜v‘ÿXíalues˜of“the˜indepSˆenden¬wt˜v‘ÿXíariable“in˜the“rst˜columnŽ¡‘ègiand–êlthe“ev‘ÿXíaluation“results“in“the“second“column:Ž¤<‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eq{r–,j=“\eqeval{"eq:series"}{x“=“-1:2:0.5}}ŽŸE&•’­jÊr‘¨à¹=ŸËY‘URÐ0ŽŸǍ‘URBŽ¤38‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽŸ®‘UR@ŽŽŸÔff‘g͹1Ž‘;+͹1Ê:¹441ŽŽ¤ÿ‘ÕT͹0Ê:¹5Ž‘8-͹0Ê:¹6487ŽŽ¡‘Ä0Ž‘5./͹0Ê:¹01357ŽŽ¡‘€0Ê:¹5Ž‘<ÈÙ0Ê:¹4639ŽŽ¡‘Ä1Ž‘<ÈÙ0Ê:¹7837ŽŽ¡‘€1Ê:¹5Ž‘<ÈÙ0Ê:¹9458ŽŽ¡‘Ä2Ž‘<ÈÙ0Ê:¹9504ŽŽŽŸËY‘júÐ1ŽŸǍ‘júCŽ¤38‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽŸ®‘júAŽŽŽŽŸC·‘ègiºThere–êlis“also“a“simplied“vš¬wersion“ÍnÇmak epspicture“ºfor“dra˜wing“a“graph“in“the“rst“quadran˜t“only:Ž¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºLoSˆcation–êlof“origin“in“the“format“(x0,“y0)Ž¤Æ>‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºLoSˆcation–êlof“lo•¬ww“er–êlleft-hand“corner“in“the“format“(x1,“y1)Ž©Æ?‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºP•¬wositiv“e–êlÊxº-axis“v›ÿXíalue,“then“pSˆositiv¬we“Êyn9º-axis“v˜alueŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊxº-axis–êlstep“v›ÿXíalue,“then“Êyn9º-axis“step“v˜alue“(for“the“axis“labSˆels)Ž¦‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊx–êlºand“Êy‘X¥ºaxis“namesŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºAn–êlarra¬wy“of“¹(Êx¹;‘
ÿþÊyn9¹)“ºv‘ÿXíalues“to“plotŽŸ<‘ègiOf–0dcourse,‘0¶if“ÍnÇmakš epspicture‘0cºand“ÍnÇmmak˜epspicture›0cºdo“not“suit˜yš¬wour“purpSˆoses,‘0¶y˜ou“can‘0cuse“theŽ¤€‘ègistandard–êlpstricš¬wks“macros“instead“or“create“y˜our“o˜wn“shortcuts.ŽŸ+ž‘ègiÆ8Ž‘^…Utilit‘ÿ{ky‘6êfunctionsŽŸJ͑ègiºIf›KVy•¬wou‘KWw“an“t˜to˜mak“e–KWa˜fresh˜start˜in“the˜middle˜of“a˜doSˆcumen¬wt,‘Kuse˜ÍnÇclearequationsº.‘LJThis˜will“delete˜allŽ¡‘ègipreviously–‰Jdened“equations“except“library“equations,‘‰cbut‘‰Kkš¬weep“the“constan˜ts“and“an˜y“functions“denedŽ¡‘ègiwith–êlthe“hin¬wt“Ölibº.ŽŸ+Ÿ‘ègiÆ9Ž‘^…PitfallsŽŸ!ʍ‘ègiÙ9.1Ž‘y¯Unexppœected‘G\newlinesŽŸ(⍑ègiºIf–—€yš¬wou“use“EQC‘—Scommands“lik˜e›—Ö\eqmul*“ºthat“proSˆduce“no“Latex“output˜inside“an“equationarraš¬wy“en˜vi-Ž¡‘ègironmen•¬wt,‘‰/follo“w–‰them“bš¬wy“a“commen˜t‘‰to“a˜v˜oid“insertion“of“a‘‰newline“in“the“output“le.‘‰úEquationarra˜yŽ¡‘ègien•¬wvironmen“ts›êlma“y˜not˜con“tain˜newlines.˜F‘ÿeor˜example:ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹¬± ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº15Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{eqnarray}Ž¤€‘ègi\eq{x–,j&“=“&“a“+“b}Ž¡‘ègi\eqsub*{"prev"}{b}Ž¡‘ègi\eqrev{"prev"}Ž¡‘ègi\end{eqnarray}ŽŸ±Ž‘ègiºwill–êlyield“the“follo¬wwing“output:Ž©±‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{eqnarray}Ž¡‘ègix&=&a+bŽ¡¡‘ègia&=&x-bŽ¡‘ègi\end{eqnarray}ŽŸ$ì荍‘ègiÙ9.2Ž‘y¯Multiple–G\ppœossible“v‘ÿÆalues“for“equationsŽŸ|‘ègiºConsider–êlthe“follo¬wwing“equation:ŽŸ^ø¡’Êp‚Êu–UR¹=“Êx–ª¨¹+“Êy‘n7¹sinŽ‘ÛSÊ'ŽŸ½ð‘ègiºNo•¬ww›êlw“e˜w“an“t˜to˜nd˜the˜v‘ÿXíalue˜ÊxŸ
ÌÌÈ1Ž‘ªpºwhic“h˜will˜mak“e˜Êu˜ºbSˆecome˜zero.Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{equation}Ž¡‘ègi\eqsubst[eq:usubst]{"prev"}{x–,j=“x_1}“=“0Ž¡‘ègi\end{equation}Ž©±Ž¡’¼9mÊu–UR¹=“ÊxŸ
ÌÌÈ1Ž‘j¬¹+‘ª¨Êy‘n5¹sinŽ‘ÛQÊ'“¹=“0Ž’
éc¼º(3)ŽŽŽŸ½ð‘ègiand–êlnd“ÊxŸ
ÌÌÈ1Ž‘ªpºto“bSˆe:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eqsubst*{"prev"}{u–,j=“0}Ž¡‘ègi\eqsub*{"prev"}{\sin\varphi‘,jy}Ž¡‘ègi$$\eqrev{"prev"}$$ŽŸ±¡’Ìú?ÊxŸ
ÌÌÈ1Ž‘V¹=‘URÍÊy‘n5¹sinŽ‘ÛQÊ'Ž©½ð‘ègiºBut–êlwhat“happSˆens“if“wš¬we“no˜w“ask“for“the“v‘ÿXíalue“of“Êuº?ŽŸ^ø‘ègiÖu–,j=“\val{u}Ž¡’ǁÊu–UR¹=“ÊxŸ
ÌÌÈ1Ž‘j¬¹+‘ª¨Êy‘n5¹sinŽ‘ÛQÊ'Ž¦‘ègiºEQC‘•prin•¬wts‘•Ca›•Bw“arning˜that˜there‘•Care˜m“ultiple˜pSˆossible˜v‘ÿXíalues‘•Cfor˜Êu˜º(i.e.,‘•nÊx–ÿ¹+‘Êy‘n7¹sinŽ‘ÛSÊ'˜ºand˜ÊxŸ
ÌÌÈ1Ž‘ß¹+“Êy‘n7¹sinŽ‘ÛSÊ'º)Ž¡‘ègiand–È~then›È}c¬whoSˆoses“the“last˜v‘ÿXíalue“(whic•¬wh˜migh“t–È~or“migh¬wt˜not“bSˆe“what˜wš¬we“w˜an˜ted).‘ȱThere“are“t˜w˜o‘È}w˜a˜ys“toŽ¡‘ègia•¬wv“oid‘êlthis:ŽŸ鍍‘ègiïcolor push gray 0‘\õ1.Ž‘ðï	color popŽŽ‘ÇUDo–êlbSˆoth“substitutions“at“the“same“time:“Ö\eqsubst{"prev"}{x–,j=“x_1;“u“=“0}º.ŽŸ¥‘ègiïcolor push gray 0‘\õ2.Ž‘ðï	color popŽŽ‘ÇUIn¬wtrošSˆduce–êla“temp˜orary“for“Êuº:“Ö\eqsubst{"prev"}{x–,j=“x_1;“u“=“u_{temp}}º.ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹¸ñ ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº16Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÙ9.3Ž‘y¯Multiple‘G\substitutionsŽŸ?„‘ègiºIf–Ûyš¬wou‘Üdo“m˜ultiple›Üsubstitutions“at˜the“same˜time,‘åthe“order˜in“whic¬wh˜the“substitutions˜are“done˜is“notŽ¤€‘ègidened.‘•By›”using–”the“k•¬weyw“ord–”ÍnÇeqsubstcº,‘”>substitutions“are“done“consecutiv¬wely“in“the˜order“they“areŽ¡‘ègilisted.–êlF‘ÿeor“example,“compare:Ž©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqsubst{x–,j=“3y}{y“=“4z;“z“=“3}$$Ž¡‘ègi$$\eqsubstc{x–,j=“3y}{y“=“4z;“z“=“3}$$Ž¦’ÛJ
Êx–UR¹=“12‘
ÿþÊzŽ¤’ßI'x–UR¹=“36Ž¡‘ègiºOnly–êluse“ÍnÇeqsubstc“ºwhen“order“is“impšSˆortan¬wt“b˜ecause“it“is“less“ecien¬wt“than“ÍnÇeqsubstº.Ž©'ɞ‘ègiÙ9.4Ž‘y¯Output–G\of“n•cum“bpœersŽŸ?„‘ègiºNot–Қall›ҙcom¬wbinations“of˜Ö\precisionº,‘ÓÖ\precision_type“ºand˜Ö\scientific_limits“ºmak¬we˜sense.‘ՇF‘ÿeorŽ¤€‘ègiexample,‘ú'y•¬wou›ùámigh“t˜set˜the‘ùâlo“w“er˜scien“tic˜limit–ùâto˜¹0Ê:¹001˜ºso˜that“n•¬wum“bSˆers˜lik“e˜¹0Ê:¹005˜ºwill‘ùâbSˆe˜prin“tedŽ¡‘ègiwithout–	ïexpSˆonenš¬wt.‘FIf“y˜ou‘	îno˜w“set“Öprecision_type“ºto“Öfixed_marker‘	îºand“precision“to“¹2º,‘
(n˜um˜bSˆers“smallerŽ¡‘ègithan–êl¹0Ê:¹01“ºwill“bSˆe“prin¬wted“as“zero!Ž¦‘ègiÙ9.5Ž‘y¯Library‘G\equationsŽŸ?„‘ègiºNote–§Öthat“library›§×equations“are“not“automatically˜used“for“nding“v‘ÿXíalues˜of“v‘ÿXíariables.‘©ÄThey“are“consideredŽ¡‘ègito–êlbšSˆe“purely“for“reference“purp˜oses.“An“example:Ž©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eq[lib:myeq]{x–,j=“7y}Ž¡‘ègi\eq{y–,j=“2}Ž¡‘ègix–,j=“\val{x}Ž¦‘ègiºThis–=
will“×not‘a=ºreturn“¹14“ºas“the›=v‘ÿXíalue“of“Êxº.‘=‰If“y•¬wou˜w“an“t–=
to“use“a˜library“equation“for“nding˜v‘ÿXíalues,‘=y¬wouŽ¡‘ègineed–êlto“activ‘ÿXíate‘åit.“There“are“t•¬ww“o›êldieren“t˜pSˆossibilities:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eq{"lib:myeq"}Ž¡‘ègi\eqsubst{"lib:myeq"}{x–,j=“x_1}Ž¦‘ègiºThe–
8last›
9case“is“really˜what“the“library˜equations“are˜there“for:‘They“need˜to“bSˆe“adapted˜to“spSˆecialŽ¡‘ègipurpSˆoses–êlbš¬wy“substituting“custom“v‘ÿXíariables“in˜to“them.ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹Àä ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº17Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÙ9.6Ž‘y¯Pstric•cks‘G\headac“hesŽŸ?„‘ègiºpstricš¬wks–Fìcan“giv˜e“y˜ou‘Fëheadac˜hes“when“the“function“that“is“to‘FëbSˆe“plotted“has“Êx“ºor“Êy‘µ$ºv‘ÿXíalues“of“less“thanŽ¤€‘ègi¹1Ê:¹0º,–êlfor“example,“¹sinŽ‘WˆÊxº:Ž©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{figure}[!htb]Ž¡‘ègi\begin{center}Ž¡‘ègi\pspicture(\val{-180\degree},-1)(\val{180\degree},1)Ž¡‘ôÀ=\psgrid(\val{-180\degree},-1)(\val{180\degree},1)Ž¡‘ôÀ=\psaxes[linewidth=2pt,‘,jlabels=none]{->}(0,0)(\val{-180\degree},-1)(\val{180\degree},1)Ž¡‘ôÀ=\pscurve{-}\printvector{\val{Ž¡‘
qå\eqeval{y–,j=“\sin“x}{x“=“-180\degree:180\degree:30\degree}}}Ž¡‘ègi\endpspictureŽ¡‘ègi\end{center}Ž¡‘ègi\caption{Function–,j$y“=“\sin“x$,“na€ïve“version}Ž¡‘ègi\end{figure}ŽŸ{g¢‘ègiŸŸBÞïcolor push gray 0Ÿ`½"ŸØ*¬’¨¹{ïcolor push gray 0ï	color popŽŽ’
ÿyŸãŒò
¥"  tx@Dict begin STP newpath 0.8 SLW 0  setgray   gsave 0.4 SLW 0.5  setgray -89.39847 -28.45274 89.39847 28.45274 -89.39847 -28.45274 28.45274 abs 28.45274 abs 5 0 {} 0 /Helvetica findfont 0 scalefont setfont Grid grestore   gsave 0.8 SLW 0  setgray -89.39847 -28.45274 89.39847 28.45274 -89.39847 -28.45274 28.45274 abs 28.45274 abs 1 0 { 0  setgray } 10.0 /Helvetica findfont 10.0 scalefont setfont Grid grestore  end ò
A"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { BeginArrow 1.  1.  scale  false 0.4 1.4 1.5 2.   1. .setopacityalpha  Arrow  EndArrow  } def  89.39847  0  -89.39847  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 2.0 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
I"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { BeginArrow 1.  1.  scale  false 0.4 1.4 1.5 2.   1. .setopacityalpha  Arrow  EndArrow  } def  28.45274  0 exch -28.45274  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 2.0 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïò"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  0 rotate /n 2 def /dx 28.45274  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ïó"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  0 rotate /n -3 def /dx 28.45274  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ïô"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  90 rotate /n -1 def /dx 28.45274  def n 0 lt { /dx dx neg def /n n neg def } if /y2 3.0 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ŽŽŽò
º"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  [ 89.38675 0.0 74.48917 14.22636 59.59116 24.64085 44.69359 28.45274 29.79558 24.64085 14.898 14.22636 0.0 0.0 -14.898 -14.22636 -29.79558 -24.64085 -44.69359 -28.45274 -59.59116 -24.64085 -74.48917 -14.22636 -89.38675 0.0   1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ŽŽŽŸ%€’š¢”ïcolor push gray 0ºFigure–êl2:“F‘ÿeunction“Êy‘˹=‘URsinŽ‘ÂnÊxº,“na€ïvš¬we“v˜ersionï	color popŽŽŽï	color popŽŸՀ‘ègiNote–Î4that“some“ÍnÇv‘ÿ@al›Î5ºstatemen¬wts“are“required“to“ev‘ÿXíaluate“the“units˜to“oating“pSˆoinš¬wt“n˜um˜bšSˆers.‘ÏçThe“metho˜dŽ¡‘ègiabSˆo•¬wv“e›!Žhas‘!t“w“o˜ma‘§jor˜dra“wbac“ks:‘!¸The–!Êxº-axis˜is˜scaled“in˜radians,‘!œnot˜in“degrees,‘!œand˜the“n•¬wum“bSˆering˜ofŽ¡‘ègithe–êlÊyn9º-axis“is“only“in“inš¬wteger“n˜um˜bSˆers.ŽŸ€‘ègiThe–êlnon-na€ïvš¬we“v˜ersion“of“the“diagram“is:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{figure}[!htb]Ž¡‘ègi\begin{center}Ž¡‘ègi\psset{xunit=0.35mm,‘,jyunit=40mm}Ž¡‘ègi\pspicture(-190,-1.05)(200,1.1)Ž¡‘ôÀ=\psaxes[linewidth=2pt,–,jticks=none,“labels=none]{->}(0,0)(-190,-1.05)(200,1.1)Ž¡‘ôÀ=\psaxes[Dx=30,–,jticks=x,“labels=x,“ticksize=40mm]{-}(0,0)(-180,-1)(180,1)Ž¡‘ôÀ=\psaxes[Dy=0.2,–,jticks=y,“labels=y,“ticksize=54mm]{-}(0,0)(-180,-1.05)(180,1.1)Ž¡‘ôÀ=\pscurve{-}\printvector{\val{\eqeval{Ž¡‘ʹ\eqsubst{y–,j=“\sin“x}{x“=“x“\degree}}{x“=“-180:180:30}}}Ž¡‘ègi\endpspictureŽ¡‘ègi\end{center}Ž¡‘ègi\caption{Function–,j$y“=“\sin“x$}Ž¡‘ègi\end{figure}ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹ɚ ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº18Ž’þñï	color popŽŽ ¦Æs ÿG`]‘ègi þãxÓïcolor push gray 0 
‡-ŸØ*¬‘?î­ïcolor push gray 0ï	color popŽŽŸˆƒ’
Éò
C"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { BeginArrow 1.  1.  scale  false 0.4 1.4 1.5 2.   1. .setopacityalpha  Arrow  EndArrow  } def  199.16992  0  -189.21143  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 2.0 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò
J"  tx@Dict begin STP newpath 2.0 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { BeginArrow 1.  1.  scale  false 0.4 1.4 1.5 2.   1. .setopacityalpha  Arrow  EndArrow  } def  125.19281  0 exch -119.5019  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 2.0 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ŽŽŽŸˆƒ’
Éïì"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  179.25293  0  -179.25293  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïô"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  113.81102  0 exch -113.81102  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïø"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  0 rotate /n 6 def /dx 29.87549  def n 0 lt { /dx dx neg def /n n neg def } if /y2 113.81102 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end Ÿ~ñÀ‘$¹30Ž‘5àD60Ž‘SÀd90Ž‘n°†120Ž’Œ¦150Ž’ªpÆ180ŽŽŽïù"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  0 rotate /n -6 def /dx 29.87549  def n 0 lt { /dx dx neg def /n n neg def } if /y2 113.81102 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end Ÿ~ñÀ‘ו8͹30Ž‘¹µ͹60Ž‘›Ôø͹90Ž’ÿ{Ú͹120Ž’ÿ]$ºÍ¹150Ž’ÿ?DšÍ¹180ŽŽŽŽŽŽŸˆƒ’
Éïì"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  179.25293  0  -179.25293  0  ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïó"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  125.19281  0 exch -119.5019  0 exch ArrowA CP 4 2 roll ArrowB L pop pop gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ïù"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  90 rotate /n 5 def /dx 22.76186  def n 0 lt { /dx dx neg def /n n neg def } if /y2 179.25235 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ¤é<÷ŸÝݍ’ÿ8U|0Ê:¹2ŽŽ¡ŸÝݍ’ÿ8U|0Ê:¹4ŽŽ¡ŸÝݍ’ÿ8U|0Ê:¹6ŽŽ¡ŸÝݍ’ÿ8U|0Ê:¹8ŽŽ¡ŸÝݍ’ÿ8U|1Ê:¹0ŽŽŽïú"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  90 rotate /n -5 def /dx 22.76186  def n 0 lt { /dx dx neg def /n n neg def } if /y2 179.25235 CLW 2 div add def /y1 y2 neg def /x dx def n { x y1 moveto x y2 lineto stroke /x x dx add def } repeat end ¤Ã	Ÿ]ݍ’ÿ/%͹0Ê:¹2ŽŽ¡Ÿ]ݍ’ÿ/%͹0Ê:¹4ŽŽ¡Ÿ]ݍ’ÿ/%͹0Ê:¹6ŽŽ¡Ÿ]ݍ’ÿ/%͹0Ê:¹8ŽŽ¡Ÿ]ݍ’ÿ/%͹1Ê:¹0ŽŽŽŽŽŽ’
ɟˆƒò
¾"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  [ 179.25293 0.0 149.37744 56.9055 119.50195 98.5635 89.62646 113.81102 59.75098 98.5635 29.87549 56.9055 0.0 0.0 -29.87549 -56.9055 -59.75098 -98.5635 -89.62646 -113.81102 -119.50195 -98.5635 -149.37744 -56.9055 -179.25293 0.0   1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ŽŽŽŸ%€’¿Ýïïcolor push gray 0ºFigure–êl3:“F‘ÿeunction“Êy‘˹=‘URsinŽ‘ÂnÊxï	color popŽŽŽï	color popŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒøÛ
ƒ’À;è¹ðü
øfZó.&Lt$ffffecbx1440ó-]Kdïecsl1200ó,둇¡ecti1200ó+D7`±ectt1200ó%ú±u
cmex10ó#¾KÈcmsy8ó"!",š
cmsy10ó ×
2cmmi8ó·ág£cmmi12ó|{Ycmr8ó¥!¢Necbx1200ós©^dG®G®ecbx1728ó·ág£ffcmmi12óX«Qffcmr12ó& Šffffecrm1440óÆîê½q½qecrm2074óÓ·åecrm1200óX«Qcmr12ùê;ßßßßßß