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"MATH 149" }}{PARA 257 "" 0 "" {TEXT 318 0 "" }{TEXT 320 12 "LABORATORY 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 321 0 "" }{TEXT 322 21 "INTRO DUCTION TO MAPLE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 " In this laboratory we \+ will define functions and analyze their domains, ranges and zeros usin g " }{TEXT 274 5 "Maple" }{TEXT -1 48 ". \n We will also consider \+ composite functions." }}{PARA 0 "" 0 "" {TEXT -1 29 " To define the f unction " }{TEXT 272 1 "f" }{TEXT -1 3 " ( " }{TEXT 275 1 "x" } {TEXT -1 6 " ) = " }{XPPEDIT 18 0 "(x^3-7*x^2-x+7)/50:" "6#*&,**$%\"x G\"\"$\"\"\"*&\"\"(F(*$F&\"\"#F(!\"\"F&F-F*F(F(\"#]F-" }{TEXT -1 7 " \+ in " }{TEXT 276 7 "Maple, " }{TEXT -1 8 "we enter" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 27 "f:=x->(x^3-7*x^2-x+7)/(50);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,**&#\"\"\"\"#]F/*$)9$\"\"$F/F/F/*&#\"\"(F0F/*$)F3\"\"#F/F/!\"\"*& #F/F0F/F3F/F;#F7F0F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " Note that since " }{TEXT 256 1 "f" }{TEXT -1 31 " is a polynomial, we know that" }{TEXT 281 7 " Domain" }{TEXT -1 3 "( " }{TEXT 278 1 "f" }{TEXT -1 9 " ) = ( " }{XPPEDIT 18 0 "-i nfinity" "6#,$%)infinityG!\"\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "inf inity" "6#%)infinityG" }{TEXT -1 18 " ), and because " }{TEXT 257 2 "f " }{TEXT -1 11 " is of odd " }}{PARA 0 "" 0 "" {TEXT -1 25 " order it follows that " }{TEXT 279 5 "Range" }{TEXT -1 3 "( " }{TEXT 280 3 "f " }{TEXT -1 6 ") = ( " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinit yG!\"\"" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "infinity" "6#%)infinityG " }{TEXT -1 20 " ). We may plot " }{TEXT 258 1 "f" }{TEXT -1 36 " \+ over the the interval [-10 , 10] " }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ with the following command:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(f,-10..10,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$!#5\"\"!$!3m***********fO$!#;7$$!3 !pmmm\"p0k&*!#<$!3A+DZ+'4#F-7$$!3OL L$3i.9!zF1$!3@Yn^%R^3$=F-7$$!3fmm;/R=0vF1$!3[6/]0%z]g\"F-7$$!3k++]P8# \\4(F1$!39o9d')y#3R\"F-7$$!3Kmm;/siqmF1$!3[!pL7&)p#*=\"F-7$$!3Q****\\( y$pZiF1$!3V4qcIqr25F-7$$!3jKLL$yaE\"eF1$!3R8ksx)es%Ha F1$!3+-9e.1izqF17$$!3]******\\$*4)*\\F1$!3_k'\\trFXv&F17$$!3o******\\_ &\\c%F1$!3)Rtfb[+()e%F17$$!3%)******\\1aZTF1$!3@R&3b!*pAh$F17$$!3Imm;/ #)[oPF1$!3]vd%4m/K%GF17$$!3%HLLL=exJ$F1$!33RZ1'H.^1#F17$$!3lKLLL2$f$HF 1$!3F6*4h4uT^\"F17$$!3%)****\\PYx\"\\#F1$!3A\\4V=&=%)))*!#=7$$!3gLLLL7 i)4#F1$!3]ND'[WDZ>'Fcq7$$!3o)***\\P'psm\"F1$!3M6O@v0=&3$Fcq7$$!3?**** \\74_c7F1$!3)*fA?^%)[e&*!#>7$$!3M:LL$3x%z#)Fcq$\"3K[0y#p>Q#\\Fcr7$$!3( )HLL3s$QM%Fcq$\"3e[ekN\">j?\"Fcq7$$!3]^omm;zr)*!#?$\"3rO'[FtP=S\"Fcq7$ $\"3fVLLezw5VFcq$\"3'Gr#\\9xkp5Fcq7$$\"3-.++v$Q#\\\")Fcq$\"3M;^=;N7bTF cr7$$\"3%\\LL$e\"*[H7F1$!35\"HG;x*)[!fFcr7$$\"3=++++dxd;F1$!3'z=VK'p(y '=Fcq7$$\"3e+++D0xw?F1$!37/d+N!3@E$Fcq7$$\"35,+]i&p@[#F1$!3k^9#>LaMm%F cq7$$\"3++++vgHKHF1$!3q7n&*zfd\"='Fcq7$$\"3ElmmmZvOLF1$!34o'p>*ehCuFcq 7$$\"3%4+++v+'oPF1$!3/OE.?QTK&)Fcq7$$\"3UKL$eR<*fTF1$!3)p%\\:2)f9E*Fcq 7$$\"3K-++])Hxe%F1$!3Kw,#\\(Q)=n*Fcq7$$\"3!fmm\"H!o-*\\F1$!33D?+BUj2'* Fcq7$$\"3X,+]7k.6aF1$!3m%4I6Qgp)*)Fcq7$$\"3#emmmT9C#eF1$!3S9s8nJj[xFcq 7$$\"33****\\i!*3`iF1$!3Q[\"*et!H;p&Fcq7$$\"3;NLLL*zym'F1$!3]yup))*Go) GFcq7$$\"3'eLL$3N1#4(F1$\"3A>-I6n(p2*Fcr7$$\"3,pm;HYt7vF1$\"3oym\\QFK& o&Fcq7$$\"37-+++xG**yF1$\"3%fyuWu-V5\"F17$$\"3gpmmT6KU$)F1$\"3Oq>?t@^T =F17$$\"3qNLLLbdQ()F1$\"3%*oMp3]Z?EF17$$\"3[++]i`1h\"*F1$\"3ogerG?8%e$ F17$$\"3A-+]P?Wl&*F1$\"3b9)*z@6LVYF17$$\"#5F*$\"3s(***********RfF1-%'C OLOURG6&%$RGBG$F*F*F`[l$\"*++++\"!\")-%+AXESLABELSG6$Q!6\"Fg[l-%%VIEWG 6$;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 " To locate the zeros of " }{TEXT 283 1 "f" } {TEXT -1 40 " we factor the polynomial expression " }{TEXT 277 1 "f " }{TEXT -1 3 " ( " }{TEXT 282 1 "x" }{TEXT -1 3 " ):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#]!\"\",&%\"xG\"\"\"F)F&F),&F(F)\"\"(F&F),&F(F)F) F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " Evidently, " }{TEXT 284 1 "f" }{TEXT -1 21 " has three zeros : " }{TEXT 285 1 "x" }{TEXT -1 66 " = -1, 1, 7. Alternatively, we ca n find these zeros by writing a " }{TEXT 268 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 314 5 "Maple" }{TEXT -1 33 " command to solve \+ the equation " }{TEXT 286 1 "f" }{TEXT -1 3 " ( " }{TEXT 287 1 "x" } {TEXT -1 12 " ) = 0 for " }{TEXT 288 2 " x" }{TEXT -1 2 " ." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "S:=solve(f(x)=0,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG6%\"\"\"\"\"(!\"\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 " In order to refe r to the first element in the list of solutions we may enter S [ 1 ]. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "S[1];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 68 " As you would expect, the third solution is denot ed by S[ 3 ] in " }{TEXT 313 6 "Maple " }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "S[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " We may sketch a graph of " }{TEXT 267 1 "f" }{TEXT -1 68 " \+ over a smaller interval to show detailed behavior of the function." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(f,-4..8,color=black); " }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6&-%'CURVESG6# 7U7$$!\"%\"\"!$!3E+++++++L!#<7$$!3!******\\2<#pQF-$!3ukCu@?/PIF-7$$!3z ******\\TVQPF-$!3u`HE$y9oy#F-7$$!3s***\\iiSYi$F-$!3#*[x/Q>CzDF-7$$!34+ +]-r%3^$F-$!37[U8Xq#4Q#F-7$$!3A+++l;!\\D$F-$!39`i@j%*yn>F-7$$!3o***** \\lfs*HF-$!3!zXDM`vif\"F-7$$!3%)****\\s@%3u#F-$!31#oc)G$*oo7F-7$$!3J++ ]U.6.DF-$!3@RhYY,+]&=aFen7$$!33++]sih[dL.g$Fen7$$!3%)****** pGf([\"F-$!3+h\"Ht_v*e?Fen7$$!3)******\\J$od7F-$!3z*>]of+\"3'*!#>7$$!3 'y******4'f))**Fen$\"3Sz%4zJZmk$!#@7$$!3y++++:t*Q(Fen$\"3Q`O+4!Hd-(Fjo 7$$!33)*******QC&)[Fen$\"3,t9Q.&o-9\"Fen7$$!3Y+++D#H4h#Fen$\"3mh?6\\:A `8Fen7$$\"3'e0+++!4X$*!#?$\"3?7FB#))3!)R\"Fen7$$\"3_++++cT%Q#Fen$\"3'* Q^u[oUv7Fen7$$\"3A*****\\<_$\\]Fen$\"3Q\"\\olmr\"y'*Fjo7$$\"3a'******f s#3uFen$\"3kB_\"p:kzk&Fjo7$$\"3e-++v@Q'***Fen$\"37G4n7Gt\"o)!#A7$$\"3I ++]_u3Y7F-$!3[()3G(\\l2O'Fjo7$$\"3P+++v8B.:F-$!3CKR*y!4'[Q\"Fen7$$\"3R ++]n(p$RF-$!3]AJ7.'o)zHFen7$$ \"3!*****\\xgkeAF-$!37Y'4=l:$*)QFen7$$\"3%)****\\-V&*)[#F-$!3g8JT7/)oo %Fen7$$\"3E+++&\\$pPFF-$!3up`=()oqObFen7$$\"3e******>am%*HF-$!3a4XE\"* z\"HQ'Fen7$$\"3k*****\\JigC$F-$!354)[C4,-;(Fen7$$\"3%*****\\P^'oDzcFen7$ $\"3q++]x2k2lF-$!3m3qX`0vrSFen7$$\"3d+++?EdRnF-$!3&H7KVjKPJ#Fen7$$\"3M +++&o#R0qF-$\"3wB4#4VB^=&Fcq7$$\"3++++?`9VsF-$\"38Ohm:]g-DFen7$$\"3G++ ]<#Rm\\(F-$\"3o?g4'=dG[&Fen7$$\"3F++]A_ERxF-$\"3%yH%*3N')zq)Fen7$$\"\" )F*$\"3M************f7F--%'COLOURG6&%$RGBG$F*F*F[\\lF[\\l-%+AXESLABELS G6$Q!6\"F_\\l-%%VIEWG6$;F(Fc[l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " Now consider the function \+ " }{TEXT 265 2 "g " }{TEXT -1 2 "( " }{TEXT 289 1 "x" }{TEXT -1 6 " ) \+ = " }{XPPEDIT 18 0 "x^4-sqrt(1+x)" "6#,&*$%\"xG\"\"%\"\"\"-%%sqrtG6#, &F'F'F%F'!\"\"" }{TEXT -1 27 " . We begin by defining " }{TEXT 310 1 "g" }{TEXT -1 5 " in " }{TEXT 259 5 "Maple" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g:=x->x^4-sqrt(1+x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*$)9$\"\"%\"\"\"F1-%%sqrtG6#,&F/F1F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " Note that " } {TEXT 290 6 "Domain" }{TEXT -1 2 " (" }{TEXT 291 2 " g" }{TEXT -1 15 " ) = [ -1 , +" }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 9 " ) since " }{XPPEDIT 18 0 "sqrt(1+x);" "6#-%%sqrtG6#,&\"\"\"F'%\"xG F'" }{TEXT -1 35 " is a real number if and only if " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{TEXT -1 28 " . \n If we try pl otting " }{TEXT 266 1 "g" }{TEXT -1 62 " over the interval [ -1 , 10 ] we obtain the following graph:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(g,-1..10,color=blue);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6&-%'CUR VESG6#7S7$$!\"\"\"\"!$\"\"\"F*7$$!3AmmmTIJ-w!#=$!3-ZIeWsLc:F07$$!3YLLe R%)4;bF0$!3oLj\\s4PqdF07$$!3Anm;H>$*pJF0$!3c9Z\")p3Xj\")F07$$!3UgmmT]8 #3)!#>$!3Y7+ZBQ&pe*F07$$\"3nj3Fu(FH7$$\"3g****\\7YF*)>FH$\"3Y[3[n'eIR\"!#;7$$\"3#***** \\UE&)=AFH$\"3`NlE=O[WAF[p7$$\"3'HL3x[JtU#FH$\"3'Q*Q3$)HO'G$F[p7$$\"3c mm;**HBvEFH$\"3\"=yCv8s.$\\F[p7$$\"3Ummm'4Q_)GFH$\"3K0e$)oXyKnF[p7$$\" 3y**\\P\\R_HJFH$\"31E8O))[())Q*F[p7$$\"3wlmm@$edM$FH$\"3g+s3&)>BK7!#:7 $$\"3-+]P*p,Ie$FH$\"3^O(p+74ni\"Fjq7$$\"3N+]7)\\8*3QFH$\"3sf;Hn(RG3#Fj q7$$\"3'om;/wGY/%FH$\"3yMFk%p:Pl#Fjq7$$\"3%pmTN&*)3hUFH$\"3E?J$=^&ytKF jq7$$\"3yKLe90d%\\%FH$\"3Gb%zR\")Hu0%Fjq7$$\"3mK$3xB#4PZFH$\"3T3lZ6Ig6 ]Fjq7$$\"3)***\\i5\"3#[\\FH$\"3gS3$4\\Z1(fFjq7$$\"3ULL3P!>i<&FH$\"3QiJ :$)p#R:(Fjq7$$\"3&*)****\\jwq3,\"!#97$$\"3O**\\PfK>leFH$\"3K/XQ'pt2=\"F]u7$$\"3Z*** \\7%Gw7hFH$\"3kN68J7a$R\"F]u7$$\"3*emm;7:_L'FH$\"3x.2Z-f53;F]u7$$\"3Y* ***\\7/tslFH$\"3gbZorfbj=F]u7$$\"3%GL3xcazy'FH$\"3#**Hn(QsB?@F]u7$$\"3 $4++vT^K-(FH$\"3yAeP2xAICF]u7$$\"3il;/;ukWsFH$\"3mPWIx]z^FF]u7$$\"3++] (o-qgZ(FH$\"3M\"Q+x:i47$F]u7$$\"3vlm;HzK-xFH$\"3$fvVL;2m^$F]u7$$\"3g)* \\P%)*)>RzFH$\"31ZD)*[?!*pRF]u7$$\"3;MLLjRLn\")FH$\"3CE-wm^dYWF]u7$$\" 38LLeH\\j+%)FH$\"3UoF ]u7$$\"3/LLLVl@1$*FH$\"3q0#4X6Jt\\(F]u7$$\"3P**\\P\\feQ&*FH$\"3_+G$f5b \\F)F]u7$$\"3o+]i?J*4w*FH$\"39SB`*zBW2*F]u7$$\"#5F*$\"3]W'4_P$o'***F]u -%'COLOURG6&%$RGBG$F*F*Fa[l$\"*++++\"!\")-%+AXESLABELSG6$Q!6\"Fh[l-%%V IEWG6$;F(Fiz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 32 " It appears \+ that the zeros of " }{TEXT 309 1 "g" }{TEXT -1 32 " occur in the int erval ( -1 , " }{XPPEDIT 18 0 "5/2" "6#*&\"\"&\"\"\"\"\"#!\"\"" } {TEXT -1 43 " ) so it makes sense to redraw the graph " }}{PARA 0 " " 0 "" {TEXT -1 90 " over a shorter interval. Doing this will help us to estimate the locations of the zeros." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(g,-1..5/2,-5.. 10,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6&-%'CURVESG6#7V7$$!\"\"\"\"!$\"\"\"F*7$$!3Umm;/'*4P#*!#=$\"3?./em87= XF07$$!3UL$e*[SIt&)F0$\"3&*yj6JQJD;F07$$!3%pm;H_'zEyF0$!3[#3Yfg$H\"4*! #>7$$!3)om;/mS`2(F0$!37Z1hjP'>!HF07$$!3]K$ekLcuK'F0$!3u5sL$*Q@dWF07$$! 3>n;HK=2McF0$!37c\"QC!R\"**f&F07$$!3e**\\iSB6;\\F0$!3q5]bD'Qga'F07$$!3 %em\"H2wftTF0$!3xCd$\\xo'HtF07$$!3,+]7GTYLMF0$!3Qo'=WgVW'zF07$$!3KKL$3 (e9sEF0$!3MUhGIQI4&)F07$$!3-mmTNjd,?F0$!3V-#**e;St#*)F07$$!3Y)***\\iQn Y7F0$!3uTG&*oc]`$*F07$$!3u'****\\(or')[F=$!3?c.m*)oa_(*F07$$\"3W2++D'Q !=CF=$!3n7HPNw,75!#<7$$\"3aPL3FkX^!*F=$!3i#e>+$*4U/\"F_p7$$\"3cnm;zJ#R p\"F0$!3m78I86c!3\"F_p7$$\"3Iomm;77iBF0$!3Ay\"H>bP(36F_p7$$\"3^+]P%Q%R RJF0$!3f2rP+$el8\"F_p7$$\"3SmmmTGTFQF0$!33%>)e?1Wa6F_p7$$\"3'=+vV8yAe% F0$!3IyUu\"\\#[j6F_p7$$\"3t-]7.%)3,`F0$!3UFkuPg+e6F_p7$$\"3[omT5:4^gF0 $!3-QB-8!eG8\"F_p7$$\"3;o;a)[G)RnF0$!3ux&o))F07$$\"3)4+Dcr;h#*)F 0$!3FBZ\\iv-4uF07$$\"3#[L$3Fgg^'*F0$!3+](=%[V86F_p$\"3_#Q0<([C=$)F=7$$\"3G+vVt'zV=\"F _p$\"3a9OZ^Zd(*[F07$$\"3#***\\78=:j7F_p$\"3o=DxR-TT5F_p7$$\"3Umm;%3KRL \"F_p$\"3tp\\(4\\`%Q;F_p7$$\"3V++DJ^]49F_p$\"3?4At\\vt%R#F_p7$$\"3=L3F Wb)zZ\"F_p$\"3Wwt\\.=i(>$F_p7$$\"3]++vBF&Gb\"F_p$\"3[h\"=``^o@%F_p7$$ \"3emT50pHB;F_p$\"3gIrw$yaSK&F_p7$$\"35+v=s8$pp\"F_p$\"3))HW=eZs\\mF_p 7$$\"3umm\"H_A*oF_p$\"3eizN?7Nz6!#;7$$\"3_L$eR666*>F_p$\"3g %zqDM'z)R\"Fbx7$$\"3;nT5g&GZ1#F_p$\"3uf!fG$>MU;Fbx7$$\"3Y++]Z`PK@F_p$ \"3C8\"*=8\"e0*=Fbx7$$\"3\"pm\"z*>1*4AF_p$\"3(fju#4i(e?#Fbx7$$\"3[LLL= 2DzAF_p$\"3-#G\">\"p)psE#zA42$Fbx7$$\" 3I+DccB&RU#F_p$\"3#*G&yvojrE$Fbx7$$\"39]7Gyh(>Y#F_p$\"3mvibuU!z[$Fbx7$ $\"3++++++++DF_p$\"39.8mIr;>PFbx-%'COLOURG6&%$RGBG$F*F*F^\\l$\"*++++\" !\")-%+AXESLABELSG6$Q!6\"Fe\\l-%%VIEWG6$;F($\"+++++D!\"*;$!\"&F*$\"#5F *" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " Note that the second interval of values in the above " }{TEXT 315 4 "plot " }{TEXT -1 12 " command : " }{TEXT 316 9 "-5 . . 10" }{TEXT -1 22 ", specifies the range " }}{PARA 0 "" 0 "" {TEXT -1 35 " of values tha t appear along the " }{TEXT 317 2 " y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 25 " It is now evident that " }{TEXT 260 2 " g" } {TEXT -1 40 " has one zero in the interval ( -1 , " }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 38 " ) and one in the \+ interval ( 1 , " }{XPPEDIT 18 0 "3/2" "6#*&\"\"$\"\"\"\"\"#!\"\"" } {TEXT -1 5 " ). " }}{PARA 0 "" 0 "" {TEXT -1 47 " We will attempt to find these values using a " }{TEXT 261 5 "solve" }{TEXT -1 18 " comma nd as above:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(g(x)=0 ,x);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&%\"xGF#F#F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " This time the " }{TEXT 325 5 "solve" }{TEXT -1 54 " c ommand does not give us a result that we can use.. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 " In such a case we ca n utilize the " }{TEXT 262 7 "fsolve " }{TEXT -1 83 " command to appr oximate each of the roots to any desired number of decimal places. " } }{PARA 0 "" 0 "" {TEXT -1 71 " (The default number of decimal places \+ is 10, which we will use here.)" }}{PARA 0 "" 0 "" {TEXT -1 47 " Whe n approximating the zeros of a function " }{TEXT 328 1 "g" }{TEXT -1 20 " we use the command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 326 59 " fso lve ( g ( " }{TEXT 329 1 "x" }{TEXT 330 11 " ) = 0 , " }{TEXT 331 1 "x" }{TEXT 332 14 " , a . . b ) ;" }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 " where ( a , b ) is a n interval known to contain a single root to the equation " }{TEXT 327 2 "g " }{TEXT -1 1 "(" }{TEXT 333 3 " x " }{TEXT -1 7 ") = 0.\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fsolve(g(x)=0, x, -1..-1/2) ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ !++K_;\")!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(g(x)=0, x, 1..3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e:)p4\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 " Finally, to determine the range of " } {TEXT 271 1 "g" }{TEXT -1 33 ", we observe from our graph that " } {TEXT 269 3 "Ran" }{TEXT -1 1 "(" }{TEXT 270 3 " g " }{TEXT -1 182 ") \+ contains all numbers larger than the minimum value of g(x).. In class \+ we will develop an analytic method for finding this minimum . For now , we can estimate the minimum value of " }{TEXT 298 1 "g" }{TEXT -1 3 " ( " }{TEXT 299 1 "x" }{TEXT -1 30 " ) by redrawing the graph of \+ " }{TEXT 300 1 "g" }{TEXT -1 28 " over the short interval \n" }} {PARA 0 "" 0 "" {TEXT -1 24 " [0.45 , 0.60 ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(g,0. 45..0.60,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$\"35+++++++X!#=$!3qH#zy?`J;\"!#<7$$\"3 #****\\7t&pKXF*$!3UDw&)[_Ij6F-7$$\"3;+v=7T9hXF*$!3;r0Io[Tj6F-7$$\"3=+] (=HPJf%F*$!3yA[<@E^j6F-7$$\"3/+]7VDMDYF*$!3_\\1%RQ$ej6F-7$$\"3-+vVGZRd YF*$!39vcp*4HO;\"F-7$$\"37+vofHq\\ZF*$!3c)*)eIk*ej6F-7$$\"3;+v$f'HU \"y%F*$!3Ah7,D;_j6F-7$$\"3>++D\"*309[F*$!3W&p)4'R@M;\"F-7$$\"3H+]i&e*y U[F*$!3GL_v\"Q2L;\"F-7$$\"3)****\\([D9v[F*$!3Q`R4O)\\J;\"F-7$$\"30++Dc $Gw!\\F*$!3gW8ma,'H;\"F-7$$\"3(****\\7XM*Q\\F*$!3Scn\"[8ZF;\"F-7$$\"3# )*\\(o%Qjt'\\F*$!3=1uc%oFD;\"F-7$$\"3X++DO\"o6+&F*$!3=e?9KTBi6F-7$$\"3 ,+++&>0)H]F*$!3sP!3:Ud>;\"F-7$$\"3++v=-p6j]F*$!3Wsv\\(z-;;\"F-7$$\"3S+ +]2Mg#4&F*$!3[\"4$)f6J&F*$!3!=e;u,7y:\"F-7$$\" 3E+](oo6AM&F*$!3]uc&F*$!3%\\]\"R))\\x^6F-7$$\"3 k***\\(Q(zSf&F*$!3Mzq2NG$3:\"F-7$$\"3o*\\(=-,FCcF*$!31$4Y5%)4*\\6F-7$$ \"3O*\\P4tFel&F*$!3#R\"pE4o!*[6F-7$$\"3;++D\"3\"o'o&F*$!31R&4'yz)y9\"F -7$$\"3Q+voz;)*=dF*$!3!p8IAZ!yY6F-7$$\"35+++&*44]dF*$!3kkJtaQnX6F-7$$ \"3!)**\\7jZ!>y&F*$!3&G-%z'>,X9\"F-7$$\"3()*\\(=(4bM\"eF*$!3S&f,+*oHV6 F-7$$\"32++]xlWUeF*$!3)3InDd`@9\"F-7$$\"3/+]i&3uc(eF*$!3\">$\\DW&*zS6F -7$$\"36*****\\;$R0fF*$!3#=Fk5e[&R6F-7$$\"3\")*\\(=-*zq$fF*$!3weGZeG " 0 "" {MPLTEXT 1 0 20 "x[0]:= 0.4689591882;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!$\"+#)=f*o%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 57 "\n To ten significant digits the minimum value, gMin, is " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gMin:=g(x[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gMinG$!+j-kj6 !\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ Thus, " }{TEXT 301 3 "Ran" }{TEXT -1 2 "( " }{TEXT 302 1 "g" }{TEXT -1 15 " ) = [ gMin , " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" } {TEXT -1 4 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " We can define the composite funtion " }{TEXT 305 1 "h " }{TEXT -1 5 " = " }{TEXT 303 1 "f" }{TEXT -1 3 " o " }{TEXT 304 1 "g" }{TEXT -1 6 " in " }{TEXT 306 5 "Maple" }{TEXT -1 29 " with the following command:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "h:=x->(f@g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(--%\"@G6$%\"fG %\"gG6#9$F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 " To see the value of " }{TEXT 307 1 "h" }{TEXT -1 3 " ( " }{TEXT 308 1 "x" }{TEXT -1 12 " ) we enter" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"#]!\"\",&*$)%\"xG\"\"%\"\"\"F,*$,&F*F, F,F,#F,\"\"#F&\"\"$F,*(\"\"(F,F%F&F'F0F&*&F%F&F*F+F&*&F%F&F.F/F,#F3F%F ," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " \nIf we would like to expan d the above expression into a sum of simpler terms we may enter:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&\"#]!\"\"%\"xG\"#7\"\"\"**\"\"$F)F%F&,&F'F)F)F)# F)\"\"#F'\"\")F&*(F+F)F%F&F'\"\"&F)*&\"#DF&F'\"\"%F)*(F%F&F,F-F'F)F&*( \"\"(F)F%F&F'F/F&**F7F)F3F&F,F-F'F4F)*(F7F)F%F&F'F)F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Note that the symb ol \" % \" stands for \"the last output\"." }}{PARA 0 "" 0 "" {TEXT -1 111 " Following is first the graph of h, and the graphs of f , g , and h plotted on the same coordinate system." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(h,-1..2,-2..5,color=blue);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7]o7$!\"\"$\" 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39 "can also be used to so lve inequalities." }}{PARA 0 "" 0 "" {TEXT -1 41 " As an example, con sider the inequality " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "x^2 < sin(x) *cos(x);" "6#2*$%\"xG\"\"#*&-%$sinG6#F%\"\"\"-%$cosG6#F%F+" }{TEXT -1 11 ", x in [-" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 "]." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 " Graph the functions together:" }{MPLTEXT 1 0 2 " \+ " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f:=x->x^ 2; g:=x->sin(x)*cos(x); plot([f,g],-Pi..Pi,color=[red,green]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(*&-%$sinG6#9$\"\"\"-%$cosGF/F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 302 227 227 {PLOTDATA 2 "6&-%'CURVESG6$7S 7$$!3*****4tk#fTJ!#<$\"3u`AjhVgp)*F*7$$!3w\"*pr)3PY+$F*$\"3*)>P*[.Wy-* 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To find this point we w ill solve the equation f(x)=g(x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(f(x)=g(x),x);" }}{PARA 7 "" 1 "" {TEXT -1 38 "W arning, solutions may have been lost\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "\nWe again need to use \"fsolve\":" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 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 0 "" 0 "" {TEXT -1 52 "Therefore, th e solution set is \{ 0, 0.7022074120 \}." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "98 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }