# LaTeX2HTML 96.1 (Feb 5, 1996)
# Associate images original text with physical files.

$key = q/{_inline}$4cdotn+2=62${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=102 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline72648" SRC="img30.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;exp-1;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=71 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72888" SRC="img68.gif"  > '; 
$key = q/{displaymath}arrayl|r|r|r|r|r|r|r|r|r|r|rmboxitindex;SPMamp;0;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;ldotshlinemboxarithmeticsequence;SPMamp;8;SPMamp;13;SPMamp;18;SPMamp;23;SPMamp;ldotsarray{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=434 HEIGHT=44 ALIGN=BOTTOM ALT="displaymath73364" SRC="img163.gif"  > '; 
$key = q/{tabular}c|c|c|c|c|c|cA;SPMamp;B;SPMamp;C;SPMamp;D;SPMamp;E;SPMamp;F;SPMamp;Ghline0;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=218 HEIGHT=44 ALIGN=BOTTOM ALT="tabular38462" SRC="img285.gif"  > '; 
$key = q/{displaymath}qquadsum_i=0^i=nmboxitmake-even(i)qquadqquadqquadqquadsum_i=0^i=nmboxitmake-odd(i)qquad{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=433 HEIGHT=49 ALIGN=BOTTOM ALT="displaymath73352" SRC="img159.gif"  > '; 
$key = q/{_inline}$^3${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=6 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline73624" SRC="img211.gif"  > '; 
$key = q/{tabular}l|c|c|c|c|c|citfrom;SPMamp;A;SPMamp;B;SPMamp;C;SPMamp;D;SPMamp;E;SPMamp;Fhlineitto;SPMamp;B;SPMamp;C;SPMamp;E;SPMamp;E;SPMamp;B;SPMamp;F{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=239 HEIGHT=44 ALIGN=BOTTOM ALT="tabular59753" SRC="img369.gif"  > '; 
$key = q/{displaymath}fracn3+2{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=271 HEIGHT=32 ALIGN=BOTTOM ALT="displaymath72438" SRC="img8.gif"  > '; 
$key = q/{_inline}$y_4(x)=10-x${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=111 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73464" SRC="img196.gif"  > '; 
$key = q/{}fboxitop1{}/;
$cached_env_img{$key} = ' <IMG WIDTH=37 HEIGHT=28 ALIGN=BOTTOM ALT="tex2html_wrap73332" SRC="img150.gif"  > '; 
$key = q/{displaymath}frac10{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=254 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath72874" SRC="img65.gif"  > '; 
$key = q/{_inline}$beta${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline74246" SRC="img355.gif"  > '; 
$key = q/{displaymath}110cdot10^0,,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=285 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath74058" SRC="img318.gif"  > '; 
$key = q/{}fboxsf'car{}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=24 ALIGN=BOTTOM ALT="tex2html_wrap73196" SRC="img131.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;3;SPMamp;3;SPMamp;8);SPMamp;(sflist;SPMamp;2;SPMamp;3;SPMamp;-2;SPMamp;3);SPMamp;(sflist;SPMamp;4;SPMamp;-2;SPMamp;2;SPMamp;4);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=362 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73784" SRC="img238.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfandmboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bformboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=274 HEIGHT=39 ALIGN=BOTTOM ALT="tabular8440" SRC="img67.gif"  > '; 
$key = q/{_inline}$y_1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=15 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73470" SRC="img198.gif"  > '; 
$key = q/{}fboxanITEM{}/;
$cached_env_img{$key} = ' <IMG WIDTH=78 HEIGHT=24 ALIGN=BOTTOM ALT="tex2html_wrap73224" SRC="img142.gif"  > '; 
$key = q/{_inline}$equiv${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=12 HEIGHT=8 ALIGN=BOTTOM ALT="tex2html_wrap_inline72892" SRC="img70.gif"  > '; 
$key = q/{picture}(140,140)(+0,0)put(010,095)mboxnumberput(010,118)mboxnameput(010,105)vector(1,0)40put(010,115)vector(1,0)40put(050,100)picture(100,20)put(000,000)line(1,0)100put(100,000)line(0,1)20put(100,020)line(-1,0)100put(000,020)line(0,-1)20put(20,5)mboxaddentrypictureput(099,100)vector(0,-1)36put(104,064)vector(0,+1)36put(100,060)picture(20,20)put(00,00)circle10put(10,-2)mboxphonebookpictureput(100,056)vector(0,-1)36put(050,000)picture(100,20)put(000,000)line(1,0)100put(100,000)line(0,1)20put(100,020)line(-1,0)100put(000,020)line(0,-1)20put(20,5)mboxlookupnumberpictureput(010,013)mboxnameput(010,010)vector(1,0)40{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=254 HEIGHT=196 ALIGN=BOTTOM ALT="picture45092" SRC="img327.gif"  > '; 
$key = q/{_inline}$5cdotx+17cdoty${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=91 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline73096" SRC="img117.gif"  > '; 
$key = q/{}fboxitf{}/;
$cached_env_img{$key} = ' <IMG WIDTH=16 HEIGHT=29 ALIGN=BOTTOM ALT="tex2html_wrap73324" SRC="img149.gif"  > '; 
$key = q/{}framebox[textwidth]picture(300,220)(0,-200)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mboxrelaxput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(0,0)line(0,-1)175multiput(-5,0)(0,-25)7line(1,0)10put(-18,+03)mbox_inline$0$_inlineput(-18,-123)mbox_inline$5$_inlineput(150,-50)circle40put(150,-50)circle*3put(137,-60)mboxslcenterput(150,-50)vector(-1,+1)14put(139,-50)mbox_inline$r$_inlineput(050,-75)line(1,00)75put(125,-75)line(0,-1)50put(125,-125)line(-1,0)75put(050,-125)line(0,+1)50put(050,-75)circle*3put(+5,-76)mboxslupperleftput(75,-72)mboxslwidthput(052,-100)mboxslheightput(153,-40)circle*3put(154,-45)mboxitA(6,1.5)put(200,-150)circle*3put(207,-145)mboxitB(8,6)put(100,-100)circle*3put(102,-95)mboxitC(4,4)picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=578 HEIGHT=361 ALIGN=BOTTOM ALT="tex2html_wrap72774" SRC="img50.gif"  > '; 
$key = q/{picture}(180,30)put(00,00)line(+1,0)180put(180,00)line(0,+1)30put(180,30)line(-1,0)180put(00,30)line(0,-1)30put(03,22)mboxfootnotesizeemphname:put(60,00)line(0,+1)30put(63,22)mboxfootnotesizeemphzip:put(120,00)line(0,+1)30put(123,22)mboxfootnotesizeemphnumber:put(10,05)mboxsmallttchar'047ttPeterLeeput(70,05)mboxsmalltt15270put(130,05)mboxsmalltt606-7771{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=287 HEIGHT=48 ALIGN=BOTTOM ALT="picture4260" SRC="img47.gif"  > '; 
$key = q/{}fbox_inline$;SPMlt;_ir$_inline{}/;
$cached_env_img{$key} = ' <IMG WIDTH=37 HEIGHT=23 ALIGN=BOTTOM ALT="tex2html_wrap73222" SRC="img140.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;vdf;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefine_inline$mboxtt;SPMlt;var;SPMgt;$_inline_inline$mboxtt;SPMlt;val;SPMgt;$_inline);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefine-struct_inline$mboxtt;SPMlt;var;SPMgt;$_inline(_inline$mboxtt;SPMlt;var;SPMgt;$_inlineldots_inline$mboxtt;SPMlt;var;SPMgt;$_inline))mboxtt;SPMlt;val;SPMgt;;SPMamp;=;SPMamp;mboxtt;SPMlt;con;SPMgt;_inline$,,vert,,$_inlinemboxtt;SPMlt;lst;SPMgt;_inline$,,vert,,$_inlinemboxtt;SPMlt;prm;SPMgt;_inline$,,vert,,$_inlinemboxtt;SPMlt;fun;SPMgt;_inline$,,vert,,$_inlinemboxtt;SPMlt;void;SPMgt;mboxtt;SPMlt;lst;SPMgt;;SPMamp;=;SPMamp;sfempty_inline$,,vert,,$_inline(sfconsmboxtt;SPMlt;val;SPMgt;mboxtt;SPMlt;lst;SPMgt;)mboxtt;SPMlt;fun;SPMgt;;SPMamp;=;SPMamp;(bflambda(mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=470 HEIGHT=168 ALIGN=BOTTOM ALT="tabular49438" SRC="img348.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;exp-2;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=70 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72890" SRC="img69.gif"  > '; 
$key = q/{_inline}$x=yquadmboxorquadx;SPMlt;yquadmboxorquadx;SPMgt;y${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=247 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline72542" SRC="img17.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(+tt11tt11)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=82 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74152" SRC="img336.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;val;SPMgt;;SPMamp;=;SPMamp;(sfmake-c_inline$mboxtt;SPMlt;val;SPMgt;$_inline..._inline$mboxtt;SPMlt;val;SPMgt;$_inline){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=298 HEIGHT=17 ALIGN=BOTTOM ALT="tabular8797" SRC="img75.gif"  > '; 
$key = q/{_inline}$underlinemboxttcounter1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=84 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline74158" SRC="img339.gif"  > '; 
$key = q/{displaymath}N,,,N^2,,,mboxor,,2^N,,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=308 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath73868" SRC="img268.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(hellostt1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=99 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline73022" SRC="img93.gif"  > '; 
$key = q/{}fboxitrel-op{}/;
$cached_env_img{$key} = ' <IMG WIDTH=52 HEIGHT=29 ALIGN=BOTTOM ALT="tex2html_wrap73204" SRC="img135.gif"  > '; 
$key = q/{displaymath}fracN2{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=258 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73828" SRC="img254.gif"  > '; 
$key = q/{picture}(200,150)(5,-30)put(60,110)vector(-4,-1)10qbezier[500](90,-15)(260,140)(60,110)put(50,40)vector(0,-1)14put(60,30)vector(-4,-1)10qbezier[500](60,30)(140,60)(240,-15)put(-15,40)minipage5incenterminipage[t]8intabbingtt(d=ttefinett(fun-parenttta-parent)>tt...tt(parent-nametta-parent)tt...>tt...tt(parent-datetta-parent)tt...>tt...tt(parent-eyestta-parent)tt...>tt...tt(fun-childrentt(parent-childrentta-parent))tt...)tabbingminipagecentercenterminipage[t]8intabbingtt(d=ttefinett(fun-childrenttaloc)>tt(c=ttond>>ttchar'133tt(empty?ttaloc)tt...char'135>>ttchar'133ttelsett...tt(fun-parenttt(firstttaloc))tt...tt(fun-childrentt(restttaloc))tt...char'135tt))tabbingminipagecenterminipage{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=770 HEIGHT=281 ALIGN=BOTTOM ALT="picture18684" SRC="img109.gif"  > '; 
$key = q/{_inline}$longrightarrow${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=27 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline73080" SRC="img111.gif"  > '; 
$key = q/{displaymath}arraylclcrcrr2cdotx;SPMamp;+;SPMamp;2cdoty;SPMamp;+;SPMamp;2cdotz;SPMamp;=;SPMamp;62cdotx;SPMamp;+;SPMamp;2cdoty;SPMamp;+;SPMamp;4cdotz;SPMamp;=;SPMamp;82cdotx;SPMamp;+;SPMamp;2cdoty;SPMamp;+;SPMamp;zcdotz;SPMamp;=;SPMamp;2array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=367 HEIGHT=59 ALIGN=BOTTOM ALT="displaymath73806" SRC="img243.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttftt(lambdatt(x)ttchar'047ttouch))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=258 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74238" SRC="img352.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(maxtt(listtt1tt2tt3))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=164 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline73846" SRC="img258.gif"  > '; 
$key = q/{_inline}$b^2;SPMgt;4cdotacdotc${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=89 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline72724" SRC="img42.gif"  > '; 
$key = q/{displaymath}56cdot10^0+56cdot10^0=(56+56)cdot10^0=112cdot10^0{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=423 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath74064" SRC="img320.gif"  > '; 
$key = q/{picture}(350,180)put(000,000)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttTS~(DIR)pictureput(060,010)line(1,0)260put(320,010)vector(0,1)14put(030,020)vector(0,1)29put(030,035)line(1,0)200put(230,035)vector(0,1)14put(000,050)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttText~(DIR)pictureput(060,060)line(1,0)35put(095,060)vector(0,1)39put(095,085)line(+1,0)60put(155,085)vector(0,1)14put(095,085)line(-1,0)60put(035,085)vector(0,1)14put(000,100)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttpart1~(99)pictureput(065,100)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttpart2~(52)pictureput(130,100)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttpart3~(17)pictureput(200,050)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttLibs~(DIR)pictureput(230,070)vector(0,1)29put(230,085)line(1,0)60put(290,085)vector(0,1)14put(200,100)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttCode~(DIR)pictureput(260,110)line(1,0)1put(261,110)line(0,1)20put(261,130)line(+1,0)20put(281,130)vector(0,1)19put(261,130)line(-1,0)20put(241,130)vector(0,1)19put(200,150)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxtthang~(8)pictureput(265,150)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttdraw~(2)pictureput(265,100)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttDocs~(DIR)pictureput(325,110)line(1,0)10put(335,110)vector(0,1)14put(300,125)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttread!~(19)pictureput(300,025)picture(58,20)put(00,00)line(+1,+0)58put(58,00)line(+0,+1)20put(58,20)line(-1,+0)58put(00,20)line(+0,-1)20put(02,05)mboxttread!~(10)picture{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=571 HEIGHT=287 ALIGN=BOTTOM ALT="picture18868" SRC="img110.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinemboxtt;SPMlt;var;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;prm;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=317 HEIGHT=57 ALIGN=BOTTOM ALT="tabular27690" SRC="img145.gif"  > '; 
$key = q/{tabular}lll1.tt(definett(fttchar'047ttx)ttx);SPMamp;2.tt(definett(fttxttyttz)tt(x));SPMamp;3.tt(definett(f)tt10)htmladdimg..iconshand.right.gifhtmladdnormallinkSolution..Solutionslegal-def1.htmlrule3pt5pt{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=684 HEIGHT=22 ALIGN=BOTTOM ALT="tabular7512" SRC="img59.gif"  > '; 
$key = q/{_inline}$underlinettx${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=9 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline74144" SRC="img334.gif"  > '; 
$key = q/{}fboxbox3{}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap73176" SRC="img127.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(increment1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=126 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74156" SRC="img338.gif"  > '; 
$key = q/{_inline}$mboxwithall${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=58 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline74242" SRC="img353.gif"  > '; 
$key = q/{displaymath}x^2+2cdotx+1=0{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=315 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath72634" SRC="img26.gif"  > '; 
$key = q/{picture}(125,100)thicklinesput(000,000)line(0,+1)100put(000,100)line(1,+0)125put(125,100)line(0,-1)100put(125,000)line(-1,0)125put(10,90)circle*3put(10,90)line(1,-1)50put(15,87)_inline$(10,10)$_inlineput(60,40)circle*3put(60,40)line(-1,0)50put(65,37)_inline$(60,60)$_inlineput(10,40)circle*3put(10,30)_inline$(10,60)$_inline{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=200 HEIGHT=161 ALIGN=BOTTOM ALT="picture14653" SRC="img99.gif"  > '; 
$key = q/{displaymath}1cdotx^2+0cdotx+(-1)=0.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=344 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath72700" SRC="img40.gif"  > '; 
$key = q/{displaymath}120cdot10^-1{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=285 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74034" SRC="img308.gif"  > '; 
$key = q/{displaymath}fracrmdrmdx{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=259 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73500" SRC="img203.gif"  > '; 
$key = q/{_inline}$(-1)^i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73392" SRC="img174.gif"  > '; 
$key = q/{tabular}l|lmboxoriginalexpression;SPMamp;mboxrequires2evaluationsofhlinett(maxtt(listtt0tt1tt2tt3));SPMamp;tt(maxtt(listtt1tt2tt3))tt(maxtt(listtt1tt2tt3));SPMamp;tt(maxtt(listtt2tt3))tt(maxtt(listtt2tt3));SPMamp;tt(maxtt(listtt3)){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=406 HEIGHT=87 ALIGN=BOTTOM ALT="tabular38048" SRC="img259.gif"  > '; 
$key = q/{displaymath}y(x)=acdotx+b.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=312 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73456" SRC="img192.gif"  > '; 
$key = q/{}fboxitop2{}/;
$cached_env_img{$key} = ' <IMG WIDTH=36 HEIGHT=28 ALIGN=BOTTOM ALT="tex2html_wrap73334" SRC="img151.gif"  > '; 
$key = q/{tabular}|l|l|l|hlinePhase;SPMamp;Goal;SPMamp;Activityhlinehlineminipage[t].60inData~~Analysis~~&Designminipage;SPMamp;minipage[t]1.6inraggedrighttoformulateagroupofdatadefinitionsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdevelopagroupofmutuallyrecursivedatadefinitions_inline$bullet$_inlineatleastonedefinitionoronealternativeinadefinitionmustrefertobasicdata_inline$bullet$_inlineemexplicitlyidentifyallreferencesamongthedatadefinitionsrule0pt6ptminipagehlineTemplate;SPMamp;minipage[t]1.6inraggedrighttoformulateanoutlineforagroupofprogramsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdevelopemasmanytemplatesastherearedatadefinitionsemsimultaneously_inline$bullet$_inlinedevelopeachtemplatesaccordingtotherulesforcompoundandormixeddatadefinitionsasappropriate_inline$bullet$_inlineannotatethetemplateswithrecursionsandcross-applicationstomatchthe(cross-)referencesinthedatadefinitionsrule0pt6ptminipagehlineBody;SPMamp;minipage[t]1.6inraggedrighttodefinetheprogramrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightformulateaSchemeexpressionforeachtemplate,andforeachbfcond-clauseinatemplate_inline$bullet$_inlinestartbyexplainingwhateachexpressionineachtemplatecomputes_inline$bullet$_inlineuseadditionalauxiliaryfunctionswherenecessaryrule0pt6ptminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=571 HEIGHT=582 ALIGN=BOTTOM ALT="tabular18635" SRC="img108.gif"  > '; 
$key = q/{_inline}$2^N${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=20 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline73856" SRC="img263.gif"  > '; 
$key = q/{_inline}$x+mboxiteps${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=53 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline73446" SRC="img187.gif"  > '; 
$key = q/{displaymath}12cdoth.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=274 HEIGHT=12 ALIGN=BOTTOM ALT="displaymath72464" SRC="img12.gif"  > '; 
$key = q/{picture}(40,40)multiput(00,00)(10,0)4circle6multiput(05,10)(10,0)3circle6multiput(10,20)(10,0)2circle6multiput(15,30)(10,0)1circle6put(25,10)circle*3put(-18,06)mbox1put(-10,10)vector(1,0)10put(-08,26)mbox2put(-00,30)vector(1,0)10{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=81 HEIGHT=58 ALIGN=BOTTOM ALT="picture42941" SRC="img303.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;def;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefine(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefinemboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefine-structmboxtt;SPMlt;var;SPMgt;(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;))mboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinemboxtt;SPMlt;var;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;boo;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;sym;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;prm;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinesfempty;SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(ttelsemboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bflocal(mboxtt;SPMlt;def;SPMgt;ldotsmboxtt;SPMlt;def;SPMgt;)mboxtt;SPMlt;exp;SPMgt;)mboxtt;SPMlt;var;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinettx_inline$,,vert,,$_inlinettarea-of-disk_inline$,,vert,,$_inlinettcircumference_inline$,,vert,,$_inlineldotsmboxtt;SPMlt;boo;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinesftrue_inline$,,vert,,$_inlinesffalsemboxtt;SPMlt;sym;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinettchar'047tta_inline$,,vert,,$_inlinettchar'047ttdoll_inline$,,vert,,$_inlinettchar'047ttsum_inline$,,vert,,$_inlineldotsmboxtt;SPMlt;num;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinett1_inline$,,vert,,$_inlinett-1_inline$,,vert,,$_inlinett35_inline$,,vert,,$_inlinett1.22_inline$,,vert,,$_inlineldotsmboxtt;SPMlt;prm;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinett+_inline$,,vert,,$_inlinett-_inline$,,vert,,$_inlinettcons_inline$,,vert,,$_inlinettfirst_inline$,,vert,,$_inlinettrest_inline$,,vert,,$_inlineldots{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=489 HEIGHT=385 ALIGN=BOTTOM ALT="tabular27613" SRC="img144.gif"  > '; 
$key = q/{displaymath}a+2cdotW+S,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=306 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath73420" SRC="img183.gif"  > '; 
$key = q/{_inline}$9cdot2-21${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=66 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline73826" SRC="img252.gif"  > '; 
$key = q/{displaymath}frac11200,.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=272 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath74036" SRC="img309.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(125,-4)mboxhspace-1ptLARGE(multiput(125,1)(0,-.1)25line(1,0)25invcnt=125advanceinvcntby25put(invcnt,-4)mboxhspace-4.5ptLARGE)put(250,-4)mboxhspace-1.5ptLARGE[multiput(250,1)(0,-.1)25vector(+1,0)52picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72656" SRC="img20.gif"  > '; 
$key = q/{displaymath}y(x)=x^2-4cdotx+7{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=328 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath73566" SRC="img204.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(symbol=?ttstatus-letterttchar'047ttchar'137tt)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=270 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74098" SRC="img330.gif"  > '; 
$key = q/{displaymath}arraylclcrcrr2cdotx;SPMamp;+;SPMamp;2cdoty;SPMamp;+;SPMamp;3cdotz;SPMamp;=;SPMamp;10;SPMamp;;SPMamp;3cdoty;SPMamp;+;SPMamp;9cdotz;SPMamp;=;SPMamp;21;SPMamp;qquad(ddagger);SPMamp;-;SPMamp;3cdoty;SPMamp;-;SPMamp;8cdotz;SPMamp;=;SPMamp;-19array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=414 HEIGHT=59 ALIGN=BOTTOM ALT="displaymath73768" SRC="img231.gif"  > '; 
$key = q/{_inline}$underlinemboxttf${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74228" SRC="img350.gif"  > '; 
$key = q/{_inline}$mboxiteps2${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline73494" SRC="img201.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;3;SPMamp;9;SPMamp;21);SPMamp;(sflist;SPMamp;-3;SPMamp;-8;SPMamp;-19);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=361 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath73774" SRC="img234.gif"  > '; 
$key = q/{}fbox_inline$;SPMgt;$_inline{}/;
$cached_env_img{$key} = ' <IMG WIDTH=25 HEIGHT=21 ALIGN=BOTTOM ALT="tex2html_wrap73202" SRC="img134.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;val;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinemboxtt;SPMlt;boo;SPMgt;_inline$,,vert,,$_inlinemboxtt;SPMlt;sym;SPMgt;_inline$,,vert,,$_inlinemboxtt;SPMlt;num;SPMgt;_inline$,,vert,,$_inlinesfempty_inline$,,vert,,$_inlinemboxtt;SPMlt;lst;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;var;SPMgt;mbox(namesofttdefinedfunctions);SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;prm;SPMgt;mboxtt;SPMlt;lst;SPMgt;;SPMamp;=;SPMamp;sfempty_inline$,,vert,,$_inline(sfconsmboxtt;SPMlt;val;SPMgt;mboxtt;SPMlt;lst;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=476 HEIGHT=82 ALIGN=BOTTOM ALT="tabular27704" SRC="img146.gif"  > '; 
$key = q/{displaymath}x^2+2cdotx+1=(-1)^2+2cdot(-1)+1=1-2+1=0.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=454 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath72640" SRC="img27.gif"  > '; 
$key = q/{displaymath}sum_i=0^i=R-1mboxitarea-of-rectangle(i).{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=345 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath73426" SRC="img185.gif"  > '; 
$key = q/{_inline}$1.25=0.25cdot1100cdot500${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=178 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline72686" SRC="img36.gif"  > '; 
$key = q/{picture}(150,110)put(000,000)line(0,+1)100put(000,100)line(1,+0)125put(125,100)line(0,-1)100put(125,000)line(-1,0)125put(-05,105)circle*5put(-05,100)vector(0,-1)60put(-15,80)mboxityput(000,105)vector(+1,0)60put(040,110)mboxitxput(30,020)circle*4put(34,024)mboxapixelput(25,020)line(+1,0)10put(30,015)line(0,+1)10{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=223 HEIGHT=184 ALIGN=BOTTOM ALT="picture4003" SRC="img46.gif"  > '; 
$key = q/{picture}(350,100)put(0,0)picture(100,100)put(0,0)line(1,0)100put(100,0)line(0,1)100put(100,100)line(-1,0)100put(0,100)line(0,-1)100put(10,50)InterfaceProgramput(30,36)(View)pictureput(250,0)picture(100,100)put(0,0)line(1,0)100put(100,0)line(0,1)100put(100,100)line(-1,0)100put(0,100)line(0,-1)100put(20,50)CoreProgramput(30,36)(Model)pictureput(240,90)vector(-1,0)140put(120,93)tt(draw-messagetta-message)put(240,90)circle*3put(240,70)vector(-1,0)140put(120,73)tt(text-contentstta-text-field)put(240,70)circle*3put(100,30)vector(1,0)150put(120,33)tt(call-backttan-event)put(240,30)line(0,1)60put(240,30)circle*3put(250,10)vector(-1,0)150put(120,13)tt(make-buttonttlabelttcall-back){picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=562 HEIGHT=161 ALIGN=BOTTOM ALT="picture30302" SRC="img152.gif"  > '; 
$key = q/{tabular}|l|l|l|hlinePhase;SPMamp;Goal;SPMamp;Activityhlinehlineminipage[t].60inData~~Analysis~~&Designminipage;SPMamp;minipage[t]1.6inraggedrighttoformulateagroupofdatadefinitionsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdevelopadatadefinitionformixeddatawithatleasttwoalternatives_inline$bullet$_inlineonealternativemustnotrefertothedefinition_inline$bullet$_inlineexplicitlyidentifyallself-referencesinthedatadefinitionrule0pt6ptminipagehlineminipage[t].60inContractHeader~&Purposeminipage;SPMamp;minipage[t]1.6inraggedrighttonametheprogram,tospecifyitsinputs&~~outputs,todescribeitspurpose,toformulateaprogram~~headerrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightnamethefunction,itsinputs,itsoutput,andspecifyitspurpose:~;;itname:itin1itin2ldots_inline$rightarrow$_inlineitout~;;bfPurpose:tocomputeldotsitx1ldotsitx2~(bfdefine(itnamex1x2ldots)ldots)rule0pt6ptminipagehlineExamples;SPMamp;minipage[t]1.6inraggedrighttocharacterizetheinput-outputrelationshipviaexamplesrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightcreateexamplesoftheinput-outputrelationship_inline$bullet$_inlinemakesurethereisatleastoneexampleperalternativerule0pt6ptminipagehlineTemplate;SPMamp;minipage[t]1.6inraggedrighttoformulateanoutlineforagroupofprogramsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdevelopabfcond-expressionwithoneclauseperalternative_inline$bullet$_inlineaddselectorexpressionstoeachclause_inline$bullet$_inlineannotatethebodywithemnaturalrecursions_inline$bullet$_inlinescTest:theself-referencesinthistemplateandthedatadefinitionmatch!rule0pt6ptminipagehlineBody;SPMamp;minipage[t]1.6inraggedrighttodefinetheprogramrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightformulateaSchemeexpressionforeachsimplebfcond-line_inline$bullet$_inlineexplainforallotherbfcond-clauseswhateachnaturalrecursionwillcompute_inline$bullet$_inlineformulatetheanswer,usingselectorexpressions,parameters,itetc.rule0pt6ptminipagehlineTest;SPMamp;minipage[t]1.6inraggedrighttoexposemistakes~~(``typos''andlogic)rule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightapplythefunctiontotheinputsoftheexamples_inline$bullet$_inlinecheckthattheoutputsareaspredictedaboverule0pt6ptminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=575 HEIGHT=763 ALIGN=BOTTOM ALT="tabular10348" SRC="img84.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;def;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefine(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;)mboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinemboxtt;SPMlt;var;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;con;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;prm;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(ttelsemboxtt;SPMlt;exp;SPMgt;)){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=440 HEIGHT=147 ALIGN=BOTTOM ALT="tabular7413" SRC="img55.gif"  > '; 
$key = q/{picture}(200,110)put(100,75)vector(-4,-1)30qbezier[100](260,25)(350,155)(100,75)put(-15,40)minipage5inquoteAsllistofsymbols(ttlist-of-symbols)iseitherenumerateitemtheemptylist,ttempty,oritemtt(consttsttlof)wherettsisasymbolandttlofisafboxlistofsymbols.enumeratequoteminipage{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=492 HEIGHT=156 ALIGN=BOTTOM ALT="picture10182" SRC="img82.gif"  > '; 
$key = q/{_inline}$35^oC${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=37 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline72414" SRC="img1.gif"  > '; 
$key = q/{displaymath}3+4cdot5,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=284 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath72416" SRC="img2.gif"  > '; 
$key = q/{_inline}$ncdot(n-1)cdotldotscdot2cdot1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=151 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73028" SRC="img94.gif"  > '; 
$key = q/{displaymath}arrayl|r|r|r|r|r|r|r|r|r|r|rmboxindex;SPMamp;0;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;5;SPMamp;6;SPMamp;7;SPMamp;8;SPMamp;9;SPMamp;ldotshlinemboxevens;SPMamp;0;SPMamp;2;SPMamp;4;SPMamp;6;SPMamp;8;SPMamp;10;SPMamp;12;SPMamp;14;SPMamp;16;SPMamp;18;SPMamp;ldotsmboxodds;SPMamp;1;SPMamp;3;SPMamp;5;SPMamp;7;SPMamp;9;SPMamp;11;SPMamp;13;SPMamp;15;SPMamp;17;SPMamp;19;SPMamp;ldotsarray{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=443 HEIGHT=66 ALIGN=BOTTOM ALT="displaymath73340" SRC="img153.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;3;SPMamp;3;SPMamp;8);SPMamp;(sflist;SPMamp;;SPMamp;0;SPMamp;-5;SPMamp;-5);SPMamp;(sflist;SPMamp;;SPMamp;-8;SPMamp;-4;SPMamp;-12);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=373 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73792" SRC="img239.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(define-structttaaatt(xxttyy))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=264 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline74272" SRC="img365.gif"  > '; 
$key = q/{_inline}$2cdoti${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=26 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline73342" SRC="img154.gif"  > '; 
$key = q/{tabular}|l|l|l|hlinePhase;SPMamp;Goal;SPMamp;Activityhlinehlineminipage[t].69inDataAnalysisminipage;SPMamp;minipage[t]1.6inraggedrighttodeterminethedistinctsituationsafunctiondealswithrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightinspecttheproblemstatementfordistinctsituations_inline$bullet$_inlineenumerateallpossiblesituationsrule0pt6ptminipagehlineExamples;SPMamp;minipage[t]1.6inraggedrighttoprovideanexamplepersituationrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightchooseatleastoneexamplepersituation_inline$bullet$_inlineforintervals,theexamplesmustincludeborderlinecasesrule0pt6ptminipagehlineminipage[t].69inBody(1)Conditionsminipage;SPMamp;minipage[t]1.6inraggedrighttoformulateaconditionalexpressionrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightwritedowntheskeletonofabfcondexpression,withoneclausepersituation_inline$bullet$_inlineformulateoneconditionpersituation,usingtheparameters_inline$bullet$_inlineshowthattheconditionsdistinguishtheexamplesappropriatelyrule0pt6ptminipagehlineminipage[t].69inBody(2)Answersminipage;SPMamp;minipage[t]1.6inraggedrighttoformulatethebodyofeachbfcond-linerule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdealwitheachbfcond-lineemseparately_inline$bullet$_inlineassumetheconditionholdsanddevelopaSchemeexpressionthatcomputestheappropriateanswerrule0pt6ptminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=579 HEIGHT=429 ALIGN=BOTTOM ALT="tabular3212" SRC="img34.gif"  > '; 
$key = q/{displaymath}1,frac12,frac14,frac18,ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=298 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73346" SRC="img156.gif"  > '; 
$key = q/{tabular}lcrlcrlmulticolumn4cEnglish;SPMamp;multicolumn2cmetrichline1bfinch;SPMamp;;SPMamp;;SPMamp;;SPMamp;=;SPMamp;2.54;SPMamp;rmcm1bffoot;SPMamp;=;SPMamp;12;SPMamp;in.1bfyard;SPMamp;=;SPMamp;3;SPMamp;ft.1bfrod;SPMamp;=;SPMamp;5_inline$frac12$_inline;SPMamp;yd.1bffurlong;SPMamp;=;SPMamp;40;SPMamp;rd.1bfmile;SPMamp;=;SPMamp;8;SPMamp;fl.{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=326 HEIGHT=143 ALIGN=BOTTOM ALT="tabular1637" SRC="img14.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(firsttt(constt10ttempty))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=223 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline72946" SRC="img78.gif"  > '; 
$key = q/{picture}(300,30)put(0,0)tt(definett(fttx)tt(+tt(*ttxttx)tt25))put(52,10)line(0,1)12put(52,10)circle*3put(52,22)line(1,0)45put(90,22)vector(0,-1)12put(97,22)vector(0,-1)12{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=251 HEIGHT=38 ALIGN=BOTTOM ALT="picture24914" SRC="img123.gif"  > '; 
$key = q/{_inline}$y_2(x)=4-x${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=102 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73460" SRC="img194.gif"  > '; 
$key = q/{displaymath}y(x)=acdotx+b{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=308 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73734" SRC="img224.gif"  > '; 
$key = q/{_inline}$a_n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=17 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73362" SRC="img162.gif"  > '; 
$key = q/{tabular}lll1.tt(definett(fttx)ttchar'047ttx);SPMamp;2.tt(definett(fttchar'047ttx)ttx);SPMamp;3.tt(definett(fttxtty)tt(+ttchar'047ttytt(notttx))){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=707 HEIGHT=22 ALIGN=BOTTOM ALT="tabular7555" SRC="img61.gif"  > '; 
$key = q/{_inline}$beta_v${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=16 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline74258" SRC="img356.gif"  > '; 
$key = q/{displaymath}12cdot10^2,.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=280 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74032" SRC="img307.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;a_11;SPMamp;ldots;SPMamp;ldots;SPMamp;b_1);SPMamp;(sflist;SPMamp;;SPMamp;a_21;SPMamp;ldots;SPMamp;b_2);SPMamp;(sflist;SPMamp;;SPMamp;vdots;SPMamp;vdots;SPMamp;vdots);SPMamp;(sflist;SPMamp;;SPMamp;;SPMamp;a_nn;SPMamp;b_n);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=376 HEIGHT=91 ALIGN=BOTTOM ALT="displaymath73812" SRC="img245.gif"  > '; 
$key = q/{picture}(140,140)(+20,0)put(050,100)picture(100,20)put(000,000)line(1,0)100put(100,000)line(0,1)20put(100,020)line(-1,0)100put(000,020)line(0,-1)20put(20,5)mboxnextpictureput(099,100)vector(0,-1)36put(102,064)vector(0,+1)36put(100,060)picture(20,20)put(00,00)circle10put(10,-2)mboxcurrentcolorpicture{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=199 HEIGHT=106 ALIGN=BOTTOM ALT="picture45123" SRC="img328.gif"  > '; 
$key = q/{}fboxsf'doll{}/;
$cached_env_img{$key} = ' <IMG WIDTH=42 HEIGHT=24 ALIGN=BOTTOM ALT="tex2html_wrap73194" SRC="img130.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;def;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefine(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefinemboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefine-struct_inline$mboxtt;SPMlt;var0;SPMgt;$_inline(_inline$mboxtt;SPMlt;var-1;SPMgt;ldotsmboxtt;SPMlt;var-n;SPMgt;$_inline))mboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinemboxtt;SPMlt;var;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;con;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;prm;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(ttelsemboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bflocal(mboxtt;SPMlt;def;SPMgt;ldotsmboxtt;SPMlt;def;SPMgt;)mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bflambda(mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfset!mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfmboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=486 HEIGHT=299 ALIGN=BOTTOM ALT="tabular48417" SRC="img332.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(make-countertt0)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=161 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74154" SRC="img337.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttxttanotherValue)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=208 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74142" SRC="img333.gif"  > '; 
$key = q/{displaymath}3cdotx+9cdoty=21{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=310 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath73820" SRC="img249.gif"  > '; 
$key = q/{displaymath}arrayl|r|r|r|r|r|r|r|r|r|r|rmboxitindex;SPMamp;0;SPMamp;1;SPMamp;2;SPMamp;3;SPMamp;4;SPMamp;ldotshlinemboxgeometricsequence;SPMamp;3;SPMamp;15;SPMamp;75;SPMamp;375;SPMamp;1875;SPMamp;ldotsarray{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=444 HEIGHT=44 ALIGN=BOTTOM ALT="displaymath73372" SRC="img167.gif"  > '; 
$key = q/{_inline}$pi${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=8 ALIGN=BOTTOM ALT="tex2html_wrap_inline72424" SRC="img5.gif"  > '; 
$key = q/{_inline}$2-frac1n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline72456" SRC="img11.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;structure-tag;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=155 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74266" SRC="img362.gif"  > '; 
$key = q/{displaymath}55cdot10^0{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=276 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74056" SRC="img317.gif"  > '; 
$key = q/{_inline}$[fraca+b2,b]${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=50 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline73406" SRC="img177.gif"  > '; 
$key = q/{_inline}$b^2;SPMlt;4cdotacdotc${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=89 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline72728" SRC="img44.gif"  > '; 
$key = q/{_inline}$y_1(x)=x+4${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=102 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73458" SRC="img193.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;10);SPMamp;(sflist;SPMamp;0;SPMamp;3;SPMamp;9;SPMamp;21);SPMamp;(sflist;SPMamp;0;SPMamp;-3;SPMamp;-8;SPMamp;-19);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=373 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73766" SRC="img230.gif"  > '; 
$key = q/{displaymath}acdotx^2+bcdotx+c=0.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=330 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath72690" SRC="img38.gif"  > '; 
$key = q/{_inline}$x_1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=16 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73416" SRC="img181.gif"  > '; 
$key = q/{displaymath}99cdot10^500{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=283 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74072" SRC="img323.gif"  > '; 
$key = q/{_inline}$(x,f(x-mboxiteps))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=106 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline73450" SRC="img189.gif"  > '; 
$key = q/{displaymath}epsfboxiconsrootf.eps{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=528 HEIGHT=339 ALIGN=BOTTOM ALT="displaymath73622" SRC="img217.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;var;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinettx_inline$,,vert,,$_inlinettarea-of-disk_inline$,,vert,,$_inlinettperimeter_inline$,,vert,,$_inlineldotsmboxtt;SPMlt;con;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinesftrue_inline$,,vert,,$_inlinesffalse;SPMamp;;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinettchar'047tta_inline$,,vert,,$_inlinettchar'047ttdoll_inline$,,vert,,$_inlinettchar'047ttsum_inline$,,vert,,$_inlineldots;SPMamp;;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinett1_inline$,,vert,,$_inlinett-1_inline$,,vert,,$_inlinett35_inline$,,vert,,$_inlinett1.22_inline$,,vert,,$_inlineldotsmboxtt;SPMlt;prm;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinett+_inline$,,vert,,$_inlinett-_inline$,,vert,,$_inlineldots{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=438 HEIGHT=104 ALIGN=BOTTOM ALT="tabular7366" SRC="img54.gif"  > '; 
$key = q/{_inline}$5cdotx+17cdoty+3cdotz${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=142 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline73106" SRC="img118.gif"  > '; 
$key = q/{displaymath}m=fraca+b2.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=294 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73674" SRC="img218.gif"  > '; 
$key = q/{displaymath}sqrtx^2+y^2.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=289 HEIGHT=29 ALIGN=BOTTOM ALT="displaymath72736" SRC="img45.gif"  > '; 
$key = q/{_inline}$,,vert,,${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=2 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline72808" SRC="img53.gif"  > '; 
$key = q/{tabular}c|r|r|r|r|r|rmbox#step;SPMamp;itleft;SPMamp;_inline$(fmboxitleft)$_inline;SPMamp;itright;SPMamp;_inline$(fmboxitright)$_inline;SPMamp;itmid;SPMamp;_inline$(fmboxitmid)$_inlinehline_inline$n=1$_inline;SPMamp;3;SPMamp;-1;SPMamp;6.00;SPMamp;8.00;SPMamp;4.5;SPMamp;1.25_inline$n=2$_inline;SPMamp;3;SPMamp;-1;SPMamp;4.25;SPMamp;1.25;SPMamp;?;SPMamp;?{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=417 HEIGHT=66 ALIGN=BOTTOM ALT="tabular35721" SRC="img219.gif"  > '; 
$key = q/{_inline}$f(n)=n^2+n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=106 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73912" SRC="img279.gif"  > '; 
$key = q/{_inline}$sf!${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=3 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline74270" SRC="img364.gif"  > '; 
$key = q/{_inline}$(dagger)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=19 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73770" SRC="img232.gif"  > '; 
$key = q/{displaymath}arraylclcrcrr2cdot1;SPMamp;+;SPMamp;2cdot1;SPMamp;+;SPMamp;3cdot2;SPMamp;=;SPMamp;102cdot1;SPMamp;+;SPMamp;5cdot1;SPMamp;+;SPMamp;12cdot2;SPMamp;=;SPMamp;31;SPMamp;4cdot1;SPMamp;+;SPMamp;1cdot1;SPMamp;-;SPMamp;2cdot2;SPMamp;=;SPMamp;1array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=376 HEIGHT=57 ALIGN=BOTTOM ALT="displaymath73754" SRC="img227.gif"  > '; 
$key = q/{tabular}lll1.ttx;SPMamp;2.tt(=ttyttz);SPMamp;3.tt(=tt(=ttyttz)tt0){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=270 HEIGHT=15 ALIGN=BOTTOM ALT="tabular7469" SRC="img56.gif"  > '; 
$key = q/{displaymath}frac-b2cdota;{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=270 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath72992" SRC="img88.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;def;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefinemboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=293 HEIGHT=17 ALIGN=BOTTOM ALT="tabular8550" SRC="img71.gif"  > '; 
$key = q/{picture}(300,250)(0,0)put(-15,-15)mbox0put(0,0)vector(0,1)200put(-25,180)mboxityput(0,0)vector(1,0)300put(290,-30)mboxitxput(0,0)line(5,3)300qbezier[500](0,0)(200,0)(300,250)put(250,-10)line(0,1)250put(232,-30)mboxsmall1000{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=518 HEIGHT=446 ALIGN=BOTTOM ALT="picture38175" SRC="img264.gif"  > '; 
$key = q/{picture}(300,200)(0,+20)put(000,180)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Carl~~(1926)put(00,05)mbox~Eyes:~~greenpictureput(100,180)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Bettina~~(1926)put(00,05)mbox~Eyes:~~greenpictureput(000,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Adam~~(1950)put(00,05)mbox~Eyes:~~yellowpictureput(035,130)vector(0,1)50put(035,130)vector(2,1)100put(100,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Dave~~(1955)put(00,05)mbox~Eyes:~~blackpictureput(135,130)vector(-2,1)100put(135,130)vector(+0,1)50put(200,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Eva~~(1965)put(00,05)mbox~Eyes:~~bluepictureput(235,130)vector(-2,1)100put(235,130)vector(-4,1)200put(290,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Fred~~(1966)put(00,05)mbox~Eyes:~~pinkpictureput(255,020)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Gustav~~(1988)put(00,05)mbox~Eyes:~~blushpictureput(290,050)vector(-1,1)50put(290,050)vector(+1,1)50{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=574 HEIGHT=304 ALIGN=BOTTOM ALT="picture15893" SRC="img103.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(75,-4)mboxhspace-1.5ptLARGE[multiput(75,1)(0,-.1)25line(1,0)100invcnt=75advanceinvcntby100put(invcnt,-4)mboxhspace-4ptLARGE]picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72660" SRC="img22.gif"  > '; 
$key = q/{_inline}$x^2+2cdotx+1=0${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=131 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline72646" SRC="img29.gif"  > '; 
$key = q/{}fboxldots(sfnode-leftitabt)ldotsitaccumulator{}/;
$cached_env_img{$key} = ' <IMG WIDTH=251 HEIGHT=28 ALIGN=BOTTOM ALT="tex2html_wrap74016" SRC="img297.gif"  > '; 
$key = q/{_inline}$mboxreplacedby${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=87 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline74244" SRC="img354.gif"  > '; 
$key = q/{displaymath}N^2+N{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=280 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath73842" SRC="img257.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(hellostt0)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=99 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline73020" SRC="img92.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttnew-knowledgettfalse)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=260 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74096" SRC="img329.gif"  > '; 
$key = q/{_inline}$y_5(x)=12${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=80 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73466" SRC="img197.gif"  > '; 
$key = q/{_inline}$10ldots0${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=51 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline74050" SRC="img314.gif"  > '; 
$key = q/{displaymath}pi=4cdot[1-frac13+frac15-frac17ldots].{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=353 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73396" SRC="img175.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(125,-4)mboxhspace-1ptLARGE(multiput(125,1)(0,-.1)25line(1,0)25invcnt=125advanceinvcntby25put(invcnt,-4)mboxhspace-4.5ptLARGE)picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72654" SRC="img19.gif"  > '; 
$key = q/{displaymath}3.14cdotr^2.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=283 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath72428" SRC="img6.gif"  > '; 
$key = q/{picture}(300,20)put(000,0)ifttcurrent-coloris50andweevaluatett(next),thenttcurrent-coloris20put(050,0)ifttcurrent-coloris40andweevaluatett(next),thenttcurrent-coloris15put(090,0)ifttcurrent-coloris70andweevaluatett(next),thenttcurrent-coloris30put(160,0)ifttcurrent-coloris30andweevaluatett(next),thenttcurrent-coloris10put(190,0)ifttcurrent-coloris30andweevaluatett(next),thenttcurrent-coloris10{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=788 HEIGHT=16 ALIGN=BOTTOM ALT="picture39543" SRC="img289.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttxtt3)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=93 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74178" SRC="img342.gif"  > '; 
$key = q/{_inline}$O(N^2)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=48 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73902" SRC="img276.gif"  > '; 
$key = q/{displaymath}3cdotx+9cdot2=21.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=315 HEIGHT=14 ALIGN=BOTTOM ALT="displaymath73824" SRC="img251.gif"  > '; 
$key = q/{_inline}$ncdot(n-1)cdotldotscdot(mboxitlimit+1)cdot1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=220 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73038" SRC="img97.gif"  > '; 
$key = q/{_inline}$n^2+10${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=56 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline72452" SRC="img9.gif"  > '; 
$key = q/{_inline}$sqrtA${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=27 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline72418" SRC="img3.gif"  > '; 
$key = q/{_inline}$ngeq1000${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=68 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline73890" SRC="img273.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;3;SPMamp;3;SPMamp;8);SPMamp;(sflist;SPMamp;;SPMamp;-8;SPMamp;-4;SPMamp;-12);SPMamp;(sflist;SPMamp;;SPMamp;;SPMamp;-5;SPMamp;-5);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=373 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73798" SRC="img242.gif"  > '; 
$key = q/{_inline}$mboxiteps4${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline73496" SRC="img202.gif"  > '; 
$key = q/{picture}(300,200)(0,+20)put(000,180)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Carl~~(1926)put(00,05)mbox~Eyes:~~greenpictureput(035,180)vector(0,-1)50put(035,180)vector(2,-1)100put(035,180)vector(4,-1)200put(100,180)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Bettina~~(1926)put(00,05)mbox~Eyes:~~greenpictureput(135,180)vector(0,-1)50put(135,180)vector(-2,-1)100put(135,180)vector(+2,-1)100put(000,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Adam~~(1950)put(00,05)mbox~Eyes:~~yellowpictureput(100,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Dave~~(1955)put(00,05)mbox~Eyes:~~blackpictureput(200,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Eva~~(1965)put(00,05)mbox~Eyes:~~bluepictureput(235,100)vector(+1,-1)50put(290,100)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Fred~~(1966)put(00,05)mbox~Eyes:~~pinkpictureput(335,100)vector(-1,-1)50put(255,020)picture(75,30)put(00,00)line(+1,+0)70put(70,00)line(+0,+1)30put(70,30)line(-1,+0)70put(00,30)line(+0,-1)30put(00,19)mbox~Gustav~~(1988)put(00,05)mbox~Eyes:~~blushpicture{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=574 HEIGHT=304 ALIGN=BOTTOM ALT="picture17826" SRC="img106.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(75,-4)mboxhspace-1ptLARGE(multiput(75,1)(0,-.1)25vector(+1,0)227put(25,-4)mboxhspace-4.5ptLARGE)multiput(25,1)(0,-.1)25vector(-1,0)27picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72666" SRC="img25.gif"  > '; 
$key = q/{displaymath}arraycl;SPMamp;(acdot10^n)cdot(bcdot10^m)=;SPMamp;(acdotb)cdot10^ncdot10^m=;SPMamp;(acdotb)cdot10^(n+m)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=332 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath74068" SRC="img321.gif"  > '; 
$key = q/{tabular}|l|l|l|hlinePhase;SPMamp;Goal;SPMamp;Activityhlinehlineminipage[t].60inContractHeader~&Purposeminipage;SPMamp;minipage[t]1.6inraggedrighttonametheprogram,tospecifyitsinputs&~~outputs,todescribeitspurpose,toformulateaprogram~~headerrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightchooseaitnamethatfitstheproblem_inline$bullet$_inlinestudytheproblemforcluesonhowmanynumbersthefunctionconsumesmedskip_inline$bullet$_inlinepickonevariableperinput;ifpossible,usenamesthatarementionedforunknown``givens''intheproblemstatementmedskip_inline$bullet$_inlinedescribewhatthefunctionshouldproduceusingthechosenvariablesnamesmedskip_inline$bullet$_inlineformulatethecontract&header:~;;itname:itnumberldots_inline$rightarrow$_inlineitnumber~;;bfPurpose:tocomputeldotsitx1ldots~(bfdefine(itnamex1ldots)ldots)rule0pt6ptminipagehlineExamples;SPMamp;minipage[t]1.6inraggedrighttocharacterizetheinput-outputrelationshipviaexamplesrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightsearchtheproblemforexamples_inline$bullet$_inlineaddexamplestofunction(input(s),output)_inline$bullet$_inlinevalidatetheexamplesusingtheproblemstatement,ifpossiblerule0pt6ptminipagehlineBody;SPMamp;minipage[t]1.6inraggedrighttodefinetheprogramrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightformulatehowthefunctioncomputesitsresults_inline$bullet$_inlinedevelopaSchemeexpressionthatusesScheme'sprimitiveoperations,otherfunctions,andthevariables_inline$bullet$_inlinetranslatethemathematicalexpressionsintheproblemstatement,ifpossiblerule0pt6ptminipagehlineTest;SPMamp;minipage[t]1.6inraggedrighttoexposemistakes~~(``typos''andlogic)rule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightapplythefunctiontotheinputsoftheexamples_inline$bullet$_inlinecheckthattheoutputsareaspredictedaboverule0pt6ptminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=574 HEIGHT=764 ALIGN=BOTTOM ALT="tabular1208" SRC="img13.gif"  > '; 
$key = q/{_inline}$5cdotx${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=30 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline73088" SRC="img116.gif"  > '; 
$key = q/{displaymath}arraylclcrcrr2cdotx;SPMamp;+;SPMamp;2cdoty;SPMamp;+;SPMamp;3cdotz;SPMamp;=;SPMamp;102cdotx;SPMamp;+;SPMamp;5cdoty;SPMamp;+;SPMamp;12cdotz;SPMamp;=;SPMamp;31;SPMamp;qquad(dagger)4cdotx;SPMamp;+;SPMamp;1cdoty;SPMamp;-;SPMamp;2cdotz;SPMamp;=;SPMamp;1array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=412 HEIGHT=59 ALIGN=BOTTOM ALT="displaymath73746" SRC="img226.gif"  > '; 
$key = q/{_inline}$mboxsfset${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=24 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline74264" SRC="img361.gif"  > '; 
$key = q/{_inline}$underlinemboxttx${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline73164" SRC="img122.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;mboxtt;SPMlt;var;SPMgt;_inline$,,vert,,$_inline(bflambda(mboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;)_inline$,,vert,,$_inline(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=518 HEIGHT=18 ALIGN=BOTTOM ALT="tabular42683" SRC="img301.gif"  > '; 
$key = q/{displaymath}arraycf(x+mboxiteps)-f(x-mboxiteps)hline2cdotmboxitepsarray{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=346 HEIGHT=40 ALIGN=BOTTOM ALT="displaymath73454" SRC="img191.gif"  > '; 
$key = q/{displaymath}10^-500{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=273 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74074" SRC="img324.gif"  > '; 
$key = q/{_inline}$95^oC${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=37 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline72984" SRC="img87.gif"  > '; 
$key = q/{_inline}$O(n^2)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=43 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73914" SRC="img280.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(resttt(constt10ttempty))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=212 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline72948" SRC="img79.gif"  > '; 
$key = q/{picture}(200,130)put(100,75)vector(-4,-1)25qbezier[100](260,30)(350,155)(100,75)put(-15,40)minipage5incenterminipage[t]8intabbingtt(d=ttefinett(fun-for-lostta-list-of-symbols)>tt(c=ttond>>ttchar'133tt(empty?tta-list-of-symbols)tt...char'135>>ttchar'133ttelsett...tt(firsttta-list-of-symbols)tt...hbox_inline$fboxmboxtt(resttta-list-of-symbols)$_inlinett...char'135tt))tabbingminipagecenterminipage{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=771 HEIGHT=160 ALIGN=BOTTOM ALT="picture10207" SRC="img83.gif"  > '; 
$key = q/{_inline}$2cdoti+1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=55 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline73344" SRC="img155.gif"  > '; 
$key = q/{_inline}$lambda${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline73588" SRC="img206.gif"  > '; 
$key = q/{tabular}c|rmboxttN;SPMamp;mboxtimeofevaluationhline100;SPMamp;220200;SPMamp;880300;SPMamp;2050400;SPMamp;5090500;SPMamp;7410600;SPMamp;10420700;SPMamp;14070800;SPMamp;18530{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=205 HEIGHT=196 ALIGN=BOTTOM ALT="tabular39702" SRC="img291.gif"  > '; 
$key = q/{tabular}ccccc1.10;SPMamp;1.12;SPMamp;1.08;SPMamp;1.09;SPMamp;1.11{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=225 HEIGHT=12 ALIGN=BOTTOM ALT="tabular58900" SRC="img367.gif"  > '; 
$key = q/{displaymath}arrayl|c|c|c|cmboxitindex;SPMamp;0;SPMamp;1;SPMamp;2;SPMamp;ldotshlinemboxitM;SPMamp;a+S;SPMamp;a+1cdotW+S;SPMamp;a+2cdotW+S;SPMamp;ldotsmboxitfatM;SPMamp;f(a+S);SPMamp;f(a+1cdotW+S);SPMamp;f(a+2cdotW+S);SPMamp;ldotsmboxitarea;SPMamp;Wcdotf(a+S);SPMamp;Wcdotf(a+1cdotW+S);SPMamp;Wcdotf(a+2cdotW+S);SPMamp;ldotsarray{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=560 HEIGHT=88 ALIGN=BOTTOM ALT="displaymath73422" SRC="img184.gif"  > '; 
$key = q/{_inline}$10^16${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=29 HEIGHT=17 ALIGN=BOTTOM ALT="tex2html_wrap_inline74076" SRC="img325.gif"  > '; 
$key = q/{displaymath}17.05+1.22+2.59=20.86.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=357 HEIGHT=14 ALIGN=BOTTOM ALT="displaymath72980" SRC="img85.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;(ttlambda(mboxtt;SPMlt;var;SPMgt;...mboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=372 HEIGHT=17 ALIGN=BOTTOM ALT="tabular32227" SRC="img205.gif"  > '; 
$key = q/{displaymath}fracncdot(n-1)cdotldots21cdot20cdotldots120cdotldots1=ncdot(n-1)cdotldotscdot21cdotfrac20cdotldots120cdotldots1=ncdot(n-1)cdotldotscdot21.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=570 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath73026" SRC="img95.gif"  > '; 
$key = q/{displaymath}itln(x)=2cdot[(fracx-1x+1)+frac13cdot(fracx-1x+1)^3+frac15cdot(fracx-1x+1)^5ldots]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=447 HEIGHT=37 ALIGN=BOTTOM ALT="displaymath73388" SRC="img172.gif"  > '; 
$key = q/{picture}(40,40)multiput(00,00)(10,0)4circle6multiput(05,10)(10,0)3circle6multiput(10,20)(10,0)2circle6multiput(15,30)(10,0)1circle6put(15,30)circle*3put(20,20)circle*3{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=58 HEIGHT=58 ALIGN=BOTTOM ALT="picture42958" SRC="img304.gif"  > '; 
$key = q/{_inline}$4cdotn^2+6cdotn+2=462${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=171 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline72652" SRC="img32.gif"  > '; 
$key = q/{picture}(200,120)(65,-20)put(40,90)vector(-4,-1)10qbezier[500](180,-5)(240,120)(40,90)put(70,55)vector(-2,-1)34put(60,40)vector(-4,-1)10qbezier[500](260,-5)(140,60)(60,40)put(-15,40)minipage5inquoteAslparentisastructure:centerlinett(make-parentttlocttnttdtte)wherettlocisalistofchildren,ttnandttearesymbols,andttdisanumber.bigskipAsllistofchildreniseitherenumerateitemttemptyoritemtt(consttpttloc)wherettpisaparentandttlocisalistofchildren.enumeratequoteminipage{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=585 HEIGHT=182 ALIGN=BOTTOM ALT="picture18600" SRC="img107.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;0;SPMamp;-5;SPMamp;-5);SPMamp;(sflist;SPMamp;-8;SPMamp;-4;SPMamp;-12);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=361 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath73794" SRC="img240.gif"  > '; 
$key = q/{displaymath}fracx^ii!.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=263 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath73384" SRC="img171.gif"  > '; 
$key = q/{}fboxanttITEM{}/;
$cached_env_img{$key} = ' <IMG WIDTH=76 HEIGHT=21 ALIGN=BOTTOM ALT="tex2html_wrap73226" SRC="img143.gif"  > '; 
$key = q/{displaymath}e^1=1+frac11!+frac1^22!+frac1^33!+ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=354 HEIGHT=41 ALIGN=BOTTOM ALT="displaymath73382" SRC="img170.gif"  > '; 
$key = q/{displaymath}mboxitTOLERANCEcdot(mboxitright-mboxitleft).{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=358 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath73714" SRC="img221.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;def;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=49 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline72910" SRC="img72.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;3;SPMamp;9;SPMamp;21);SPMamp;(sflist;SPMamp;;SPMamp;1;SPMamp;2);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=340 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath73776" SRC="img235.gif"  > '; 
$key = q/{tabular}|l|c|hlinettempty;SPMamp;picture(100,10)put(0,0)circle*3picturehlinett(make-nodettemptyttempty);SPMamp;picture(100,30)btnode(00,25)(+1,-1)btnode(50,25)(-1,-1)put(25,00)circle*3picturehlineparbox[t]2.5inminipage[t]8intabbingtt(m=ttake-node>tt(m=ttake-node>>ttempty>>tt(make-nodettemptyttempty))>ttempty)tabbingminipage;SPMamp;picture(100,80)btnode(25,75)(+1,-1)btnode(75,75)(-1,-1)btnode(00,50)(+1,-1)btnode(50,50)(-1,-1)btnode(25,25)(+1,-1)btnode(75,25)(-1,-1)put(50,00)circle*3picturehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=486 HEIGHT=321 ALIGN=BOTTOM ALT="tabular41617" SRC="img295.gif"  > '; 
$key = q/{displaymath}1cdot8^2+0cdot8^1+2cdot8^0=((1cdot8)+0)cdot8)+2.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=415 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath73996" SRC="img299.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;var-1;SPMgt;ldotsmboxtt;SPMlt;var-n;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=170 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline72914" SRC="img74.gif"  > '; 
$key = q/{_inline}$mboxiteps${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=22 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73448" SRC="img188.gif"  > '; 
$key = q/{displaymath}KcdotfracN2=fracK2cdotN{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=310 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73834" SRC="img255.gif"  > '; 
$key = q/{displaymath}01cdot10^-99,,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=289 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath74042" SRC="img312.gif"  > '; 
$key = q/{picture}(40,40)multiput(00,00)(10,0)4circle6multiput(05,10)(10,0)3circle6multiput(10,20)(10,0)2circle6multiput(15,30)(10,0)1circle6put(15,10)circle*3{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=58 HEIGHT=58 ALIGN=BOTTOM ALT="picture42928" SRC="img302.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;def;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefine(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefinemboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefine-structmboxtt;SPMlt;var0;SPMgt;(mboxtt;SPMlt;var-1;SPMgt;ldotsmboxtt;SPMlt;var-n;SPMgt;))mboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inlinemboxtt;SPMlt;var;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inlinemboxtt;SPMlt;con;SPMgt;;SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;prm;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;ldotsmboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfcond(mboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;)ldots(ttelsemboxtt;SPMlt;exp;SPMgt;));SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfandmboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bformboxtt;SPMlt;exp;SPMgt;mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=485 HEIGHT=234 ALIGN=BOTTOM ALT="tabular9083" SRC="img77.gif"  > '; 
$key = q/{picture}(130,30)put(00,00)line(+1,0)130put(130,00)line(0,+1)30put(130,30)line(-1,0)130put(00,30)line(0,-1)30put(03,22)mboxfootnotesizeemphx:put(60,00)line(0,+1)30put(63,22)mboxfootnotesizeemphy:put(10,05)mboxsmalltt3put(70,05)mboxsmalltt4{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=208 HEIGHT=48 ALIGN=BOTTOM ALT="picture52618" SRC="img357.gif"  > '; 
$key = q/{_inline}$x-mboxiteps${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=53 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline73444" SRC="img186.gif"  > '; 
$key = q/{_inline}$g(n)=n^2${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=73 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73926" SRC="img284.gif"  > '; 
$key = q/{displaymath}f(n)leq1cdotg(n){displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=304 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73892" SRC="img274.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5,000$_inlineput(245,-18)mbox_inline$10,000$_inlineput(0,-4)mboxhspace-1.5ptLARGE[multiput(0,1)(0,-.1)25line(1,0)25invcnt=0advanceinvcntby25put(invcnt,-4)mboxhspace-4ptLARGE]put(25,-4)mboxhspace-1ptLARGE(multiput(25,1)(0,-.1)25line(1,0)100invcnt=25advanceinvcntby100put(invcnt,-4)mboxhspace-4ptLARGE]put(125,-4)mboxhspace-1ptLARGE(multiput(125,1)(0,-.1)25vector(+1,0)175picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72730" SRC="img33.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttxtt11)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=103 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74146" SRC="img335.gif"  > '; 
$key = q/{displaymath}1cdot10^2+0cdot10^1+2cdot10^0=((1cdot10+0)cdot10)+2;{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=434 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath73994" SRC="img298.gif"  > '; 
$key = q/{_inline}$mboxiteps;SPMgt;0${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=54 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline73488" SRC="img200.gif"  > '; 
$key = q/{tabular}|l|l|l|hlinePhase;SPMamp;Goal;SPMamp;Activityhlinehlineminipage[t].60inData~~Analysis~~&Designminipage;SPMamp;minipage[t]1.6inraggedrighttoformulateagroupofdatadefinitionsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdeterminehowmanydistinctclassesof``objects''makeupthecollectionofinputs_inline$bullet$_inlineenumeratethealternativesinadatadefinition_inline$bullet$_inlineformulateadatadefinitionforeachalternative,ifitisafromofcompounddatarule0pt6ptminipagehlineminipage[t].60inContractHeader~&Purposeminipage;SPMamp;minipage[t]1.6inraggedrighttonametheprogram,tospecifyitsinputs&~~outputs,todescribeitspurpose,toformulateaprogram~~headerrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightnamethefunction,itsinputs,itsoutput,andspecifyitspurpose:~;;itname:itin1itin2ldots_inline$rightarrow$_inlineitout~;;bfPurpose:tocomputeldotsitx1ldotsitx2~(bfdefine(itnamex1x2ldots)ldots)rule0pt6ptminipagehlineExamples;SPMamp;minipage[t]1.6inraggedrighttocharacterizetheinput-outputrelationshipviaexamplesrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightcreateexamplesoftheinput-outputrelationship_inline$bullet$_inlinemakesurethereisatleastoneexampleperalternativerule0pt6ptminipagehlineTemplate;SPMamp;minipage[t]1.6inraggedrighttoformulateanoutlineforagroupofprogramsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightintroduceabfcond-expressionwithoneclauseperalternative_inline$bullet$_inlineformulateaconditionforeachcase,usingbuilt-inandpredefinedpredicatesrule0pt6ptminipagehlineBody;SPMamp;minipage[t]1.6inraggedrighttodefinetheprogramrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdevelopaSchemeexpression(ananswer)foreachbfcond-line,assumingthattheconditionholdsrule0pt6ptminipagehlineTest;SPMamp;minipage[t]1.6inraggedrighttoexposemistakes~~(``typos''andlogic)rule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightapplythefunctiontotheinputsoftheexamples_inline$bullet$_inlinecheckthattheoutputsareaspredictedaboverule0pt6ptminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=575 HEIGHT=643 ALIGN=BOTTOM ALT="tabular6637" SRC="img52.gif"  > '; 
$key = q/{}framebox[3in]picture(300,220)(0,-190)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mboxrelaxput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(0,0)line(0,-1)175multiput(-5,0)(0,-25)7line(1,0)10put(-18,+03)mbox_inline$0$_inlineput(-18,-123)mbox_inline$5$_inlineput(25,0)circle*4put(22,10)mboxitCput(50,0)circle*4put(48,10)mboxitDput(75,0)circle*4put(73,10)mboxitEput(150,-150)circle*4put(154,-145)mboxitAput(25,-50)circle*4put(029,-045)mboxitBpicture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=507 HEIGHT=361 ALIGN=BOTTOM ALT="tex2html_wrap72792" SRC="img51.gif"  > '; 
$key = q/{_inline}$vdots${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=2 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline72922" SRC="img76.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;10);SPMamp;(sflist;SPMamp;0;SPMamp;3;SPMamp;9;SPMamp;21);SPMamp;(sflist;SPMamp;4;SPMamp;1;SPMamp;-2;SPMamp;1);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=360 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73764" SRC="img229.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(+tt0tt1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=61 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74160" SRC="img340.gif"  > '; 
$key = q/{displaymath}itsin(x)=fracx1!-fracx^33!+fracx^55!-fracx^77!ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=368 HEIGHT=41 ALIGN=BOTTOM ALT="displaymath73390" SRC="img173.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttcounter1tt1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=166 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74162" SRC="img341.gif"  > '; 
$key = q/{_inline}$y_3(x)=x+10${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=111 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73462" SRC="img195.gif"  > '; 
$key = q/{_inline}$0.50cdot1100cdot900=4.50${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=179 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline72688" SRC="img37.gif"  > '; 
$key = q/{displaymath}mcdot10^e{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=276 HEIGHT=14 ALIGN=BOTTOM ALT="displaymath74028" SRC="img305.gif"  > '; 
$key = q/{tabular}|l|l|l|hline;SPMamp;tt(nullchar'077ttalos);SPMamp;tt(cons?ttalos)hlinett(=ttntt1);SPMamp;rule0pt22ptminipage[t]8intabbingtt(and=tt(=ttntt1)>tt(nullchar'077ttalos))tabbingminipage;SPMamp;rule0pt22ptminipage[t]8intabbingtt(and=tt(=ttntt1)>tt(cons?ttalos))tabbingminipagehlinett(;SPMgt;ttntt1);SPMamp;rule0pt22ptminipage[t]8intabbingtt(and=tt(;SPMgt;ttntt1)>tt(nullchar'077ttalos))tabbingminipage;SPMamp;rule0pt22ptminipage[t]8intabbingtt(and=tt(;SPMgt;ttntt1)>tt(cons?ttalos))tabbingminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=789 HEIGHT=194 ALIGN=BOTTOM ALT="tabular20257" SRC="img113.gif"  > '; 
$key = q/{displaymath}arrayl|c|c|c|c|c|c|c|c|c|c|cmboxTreeA:;SPMamp;10;SPMamp;15;SPMamp;24;SPMamp;29;SPMamp;63;SPMamp;77;SPMamp;89;SPMamp;95;SPMamp;99hlinemboxTreeB:;SPMamp;87;SPMamp;15;SPMamp;24;SPMamp;29;SPMamp;63;SPMamp;33;SPMamp;89;SPMamp;95;SPMamp;99array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=436 HEIGHT=44 ALIGN=BOTTOM ALT="displaymath73066" SRC="img105.gif"  > '; 
$key = q/{_inline}$ntimesn${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=42 HEIGHT=16 ALIGN=MIDDLE ALT="tex2html_wrap_inline73958" SRC="img286.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;10);SPMamp;(sflist;SPMamp;2;SPMamp;5;SPMamp;12;SPMamp;31);SPMamp;(sflist;SPMamp;4;SPMamp;1;SPMamp;-2;SPMamp;1);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=359 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73762" SRC="img228.gif"  > '; 
$key = q/{displaymath}99cdot10^99,,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=283 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath74040" SRC="img311.gif"  > '; 
$key = q/{picture}(300,50)(0,-30)put(0,0)tt(localtt(put(36,0)picture(120,30)put(0,0)tt(definett(fttx)tt(+tt(*ttxttx)tt25))put(52,10)line(0,1)12put(52,10)circle*3put(52,22)line(1,0)45put(90,22)vector(0,-1)12put(97,22)vector(0,-1)12pictureput(22,-20)mboxtt...ttxldots){picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=312 HEIGHT=71 ALIGN=BOTTOM ALT="picture25139" SRC="img126.gif"  > '; 
$key = q/{_inline}$NcdotN${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=45 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline73866" SRC="img267.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(tt1tt0)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=61 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline72876" SRC="img66.gif"  > '; 
$key = q/{_inline}$mboxwithalloccurrencesofttx-1tt...ttx-n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=280 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73590" SRC="img207.gif"  > '; 
$key = q/{tabular}llllltabularladerdaerdearderaaedrtabular;SPMamp;tabularleadredaredraaerdeardtabular;SPMamp;tabularleraderdaadredaredraetabular;SPMamp;tabularldreaarderaderdaerdeatabular;SPMamp;tabularlaredraedreadredatabular{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=325 HEIGHT=99 ALIGN=BOTTOM ALT="tabular14871" SRC="img100.gif"  > '; 
$key = q/{_inline}$a_n+1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=34 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline73360" SRC="img161.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;variable;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=102 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline74260" SRC="img359.gif"  > '; 
$key = q/{_inline}$f(x)=x^3-27${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=113 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73638" SRC="img216.gif"  > '; 
$key = q/{tabular}|l|l|l|hline;SPMamp;tt(empty?ttalos);SPMamp;tt(cons?ttalos)hlinett(=ttntt1);SPMamp;;SPMamp;hlinett(;SPMgt;ttntt1);SPMamp;;SPMamp;hline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=369 HEIGHT=68 ALIGN=BOTTOM ALT="tabular20240" SRC="img112.gif"  > '; 
$key = q/{displaymath}array|rrr|1;SPMamp;2;SPMamp;10;SPMamp;0;SPMamp;1-1;SPMamp;9;SPMamp;1array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=294 HEIGHT=65 ALIGN=BOTTOM ALT="displaymath73964" SRC="img288.gif"  > '; 
$key = q/{displaymath}arraycl;SPMamp;1cdot10^0+;SPMamp;2cdot10^0hline;SPMamp;3cdot10^0array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=296 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath74054" SRC="img316.gif"  > '; 
$key = q/{_inline}$x=yquadmboxandquadx;SPMlt;yquadmboxandquadx;SPMgt;y${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=273 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline72540" SRC="img16.gif"  > '; 
$key = q/{picture}(130,30)put(0,0)picture(130,30)put(00,00)line(+1,0)130put(130,00)line(0,+1)30put(130,30)line(-1,0)130put(00,30)line(0,-1)30put(03,22)mboxfootnotesizeemphx:put(60,00)line(0,+1)30put(63,22)mboxfootnotesizeemphy:put(10,05)mboxsmalltt3put(70,05)mboxsmalltt4pictureput(30,05)mboxsmall5put(16,08)line(+1,+1)11put(16,08)line(+1,-1)11put(16,08)line(-1,+1)11put(16,08)line(-1,-1)11{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=208 HEIGHT=54 ALIGN=BOTTOM ALT="picture52647" SRC="img358.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttftt11)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=103 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74186" SRC="img343.gif"  > '; 
$key = q/{displaymath}10^-99{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=270 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74052" SRC="img315.gif"  > '; 
$key = q/{displaymath}a_0,a_1,a_2,a_3,ldots,a_n,a_n+1,ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=359 HEIGHT=13 ALIGN=BOTTOM ALT="displaymath73358" SRC="img160.gif"  > '; 
$key = q/{_inline}$mboxthenodetowhichttorigisconnected${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=283 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline74302" SRC="img370.gif"  > '; 
$key = q/{}fboxparbox[t]2.2inAsllistofnumbersiseitherenumerateitemttemptyoritemtt(consttsttl)whereenumerateitemttsisfboxanumberanditemttlisalistofnumbers.~rule[-8pt]0pt15ptenumerateenumerate{}/;
$cached_env_img{$key} = ' <IMG WIDTH=264 HEIGHT=230 ALIGN=BOTTOM ALT="tex2html_wrap73216" SRC="img138.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(tt(sfvoidtt)tt5)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=145 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline74198" SRC="img346.gif"  > '; 
$key = q/{_inline}$N^2${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=22 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline73838" SRC="img256.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;val;SPMgt;;SPMamp;=;SPMamp;mboxtt;SPMlt;con;SPMgt;{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=150 HEIGHT=11 ALIGN=BOTTOM ALT="tabular7623" SRC="img62.gif"  > '; 
$key = q/{}fboxparbox[t]2.2inAsllistofIRsiseitherenumerateitemttemptyoritemtt(consttsttl)whereenumerateitemttsisfboxanIRanditemttlisalistofIRs.~rule[-8pt]0pt15ptenumerateenumerate{}/;
$cached_env_img{$key} = ' <IMG WIDTH=264 HEIGHT=208 ALIGN=BOTTOM ALT="tex2html_wrap73218" SRC="img139.gif"  > '; 
$key = q/{displaymath}f(r)=0.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=286 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73636" SRC="img215.gif"  > '; 
$key = q/{displaymath}frac-b-sqrtb^2-4cdotacdotc2cdota;{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=327 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath72998" SRC="img90.gif"  > '; 
$key = q/{tabular}lcclshape;SPMamp;position;SPMamp;size(s);SPMamp;colorhlinecircle;SPMamp;(50,50);SPMamp;40;SPMamp;redrectangle;SPMamp;(30,20);SPMamp;_inline$5times5$_inline;SPMamp;bluerectangle;SPMamp;(65,20);SPMamp;_inline$5times5$_inline;SPMamp;bluerectangle;SPMamp;(40,75);SPMamp;_inline$20times10$_inline;SPMamp;redrectangle;SPMamp;(45,35);SPMamp;_inline$10times30$_inline;SPMamp;blue{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=305 HEIGHT=126 ALIGN=BOTTOM ALT="tabular12129" SRC="img91.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;10);SPMamp;(sflist;SPMamp;;SPMamp;3;SPMamp;9;SPMamp;21);SPMamp;qquad(*);SPMamp;(sflist;SPMamp;;SPMamp;;SPMamp;1;SPMamp;2);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=378 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73778" SRC="img236.gif"  > '; 
$key = q/{_inline}$(x,f(x+mboxiteps))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=106 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline73452" SRC="img190.gif"  > '; 
$key = q/{displaymath}arraylclcrcrr2cdotx;SPMamp;+;SPMamp;2cdoty;SPMamp;+;SPMamp;3cdotz;SPMamp;=;SPMamp;10;SPMamp;;SPMamp;3cdoty;SPMamp;+;SPMamp;9cdotz;SPMamp;=;SPMamp;21;SPMamp;qquad(*);SPMamp;;SPMamp;;SPMamp;;SPMamp;1cdotz;SPMamp;=;SPMamp;2array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=408 HEIGHT=55 ALIGN=BOTTOM ALT="displaymath73780" SRC="img237.gif"  > '; 
$key = q/{}fboxldots(sffirstitalon)ldotsitaccumulator{}/;
$cached_env_img{$key} = ' <IMG WIDTH=219 HEIGHT=28 ALIGN=BOTTOM ALT="tex2html_wrap74008" SRC="img293.gif"  > '; 
$key = q/{picture}(300,140)(0,-40)put(0,70)tt(localtt(put(36,0)picture(160,80)put(0,70)tt(definett(ettz)tt(ftt(*ttzttz)))put(65,57)vector(0,1)10put(42,57)line(1,0)23put(42,45)line(0,1)12put(42,45)circle*3put(0,35)tt(definett(fttx)tt(+tt(*ttxttx)tt25))put(42,30)line(0,-1)08put(42,30)circle*3put(42,22)vector(-1,-1)38put(42,22)line(1,0)90put(82,22)vector(0,-1)12put(132,22)vector(0,-1)12put(0,0)tt(definett(gtty)tt(+tt(ftt(+ttytt1))tt(ftt(-ttytt1)))pictureput(22,-24)mboxldots(ttfldots)ldotsput(00,-50)mboxldots(ttfldots)ldotsput(00,+90)mboxldots(ttfldots)ldots{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=417 HEIGHT=240 ALIGN=BOTTOM ALT="picture25089" SRC="img125.gif"  > '; 
$key = q/{displaymath}y=2,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=274 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath73822" SRC="img250.gif"  > '; 
$key = q/{displaymath}120cdot10^1,;{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=285 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath74030" SRC="img306.gif"  > '; 
$key = q/{displaymath}2^N-1=frac12cdot2^N,,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=307 HEIGHT=35 ALIGN=BOTTOM ALT="displaymath73854" SRC="img262.gif"  > '; 
$key = q/{displaymath}1.0cdot10^16+1=1.00000000000000001cdot10^16{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=411 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath74078" SRC="img326.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(gtt1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=46 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74200" SRC="img347.gif"  > '; 
$key = q/{tabular}r|rmboxplainfactorial;SPMamp;mboxaccumulator-stylefactorialhline5.760;SPMamp;5.9705.780;SPMamp;5.9405.800;SPMamp;5.9805.820;SPMamp;5.9705.870;SPMamp;6.690hline5.806;SPMamp;6.110{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=351 HEIGHT=153 ALIGN=BOTTOM ALT="tabular42156" SRC="img300.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(resttt(constt10tt(constt22ttempty)))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=304 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline72950" SRC="img80.gif"  > '; 
$key = q/{picture}(223,82)put(0,0)thicklinespicture(223,82)put(5,75)mztpMeasurethesubcollectiondesignatedbyadatadefinitionT1QQpk7fHyNXZCPbzWqt-V1AOYZevtxWpsy35Vgok6j3QMEcCndFTquCDvKput(0,0)mztpMeasurethecollectionofallSchemedataT7jhBgoZ-PpBNZde+swTVA67pIgkYuzbJOEcC7Fxqbezier[35](173,40)(189,40)(189,32)qbezier[35](189,32)(189,24)(173,24)qbezier[22](173,24)(165,24)(165,32)qbezier[22](165,32)(165,40)(173,40)qbezier[80](199,64)(223,64)(223,32)qbezier[90](223,32)(223,0)(191,0)qbezier[124](191,0)(143,0)(143,40)qbezier[121](143,40)(143,64)(199,64)put(68,10)vector(3,1)72.0put(177,70)vector(0,-1)27picture{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=381 HEIGHT=133 ALIGN=BOTTOM ALT="picture4776" SRC="img48.gif"  > '; 
$key = q/{displaymath}x^3=27.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=283 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73630" SRC="img213.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(quick-sorttt(listtt5))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=208 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline73616" SRC="img210.gif"  > '; 
$key = q/{_inline}$1000cdotN${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=63 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline73864" SRC="img266.gif"  > '; 
$key = q/{}fboxits{}/;
$cached_env_img{$key} = ' <IMG WIDTH=18 HEIGHT=19 ALIGN=BOTTOM ALT="tex2html_wrap73198" SRC="img132.gif"  > '; 
$key = q/{_inline}$b^2=4cdotacdotc${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=88 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline72726" SRC="img43.gif"  > '; 
$key = q/{tabular}|l|l|l|hlinePhase;SPMamp;Goal;SPMamp;Activityhlinehlineminipage[t].60inData~~Analysis~~&Designminipage;SPMamp;minipage[t]1.6inraggedrighttoformulateagroupofdatadefinitionsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdeterminehowmanypiecesofdatadescribethe``interesting''aspectsofatypicalobjectmentionedintheproblemstatement_inline$bullet$_inlineaddastructuredefinitionandadatadefinition(foreachclassofproblemobject)rule0pt6ptminipagehlineminipage[t].60inContractHeader~&Purposeminipage;SPMamp;minipage[t]1.6inraggedrighttonametheprogram,tospecifyitsinputs&~~outputs,todescribeitspurpose,toformulateaprogram~~headerrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightnamethefunction,itsinputs,itsoutput,andspecifyitspurpose:~;;itname:itin1itin2ldots_inline$rightarrow$_inlineitout~;;bfPurpose:tocomputeldotsitx1ldotsitx2~(bfdefine(itnamex1x2ldots)ldots)rule0pt6ptminipagehlineExamples;SPMamp;minipage[t]1.6inraggedrighttocharacterizetheinput-outputrelationshipviaexamplesrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightsearchtheproblemforexamples_inline$bullet$_inlineaddexamplestofunction(input(s),output)_inline$bullet$_inlinevalidatetheexamplesusingtheproblemstatement,ifpossiblerule0pt6ptminipagehlineTemplate;SPMamp;minipage[t]1.6inraggedrighttoformulateanoutlineforagroupofprogramsrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightforthoseparametersthatstandforcompoundvalues,annotatethebodywithselectorexpressions_inline$bullet$_inlineifthefunctionisconditional,annotateallappropriatebranchesrule0pt6ptminipagehlineBody;SPMamp;minipage[t]1.6inraggedrighttodefinetheprogramrule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightdevelopaSchemeexpressionthatusesScheme'sprimitiveoperations,otherfunctions,selectorexpressions,andthevariablesrule0pt6ptminipagehlineTest;SPMamp;minipage[t]1.6inraggedrighttoexposemistakes~~(``typos''andlogic)rule0pt6ptminipage;SPMamp;minipage[t]2.2inraggedrightapplythefunctiontotheinputsoftheexamples_inline$bullet$_inlinecheckthattheoutputsareaspredictedaboverule0pt6ptminipagehline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=574 HEIGHT=724 ALIGN=BOTTOM ALT="tabular5123" SRC="img49.gif"  > '; 
$key = q/{_inline}$g(n)=1000cdotn${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=114 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73920" SRC="img282.gif"  > '; 
$key = q/{picture}(300,80)put(0,70)tt(definett(ettz)tt(ftt(*ttzttz)))put(65,57)vector(0,1)10put(42,57)line(1,0)23put(42,45)line(0,1)12put(42,45)circle*3put(0,35)tt(definett(fttx)tt(+tt(*ttxttx)tt25))put(42,30)line(0,-1)08put(42,30)circle*3put(42,22)line(1,0)90put(82,22)vector(0,-1)12put(132,22)vector(0,-1)12put(0,0)tt(definett(gtty)tt(+tt(ftt(+ttytt1))tt(ftt(-ttytt1))){picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=352 HEIGHT=126 ALIGN=BOTTOM ALT="picture24969" SRC="img124.gif"  > '; 
$key = q/{_inline}$n^2${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=17 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline73192" SRC="img137.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;var0;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=60 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline72912" SRC="img73.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(25,-4)mboxhspace-1ptLARGE(multiput(25,1)(0,-.1)25line(1,0)50invcnt=25advanceinvcntby50put(invcnt,-4)mboxhspace-4.5ptLARGE)put(225,-4)mboxhspace-1ptLARGE(multiput(225,1)(0,-.1)25line(1,0)50invcnt=225advanceinvcntby50put(invcnt,-4)mboxhspace-4.5ptLARGE)picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72664" SRC="img24.gif"  > '; 
$key = q/{displaymath}frac-b+sqrtb^2-4cdotacdotc2cdota{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=322 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath72996" SRC="img89.gif"  > '; 
$key = q/{tabular}lll1.tt(3tt+tt4);SPMamp;2.ttempty?(l);SPMamp;3.tt(x)htmladdimg..iconshand.right.gifhtmladdnormallinkSolution..Solutionslegal-ex1.htmlrule3pt5pt{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=353 HEIGHT=16 ALIGN=BOTTOM ALT="tabular7481" SRC="img57.gif"  > '; 
$key = q/{tabular}lll1.tt(definett(fttx)ttx);SPMamp;2.tt(definett(fttx)tty);SPMamp;3.tt(definett(fttxtty)tt3){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=569 HEIGHT=22 ALIGN=BOTTOM ALT="tabular7496" SRC="img58.gif"  > '; 
$key = q/{displaymath}a+W+S,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=295 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath73418" SRC="img182.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;field;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=70 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline74268" SRC="img363.gif"  > '; 
$key = q/{_inline}$x^2;SPMgt;c${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=49 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline73190" SRC="img136.gif"  > '; 
$key = q/{}fboxitIR-name{}/;
$cached_env_img{$key} = ' <IMG WIDTH=73 HEIGHT=23 ALIGN=BOTTOM ALT="tex2html_wrap73322" SRC="img148.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;exp;SPMgt;;SPMamp;=;SPMamp;(bflocal(mboxtt;SPMlt;def-1;SPMgt;ldotsmboxtt;SPMlt;def-n;SPMgt;)mboxtt;SPMlt;exp;SPMgt;){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=391 HEIGHT=17 ALIGN=BOTTOM ALT="tabular22688" SRC="img121.gif"  > '; 
$key = q/{_inline}$a_ij${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=19 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73814" SRC="img246.gif"  > '; 
$key = q/{_inline}$g_n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=17 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73370" SRC="img166.gif"  > '; 
$key = q/{_inline}$2^10=1024${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=80 HEIGHT=17 ALIGN=BOTTOM ALT="tex2html_wrap_inline73910" SRC="img278.gif"  > '; 
$key = q/{_inline}$ngeqmboxitbigEnough${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=110 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline73880" SRC="img269.gif"  > '; 
$key = q/{displaymath}0cdot10^0,,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=276 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath74038" SRC="img310.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(definett(fttx)ttx)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=149 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74226" SRC="img349.gif"  > '; 
$key = q/{_inline}$;SPMlt;_ir${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=23 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline73210" SRC="img141.gif"  > '; 
$key = q/{_inline}$x=yquadmboxorquadx;SPMlt;y.${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=153 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline72544" SRC="img18.gif"  > '; 
$key = q/{displaymath}3.14cdot5^2=3.14cdot25=78.5.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=352 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath72436" SRC="img7.gif"  > '; 
$key = q/{picture}(150,80)(0,20)put(0,80)tabularccccc1.10;SPMamp;1.12;SPMamp;1.08;SPMamp;1.09;SPMamp;1.11tabularput(5,75)line(1,0)85put(5,75)line(0,1)5put(90,75)line(0,1)5put(47,45)line(0,1)30put(40,25)mbox1.10put(35,70)line(1,0)85put(35,70)line(0,1)10put(120,70)line(0,1)10put(77,45)line(0,1)25put(68,25)_inline$frac329300$_inlineput(65,65)line(1,0)85put(65,65)line(0,1)15put(150,65)line(0,1)15put(105,45)line(0,1)20put(96,25)_inline$frac328300$_inline{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=232 HEIGHT=106 ALIGN=BOTTOM ALT="picture58905" SRC="img368.gif"  > '; 
$key = q/{picture}(200,200)multiput(0,0)(25,0)9line(0,1)200multiput(0,0)(0,25)9line(1,0)200put(37,62)circle*10multiput(00,61)(0,.5)6line(1,0)200multiput(00,24)(.5,.5)6line(1,1)173multiput(99.8,00)(.1,.1)6line(-1,1)100multiput(36,00)(.5,0)6line(0,1)200{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=320 HEIGHT=320 ALIGN=BOTTOM ALT="picture37267" SRC="img253.gif"  > '; 
$key = q/{_inline}$6cdotx^2${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=37 HEIGHT=17 ALIGN=BOTTOM ALT="tex2html_wrap_inline73632" SRC="img214.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;val;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=49 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline73058" SRC="img102.gif"  > '; 
$key = q/{}fboxmboxitCtt-;SPMgt;mboxitF{}/;
$cached_env_img{$key} = ' <IMG WIDTH=54 HEIGHT=24 ALIGN=BOTTOM ALT="tex2html_wrap73320" SRC="img147.gif"  > '; 
$key = q/{displaymath}fracb_na_nn.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=268 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath73818" SRC="img248.gif"  > '; 
$key = q/{_inline}$Sigma${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline73350" SRC="img158.gif"  > '; 
$key = q/{_inline}$underlinettf${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=9 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74236" SRC="img351.gif"  > '; 
$key = q/{}fboxldots(sffirstitalox)ldotsitaccumulator{}/;
$cached_env_img{$key} = ' <IMG WIDTH=218 HEIGHT=28 ALIGN=BOTTOM ALT="tex2html_wrap73970" SRC="img292.gif"  > '; 
$key = q/{_inline}$e^x${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=14 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline73380" SRC="img169.gif"  > '; 
$key = q/{displaymath}r-fracf(r0)f'(r0).{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=294 HEIGHT=41 ALIGN=BOTTOM ALT="displaymath73738" SRC="img225.gif"  > '; 
$key = q/{tabular}l|c|c|c|c|c|citN;SPMamp;1;SPMamp;10;SPMamp;50;SPMamp;100;SPMamp;500;SPMamp;1000hlinettF(_inline$1000cdotN$_inline);SPMamp;1000;SPMamp;10000;SPMamp;50000;SPMamp;100000;SPMamp;500000;SPMamp;1000000hlinettG(_inline$NcdotN$_inline);SPMamp;1;SPMamp;100;SPMamp;2500;SPMamp;10000;SPMamp;250000;SPMamp;1000000{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=511 HEIGHT=67 ALIGN=BOTTOM ALT="tabular38195" SRC="img265.gif"  > '; 
$key = q/{picture}(300,20)put(0,3)mbox0put(000,0)picture(50,10)put(50,3)mboxhspace-7pt50put(0,0)circle*3put(0,0)line(1,0)50put(50,0)circle*3pictureput(050,0)picture(40,10)put(40,3)mboxhspace-7pt90put(0,0)circle*3put(0,0)line(1,0)40put(40,0)circle*3pictureput(090,0)picture(70,10)put(70,3)mboxhspace-7pt160put(0,0)circle*3put(0,0)line(1,0)70put(70,0)circle*3pictureput(160,0)picture(30,10)put(30,3)mboxhspace-7pt190put(0,0)circle*3put(0,0)line(1,0)30put(30,0)circle*3pictureput(190,0)picture(30,10)put(30,3)mboxhspace-7pt220put(0,0)circle*3put(0,0)line(1,0)30put(30,0)circle*3picture{picture}/;
$cached_env_img{$key} = ' <IMG WIDTH=368 HEIGHT=20 ALIGN=BOTTOM ALT="picture39562" SRC="img290.gif"  > '; 
$key = q/{displaymath}1,frac12,frac13,frac14,ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=297 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath73348" SRC="img157.gif"  > '; 
$key = q/{}fboxldots(sfnode-rightitabt)ldotsitaccumulator{}/;
$cached_env_img{$key} = ' <IMG WIDTH=262 HEIGHT=28 ALIGN=BOTTOM ALT="tex2html_wrap74014" SRC="img296.gif"  > '; 
$key = q/{displaymath}g(N)=N^2,;{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=296 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath73886" SRC="img272.gif"  > '; 
$key = q/{_inline}$5^oC${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=28 HEIGHT=14 ALIGN=BOTTOM ALT="tex2html_wrap_inline72982" SRC="img86.gif"  > '; 
$key = q/{}fboxbox1{}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap73172" SRC="img127.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttftt(lambdatt(x)tt22))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=227 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74196" SRC="img345.gif"  > '; 
$key = q/{tabular}lll1.tt(x);SPMamp;2.tt(+tt1tt(notttx));SPMamp;3.tt(+tt1tt2tt3){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=312 HEIGHT=14 ALIGN=BOTTOM ALT="tabular7536" SRC="img60.gif"  > '; 
$key = q/{_inline}$x^3${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=16 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline73628" SRC="img212.gif"  > '; 
$key = q/{_inline}$b_i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline73816" SRC="img247.gif"  > '; 
$key = q/{displaymath}2cdot10^+4*8cdot10^+10=16cdot10^+14{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=364 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74070" SRC="img322.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;-8;SPMamp;-4;SPMamp;-12);SPMamp;(sflist;SPMamp;0;SPMamp;-5;SPMamp;-5);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=361 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath73796" SRC="img241.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(firsttt(constt22ttempty))${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=223 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline72952" SRC="img81.gif"  > '; 
$key = q/{displaymath}array|rrr|1;SPMamp;0;SPMamp;-12;SPMamp;0;SPMamp;91;SPMamp;1;SPMamp;1array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=294 HEIGHT=65 ALIGN=BOTTOM ALT="displaymath73962" SRC="img287.gif"  > '; 
$key = q/{_inline}$120cdot10^1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=61 HEIGHT=17 ALIGN=BOTTOM ALT="tex2html_wrap_inline74048" SRC="img313.gif"  > '; 
$key = q/{tabular}lcltt(fttv-1tt...ttv-n);SPMamp;=;SPMamp;ttexpwithallttx-1tt...ttx-nreplacedbyttv-1tt...ttv-n{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=617 HEIGHT=22 ALIGN=BOTTOM ALT="tabular7693" SRC="img63.gif"  > '; 
$key = q/{_inline}$g_n+1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=33 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline73368" SRC="img165.gif"  > '; 
$key = q/{displaymath}arrayll5cdotx;SPMamp;5cdotx;SPMamp;+17cdoty5cdotx;SPMamp;+17cdoty+3cdotzarray{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=324 HEIGHT=59 ALIGN=BOTTOM ALT="displaymath73084" SRC="img115.gif"  > '; 
$key = q/{displaymath}11cdot10^1,.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=280 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath74062" SRC="img319.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;prm;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=49 HEIGHT=19 ALIGN=MIDDLE ALT="tex2html_wrap_inline73056" SRC="img101.gif"  > '; 
$key = q/{displaymath}mboxitstep=fracmboxitwidth2,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=304 HEIGHT=37 ALIGN=BOTTOM ALT="displaymath73412" SRC="img179.gif"  > '; 
$key = q/{displaymath}y(x)=x^2-4cdotx+7.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=332 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath73484" SRC="img199.gif"  > '; 
$key = q/{displaymath}arraylclt(x);SPMamp;=;SPMamp;f'(r0)cdot(x-r0)+f(r0);SPMamp;=;SPMamp;f'(r0)cdotx+[f(r0)-f'(r0)cdotr0]array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=405 HEIGHT=40 ALIGN=BOTTOM ALT="displaymath73720" SRC="img223.gif"  > '; 
$key = q/{_inline}$r_i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=12 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline73718" SRC="img222.gif"  > '; 
$key = q/{displaymath}2^N{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=260 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath73850" SRC="img260.gif"  > '; 
$key = q/{_inline}$2^N-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=36 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline73852" SRC="img261.gif"  > '; 
$key = q/{_inline}$underlinex${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline73594" SRC="img209.gif"  > '; 
$key = q/{_inline}$O(2^N)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=46 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline73904" SRC="img277.gif"  > '; 
$key = q/{displaymath}3cdoty+9cdot2=21.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=315 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath73810" SRC="img244.gif"  > '; 
$key = q/{_inline}$[a,fraca+b2]${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=51 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline73404" SRC="img176.gif"  > '; 
$key = q/{_inline}$A^B${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=23 HEIGHT=16 ALIGN=BOTTOM ALT="tex2html_wrap_inline72420" SRC="img4.gif"  > '; 
$key = q/{}fboxldotsnldotsitaccumulator{}/;
$cached_env_img{$key} = ' <IMG WIDTH=156 HEIGHT=24 ALIGN=BOTTOM ALT="tex2html_wrap74010" SRC="img294.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(=tt1tt0)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=61 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline72860" SRC="img64.gif"  > '; 
$key = q/{_inline}$gcdott${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=28 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline72498" SRC="img15.gif"  > '; 
$key = q/{displaymath}arrayllrrrrr(sflist;SPMamp;(sflist;SPMamp;2;SPMamp;2;SPMamp;3;SPMamp;10);SPMamp;(sflist;SPMamp;;SPMamp;3;SPMamp;9;SPMamp;21);SPMamp;(sflist;SPMamp;;SPMamp;-3;SPMamp;-8;SPMamp;-19);SPMamp;)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=373 HEIGHT=60 ALIGN=BOTTOM ALT="displaymath73772" SRC="img233.gif"  > '; 
$key = q/{displaymath}mboxitwidth=frac(b-a)R.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=313 HEIGHT=38 ALIGN=BOTTOM ALT="displaymath73408" SRC="img178.gif"  > '; 
$key = q/{tabular}lclmboxtt;SPMlt;def;SPMgt;;SPMamp;=;SPMamp;_inline$newdimenpsetbox4=hbox$,,vert,,$p=wd4kernp$_inline(bfdefine(mboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefinemboxtt;SPMlt;var;SPMgt;mboxtt;SPMlt;exp;SPMgt;);SPMamp;;SPMamp;_inline$,,vert,,$_inline(bfdefine-structmboxtt;SPMlt;var;SPMgt;(mboxtt;SPMlt;var;SPMgt;ldotsmboxtt;SPMlt;var;SPMgt;)){tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=439 HEIGHT=61 ALIGN=BOTTOM ALT="tabular22665" SRC="img120.gif"  > '; 
$key = q/{tabular}l|l|l|l|;SPMamp;multicolumn3|cttaloshline;SPMamp;;SPMamp;tt(empty?ttalos);SPMamp;tt(cons?ttalos)cline2-4ttn;SPMamp;tt(=ttntt1);SPMamp;;SPMamp;cline2-4;SPMamp;tt(;SPMgt;ttntt1);SPMamp;;SPMamp;cline2-4{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=398 HEIGHT=88 ALIGN=BOTTOM ALT="tabular21053" SRC="img114.gif"  > '; 
$key = q/{displaymath}e^x=1+fracx1!+fracx^22!+fracx^33!+ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=356 HEIGHT=41 ALIGN=BOTTOM ALT="displaymath73378" SRC="img168.gif"  > '; 
$key = q/{_inline}$ncdot(n-1)cdotldotscdot21cdot1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=160 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73030" SRC="img96.gif"  > '; 
$key = q/{displaymath}x^n.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=263 HEIGHT=13 ALIGN=BOTTOM ALT="displaymath73046" SRC="img98.gif"  > '; 
$key = q/{_inline}$0.25cdot1100cdot400${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=125 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline72684" SRC="img35.gif"  > '; 
$key = q/{displaymath}mboxminipage4.5inAttchildisastructure:centerlinett(make-childttfttmttnattdattec)wherettfandttmarettchildstructures;ttnaandttecaresymbols;andttdaisanumber.minipage{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=517 HEIGHT=77 ALIGN=BOTTOM ALT="displaymath73064" SRC="img104.gif"  > '; 
$key = q/{_inline}$mboxreplacedbyttval-1tt...ttval-n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=237 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73592" SRC="img208.gif"  > '; 
$key = q/{_inline}$itbigEnough=1000${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=135 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline73894" SRC="img275.gif"  > '; 
$key = q/{displaymath}2cdot(-1)^2+4cdot(-1)+2=2-4+2=0.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=402 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath72706" SRC="img41.gif"  > '; 
$key = q/{_inline}$frac12cdotn^2+20${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=77 HEIGHT=32 ALIGN=MIDDLE ALT="tex2html_wrap_inline72454" SRC="img10.gif"  > '; 
$key = q/{}fbox_inline$;SPMlt;$_inline{}/;
$cached_env_img{$key} = ' <IMG WIDTH=25 HEIGHT=21 ALIGN=BOTTOM ALT="tex2html_wrap73200" SRC="img133.gif"  > '; 
$key = q/{_inline}$mboxtt;SPMlt;expression;SPMgt;${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=123 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline74262" SRC="img360.gif"  > '; 
$key = q/{displaymath}(mboxitright-mboxitleft)cdotfracf(mboxitright)+f(mboxitleft)2.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=371 HEIGHT=37 ALIGN=BOTTOM ALT="displaymath73708" SRC="img220.gif"  > '; 
$key = q/{displaymath}x^2+2cdotx+1=(1)^2+2cdot(1)+1=1+2+1=4,{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=440 HEIGHT=20 ALIGN=BOTTOM ALT="displaymath72644" SRC="img28.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttftt10)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=103 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74194" SRC="img344.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(75,-4)mboxhspace-1ptLARGE(multiput(75,1)(0,-.1)25line(1,0)100invcnt=75advanceinvcntby100put(invcnt,-4)mboxhspace-4ptLARGE]picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72658" SRC="img21.gif"  > '; 
$key = q/{displaymath}g_0,g_1,g_2,g_3,ldots,g_n,g_n+1,ldots{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=356 HEIGHT=13 ALIGN=BOTTOM ALT="displaymath73366" SRC="img164.gif"  > '; 
$key = q/{displaymath}f(N)=1000cdotN,;{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=316 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73884" SRC="img271.gif"  > '; 
$key = q/{displaymath}Wcdotf(a+S),{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=302 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath73414" SRC="img180.gif"  > '; 
$key = q/{_inline}$f(n)=2^n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=74 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73918" SRC="img281.gif"  > '; 
$key = q/{_inline}$f(n)=nlogn${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=106 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73924" SRC="img283.gif"  > '; 
$key = q/{}fboxbox2{}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=11 ALIGN=BOTTOM ALT="tex2html_wrap73174" SRC="img127.gif"  > '; 
$key = q/{_inline}$f(n)leqccdotg(n).${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=114 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline73882" SRC="img270.gif"  > '; 
$key = q/{}fboxitaccumulator{}/;
$cached_env_img{$key} = ' <IMG WIDTH=99 HEIGHT=24 ALIGN=BOTTOM ALT="tex2html_wrap74276" SRC="img366.gif"  > '; 
$key = q/{}framebox[numwidth]picture(300,30)(0,-20)put(0,0)line(1,0)300multiput(0,-5)(25,0)12line(0,1)10put(-03,-18)mbox_inline$0$_inlineput(123,-18)mbox_inline$5$_inlineput(245,-18)mbox_inline$10$_inlineput(75,-4)mboxhspace-1.5ptLARGE[multiput(75,1)(0,-.1)25line(1,0)150invcnt=75advanceinvcntby150put(invcnt,-4)mboxhspace-4.5ptLARGE)picture{}/;
$cached_env_img{$key} = ' <IMG WIDTH=510 HEIGHT=59 ALIGN=BOTTOM ALT="tex2html_wrap72662" SRC="img23.gif"  > '; 
$key = q/{_inline}$underlinemboxtt(set!ttnew-knowledgetttrue)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=249 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline74100" SRC="img331.gif"  > '; 
$key = q/{_inline}$2cdotn^2=102${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=87 HEIGHT=17 ALIGN=BOTTOM ALT="tex2html_wrap_inline72650" SRC="img31.gif"  > '; 
$key = q/{tabular}|l|l|l|hline;SPMamp;tt(empty?tta-list);SPMamp;tt(cons?tta-list)hlinett(empty?ttanother-list);SPMamp;;SPMamp;hlinett(cons?ttanother-list);SPMamp;;SPMamp;hline{tabular}/;
$cached_env_img{$key} = ' <IMG WIDTH=567 HEIGHT=68 ALIGN=BOTTOM ALT="tabular21729" SRC="img119.gif"  > '; 
$key = q/{displaymath}2cdotx^2+4cdotx+2=0{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=326 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath72698" SRC="img39.gif"  > '; 

1;