{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warni ng" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 " " 0 "" {TEXT -1 73 "Lesson 16: Analysing the Graphs of Functions 2 - \+ \nExtrema and Asymptotes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "For each function, we find: " }}{PARA 0 " " 0 "" {TEXT -1 42 " - intervals of increase or decrease " }} {PARA 0 "" 0 "" {TEXT -1 16 " - extrema " }}{PARA 0 "" 0 "" {TEXT -1 31 " - intervals of concavity " }}{PARA 0 "" 0 "" {TEXT -1 29 " - points of inflection " }}{PARA 0 "" 0 "" {TEXT -1 19 " \+ - asymptotes " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 9 "Example 1" }{TEXT -1 41 "\nf(x) = sin(2x) - 2 s in(x), x in [ -" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 15 " ] " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f1:= x -> sin (2 * x) - 2 * sin(x);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#f1GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$sinG6#,$9$\"\"#\"\"\"*& F2F3-F.6#F1F3!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(f1(x), x=-Pi- .5..Pi + .5, color = red);" }{TEXT -1 0 "" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7ho7 $$!3&****4ok#fTO!#<$!3*)p.zU?K+=F*7$$!37A@2M*RG[$F*$!3-[F(>>c+I\"F*7$$ !3rz&=3*\\x8MF*$!3[2RiZGab5F*7$$!3JP]cZ+rWLF*$!3s%3Yb%e+')z!#=7$$!3e.& *f&HRqE$F*$!3wt:.180&)\\F<7$$!3EqRjV&o$*=$F*$!3^))>)R!)=#4>F<7$$!3s#[1 &RH=6JF*$\"34%RHp%*>f@\"F<7$$!3j&**y`L(*H.$F*$\"3*Q%zrm6^AVF<7$$!3m')* zPL$=bHF*$\"3?>\"*f180\\tF<7$$!37y4=K$pt(GF*$\"3zN!p$H<^E5F*7$$!38^J\" =zD_!GF*$\"3-ip8(*eC$G\"F*7$$!38C`W^A3LFF*$\"3\">@Uwg&fB:F*7$$!3gRuxO8 o$e#F*$\"3Em/yuL5d>F*7$$!3&pp2rpq\"HCF*$\"3I\"G?V*\\q'H#F*7$$!3S=L1OJ; _BF*$\"36yYSZ2()>CF*7$$!3%)R*=]db^F#F*$\"3bk^skt%4^#F*7$$!3YUc./(\\bB# F*$\"3Kk%)e;&o^a#F*7$$!33XB0LQ%f>#F*$\"3[s#\\C@J3d#F*7$$!3;[!p?'zLc@F* $\"3%)RT8V%)*ze#F*7$$!3y]d3\"4Kn6#F*$\"3+\")=S9Xx'f#F*7$$!3sPdO=s%=3#F *$\"3I^.s\\\"owf#F*7$$!3ACdkXB'p/#F*$\"3Up5_-vG#f#F*7$$!3:6d#HZx?,#F*$ \"3K&o?[Pz2e#F*7$$!3)yp0-g#>x>F*$\"3'fS09\\7Lc#F*7$$!3i\"\\fqU36xjT#F*7$$!3co$)GY2Pi;F*$\"3k l`UImut@F*7$$!3q?(>CNl.^\"F*$\"37P8p&H\"yv=F*7$$!3C3+T#*)HBP\"F*$\"3WR hH&p[Td\"F*7$$!3,DHQzB>37F*$\"3Eq9!*e;m17F*7$$!3'p&f*ePY\"p5F*$\"3F82O OLJ.\"*F<7$$!3+U(Rc=GS2*F<$\"3sT<`C*pV0'F<7$$!3-S3MROKUwF<$\"3K'GCE>>' [QF<7$$!3U&=(3Ro^rgF<$\"3Q>CQ(RX$R?F<7$$!39Gdz,ttvXF<$\"3Av()=JwH*3*!# >7$$!3$\\t4AF[],$F<$\"3%>@fj$)3\"zEFbv7$$!3g;Em[&[=e\"F<$\"39;+(QVtM$R !#?7$$!3];u'f#G!\\f$F]w$\"3[mr&RA&zXY!#D7$$\"3_pQH1h!)p:F<$!3p)zsS^&oW QF]w7$$\"3/;'H8n?w'HF<$!3;%Gc*\\'ykb#Fbv7$$\"3'3e\\fp)HxWF<$!3Hg-:B\"* RM&)Fbv7$$\"3N+I[H_Fc(F<$!3&)e`J(f% HTPF<7$$\"3f,E`Y/0R!*F<$!39d(ow6KX*fF<7$$\"3k_$HXyAy1\"F*$!3Yd'yLvoj2* F<7$$\"3>DSMk,6:7F*$!3OL+^`;%>A\"F*7$$\"3_Nz:!)3Ps8F*$!39(fC(f/Cu:F*7$ $\"3rtMAiC([^\"F*$!3%=4OBme_)=F*7$$\"3/[8-JVmq;F*$!3_$Q\")HEX%)=#F*7$$ \"303fTQBD<=F*$!3(4Q&yE&fFT#F*7$$\"3Oyn)4plQ*=F*$!3q<>!R')Q')\\#F*7$$ \"3q[wbV!z/(>F*$!3y'e#)o%GGfDF*7$$\"3@8'\\9#3$z+#F*$!3M%4]$z;,zDF*7$$ \"3Ix:M*f#QX?F*$!3Ak&4yi(*=f#F*7$$\"3OTNBxV$G3#F*$!3wk(R;wHxf#F*7$$\"3 *e]D^:'G?@F*$!3_;)z]@Ejf#F*7$$\"3#REGdp%\\f@F*$!3!zv-p`Rpe#F*7$$\"3#>- JjB.()>#F*$!3CFvaR&>$pDF*7$$\"3'*zP$px6zB#F*$!3;A/H')zOVDF*7$$\"3)z`Ov J?rF#F*$!3%)\\?Z!QE!4DF*7$$\"3sG'GoGXEN#F*$!3!olzCy)>>CF*7$$\"3Y>27c-< GCF*$!3yq`K'e2&)H#F*7$$\"3!Q2\\\\jSEe#F*$!3![rT#*G\"yf>F*7$$\"3O=.0@>$ et#F*$!3mPoAe5x9:F*7$$\"3(p3)4i`@1GF*$!3?C#ev>G)z7F*7$$\"3ebe9.))fwGF* $!3luR\"fRH$H5F*7$$\"3oYQNphEdHF*$!3u&e_`n\"GpsF<7$$\"3wP=cNN$z.$F*$!3 v')3g5Y$y7%F<7$$\"3,>4a*Q$35JF*$!37$[]*e#\\)f7F<7$$\"3C++_VKB#=$F*$\"3 b6%HB<0Xi\"F<7$$\"3%HV:j.g\"fKF*$\"3K)=iy3vcn%F<7$$\"3il36Ho3OLF*$\"3G qK&)>3$zl(F<7$$\"39$o`ae:(4MF*$\"3s!>w$HivS5F*7$$\"3m+lzTVM$[$F*$\"3iN pNt#*y,8F*7$$\"3U]K(e\\oCc$F*$\"3WB**>)>JHc\"F*7$$\"3@++&*\\EfTOF*$\"3 i(o%p^?K+=F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!FjclFicl-%+AXESLABEL SG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+aEfTO!\"*$\"+aEfTOFhdl%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "D(f1);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&-% $cosG6#,$9$\"\"#F0*&F0\"\"\"-F,6#F/F2!\"\"F&F&F&" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 24 "We have: f1' (x) = " }{XPPEDIT 18 0 "2 *cos ( 2*x) - 2*cos (x)" "6#,&*&\"\"#\"\"\"-%$cosG6#*&F%F&%\"xGF&F&F&*&F%F&-F (6#F+F&!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ = " }{XPPEDIT 18 0 "2 *(2 cos(x)^2 - 1)- 2 *cos (x)" "6#,&*&\"\"#\"\"\",&*&F%F&*$-%$cosG6#%\"xGF%F&F&F&!\"\"F&F&*&F%F &-F+6#F-F&F." }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 31 " \+ = " }{XPPEDIT 18 0 "2 *( 2 cos(x) + 1) (cos(x) \+ -1)" "6#*&\"\"#\"\"\"-,&*&F$F%-%$cosG6#%\"xGF%F%F%F%6#,&-F*6#F,F%F%!\" \"F%" }}{PARA 0 "" 0 "" {TEXT -1 48 "Hence we have critical values whe n cos (x) = " }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" } {TEXT -1 21 " OR cos (x) = 1. " }}{PARA 0 "" 0 "" {TEXT -1 24 "Rec all plot of cos (x). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "pl ot(cos(x), x= -Pi..Pi, color = red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7en7$$!3*****4tk#fT J!#<$!\"\"\"\"!7$$!35'\\8!o[6tIF*$!3EdR#\\7jl(**!#=7$$!3w\"*pr)3PY+$F* $!3ng@plBO1**F37$$!3)**z*4R\\0XHF*$!3y?$G=_&[2)*F37$$!3?3E[*ysa)GF*$!3 -/?(R3-Qn*F37$$!3gX&))>2g9v#F*$!3wsjRdke[#*F37$$!3)4577.flh#F*$!3m\")) 3;*32`')F37$$!3u\"QM::*H#[#F*$!3Xt%*)4+_U!zF37$$!3y\\AhcI#yN#F*$!3[dVK LTd#3(F37$$!3&e=d-FN*GAF*$!3gz^S3dX;hF37$$!3u%*GBP$Rc4#F*$!3HBRr;:x5]F 37$$!3mCj&f)3xi>F*$!3WB/R/'R,#QF37$$!3(or4AN*4E=F*$!3E5L?OiQDDF37$$!3Q QQ*3**=dq\"F*$!3IXjd$yO^M\"F37$$!3e!>jg@*>q:F*$\"3+_hBip5rf!#@7$$!3`;m ,%)H7M9F*$\"3g*)e)QR#[i8F37$$!3v(4F,K))HI\"F*$\"3[\"4??=#=YEF37$$!3S*4 !R#[0R=\"F*$\"3%Rft'f*3Jx$F37$$!3pY@QqWIU5F*$\"3#[Z2$)H:B/&F37$$!3Wp3w $R)\\B#*F3$\"3I-2e(\\*[RgF37$$!3EJc8o39GyF3$\"3GV=%RO;$*3(F37$$!3'=[BP -8If'F3$\"3:P!R#>U?/zF37$$!3CB3<@?)yB&F3$\"3ce2))\\nIf')F37$$!3_*)RF 3$\"3w5p#*ya0/)*F37$$!3qTJY)ocYO\"F3$\"3x_tH@+.2**F37$$!3c.@Z/\"\\$yp! #>$\"3eOID,7mv**F37$$!38,z7VKJ,J!#?$\"3[%Q>$4>&*****F37$$\"3p0t!3)GF;m F_t$\"3Si.#)\\/7y**F37$$\"39SFf3xEa8F3$\"3$y;bu,Q%3**F37$$\"312**yIK@d >F3$\"3Pq7lgk24)*F37$$\"3'R2()Hve,c#F3$\"3c+KPla1u'*F37$$\"3IcwR>CZ'eYk> F*$!3_-=qm$)zNQF37$$\"3kYuyfix%4#F*$!3#oH!ec0I.]F37$$\"3T4q))fu.GAF*$! 3$>4B*z.N4hF37$$\"3y?w'**=&>gBF*$!3!*Gdan.I*4(F37$$\"3!*>3a=Wj\"[#F*$! 3Yyl.w$y,!zF37$$\"3A?c*)yu\"3i#F*$!333*G2]PVn)F37$$\"3-i-HnWIXFF*$!3G: DUS4+D#*F37$$\"3u=RWhN.yGF*$!3'[\"[-g))oa'*F37$$\"3tX=#4!HbTHF*$!3n!=4 Vz'e+)*F37$$\"3ss(*RSA20IF*$!3(Qny3`bp!**F37$$\"3/')[UXCLtIF*$!3Qtebk< rw**F37$$\"3!)***\\/l#fTJF*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F-F-Fi]l -%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJFg^l%(DEFA ULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(cos(x) = -1/2, x );" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"#\"\" $" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We see that our critical val ues are: x = 0, " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 7 " , and " } {XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{XPPEDIT 18 0 "P i;" "6#%#PiG" }{TEXT -1 44 " . Let's look at the plot of f1' (x). \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Thus f1 is increasing on ( \+ -" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 9 " ) and (" }{XPPEDIT 18 0 " 2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 3 " , " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 36 "Hence f1 has a local max when \+ x = " }{XPPEDIT 18 0 "-2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 26 " and a local min \+ when x = " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "We now turn to concavity. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "D(D(f1));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&-%$sinG6#,$9$\"\"#!\"%*&F0\" \"\"-F,6#F/F3F3F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Thus, f 1'' (x) = -4 sin(2x) + 2sin(x) " }}{PARA 0 "" 0 "" {TEXT -1 54 " \+ = -4(2sin(x) cos(x))+ 2sin(x) " }}{PARA 0 "" 0 "" {TEXT -1 50 " = 2sin(x) (-4cos(x) + 1)" }} {PARA 0 "" 0 "" {TEXT -1 54 "Hence, f1'' (x) = 0 when sin(x) = 0 O R cos(x) = " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT -1 81 " does not come from our standar d triangles, we can only get a numerical estimate " }}{PARA 0 "" 0 "" {TEXT -1 29 "for the solution to cos(x) = " }{XPPEDIT 18 0 "1/4" "6#*& \"\"\"F$\"\"%!\"\"" }{TEXT -1 25 ". Lets plot f '' (x). " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fsolve(cos(x) = 1/4, x = -Pi ..Pi);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+sg6=8!\"* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Conclusion, f 1 is concave u p on ( " }{XPPEDIT 18 0 "1.318116072;" "6#$\"+sg6=8!\"*" }{TEXT -1 12 ",0) and ( " }{XPPEDIT 18 0 "1.318116072;" "6#$\"+sg6=8!\"*" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " ) " }} {PARA 0 "" 0 "" {TEXT -1 31 "and f1 is concave down on ( -" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "1.31811 6072;" "6#$\"+sg6=8!\"*" }{TEXT -1 12 ") and (0, " }{XPPEDIT 18 0 "1 .318116072;" "6#$\"+sg6=8!\"*" }{TEXT -1 4 "). " }}{PARA 0 "" 0 "" {TEXT -1 43 "There are points of inflection when x = - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 " , -" }{XPPEDIT 18 0 "1.318116072; " "6#$\"+sg6=8!\"*" }{TEXT -1 7 ", 0, " }{XPPEDIT 18 0 "1.318116072; " "6#$\"+sg6=8!\"*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 76 "There are no asymptot es. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Example 2 " }{TEXT -1 2 "\n " }{XPPEDIT 18 0 "f(x) = x^(5/3) - 5* x^(2/3) " " 6#/-%\"fG6#%\"xG,&)F'*&\"\"&\"\"\"\"\"$!\"\"F,*&F+F,)F'*&\"\"#F,F-F.F, F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f2:= x -> surd(x,3)^5 - 5 * surd(x,3)^2;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2GR6#%\"xG6\"6$%)operatorG%&arrowG F(,&*$)-%%surdG6$9$\"\"$\"\"&\"\"\"F5*&F4F5)F/\"\"#F5!\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(f2(x), x = -10..10, col or = red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7V7$$!#5\"\"!$!3mp\">/D$Qip!#;7$$!3!pmmm\"p 0k&*!#<$!392vfsW6ilF-7$$!3uKL$36,c8%F-7$$!3Q****\\(y$pZiF1$!3EL%G:SG a\"QF-7$$!3jKLL$yaE\"eF1$!3W]zz1([b\\$F-7$$!3s%HaF1$!3z]#=\"Rg \"=A$F-7$$!3]******\\$*4)*\\F1$!3\\:**4u5sAHF-7$$!3o******\\_&\\c%F1$! 3A5lwR(H@j#F-7$$!3%)******\\1aZTF1$!3))e4*y-u8O#F-7$$!3Imm;/#)[oPF1$!3 .&y&pk7UB@F-7$$!3%HLLL=exJ$F1$!3cGkIqUF]=F-7$$!3lKLLL2$f$HF1$!338yq_I: F;F-7$$!3%)****\\PYx\"\\#F1$!33S+'))4opP\"F-7$$!3gLLLL7i)4#F1$!3q4&)>W 7ej6F-7$$!3o)***\\P'psm\"F1$!3_H**4E?eu$*F17$$!3?****\\74_c7F1$!39R*QR kw_G(F17$$!3M:LL$3x%z#)!#=$!3PMt:[AnQ^F17$$!3()HLL3s$QM%Fer$!3****\\ivF@AFer$!3odT=PaN:>F17$$!3]^omm;zr)*!#?$!3r/0,t#eaI# Fer7$$\"3aPL$3-Dg5#Fer$!3)42#*p0R`p\"F17$$\"3fVLLezw5VFer$!3)*y@5\"obs g#F17$$\"3JtmmmJ+IiFer$!3!*yMK)yxF>$F17$$\"3-.++v$Q#\\\")Fer$!3U`$4]X/ 8l$F17$$\"3%\\LL$e\"*[H7F1$!34SZ)Rf'HFVF17$$\"3=++++dxd;F1$!3kG8w;;^\" o%F17$$\"3e+++D0xw?F1$!3;UD`\"eP$eZF17$$\"35,+]i&p@[#F1$!3iSe'H*[!eh%F 17$$\"3++++vgHKHF1$!3Ub@0Z%G!#>7$$ \"3X,+]7k.6aF1$\"3E63r[!=#=F-7 $$\"3A-+]P?Wl&*F1$\"3'3V&oceCd?F-7$$\"#5F*$\"3O!R1oT%z?BF--%'COLOURG6& %$RGBG$\"*++++\"!\")$F*F*Fb\\l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F (Fg[l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "D(f2);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operato rG%&arrowGF&,&*&*$)-%%surdG6$9$\"\"$\"\"&\"\"\"F4F1!\"\"#F3F2*&#\"#5F2 F4*&*$)F.\"\"#F4F4F1F5F4F5F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Too complcated!! We can do better ourselves! " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "f2 '(x) = " }{XPPEDIT 18 0 "(5/3) *x ^(2/3) - (10/3)* x ^(-1/3)" "6#,&*(\"\"&\"\"\"\"\"$! \"\")%\"xG*&\"\"#F&F'F(F&F&*(\"#5F&F'F()F*,$*&F&F&F'F(F(F&F(" }}{PARA 0 "" 0 "" {TEXT -1 14 " = " }{XPPEDIT 18 0 "(5*x-10) / ( 3* x^(1/3) )" "6#*&,&*&\"\"&\"\"\"%\"xGF'F'\"#5!\"\"F'*&\"\"$F')F(*&F'F'F ,F*F'F*" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 67 "Thus, f2 '(x) = 0 when x = 2 and f2 '(x) is not de fined for x = 0. " }}{PARA 0 "" 0 "" {TEXT -1 99 "Hence we have two cr itical points: x = 0, 2. Let's plot f2 '(x). Since f2 '(x) is not defined " }}{PARA 0 "" 0 "" {TEXT -1 82 "at 0, we plot first with neg ative values of x and then with positive values of x. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Conclusion: f2 is increasing on (-" } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 14 ",0) and (2, \+ " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 4 " ). " }} {PARA 0 "" 0 "" {TEXT -1 28 "f2 is decreasing on (0,2). " }}{PARA 0 " " 0 "" {TEXT -1 66 "Hence, f2 has a local max when x = 0 and a local m in when x = 2. " }}{PARA 0 "" 0 "" {TEXT -1 47 "The function is NOT d ifferentiable at x = 0. " }}{PARA 0 "" 0 "" {TEXT -1 43 "Now we turn to concavity. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "D(D(f2));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&*&*$)-%%surdG6$9$\"\"$\"\"& \"\"\"F4*$)F1\"\"#F4!\"\"#\"#5\"\"**&*&F9F4)F.F7F4F4*$F6F4F8F4F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Again, too complicated! We can do better ourselves! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "f2 ''(x) = " }{XPPEDIT 18 0 "(10/9)* x ^(-1/3) \+ + (10/9)* x ^(-4/3) " "6#,&*(\"#5\"\"\"\"\"*!\"\")%\"xG,$*&F&F&\"\" $F(F(F&F&*(F%F&F'F()F*,$*&\"\"%F&F-F(F(F&F&" }}{PARA 0 "" 0 "" {TEXT -1 18 " = " }{XPPEDIT 18 0 "(10/9)* x^(-4/3) * (x + 1 )" "6#**\"#5\"\"\"\"\"*!\"\")%\"xG,$*&\"\"%F%\"\"$F'F'F%,&F)F%F%F%F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We see that f2 ''(x) = 0 when x = -1 and is NOT defined w hen x = 0. " }}{PARA 0 "" 0 "" {TEXT -1 60 "Let's plot f2 ''(x). Agai n we do two plots to avoid x = 0. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Conclusions: f2 is concave up on (-1,0) and (0, " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 "" {TEXT -1 29 "and f2 is concave down on (-" }{XPPEDIT 18 0 "infinity; " "6#%)infinityG" }{TEXT -1 7 " ,-1). " }}{PARA 0 "" 0 "" {TEXT -1 44 "There is a point of inflcetion when x = -1. " }}{PARA 0 "" 0 "" {TEXT -1 36 "There are no asymptotes. " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 9 "Example 3" }{TEXT -1 3 "\n " }{XPPEDIT 18 0 "f(x) = x^2 / (2*x+5)" "6#/-%\"fG6#%\"xG* &F'\"\"#,&*&F)\"\"\"F'F,F,\"\"&F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f3:= x -> x^2 / (2 * x + 5);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3GR6#%\"xG6\"6$%)operatorG%&arrowG F(*&*$)9$\"\"#\"\"\"F1,&F/F0\"\"&F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "plot(f3(x), x = -5..5,y = -25..25, color = red );" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7ao7$$!\"&\"\"!F(7$$!3YLLLe%G?y%!#<$!3lo!>v%*4/,&F.7$ $!3OmmT&esBf%F.$!3XFz_ZhqR]F.7$$!3ALL$3s%3zVF.$!3F&[Do0'e-^F.7$$!3_LL$ e/$QkTF.$!3'omd77k(4_F.7$$!3ommT5=q]RF.$!3A6Mq/2[z`F.7$$!3ILL3_>f_PF.$ !3)32x()zA6i&F.7$$!3K++vo1YZNF.$!3\")>gd5f82gF.7$$!3;LL3-OJNLF.$!3#p^s trm(emF.7$$!3p***\\P*o%Q7$F.$!3l\")Fhn7$$!3JmTgx/,EDF.$!3Et$ >x'*olA\"!#:7$$!3bCJ&pFq#>DF.$!3#G2/Kzknk\"Faq7$$!3!G3-j2ID^#F.$!3))>t cG=1>DFaq7$$!3]T5lv)*y0DF.$!3t#*\\\\![\"QAaFaq7$$!3u*****\\n\\!*\\#F.$ \"3o&)p'z>\\eG$!#97$$!3U*\\P%)z\"G#\\#F.$\"3*3ev)p#4R-%Faq7$$!3a**\\(= #R^&[#F.$\"3ig$Qq2;B8#Faq7$$!3m*\\7`/Y(yCF.$\"3%o+&>v3UX9Faq7$$!3w*** \\(o\"y>Z#F.$\"3q+5tG?M!4\"Faq7$$!3)***\\i:CWeCF.$\"3eyJ4IUwrsFhn7$$!3 x****\\im!\\W#F.$\"3S8%z1)o%\\U&Fhn7$$!3y***\\i:NyT#F.$\"3AW2Wd!Qub$Fh n7$$!3y******\\Ow!R#F.$\"3#3S?$\\6B;EFhn7$$!3#)****\\P1iOBF.$\"3i?G^Y_ *3n\"Fhn7$$!3%)*****\\ixCG#F.$\"3;\\bIO*4v>\"Fhn7$$!35+++vR7y@F.$\"3%o II&f\\kptF.7$$!3#******\\KqP2#F.$\"3a`s[PF%[/&F.7$$!39LL3-TC%)=F.$\"3O M=H!*4%H)GF.7$$!3[mmm\"4z)e;F.$\"3(=`GZ#3%ej\"F.7$$!3Mmmmm`'zY\"F.$\"3 n*3IRZ;S/\"F.7$$!3#****\\(=t)eC\"F.$\"3#)>I7?&z&)='!#=7$$!3!ommmh5$\\5 F.$\"3I<]ZT%H\\z$Fgw7$$!3S$***\\(=[jL)Fgw$\"3KH$H3B=_3#Fgw7$$!3)f***\\ iXg#G'Fgw$\"3!\\&zL.nRa5Fgw7$$!3ndmmT&Q(RTFgw$\"3nyM*42vw5%!#>7$$!3%\\ mmTg=><#Fgw$\"3h$o+d\"y?L5F\\y7$$!3vDMLLe*e$\\!#?$\"3)Rl2@&GD#)[!#B7$$ \"3!=nm\"zRQb@Fgw$\"3iP\"eG,%)Qb)Fey7$$\"3_,+](=>Y2%Fgw$\"37w^5wr:bGF \\y7$$\"3summ\"zXu9'Fgw$\"3](4O#\\H[mgF\\y7$$\"3#4+++]y))G)Fgw$\"3H].\"Fgw7$$\"3H++]i_QQ5F.$\"3CtZAI%QO_\"Fgw7$$\"3b++D\"y%3T7F.$\"396 N(*[ghe?Fgw7$$\"3+++]P![hY\"F.$\"3S^z9E;#*4FFgw7$$\"3iKLL$Qx$o;F.$\"3S d!=/*)3)QLFgw7$$\"3Y+++v.I%)=F.$\"3%G\"[UUp?\\SFgw7$$\"3?mm\"zpe*z?F.$ \"3,%*Q_D\"**Hs%Fgw7$$\"3;,++D\\'QH#F.$\"3OB\\+pJ2)[&Fgw7$$\"3%HL$e9S8 &\\#F.$\"3QCu!*z%e " 0 " " {MPLTEXT 1 0 6 "D(f3);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&*&9$\"\"\",&F,\"\"#\"\"&F-! \"\"F/*&*&F/F-)F,F/F-F-*$)F.F/F-F1F1F&F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "simplify( 2*x/(2 * x + 5) - 2 * x^2 / (2*x + 5)^2) ;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"xG\"\"\", &F&F'\"\"&F'F'F'*$),&F&\"\"#F)F'F-F'!\"\"F-" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 49 "Hence there are critical points when x = 0, -5. " }} {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "x = -5/2" "6# /%\"xG,$*&\"\"&\"\"\"\"\"#!\"\"F*" }{TEXT -1 64 " is not a critical va lue as the function is not defined there. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "To determine where f3 '(x) is p ositive and negative it suffices to consider " }{XPPEDIT 18 0 "x *(x + 5)" "6#*&%\"xG\"\"\",&F$F%\"\"&F%F%" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "(2*x+5)^5 " "6#*$,&*&\"\"#\"\" \"%\"xGF'F'\"\"&F'F)" }{TEXT -1 20 " is positive or 0. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot(x*(x+5), x = -10..10, color = \+ red);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$!#5\"\"!$\"#]F*7$$!3!pmmm\"p0k&*!#<$\" 3g<1>7+4lV!#;7$$!3uKL$3s%HaF0$\"3T+$[F07$$!3%)******\\1aZTF0$!31yvf1)4c`$F07$$!3Imm;/#)[oPF0$!3#e- !*enP4k%F07$$!3%HLLL=exJ$F0$!3im8f`:F\"e&F07$$!3lKLLL2$f$HF0$!3)HYNdRk *fgF07$$!3%)****\\PYx\"\\#F0$!3wVF07$$!3]^omm;zr)*Feo$!3s=CE11:E\\!#>7$$\"3fVLLezw5VFfr$\"3*egsI=6 7M#F07$$\"3-.++v$Q#\\\")Ffr$\"39GdU[+sQZF07$$\"3%\\LL$e\"*[H7F0$\"3?() e7#Q*3fwF07$$\"3=++++dxd;F0$\"39\\5:x)4P5\"F37$$\"3e+++D0xw?F0$\"39y3& Q%Gop9F37$$\"35,+]i&p@[#F0$\"3EX,&\\N,s&=F37$$\"3++++vgHKHF0$\"3SSgkkS )fK#F37$$\"3ElmmmZvOLF0$\"3m.1i?1x\"y#F37$$\"3%4+++v+'oPF0$\"3Dd+HOb`/ LF37$$\"3UKL$eR<*fTF0$\"3_VB$>(*\\/\"QF37$$\"3K-++])Hxe%F0$\"3U0\"eEW \"f)R%F37$$\"3!fmm\"H!o-*\\F0$\"3GHc([^6a)\\F37$$\"3X,+]7k.6aF0$\"3`*3 !*>r\\Mj&F37$$\"3#emmmT9C#eF0$\"3-;TFs\"e7I'F37$$\"33****\\i!*3`iF0$\" 396dg8olOqF37$$\"3;NLLL*zym'F0$\"3!Rg,sC-+y(F37$$\"3'eLL$3N1#4(F0$\"3) H+!zM#odd)F37$$\"3,pm;HYt7vF0$\"3cR8Tva[+%*F37$$\"37-+++xG**yF0$\"3R8P nY=&*=5!#:7$$\"3gpmmT6KU$)F0$\"3H?.Rx#fI6\"Fcy7$$\"3qNLLLbdQ()F0$\"3v' Q%=+eb+7Fcy7$$\"3[++]i`1h\"*F0$\"3es\\)QX/tH\"Fcy7$$\"3A-+]P?Wl&*F0$\" 3#>s-c\"*[KR\"Fcy7$$\"#5F*$\"$]\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F* F*Fd[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fiz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Hence, f3 is increasing on ( -" } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 14 ",-5) and (0, \+ " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 4 " ). " }} {PARA 0 "" 0 "" {TEXT -1 34 "The function f3 is decreasing on (" } {XPPEDIT 18 0 "-5,-5/2" "6$,$\"\"&!\"\",$*&F$\"\"\"\"\"#F%F%" }{TEXT -1 7 ") and (" }{XPPEDIT 18 0 "-5/2" "6#,$*&\"\"&\"\"\"\"\"#!\"\"F(" } {TEXT -1 20 ",0). Remember the " }}{PARA 0 "" 0 "" {TEXT -1 32 "func tion is NOT defined for x = " }{XPPEDIT 18 0 "-5/2" "6#,$*&\"\"&\"\"\" \"\"#!\"\"F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "Thus, f 3 has a local max when x = -5 and a local min when x = 0. \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Let's turn to concavity. \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "D(D(f3));" }{TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF& ,(*&\"\"\"F,,&9$\"\"#\"\"&F,!\"\"F/*&*&\"\")F,F.F,F,*$)F-F/F,F1F1*&*&F 4F,)F.F/F,F,*$)F-\"\"$F,F1F,F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify( 2*1/(2*x+5)-8*x/((2*x+5)^2)+8*x^2/((2*x+5)^ 3) );" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$ ),&%\"xG\"\"#\"\"&F%\"\"$F%!\"\"\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Thus, f3 ''(x) = " }{XPPEDIT 18 0 "50 /(2*x+5)^10" "6#*&\"#] \"\"\"*$,&*&\"\"#F%%\"xGF%F%\"\"&F%\"#5!\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "Conclusions: f3 is concave up on (" } {XPPEDIT 18 0 "-5/2" "6#,$*&\"\"&\"\"\"\"\"#!\"\"F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 27 ") and is conc ave down on (-" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ," }{XPPEDIT 18 0 "-5/2" "6#,$*&\"\"&\"\"\"\"\"#!\"\"F(" }{TEXT -1 4 "). " }}{PARA 0 "" 0 "" {TEXT -1 37 "Concavity chages as x passe s through " }{XPPEDIT 18 0 "-5/2" "6#,$*&\"\"&\"\"\"\"\"#!\"\"F(" } {TEXT -1 9 ", but as " }{XPPEDIT 18 0 "-5/2" "6#,$*&\"\"&\"\"\"\"\"#! \"\"F(" }{TEXT -1 22 " is NOT in the domain " }}{PARA 0 "" 0 "" {TEXT -1 58 "of the function we have no points of inflection. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a3:= plot(f3(x), x = -10..10 , y = -25..25,color = red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "b3:= plot(1/2 * x - 5/4, x = -10..10, color = magenta):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{a3,b3\});" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CUR VESG6$7co7$$!#5\"\"!$!3'pmmmmmmm'!#<7$$!3!pmmm\"p0k&*F-$!3+)>3C\")3WZ' F-7$$!3uKL$3@T.$p2_F-7$$!3jKLL$ yaE\"eF-$!3O'[C!Q&z'*4&F-7$$!3s%HaF-$!3O1Mjf6[J]F-7$$!3]****** \\$*4)*\\F-$!3t&yVIs+++&F-7$$!3o******\\_&\\c%F-$!3qX,k6w#e/&F-7$$!3%) ******\\1aZTF-$!3akeAyo`?_F-7$$!3Imm;/#)[oPF-$!3)[))oXi1yf&F-7$$!3%HLL L=exJ$F-$!3-qu)p<-.t'F-7$$!3!GLL$eW%o7$F-$!3K>!3.)*4()z(F-7$$!3lKLLL2$ f$HF-$!37(f4$>a`'))*F-7$$!3;mTN@7T!)GF-$!3?vKs3+]!4\"!#;7$$!3n**\\P4<* [#GF-$!3#zng-a.\"G7Fgq7$$!3uKeR(>s$pFF-$!3Nv.9([tNU\"Fgq7$$!3CmmT&o_Qr #F-$!3QxU*p$*y>s\"Fgq7$$!3w*\\PM'Rg*Fgq7$$!3I$e*[$RM&>DF-$!3q T-u)>S[i\"!#:7$$!3d\"z%\\:Xl0DF-$!3)3b\"R&)[e^bFet7$$!3%)****\\PYx\"\\ #F-$\"3I()\\/\"ylUx$Fet7$$!3Y\"z>Of)[zCF-$\"3w)*oZ)3X')\\\"Fet7$$!33$e R(\\D?nCF-$\"3UKu7z&>)z#*Fgq7$$!3qu$fe];\\X#F-$\"3G*=)f$ePQo'Fgq7$$!3L m\"z>YIEW#F-$\"3G&='*p75+?&Fgq7$$!3e\\(=UPe!=CF-$\"3+=Cx$e(ynNFgq7$$!3 EL$ekG'[$R#F-$\"3%[$QJ,-A*o#Fgq7$$!3A+v$46UVM#F-$\"3GroZW2Rl&H#F-$\"3W5(*GFO5'G\"Fgq7$$!3;+]P%e4p>#F-$\"3+\"y.o!***>'zF-7$$! 3gLLLL7i)4#F-$\"3)oWo,&GN'[&F-7$$!39mmTNa%H)=F-$\"3C8f%QB4H(GF-7$$!3o) ***\\P'psm\"F-$\"3?d-s903p;F-7$$!3?****\\74_c7F-$\"3sOv97w\\[j!#=7$$!3 M:LL$3x%z#)Fay$\"31@'G*H$o)\\?Fay7$$!3()HLL3s$QM%Fay$\"3cO#e93$QnX!#>7 $$!3]^omm;zr)*!#?$\"3ST\\,`@xc>!#A7$$\"3fVLLezw5VFay$\"33B`J;x%*pJF\\z 7$$\"3-.++v$Q#\\\")Fay$\"3ZM$4vV$o,5Fay7$$\"3%\\LL$e\"*[H7F-$\"33*))HZ W4m-#Fay7$$\"3=++++dxd;F-$\"3\"y?Zpt$=*H\"F-7$$\"3K-++])Hxe%F-$\"3 ]Nz&RtnZ[\"F-7$$\"3!fmm\"H!o-*\\F-$\"35)os([?Mi;F-7$$\"3X,+]7k.6aF-$\" 3S@l/\")f`]=F-7$$\"3#emmmT9C#eF-$\"3k(zIU?*pO?F-7$$\"33****\\i!*3`iF-$ \"3X%p0([8cLAF-7$$\"3;NLLL*zym'F-$\"3Z![f?%R![U#F-7$$\"3'eLL$3N1#4(F-$ \"3L8#fS#>#=i#F-7$$\"3,pm;HYt7vF-$\"3995cj)p%=GF-7$$\"37-+++xG**yF-$\" 3/cQo+_9+IF-7$$\"3gpmmT6KU$)F-$\"3r%pSt)HQ4KF-7$$\"3qNLLLbdQ()F-$\"3It Dz;zM(R$F-7$$\"3[++]i`1h\"*F-$\"3'y+^s[=&)f$F-7$$\"3A-+]P?Wl&*F-$\"3sf gF0_s\"z$F-7$$\"#5F*$\"\"%F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fcbl- F$6$7S7$F($!3+++++++]iF-7$F/$!3XLLLe%G?.'F-7$F4$!3PmmT&esB%eF-7$F9$!3A LL$3s%3HcF-7$F>$!3_LL$e/$Q9aF-7$FC$!3ommT5=q+_F-7$FH$!3uLL3_>f-]F-7$FM $!3K++vo1Y(z%F-7$FR$!3gLL3-OJ&e%F-7$FW$!3o***\\P*o%QP%F-7$Ffn$!3Kmmm\" RFj:%F-7$F[o$!35LL$e4OZ'RF-7$F`o$!3w*****\\n\\!\\PF-7$Feo$!3%)*****\\i xC`$F-7$Fjo$!3#******\\KqPK$F-7$F_p$!3:LL3-TCMJF-7$Fdp$!3Zmmm\"4z)3HF- 7$F^q$!3Lmmmm`'zr#F-7$F\\u$!3#****\\(=t)e\\#F-7$F^x$!3!ommmh5$*H#F-7$F hx$!3M***\\(=[j$3#F-7$F]y$!3g***\\iXg#y=F-7$Fcy$!3wlm;aQ(Rm\"F-7$Fhy$! 3]mmTg=>n9F-7$F^z$!3ULL$e*e$\\D\"F-7$Fez$!3#GL$3-;YM5F-7$Fjz$!3[)***\\ 73QD%)Fay7$F_[l$!3IDLL3Ub_jFay7$Fd[l$!32*******\\@6@%Fay7$Fi[l$!30(*** *\\PZh6#Fay7$F^\\l$!3'RX***\\(=_\"*)F`z7$Fc\\l$\"3++++v.[h@Fay7$Fh\\l$ \"3IELLLQx$=%Fay7$F]]l$\"3r/++]P+VjFay7$Fb]l$\"35im;zpe*H)Fay7$Fg]l$\" 3;,++D\\'Q/\"F-7$F\\^l$\"3'HL$e9S8X7F-7$Fa^l$\"3s++D1#=bX\"F-7$Ff^l$\" 3!HLL$3s?h;F-7$F[_l$\"3a***\\7`Wl(=F-7$F`_l$\"3enmmm*RR3#F-7$Fe_l$\"3% zmmTvJgH#F-7$Fj_l$\"3]MLe9tO1DF-7$F_`l$\"31,++]Qk*p#F-7$Fd`l$\"3![LL3d g6#HF-7$Fi`l$\"3&ymmmw(G>JF-7$F^al$\"3C++D\"oK0L$F-7$Fcal$\"36,+v=5sKN F-7$Fhal$\"3+++++++]PF--F]bl6&F_blF`blFcblF`bl-%+AXESLABELSG6%Q\"x6\"Q \"yF`\\m%(DEFAULTG-%%VIEWG6$;F(Fhal;$!#DF*$\"#DF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}}{MARK "1 6 0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }