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TO MAPLE" }}{PARA 208 "" 0 "" {TEXT 245 0 "" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 97 " In this laboratory we will define functions and analyze their domains, ranges and zeros \+ using " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 2 " " }{TEXT 218 5 "Maple" }{TEXT 204 45 ". We will also conside r composite functions." }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 21 " restart;with (plots):" }}}{PARA 203 "" 0 "" {TEXT 204 29 " To define \+ the function " }{TEXT 218 1 "f" }{TEXT 204 3 " ( " }{TEXT 218 1 "x " }{TEXT 204 6 " ) = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting: -mfrac(Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", exec utable = \"true\", font_style_name = \"2D Input\", mathvariant = \"ita lic\"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", exe cutable = \"true\", font_style_name = \"2D Input\", mathvariant = \"it alic\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"3\", mat hvariant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\" −\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22222 22em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn( \"7\", mathvariant = \"normal\"), Typesetting:-mo(\"⁢\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace \+ = \"0.0em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x \", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2 \", mathvariant = \"normal\"), superscriptshift = \"0\")), Typesetting :-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Typesetting:-mo(\"−\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", r space = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \"true\", ma thvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mn(\"7\", mathvariant = \"normal\")), Typesetting:-mi( \"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-m n(\"50\", mathvariant = \"normal\")), linethickness = \"1\", denomalig n = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mr owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I&mfracGF$6(-F#6%-I #miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF6/%0font_style_nameGQ)2D~ InputF'/%,mathvariantGQ'italicF'-F#6*F0-F#6#-I%msupGF$6%-F16%Q\"xF'F4F <-I#mnGF$6$Q\"3F'/F=Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q(&m inus;F'FM/%&fenceGQ&falseF'/%*separatorGFX/%)stretchyGFX/%*symmetricGF 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= ( " }{XPPEDIT 18 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true \", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typeset ting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Type setting:-mi(\"\21036\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_ style_name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+ modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%tr ueF'/%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'it alicF'-F#6$-I#moGF$6-Q*&uminus0;F'/F8Q'normalF'/%&fenceGQ&falseF'/%*se paratorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGF D/%'accentGFD/%'lspaceGQ,0.2222222emF'/%'rspaceGFS-F,6%Q(∞F'F/F7 F+" }{TEXT 204 1 " " }{TEXT 204 4 " , " }{XPPEDIT 18 0 "Typesetting:- mrow(Typesetting:-mi(\"\21036\", italic = \"true\", mathvariant = \"it alic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I# miGF$6%Q(∞F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 204 1 " " }{TEXT 204 18 " ), and because " }{TEXT 218 2 "f " }{TEXT 204 11 " is of odd " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 25 " order it follows that " }{TEXT 227 5 "Range" } {TEXT 204 3 "( " }{TEXT 218 3 "f " }{TEXT 204 6 ") = ( " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mi(\"\21036\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\", italic = \"true\", executable = \+ \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F '/%'italicGQ%trueF'/%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,m athvariantGQ'italicF'-F#6$-I#moGF$6-Q*&uminus0;F'/F8Q'normalF'/%&fence GQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%. movablelimitsGFD/%'accentGFD/%'lspaceGQ,0.2222222emF'/%'rspaceGFS-F,6% Q(∞F'F/F7F+" }{TEXT 204 1 " " }{TEXT 204 5 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\21036\", italic = \"true\", ma thvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6#-I#miGF$6%Q(∞F'/%'italicGQ%trueF'/%,mathvariantGQ'i talicF'" }{TEXT 204 1 " " }{TEXT 204 20 " ). We may plot " }{TEXT 218 1 "f" }{TEXT 204 36 " over the the interval [-10 , 10] " }} {PARA 203 "" 0 "" {TEXT 204 1 " " }}{PARA 203 "" 0 "" {TEXT 204 29 " \+ with the following command:" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 27 "plot(f,-10..10,color=blu e);" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$!$+\"!\"\"$!%mL!\"#7$$!1nmm;p0k&*!#:$! 2-q^bqRr*H!#:7$$!1LL$3DZ+'4#!#97$$!1LL$3i.9!z!#:$!2iu;X R^3$=!#:7$$!1nm;/R=0v!#:$!2:T+bSz]g\"!#:7$$!1,+]P8#\\4(!#:$!2\"o9d')y# 3R\"!#:7$$!1mm;/siqm!#:$!20pL7&)p#*=\"!#:7$$!1****\\(y$pZi!#:$!2%4qcIq r25!#:7$$!1LLL$yaE\"e!#:$!18ksx)es%Ha!#:$!1-9e.1izq!# :7$$!2&******\\$*4)*\\!#;$!2Xm\\trFXv&!#;7$$!*Db\\c%!\")$!1M(fb[+()e%! #:7$$!*lSv9%!\")$!1R&3b!*pAh$!#:7$$!2jmmT?)[oP!#;$!2bxX4m/K%G!#;7$$!1L LL$=exJ$!#:$!1RZ1'H.^1#!#:7$$!2FLLLtIf$H!#;$!28\"*4h4uT^\"!#;7$$!,vju< \\#!#5$!1\\4V=&=%)))*!#;7$$!2OLLLB@')4#!#;$!1OD'[WDZ>'!#;7$$!2()***\\P 'psm\"!#;$!28h8_d!=&3$!#<7$$!2#****\\74_c7!#;$!0E-7X)[e&*!#;7$$!1:LL$3 x%z#)!#;$\"1[0y#p>Q#\\!#<7$$!0LL$3s$QM%!#:$\"2'[ekN\">j?\"!#<7$$!1^omm ;zr)*!#=$\"2nj[FtP=S\"!#<7$$\"2OML$ezw5V!#<$\"2Hr#\\9xkp5!#<7$$\"1.++v $Q#\\\")!#;$\"1;^=;N7bT!#<7$$\"1NLLe\"*[H7!#:$!1\"HG;x*)[!f!#<7$$\"2-+ ++qvxl\"!#;$!1)=VK'p(y'=!#;7$$\"21++]_qn2#!#;$!1/d+N!3@E$!#;7$$\"1,+]i &p@[#!#:$!2;X@>LaMm%!#<7$$\"+vgHKH!\"*$!18n&*zfd\"='!#;7$$\"2`mmmwanL$ !#;$!1o'p>*ehCu!#;7$$\"1,++]2goP!#:$!1OE.?QTK&)!#;7$$\"1KL$eR<*fT!#:$! 1Z\\:2)f9E*!#;7$$\"1-++])Hxe%!#:$!1w,#\\(Q)=n*!#;7$$\"1mm;H!o-*\\!#:$! 1D?+BUj2'*!#;7$$\"29++DTO5T&!#;$!1&4I6Qgp)*)!#;7$$\"1mmm;WTAe!#:$!19s8 nJj[x!#;7$$\"1****\\i!*3`i!#:$!1[\"*et!H;p&!#;7$$\"1NLLL*zym'!#:$!2&yu p))*Go)G!#<7$$\"1OLL3N1#4(!#:$\"1>-I6n(p2*!#<7$$\"1pm;HYt7v!#:$\"1zm \\QFK&o&!#;7$$\"1-+++xG**y!#:$\"1'yuWu-V5\"!#:7$$\"0nm;9@BM)!#9$\"2/(> ?t@^T=!#;7$$\"1OLLLbdQ()!#:$\"1pMp3]Z?E!#:7$$\",DOl5;*!#5$\"22'erG?8%e $!#;7$$\"1-+]P?Wl&*!#:$\"2Y\")*z@6LVY!#;7$$\"$+\"!\"\"$\"1)*********** Rf!#:-%%VIEWG6$;$!$+\"!\"\"$\"$+\"!\"\"%(DEFAULTG-%&COLORG6&%$RGBG$\" \"!!\"\"$\"\"!!\"\"$\"#5!\"\"-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UN CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }} {TEXT 246 0 "" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 27 " To locate the zeros of " }{TEXT 218 1 "f" }{TEXT 204 40 " we factor the polynomial expression " }{TEXT 218 1 "f" } {TEXT 204 3 " ( " }{TEXT 218 1 "x" }{TEXT 204 3 " ):" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 13 "fac tor(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#]!\"\",&%\"xG\"\" \"\"\"\"!\"\"\"\"\",&%\"xG\"\"\"\"\"(!\"\"\"\"\",&%\"xG\"\"\"\"\"\"\" \"\"\"\"\"\"\"\"" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 15 " Evidently, " }{TEXT 218 1 "f" }{TEXT 204 21 " \+ has three zeros: " }{TEXT 218 1 "x" }{TEXT 204 66 " = -1, 1, 7. Alte rnatively, we can find these zeros by writing a " }{TEXT 218 1 " " }} {PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 2 " " }{TEXT 218 5 "Maple" }{TEXT 204 33 " command to solve the equation " }{TEXT 218 1 "f" }{TEXT 204 3 " ( " }{TEXT 218 1 "x" }{TEXT 204 12 " \+ ) = 0 for " }{TEXT 218 2 " x" }{TEXT 204 2 " ." }}{PARA 203 "" 0 "" {TEXT 204 1 " " }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 19 "S:=sol ve(f(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG6%!\"\"\"\"\"\" \"(" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 88 " In order to refer to the first element in the list of soluti ons we may enter S [ 1 ]." }}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 5 "S[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 68 " As you would expect, the third solution is de noted by S[ 3 ] in " }{TEXT 218 6 "Maple " }{TEXT 204 1 "." }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 5 "S[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 29 " We may sketch a graph of " }{TEXT 218 1 "f" }{TEXT 204 68 " over a smaller inter val to show detailed behavior of the function." }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 26 "plot(f, -4..8,color=black);" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7U7$$!#S!\"\"$!2.++++++I$!#;7$$!+vq @pQ!\"*$!2ZYU<-Uq.$!#;7$$!*:M%QP!\")$!2P&HE$y9oy#!#;7$$!2(***\\iiSYi$! #;$!1\\x/Q>CzD!#:7$$!,D5Z3^$!#5$!1[U8Xq#4Q#!#:7$$!+l;!\\D$!\"*$!2JD;KY *yn>!#;7$$!2(*****\\lfs*H!#;$!1eaULbF'f\"!#:7$$!,DY,+]&=a!#;7$$!,DF;'[dL.g$!#;7$$!2)******pGf( [\"!#;$!1h\"Ht_v*e?!#;7$$!+:Lod7!\"*$!/-&of+\"3'*!#:7$$!1)******4'f))* *!#;$\"2%z%4zJZmk$!#?7$$!1,+++:t*Q(!#;$\"1`O+4!Hd-(!#<7$$!1)*******QC& )[!#;$\"1t9Q.&o-9\"!#;7$$!20++]AH4h#!#<$\"2<17\"\\:A`8!#<7$$\"1c++++4X $*!#=$\"2ArKA))3!)R\"!#<7$$\"20+++g:WQ#!#<$\"1R^u[oUv7!#;7$$\"1***** \\<_$\\]!#;$\"1\"\\olmr\"y'*!#<7$$\"1(******fs#3u!#;$\"2OA:p:kzk&!#=7$ $\"1.++v@Q'***!#;$\"1G4n7Gt\"o)!#?7$$\"2.++DX(3Y7!#;$!1()3G(\\l2O'!#<7 $$\"2/++]PJK]\"!#;$!2A$R*y!4'[Q\"!#<7$$\"2/++vwp$R!\"*$!2D7BJgo)zH!#<7$$\",v2Y'eA!#5$!1Y'4=l:$*)Q!#;7$$\",DI a*)[#!#5$!2O68CT!)oo%!#<7$$\"2.++]\\$pPF!#;$!0P&=()oqOb!#:7$$\"2'***** *>am%*H!#;$!0^k7*z\"HQ'!#:7$$\"2'*****\\JigC$!#;$!14)[C4,-;(!#;7$$\",v t,$*[$!#5$!1,**GK(pl%y!#;7$$\"+XwPfP!\"*$!1vho2qw6&)!#;7$$\"1******fG0 -S!#:$!1yb.K4p.!*!#;7$$\"*Xg6E%!\")$!1iZ6^:K)R*!#;7$$\"2&****\\P/&f\\% !#;$!1VL,zkMA'*!#;7$$\"1,++5zj_Z!#:$!1#\\yZ)f,.(*!#;7$$\"1****\\<3;%* \\!#:$!0*\\8Koh/'*!#:7$$\",v%=iY_!#5$!1W*y:w'Q-$*!#;7$$\"2&******\\'[M \\&!#;$!1v?Q]Wi\"z)!#;7$$\"1****\\PM&=v&!#:$!1*oZ8Uh!4!)!#;7$$\"1,++gz s+g!#:$!1yh3=-l'*p!#;7$$\"+0\"Q_D'!\"*$!1i(>^'oDzc!#;7$$\"1,+]x2k2l!#: $!2'3qX`0vrS!#<7$$\"1,++?EdRn!#:$!1B@LMEt8B!#;7$$\"+&o#R0q!\"*$\"1C4#4 VB^=&!#=7$$\"*KXJC(!\")$\"1Ohm:]g-D!#;7$$\",v@Rm\\(!#5$\"1@g4'=dG[&!#; 7$$\",DAl#Rx!#5$\"1)H%*3N')zq)!#;7$$\"#!)!\"\"$\"2$************f7!#;-% %VIEWG6$;$!#S!\"\"$\"#!)!\"\"%(DEFAULTG-%&COLORG6&%$RGBG$\"\"!!\"\"$\" \"!!\"\"$\"\"!!\"\"-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINE DG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 246 0 " " }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 30 " Now consider the function " }{TEXT 218 2 "g " }{TEXT 204 2 "( " }{TEXT 218 1 "x" }{TEXT 204 6 " ) = " }{XPPEDIT 18 0 "Typesetting:- mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", f ont_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting: -mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", \+ font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting :-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mat hvariant = \"italic\"), Typesetting:-mn(\"4\", mathvariant = \"normal \"), superscriptshift = \"0\")), Typesetting:-mo(\"−\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \" 0.2222222em\"), Typesetting:-msqrt(Typesetting:-mrow(Typesetting:-mn( \"1\", mathvariant = \"normal\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fa lse\", symmetric = \"false\", largeop = \"false\", movablelimits = \"f alse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.222 2222em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"i talic\")))), Typesetting:-mi(\"\", italic = \"true\", executable = \"t rue\", font_style_name = \"2D Input\", mathvariant = \"italic\"));" "- I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/% 'italicGQ%trueF'/%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,math variantGQ'italicF'-F#6&F+-F#6#-I%msupGF$6%-F,6%Q\"xF'F/F7-I#mnGF$6$Q\" 4F'/F8Q'normalF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q(−F'FH/%&fe nceGQ&falseF'/%*separatorGFS/%)stretchyGFS/%*symmetricGFS/%(largeopGFS /%.movablelimitsGFS/%'accentGFS/%'lspaceGQ,0.2222222emF'/%'rspaceGF\\o -I&msqrtGF$6#-F#6%-FE6$Q\"1F'FH-FN6-Q\"+F'FHFQFTFVFXFZFfnFhnFjnF]oFAF+ " }{TEXT 204 1 " " }{TEXT 204 27 " . We begin by defining " }{TEXT 218 1 "g" }{TEXT 204 5 " in " }{TEXT 218 5 "Maple" }{TEXT 204 1 "." } }{PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 20 "g:=x->x^4-sqrt(1+x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowG6\",&*$)%\"xG\"\"%\"\"\" \"\"\"-%%sqrtG6#,&\"\"\"\"\"\"%\"xG\"\"\"!\"\"6\"6\"6\"" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 13 " Note that " }{TEXT 227 6 "Domain" }{TEXT 204 2 " (" }{TEXT 218 2 " g" }{TEXT 204 15 " ) = [ -1 , +" }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting :-mi(\"\21036\", italic = \"true\", mathvariant = \"italic\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q(∞ F'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 204 1 " " }{TEXT 204 9 " ) since " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-msqr t(Typesetting:-mrow(Typesetting:-mn(\"1\", mathvariant = \"normal\"), \+ Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"x\", \+ italic = \"true\", mathvariant = \"italic\"))));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6#-I&msqrtGF$6#-F#6%-I#mnGF$6$Q\"1F' /%,mathvariantGQ'normalF'-I#moGF$6-Q\"+F'F4/%&fenceGQ&falseF'/%*separa torGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%' accentGF=/%'lspaceGQ,0.2222222emF'/%'rspaceGFL-I#miGF$6%Q\"xF'/%'itali cGQ%trueF'/F5Q'italicF'" }{TEXT 204 1 " " }{TEXT 204 35 " is a real nu mber if and only if " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting: -mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \+ \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting :-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typesettin g:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"1\", ma thvariant = \"normal\")), Typesetting:-mo(\"≤\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778 em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"itali c\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrow G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'itali cGQ%trueF'/%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvarian tGQ'italicF'-F#6&F+-F#6$-I#moGF$6-Q*&uminus0;F'/F8Q'normalF'/%&fenceGQ &falseF'/%*separatorGFF/%)stretchyGFF/%*symmetricGFF/%(largeopGFF/%.mo vablelimitsGFF/%'accentGFF/%'lspaceGQ,0.2222222emF'/%'rspaceGFU-I#mnGF $6$Q\"1F'FB-F?6-Q%≤F'FBFDFGFIFKFMFOFQ/FTQ,0.2777778emF'/FWFjn-F,6%Q \"xF'F/F7F+" }{TEXT 204 1 " " }{TEXT 204 25 " . If we try plotting " }{TEXT 218 1 "g" }{TEXT 204 62 " over the interval [ -1 , 10 ] we o btain the following graph:" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 26 "plot(g,-1..10,color=blue );" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$!#5!\"\"$\"# 5!\"\"7$$!1mmmTIJ-w!#;$!1ZIeWsLc:!#;7$$!1LLeR%)4;b!#;$!1Mj\\s4Pqd!#;7$ $!1nm;H>$*pJ!#;$!1:Z\")p3Xj\")!#;7$$!0mm;/N@3)!#;$!17+ZBQ&pe*!#;7$$\"2 _m;a3!GU:!#<$!2P1$*HD%yt5!#;7$$\"1KL3F&)[@P!#;$!1L_'[x1A:\"!#:7$$\"1** *\\PkKz(f!#;$!2l))z5gMj8\"!#;7$$\"1KL3x.b6$)!#;$!1Qd/5Irf()!#;7$$\".vo ToP1\"!#7$!27\">nj3Fu(!#:7$$\"2 '****\\7YF*)>!#;$\"2&[3[n'eIR\"!#:7$$\",Dk_)=A!#5$\"2b`m#=O[WA!#:7$$\" 1L$3x[JtU#!#:$\"1%*Q3$)HO'G$!#97$$\"2mmm\"**HBvE!#;$\"1#yCv8s.$\\!#97$ $\"2kmmm4Q_)G!#;$\"10e$)oXyKn!#97$$\".v$\\R_HJ!#7$\"1E8O))[())Q*!#97$$ \"2emm;KedM$!#;$\"21?(3&)>BK7!#97$$\".v$*p,Ie$!#7$\"2ltp+74ni\"!#97$$ \"2/+D\")\\8*3Q!#;$\"2(f;Hn(RG3#!#97$$\"1nmTg(GY/%!#:$\"1NFk%p:Pl#!#87 $$\"1n;a`*)3hU!#:$\"07L=^&ytK!#77$$\"1LLe90d%\\%!#:$\"1b%zR\")Hu0%!#87 $$\"1L$3xB#4PZ!#:$\"2%3lZ6Ig6]!#97$$\".D163#[\\!#7$\"1T3$4\\Z1(f!#87$$ \"1LL3P!>i<&!#:$\"1iJ:$)p#R:(!#87$$\"1*****\\jwq3,\"!#87$$\"1**\\PfK>le!#:$\"2V]%Q'pt 2=\"!#87$$\"2&***\\7%Gw7h!#;$\"2c8J6BTNR\"!#87$$\"1mmm@^@Nj!#:$\"2QqqC !f53;!#87$$\"2&****\\7/tsl!#;$\"2cv%orfbj=!#87$$\"1L$3xcazy'!#:$\"/twQ sB?@!#57$$\"1,+]<9DBq!#:$\"1BeP2xAIC!#77$$\"1m;/;ukWs!#:$\"2xV/t2&z^F! #87$$\".vo-qgZ(!#7$\"28Q+x:i47$!#87$$\"1mm;HzK-x!#:$\"1cPMjrg;N!#77$$ \"1**\\P%)*)>Rz!#:$\"1ZD)*[?!*pR!#77$$\"1MLLjRLn\")!#:$\"1E-wm^dYW!#77 $$\"1LLeH\\j+%)!#:$\"1o!#8 7$$\"1LLLVl@1$*!#:$\"11#4X6Jt\\(!#77$$\".v$\\feQ&*!#7$\"/G$f5b\\F)!#57 $$\".D17$*4w*!#7$\"0MK&*zBW2*!#67$$\"$+\"!\"\"$\"1X'4_P$o'***!#7-%%VIE WG6$;$!#5!\"\"$\"$+\"!\"\"%(DEFAULTG-%&COLORG6&%$RGBG$\"\"!!\"\"$\"\"! !\"\"$\"#5!\"\"-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 246 0 "" }} }{PARA 203 "" 0 "" {TEXT 204 32 " It appears that the zeros of " } {TEXT 218 1 "g" }{TEXT 204 32 " occur in the interval ( -1 , " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow( Typesetting:-mn(\"5\", mathvariant = \"normal\")), Typesetting:-mrow(T ypesetting:-mn(\"2\", mathvariant = \"normal\")), linethickness = \"1 \", denomalign = \"center\", numalign = \"center\", bevelled = \"false \"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I&mfra cGF$6(-F#6#-I#mnGF$6$Q\"5F'/%,mathvariantGQ'normalF'-F#6#-F16$Q\"2F'F4 /%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFA/%)bevell edGQ&falseF'" }{TEXT 204 1 " " }{TEXT 204 43 " ) so it makes sense t o redraw the graph " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 90 " over a shorter interval. Doing this will help us to estimate the locations of the zeros." }}{EXCHG {PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 232 34 "plot(g,-1..5/2 ,-5..10,color=blue);" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7V7$$!#5!\"\"$\"#5!\"\"7$$!1mm;/'* 4P#*!#;$\"1./em87=X!#;7$$!1L$e*[SIt&)!#;$\"1zj6JQJD;!#;7$$!1nm\"H_'zEy !#;$!1#3Yfg$H\"4*!#<7$$!1nmTg1Mvq!#;$!1Z1hjP'>!H!#;7$$!1K$ekLcuK'!#;$! 16sL$*Q@dW!#;7$$!1n;HK=2Mc!#;$!1c\"QC!R\"**f&!#;7$$!2'**\\iSB6;\\!#<$! 16]bD'Qga'!#;7$$!1m;H2wftT!#;$!1Dd$\\xo'Ht!#;7$$!.D\"GTYLM!#8$!1o'=WgV W'z!#;7$$!2BLL3(e9sE!#<$!1UhGIQI4&)!#;7$$!1mmTNjd,?!#;$!1-#**e;St#*)!# ;7$$!2&)***\\iQnY7!#<$!1UG&*oc]`$*!#;7$$!1(****\\(or')[!#<$!1c.m*)oa_( *!#;7$$\"2u++]iQ!=C!#=$!2F\"HPNw,75!#;7$$\"1QL3FkX^!*!#<$!2Ee>+$*4U/\" !#;7$$\"2wmm\"zJ#Rp\"!#<$!2FJ,L6h03\"!#;7$$\"2$omm;77iB!#<$!2#y\"H>bP( 36!#;7$$\"20+vVQ%RRJ!#<$!2w5x.Iel8\"!#;7$$\"2kmm;%GTFQ!#<$!1%>)e?1Wa6! #:7$$\"1-]PM\"yAe%!#;$!2$yUu\"\\#[j6!#;7$$\"1.]7.%)3,`!#;$!2uUYx.1!e6! #;7$$\"1omT5:4^g!#;$!1QB-8!eG8\"!#:7$$\"1o;a)[G)Rn!#;$!2xx@;@zu3\"!#;7 $$\"1LLekVs#[(!#;$!29-#*H\\@(35!#;7$$\"1M$3FR%Qa#)!#;$!1V,(fM>&o))!#;7 $$\"1,]i:n6E*)!#;$!1BZ\\iv-4u!#;7$$\"1NL3Fgg^'*!#;$!0Lq.DW3M&!#:7$$\", vu5,/\"!#5$!1k/e9^nzD!#;7$$\"2-+v=%[V86!#;$\"1$Q0<([C=$)!#<7$$\"2.]PMn zV=\"!#;$\"2Xht9vuv*[!#<7$$\".DJ\"=:j7!#7$\"2(=DxR-TT5!#;7$$\"2kmmT3KR L\"!#;$\"2(p\\(4\\`%Q;!#;7$$\"2/+]780&49!#;$\"14At\\vt%R#!#:7$$\"2K$3F Wb)zZ\"!#;$\"2lP(\\.=i(>$!#;7$$\"20+]Ps_Gb\"!#;$\"2:;=``^o@%!#;7$$\"2m ;/^!pHB;!#;$\"1Jrw$yaSK&!#:7$$\"2,](=s8$pp\"!#;$\"0V%=eZs\\m!#97$$\"2n m;H_A*o!#:$\"2E'zN?7Nz6!#:7$$\"2NLeR666*>!#;$\"2YzqDM'z)R\"!#:7 $$\"1nT5g&GZ1#!#:$\"2(f!fG$>MU;!#:7$$\"20++vMvB8#!#;$\"2K6*=8\"e0*=!#: 7$$\"1n;z*>1*4A!#:$\"1OYF4i(e?#!#97$$\"2NLL$=2DzA!#;$\"1#G\">\"p)psE#zA42$!#97$$\"2.]ilN_RU#!#;$\"1H&yvojrE$!#97$$ \"0D\"Gyh(>Y#!#9$\"1wibuU!z[$!#97$$\"#D!\"\"$\"1.8mIr;>P!#9-%%VIEWG6$; $!#5!\"\"$\"#D!\"\";$!#]!\"\"$\"$+\"!\"\"-%&COLORG6&%$RGBG$\"\"!!\"\"$ \"\"!!\"\"$\"#5!\"\"-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAIN EDG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 246 0 "" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 56 " Note that the second interval of values in the above " }{TEXT 227 4 "plot" }{TEXT 204 12 " command : " }{TEXT 227 9 "-5 . . 10" } {TEXT 204 22 ", specifies the range " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 35 " of values that appear along th e " }{TEXT 218 2 " y" }{TEXT 204 6 " axis." }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 25 " It is now evident that " }{TEXT 218 2 " g" }{TEXT 204 40 " has one zero in the interval ( -1 \+ , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \+ \"true\", executable = \"true\", font_style_name = \"2D Input\", mathv ariant = \"italic\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rs pace = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesett ing:-mn(\"1\", mathvariant = \"normal\")), Typesetting:-mrow(Typesetti ng:-mn(\"2\", mathvariant = \"normal\")), linethickness = \"1\", denom align = \"center\", numalign = \"center\", bevelled = \"false\")), Typ esetting:-mi(\"\", italic = \"true\", executable = \"true\", font_styl e_name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF' /%+executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italic F'-F#6$-I#moGF$6-Q*&uminus0;F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separa torGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%' accentGFD/%'lspaceGQ,0.2222222emF'/%'rspaceGFS-I&mfracGF$6(-F#6#-I#mnG F$6$Q\"1F'F@-F#6#-Ffn6$Q\"2F'F@/%.linethicknessGQ\"1F'/%+denomalignGQ' centerF'/%)numalignGFco/%)bevelledGFDF+" }{TEXT 204 1 " " }{TEXT 204 38 " ) and one in the interval ( 1 , " }{XPPEDIT 18 0 "Typesettin g:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mn(\"3\", ma thvariant = \"normal\")), Typesetting:-mrow(Typesetting:-mn(\"2\", mat hvariant = \"normal\")), linethickness = \"1\", denomalign = \"center \", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6#-I&mfracGF$6(-F#6#-I#mnGF$6$Q\"3 F'/%,mathvariantGQ'normalF'-F#6#-F16$Q\"2F'F4/%.linethicknessGQ\"1F'/% +denomalignGQ'centerF'/%)numalignGFA/%)bevelledGQ&falseF'" }{TEXT 204 1 " " }{TEXT 204 5 " ). " }}{PARA 203 "" 0 "" {TEXT 204 2 " " }} {PARA 203 "" 0 "" {TEXT 204 47 " We will attempt to find these values using a " }{TEXT 227 5 "solve" }{TEXT 204 18 " command as above:" }} {PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 16 "solve(g(x)=0,x);" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {PARA 12 "" 1 "" {XPPMATH 20 "6&,&\"\"\"!\"\"*$)-%'RootOfG6$,.F$F$*&\" \"%F$)%#_ZG\"\"#F$F%*&\"\"'F$)F/F-F$F$*&F-F$)F/F2F$F%*$)F/\"\")F$F$F/F %/%&indexG\"\"$F0F$F$,&F$F%*$)-F)6$F+/F:F$F0F$F$,&F$F%*$)-F)6$F+/F:F0F 0F$F$,&F$F%*$)-F)6$F+/F:F8F0F$F$" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 122 " This is a very complicated expres sion which involves the root of an eighth degree polynnomial-- it is n ot of use to us. " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 38 " In such a case we can utilize the " }{TEXT 227 7 "fsolve " }{TEXT 204 152 " command to approximate each of the roots \+ to any desired number of decimal places. (The default number of decima l places is 10, which we will use here.)" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 28 "fsolve(g(x)=0 , x, -1..-1/2);" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++K_;\")!#5" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 26 "fsolve(g(x)=0, x, 1..3/2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e:)p4\"!\"*" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 47 " When approxi mating the zeros of a function " }{TEXT 218 1 "g" }{TEXT 204 20 " we use the command" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 211 "" 0 "" {TEXT 227 59 " fsolve \+ ( g ( " }{TEXT 239 1 "x" }{TEXT 227 11 " ) = 0 , " }{TEXT 239 1 "x" }{TEXT 227 14 " , a . . b ) ;" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {PARA 203 "" 0 "" {TEXT 204 82 " where ( a , b ) is an interval know n to contain a single root to the equation " }{TEXT 218 2 "g " }{TEXT 204 1 "(" }{TEXT 218 3 " x " }{TEXT 204 6 ") = 0." }}{PARA 203 "" 0 " " {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 38 " Finally, to deter mine the range of " }{TEXT 218 1 "g" }{TEXT 204 33 ", we observe from our graph that " }{TEXT 227 3 "Ran" }{TEXT 204 1 "(" }{TEXT 218 3 " g " }{TEXT 204 182 ") contains all numbers larger than the minimum valu e of g(x).. In class we will develop an analytic method for finding t his minimum . For now, we can estimate the minimum value of " }{TEXT 218 1 "g" }{TEXT 204 3 " ( " }{TEXT 218 1 "x" }{TEXT 204 30 " ) by red rawing the graph of " }{TEXT 218 1 "g" }{TEXT 204 42 " over the shor t interval [0.45 , 0.60 ]." }}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 30 "plot(g,0.45..0.60,color= blue);" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"#X!\"#$!2(H#zy?`J;\"!#;7$$\"-DJdpKX! #7$!2aid)[_Ij6!#;7$$\"/v=7T9hX!#9$!27d+$o[Tj6!#;7$$\".v=HPJf%!#8$!2G#[ <@E^j6!#;7$$\".DJaU`i%!#8$!2&\\1%RQ$ej6!#;7$$\"/vVGZRdY!#9$!2^n&p*4HO;\"!#; 7$$\"/vofHq\\Z!#9$!2')*)eIk*ej6!#;7$$\"/v$f'HU\"y%!#9$!27E6]i@N;\"!#;7 $$\"-D\"*309[!#7$!2ap)4'R@M;\"!#;7$$\"2.+Dce*yU[!#<$!2LBb0)H]!#5$!2x.3:Ud>;\"!#;7$$\"/v=-p6j]!#9$ !2Cd(\\(z-;;\"!#;7$$\",vS.E4&!#6$!2:4$)f6J&!#9$!2=e;u ,7y:\"!#;7$$\".voo6AM&!#8$!2Xx@$y!)=d6!#;7$$\",vc&!#9$!10:R))\\x^6!#:7$$\"-vQ(zSf&!#7 $!2$zq2NG$3:\"!#;7$$\"/v=-,FCc!#9$!1$4Y5%)4*\\6!#:7$$\"1*\\P4tFel&!#;$ !19pE4o!*[6!#:7$$\"-D\"3\"o'o&!#7$!1R&4'yz)y9\"!#:7$$\"/voz;)*=d!#9$!1 P,Bs/yY6!#:7$$\"+&*44]d!#5$!2Y;LZ&QnX6!#;7$$\".DJw/>y&!#8$!2G-%z'>,X9 \"!#;7$$\"/v=(4bM\"e!#9$!2af,+*oHV6!#;7$$\",vdYC%e!#6$!24InDd`@9\"!#;7 $$\".Dc3uc(e!#8$!1K\\DW&*zS6!#:7$$\"1*****\\;$R0f!#;$!2=Fk5e[&R6!#;7$$ \"/v=-*zq$f!#9$!2)eGZeG&%\"xG6#\"\"!$\"+#)=f*o%!#5" }{TEXT 236 1 " " }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 56 " To ten significant digits the minimum value, gMin, \+ is " }}{EXCHG {PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 232 13 "gMin:=g(x[0])" }{MPLTEXT 1 232 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gMinG,&*$)&%\"xG6#\"\"!\"\"%\"\"\"\"\"\"*$),&\" \"\"\"\"\"&%\"xG6#\"\"!\"\"\"#\"\"\"\"\"#\"\"\"!\"\"" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 9 " Thus, " }{TEXT 227 3 "Ran" }{TEXT 204 2 "( " }{TEXT 218 1 "g" }{TEXT 204 15 " ) = [ gMin , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\21036\" , italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q(∞F'/%'italicGQ%t rueF'/%,mathvariantGQ'italicF'" }{TEXT 204 1 " " }{TEXT 204 4 " )." } }{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 39 " \+ We can define the composite funtion " }{TEXT 218 1 "h" }{TEXT 204 5 " = " }{TEXT 218 1 "f" }{TEXT 204 3 " o " }{TEXT 218 1 "g" }{TEXT 204 6 " in " }{TEXT 218 5 "Maple" }{TEXT 204 29 " with the followin g command:" }}{EXCHG {PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 232 15 "h:=x->(f@g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowG6\"--%\"@G6$%\"fG%\"gG6#% \"xG6\"6\"6\"" }}}{PARA 203 "" 0 "" {TEXT 204 1 " " }}{PARA 203 "" 0 " " {TEXT 204 23 " To see the value of " }{TEXT 218 1 "h" }{TEXT 204 3 " ( " }{TEXT 218 1 "x" }{TEXT 204 12 " ) we enter" }}{EXCHG {PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 232 5 "h(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"#]!\"\"),&*$)%\"xG\"\"%\" \"\"\"\"\"*$),&%\"xG\"\"\"\"\"\"\"\"\"#\"\"\"\"\"#\"\"\"!\"\"\"\"$\"\" \"\"\"\"*(\"\"(\"\"\"\"#]!\"\"),&*$)%\"xG\"\"%\"\"\"\"\"\"*$),&%\"xG\" \"\"\"\"\"\"\"\"#\"\"\"\"\"#\"\"\"!\"\"\"\"#\"\"\"!\"\"*&\"#]!\"\")%\" xG\"\"%\"\"\"!\"\"*&\"#]!\"\"),&%\"xG\"\"\"\"\"\"\"\"\"#\"\"\"\"\"#\" \"\"\"\"\"#\"\"(\"#]\"\"\"" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }} {PARA 203 "" 0 "" {TEXT 204 91 " If we would like to expand the above expression into a sum of simpler terms we may enter:" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 10 "ex pand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&\"#]!\"\")%\"xG\"#7\" \"\"\"\"\"**\"\"$\"\"\"\"#]!\"\"),&%\"xG\"\"\"\"\"\"\"\"\"#\"\"\"\"\"# \"\"\")%\"xG\"\")\"\"\"!\"\"*(\"\"$\"\"\"\"#]!\"\")%\"xG\"\"&\"\"\"\" \"\"*&\"#D!\"\")%\"xG\"\"%\"\"\"\"\"\"*(\"#]!\"\"),&%\"xG\"\"\"\"\"\" \"\"\"#\"\"\"\"\"#\"\"\"%\"xG\"\"\"!\"\"*(\"\"(\"\"\"\"#]!\"\")%\"xG\" \")\"\"\"!\"\"**\"\"(\"\"\"\"#D!\"\"),&%\"xG\"\"\"\"\"\"\"\"\"#\"\"\" \"\"#\"\"\")%\"xG\"\"%\"\"\"\"\"\"*(\"\"(\"\"\"\"#]!\"\"%\"xG\"\"\"!\" \"" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 60 " Note that the symbol \" % \" stands for \"the last output\"." }}{PARA 203 "" 0 "" {TEXT 204 111 " Following is first the graph of h , and the graphs of f , g , and h plotted on the same coordinate s ystem." }}{PARA 203 "" 0 "" {TEXT 204 2 " " }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 31 "plot(h,-1..2,-2..5,color=blue);" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6# 7]o7$$!#5!\"\"$\"\"!!\"\"7$$!+v`3Y$*!#5$\"+\\]'Rk*!#67$$!+cx6x()!#5$\" +!#57$$!*1Bt_'! 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We \+ will look more closely at the interval [0,1]:" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 35 "plot([ f,g],0..1,color=[red,green]);" }}{PARA 13 "" 1 "" {TEXT 246 0 "" } {GLPLOT2D 300 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"!!\"\"$\"\"! !\"\"7$$\"2mmmmT:(z@!#=$\"0MawHf6v%!#=7$$\"2NLLe9ui2%!#=$\"2t#*)>\"4,; m\"!#>7$$\"1nmm\"z_\"4i!#<$\"1'>E!RyNbQ!#=7$$\"1mmmT&phN)!#<$\"1rx!4%p b#)p!#=7$$\"2KL$e*=)H\\5!#<$\"22'Gm!pE55\"!#=7$$\"2mm;z/3uC\"!#<$\"2(o s+Qo-c:!#=7$$\"2)***\\7LRDX\"!#<$\"2n>G)30()4@!#=7$$\"2lm;zR'ok;!#<$\" 1m(3M!3=rF!#<7$$\"2)***\\i5`h(=!#<$\"1B:4y/&*>N!#<7$$\"2NLL$3En$4#!#<$ \"1/`)3*\\Y$Q%!#<7$$\"2nmmT!RE&G#!#<$\"1wqo66VA_!#<7$$\"+D.&4]#!#5$\"1 0w6GDvai!#<7$$\"+vB_!#<7$$\"2NLLe9Ege%!#<$\"2QpE5ejJ5#!#<7$$ \"2LL$eR\"3Gy%!#<$\"2&ek+q`_(G#!#<7$$\"2mmmT5k]*\\!#<$\"2VVt/am]\\#!#< 7$$\"1mm\"zRQb@&!#;$\"2WJ9!yS=?F!#<7$$\"-v=>Y2a!#7$\"2D9t-WkS#H!#<7$$ \"1mm;zXu9c!#;$\"2k9G*oc`_J!#<7$$\"*&y))Ge!\"*$\"1ixycLf(R$!#;7$$\"1** **\\i_QQg!#;$\"1=x$yl4ik$!#;7$$\"1***\\7y%3Ti!#;$\"1N]nCR6&*Q!#;7$$\"1 ****\\P![hY'!#;$\"14lGWq5\"=%!#;7$$\"1LLL$Qx$om!#;$\"0:bEpDnW%!#:7$$\" +v.I%)o!#5$\"2P^A`;f$RZ!#<7$$\"1mm\"zpe*zq!#;$\"1'e?k^\"e7]!#;7$$\"+D \\'QH(!#5$\"1DXTal/?`!#;7$$\"1KLe9S8&\\(!#;$\"10kl*Q.xh&!#;7$$\"1*** \\i?=bq(!#;$\"0-&o#3,v$f!#:7$$\"1LLL3s?6z!#;$\"1H&=$\\*>(ei!#;7$$\"-DJ XaE\")!#7$\"1F*Q=gsSg'!#;7$$\"1nmmm*RRL)!#;$\"//!o`ba%p!#97$$\"1mm;a<. Y&)!#;$\"1*\\AV(eY.t!#;7$$\"1LLe9tOc()!#;$\"1M.zaoRnw!#;7$$\"*&Qk\\*)! \"*$\"1\"G%=/Dh4!)!#;7$$\"1LL$3dg6<*!#;$\"1*z+;i=5T)!#;7$$\"1mmmmxGp$* !#;$\"1l4YD`Ny()!#;7$$\"-D\"oK0e*!#7$\"1K*\\ckg'y\"*!#;7$$\"-v=5s#y*!# 7$\"1/$pI0j,d*!#;7$$\"#5!\"\"$\"#5!\"\"-%&COLORG6&%$RGBG$\"#5!\"\"$\" \"!!\"\"$\"\"!!\"\"-%'CURVESG6$7S7$$\"\"!!\"\"$\"\"!!\"\"7$$\"2mmmmT:( z@!#=$\"2%R4fq]-z@!#=7$$\"2NLLe9ui2%!#=$\"2;K^%=-wrS!#=7$$\"1nmm\"z_\" 4i!#<$\"1iuv)31K>'!#<7$$\"1mmmT&phN)!#<$\"1wX.TbK<$)!#<7$$\"2KL$e*=)H \\5!#<$\"2p`\"o'48;/\"!#<7$$\"2mm;z/3uC\"!#<$\"2Ow.cD3XB\"!#<7$$\"2)** *\\7LRDX\"!#<$\"1%)*\\'RU>K9!#;7$$\"2lm;zR'ok;!#<$\"1o0,(3-Tj\"!#;7$$ \"2)***\\i5`h(=!#<$\"1_O(3dNC$=!#;7$$\"2NLL$3En$4#!#<$\"2%zJ)H;BI.#!#< 7$$\"2nmmT!RE&G#!#<$\"2%o*H!)fEl?#!#<7$$\"+D.&4]#!#5$\"1ejOQ;'zR#!#;7$ $\"+vB_0B?-iF!#;7$$\"2mm;z* ev:J!#<$\"2MYbKu%)z\"H!#<7$$\"1LLL347TL!#;$\"2$)Gu'=a'z4$!#<7$$\"1LLLL Y.KN!#;$\"1=c0(fBbC$!#;7$$\"-D\"o7Tv$!#7$\"1Va$*\\=?6M!#;7$$\"1LLL$Q*o ]R!#;$\"2<'obZ:D_N!#<7$$\"-D\"=lj;%!#7$\"2a+nn`\"o+P!#<7$$\"-vV&RL:m)>V!#;7$$\"-v=>Y2a!#7$\"1$o;j8,LT%!#;7$$\"1mm;zXu9c!#;$ \"2&Hy[(e1p]%!#<7$$\"*&y))Ge!\"*$\"2l6vB.vaf%!#<7$$\"1****\\i_QQg!#;$ \"2&zO7ir'Rn%!#<7$$\"1***\\7y%3Ti!#;$\"1Caz(Q/@u%!#;7$$\"1****\\P![hY' !#;$\"1R_!p@D'3[!#;7$$\"1LLL$Qx$om!#;$\"2tN.sk\"4g[!#<7$$\"+v.I%)o!#5$ \"20J'GmhE1\\!#<7$$\"1mm\"zpe*zq!#;$\"2ttx()R3-%\\!#<7$$\"+D\\'QH(!#5$ \"179]A(f'o\\!#;7$$\"1KLe9S8&\\(!#;$\"1yH.l$Gr)\\!#;7$$\"1***\\i?=bq(! #;$\"1>!**H-'z(*\\!#;7$$\"1LLL3s?6z!#;$\"2LhQp_s'**\\!#<7$$\"-DJXaE\") !#7$\"2w3:iysD*\\!#<7$$\"1nmmm*RRL)!#;$\"1YD.$o\")p(\\!#;7$$\"1mm;a<.Y &)!#;$\"13JOtI=_\\!#;7$$\"1LLe9tOc()!#;$\"29@Bmz!z=\\!#<7$$\"*&Qk\\*)! \"*$\"1A9@V?V!)[!#;7$$\"1LL$3dg6<*!#;$\"1$*)=\"4]]F[!#;7$$\"1mmmmxGp$* !#;$\"2ZX5)*zO@x%!#<7$$\"-D\"oK0e*!#7$\"092wZ_[q%!#:7$$\"-v=5s#y*!#7$ \"1HlV2leKY!#;7$$\"#5!\"\"$\"14%GT8([YX!#;-%&COLORG6&%$RGBG$\"\"!!\"\" $\"#5!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"#5!\"\"%(DEFAULTG-%*AXES STYLEG6#%'NORMALG-%(SCALINGG6#%.UNCONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" }}{TEXT 246 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 204 135 "The curves intersect at 0 and at a po int with X-coordinate between 0.6 and 0.8. To find this point we ill s olve the equation f(x)=g(x):" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 19 "solve(f(x)=g(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'Ro otOfG6#,(*$)%#_ZG\"\"%\"\"\"\"\"\"*$)-%$sinG6#%#_ZG\"\"#\"\"\"!\"\"*$) -%$sinG6#%#_ZG\"\"%\"\"\"\"\"\"" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 204 30 "We again need to use \"fsolve\":" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 232 29 "fsolve(f(x)= g(x),x,0.6..0.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?T2Aq!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 204 37 "Therefore, the solution set is [ 0 , " }{XPPEDIT 18 0 " Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:- mn(\"0.7022074120\", mathvariant = \"normal\")), mathvariant = \"norma l\", open = \"[\", close = \"]\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6#-I(mfencedGF$6&-F#6#-I#mnGF$6$Q-0.7022074120F'/ %,mathvariantGQ'normalF'F4/%%openGQ\"[F'/%&closeGQ\"]F'" }{TEXT 204 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 244 0 "" }}}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 209 "" 0 "" {TEXT 237 126 " \+ \+ " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 227 11 " Exercises" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 4 " 1." }{TEXT 218 7 " M aple " }{TEXT 204 18 "has an excellent \"" }{TEXT 227 4 "Help" }{TEXT 204 12 "\" facility. " }}{PARA 203 "" 0 "" {TEXT 204 55 " a) Find th e description of the command \"ifactor\" in " }{TEXT 227 4 "Help" } {TEXT 204 1 "." }{TEXT 204 65 " Execute the command \"ifactor(2520)\". What does the command do? " }}{PARA 203 "" 0 "" {TEXT 204 38 " b) Look up the command \"solve\" in " }{TEXT 227 4 "Help" }{TEXT 204 72 ", and use it to solve the following system of 4 equations in 4 unknow ns:" }}{PARA 203 "" 0 "" {TEXT 204 3 " " }{TEXT 204 19 "x + 2y - z - w = -4" }}{PARA 203 "" 0 "" {TEXT 204 22 " x - 2y + 3z - w = 8" }} {PARA 203 "" 0 "" {TEXT 204 24 " -2x -3y - 4z - w = -9" }}{PARA 203 "" 0 "" {TEXT 204 20 " x + y + z + w = 6" }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 2 " " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 26 " 2. Let f (x) = x \+ +1 - " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-msqrt(Typesettin g:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true\" , font_style_name = \"2D Input\", mathvariant = \"italic\"), Typesetti ng:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typesetting :-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo(\"+\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \" 0.2222222em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\"))));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I&msqrtGF$6#- F#6&-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF6/%0font_style_name GQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6%-I#mnGF$6$Q\"4F'/F=Q'normal F'-I#moGF$6-Q1⁢F'FE/%&fenceGQ&falseF'/%*separatorGFM/%) stretchyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM /%'lspaceGQ&0.0emF'/%'rspaceGFfn-F16%Q\"xF'F4F<-FH6-Q\"+F'FEFKFNFPFRFT FVFX/FenQ,0.2222222emF'/FhnF`o-FB6$Q\"1F'FE" }{TEXT 204 1 " " }{TEXT 204 1 "." }}{PARA 203 "" 0 "" {TEXT 204 51 " Plot the graph of f , and use the graph and " }{TEXT 218 7 "Maple's" }{TEXT 204 61 " zoom capability to find the domain, range and zeros of f . " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 34 " 3. Write t he function p( x ) = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:- mi(\"\", italic = \"true\", executable = \"true\", font_style_name = \+ \"2D Input\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting :-mi(\"tan\", italic = \"false\", mathvariant = \"normal\"), Typesetti ng:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesett ing:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true \", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typeset ting:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"tru e\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typese tting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typesett ing:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetti ng:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"ita lic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscripts hift = \"0\")), Typesetting:-mi(\"\", italic = \"true\", executable = \+ \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), \+ Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn( \"3\", mathvariant = \"normal\")), Typesetting:-mi(\"\", italic = \"tr ue\", executable = \"true\", font_style_name = \"2D Input\", mathvaria nt = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\", i talic = \"true\", executable = \"true\", font_style_name = \"2D Input \", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+executableGF1/%0 font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'-F#6%-F,6%Q$tanF' /F0Q&falseF'/F8Q'normalF'-I#moGF$6-Q0⁡F'FA/%&fenceGF@/%* separatorGF@/%)stretchyGF@/%*symmetricGF@/%(largeopGF@/%.movablelimits GF@/%'accentGF@/%'lspaceGQ&0.0emF'/%'rspaceGFW-I(mfencedGF$6$-F#6%F+-F #6&F+-F#6&-I#mnGF$6$Q\"4F'FA-FD6-Q1⁢F'FAFGFIFKFMFOFQFSF UFX-F#6#-I%msupGF$6%-F,6%Q\"xF'F/F7-F^o6$Q\"2F'FA/%1superscriptshiftGQ \"0F'F+-FD6-Q(−F'FAFGFIFKFMFOFQFS/FVQ,0.2222222emF'/FYFfp-F^o6$Q \"3F'FAF+FAF+" }{TEXT 204 1 " " }{TEXT 204 97 " as a composite f o g \+ o h of three functions. Check your answer by defining f , g , and h \+ in " }{TEXT 218 6 "Maple " }{TEXT 204 43 "and performing the composit ion using \"@\". " }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 38 " 4. Find all real zeros of q( x ) = " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", execu table = \"true\", font_style_name = \"2D Input\", mathvariant = \"ital ic\"), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", exec utable = \"true\", font_style_name = \"2D Input\", mathvariant = \"ita lic\"), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvariant = \"norma l\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-m row(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\"), Typesetting:-mn(\"6\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typesetting:-mi(\"\", italic = \"true\", \+ executable = \"true\", font_style_name = \"2D Input\", mathvariant = \+ \"italic\")), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesett ing:-mrow(Typesetting:-mn(\"8\", mathvariant = \"normal\"), Typesettin g:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting :-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"itali c\"), Typesetting:-mn(\"3\", mathvariant = \"normal\"), superscriptshi ft = \"0\")), Typesetting:-mi(\"\", italic = \"true\", executable = \" true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Ty pesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(T ypesetting:-mn(\"5\", mathvariant = \"normal\"), Typesetting:-mo(\"&In visibleTimes;\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo(\"−\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.22 22222em\"), Typesetting:-mn(\"19\", mathvariant = \"normal\")), Typese tting:-mi(\"\", italic = \"true\", executable = \"true\", font_style_n ame = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/%+ executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF'- F#6*F+-F#6&-I#mnGF$6$Q\"3F'/F8Q'normalF'-I#moGF$6-Q1⁢F' FB/%&fenceGQ&falseF'/%*separatorGFJ/%)stretchyGFJ/%*symmetricGFJ/%(lar geopGFJ/%.movablelimitsGFJ/%'accentGFJ/%'lspaceGQ&0.0emF'/%'rspaceGFY- F#6#-I%msupGF$6%-F,6%Q\"xF'F/F7-F?6$Q\"6F'FB/%1superscriptshiftGQ\"0F' F+-FE6-Q\"+F'FBFHFKFMFOFQFSFU/FXQ,0.2222222emF'/FenFho-F#6&-F?6$Q\"8F' FBFD-F#6#-Fin6%F[oF>FaoF+-FE6-Q(−F'FBFHFKFMFOFQFSFUFgoFio-F#6%-F ?6$Q\"5F'FBFDF[oFcp-F?6$Q#19F'FBF+" }{TEXT 204 1 " " }{TEXT 204 1 "." }}{PARA 203 "" 0 "" {TEXT 244 0 "" }}{PARA 203 "" 0 "" {TEXT 204 26 " \+ 5. Solve the inequality " }}{EXCHG {PARA 203 "" 0 "" {TEXT 204 45 " \+ " }{XPPEDIT 18 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"true \", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typeset ting:-mrow(Typesetting:-mi(\"\", italic = \"true\", executable = \"tru e\", font_style_name = \"2D Input\", mathvariant = \"italic\"), Typese tting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting: -mrow(Typesetting:-mi(\"cos\", italic = \"false\", mathvariant = \"nor mal\"), Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:- mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", ma thvariant = \"italic\")), mathvariant = \"normal\")), Typesetting:-mi( \"\", italic = \"true\", executable = \"true\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Typesetting:-mo(\"<\", mathvaria nt = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0 .2777778em\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \"f alse\", mathvariant = \"normal\"), Typesetting:-mo(\"⁡\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace \+ = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi( \"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \" normal\")), Typesetting:-mi(\"\", italic = \"true\", executable = \"tr ue\", font_style_name = \"2D Input\", mathvariant = \"italic\")), Type setting:-mi(\"\", italic = \"true\", executable = \"true\", font_style _name = \"2D Input\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modul enameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q!F'/%'italicGQ%trueF'/ %+executableGF1/%0font_style_nameGQ)2D~InputF'/%,mathvariantGQ'italicF '-F#6'F+-F#6&-F,6%Q\"xF'F/F7-I#moGF$6-Q\"+F'/F8Q'normalF'/%&fenceGQ&fa lseF'/%*separatorGFI/%)stretchyGFI/%*symmetricGFI/%(largeopGFI/%.movab lelimitsGFI/%'accentGFI/%'lspaceGQ,0.2222222emF'/%'rspaceGFX-F#6%-F,6% Q$cosF'/F0FIFE-FB6-Q0⁡F'FEFGFJFLFNFPFRFT/FWQ&0.0emF'/FZF _o-I(mfencedGF$6$-F#6#F>FEF+-FB6-Q\"