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2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 59 "Use of the first and second deriv ative in graph sketching " }}{PARA 0 "" 0 "" {TEXT -1 15 "by Peter St one " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 3.10.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 62 "The sign of the derivative in connection with grap h sketching " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 262 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "Sketch the graph of " }{XPPEDIT 18 0 "y = x^4-4*x^3;" "6 #/%\"yG,&*$%\"xG\"\"%\"\"\"*&F(F)*$F'\"\"$F)!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = x^4-4*x^3;" "6#/-%\"fG6#%\"xG,&*$F'\"\"%\"\"\"*&F*F+*$F'\"\" $F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {XPPEDIT 18 0 "`f '`(x) = 4*x^3-12*x^2;" "6#/-%$f~'G6#%\"xG,&*&\"\"%\" \"\"*$F'\"\"$F+F+*&\"#7F+*$F'\"\"#F+!\"\"" }{XPPEDIT 18 0 "`` = 4*x^2* (x-3);" "6#/%!G*(\"\"%\"\"\"*$%\"xG\"\"#F',&F)F'\"\"$!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"x G\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 10 " and when " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 70 "It is useful to investi gate how the sign of the derivative changes as " }{TEXT 264 1 "x" } {TEXT -1 9 " varies. " }}{PARA 0 "" 0 "" {TEXT -1 51 "Because the deri vative is a continuous function of " }{TEXT 265 1 "x" }{TEXT -1 84 ", \+ it can only change sign from positive to negative or from negative to \+ positive as " }{TEXT 266 1 "x" }{TEXT -1 97 " increases by going throu gh the value zero. The only possible sign changes therefore occur wher e " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 11 " and where " } {XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 54 ". To say that the derivative is continuous means that " }{TEXT 270 27 "the graph of the derivative" }{TEXT -1 67 " is a continuous unbroken curve. Since this curve only crosses the " }{TEXT 267 1 "x" }{TEXT -1 12 " axis where \+ " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 11 " and where " } {XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 39 ", it must stay on the same side of the " }{TEXT 268 1 "x" }{TEXT -1 14 " axis between \+ " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 85 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 34 " From the fact that the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~ 'G6#%\"xG" }{TEXT -1 18 " is negative when " }{TEXT 271 1 "x" }{TEXT -1 39 " is just less than 3 and positive when " }{TEXT 272 1 "x" } {TEXT -1 55 " is just greater than 3, we can conclude that the point" }{XPPEDIT 18 0 " ``(3,-27)" "6#-%!G6$\"\"$,$\"#F!\"\"" }{TEXT -1 6 " i s a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 47 "Because the derivative does not change sign as " } {TEXT 273 1 "x" }{TEXT -1 34 " increases through zero, the point" } {XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 35 " is not a maxi mum or minimum point." }}{PARA 0 "" 0 "" {TEXT -1 109 "However, since \+ the gradient of the tangent line is zero at this point, the graph is v irtually horizontal for " }{TEXT 274 1 "x" }{TEXT -1 15 " close to zer o." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "A f ree-hand sketch of the graph may now be made." }}{PARA 0 "" 0 "" {TEXT -1 48 "One extra piece of information is useful. Since " } {XPPEDIT 18 0 "y=x^3*(x-4)" "6#/%\"yG*&%\"xG\"\"$,&F&\"\"\"\"\"%!\"\"F )" }{TEXT -1 22 ", the graph meets the " }{TEXT 275 1 "x" }{TEXT -1 12 " axis where " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 86 ", ( which we already knew because this is a stationary point on the gra ph ) and where " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 326 317 317 {PLOTDATA 2 "6,-%'CURVESG6%7[o7$$!\"#\"\"!$\"#[F*7$$!3NL$e*[#f\"H>!#<$\"3o]31y#Q pD%!#;7$$!3qmm\"z\\=$e=F0$\"3;Qw\\6*G&fPF37$$!3-](=#R.o'z\"F0$\"3A&=P* RU&>O$F37$$!3PL3_!=U]t\"F0$\"3O1Sd)4#[&*HF37$$!35]7`COsl;F0$\"3=O/*>?q &=EF37$$!3km;ao]S'f\"F0$\"35O]G]5(oF#F37$$!3gm;zz*[oX\"F0$\"3eLnU+PF(o \"F37$$!3VL3xwh&zJ\"F0$\"38Ww[r-W<7F37$$!3ymT&)oZ=*=\"F0$\"3&*znI!yTms )F07$$!3/+voM%\\e0\"F0$\"3e)*o$)y\"R6&fF07$$!3+o;a8%Q&z\"*!#=$\"3c^CJx m0/QF07$$!3>+]P4[+0yF^o$\"3OU2!zulHF#F07$$!3;ML$e/G6R'F^o$\"3Bt0:+^167 F07$$!31nm\"Hi%yX^F^o$\"3GQm6-CP^hF^o7$$!3.****\\()G#Qu$F^o$\"39bHF@CU &H#F^o7$$!3)4++Dc/hL#F^o$\"3-[C#e\")QuR&!#>7$$!3)))****\\7r]z*Fjp$\"3% f*y%=#=9^Q!#?7$$\"32.LekL8CDFjp$!3GD;i(Hg@R'!#A7$$\"3\\lm;/fG<Fjp7$$\"3Ammm;^AeHF^o$!3u%=u<#yE*e*Fjp7$$\"3)e*\\7GCt,WF^o$! 3A8_c7h)f.$F^o7$$\"3Kjmm\"*4[zcF^o$!3YRw;dg_(G'F^o7$$\"31)*\\7yOP\"3(F ^o$!3eR?zRa%*o6F07$$\"3F,]PMqI;%)F^o$!3erL#Qe,H)=F07$$\"3(\\m;z%*p\"4) *F^o$!3i@nC#z?&\\GF07$$\"3Zm\"H2HD)36F0$!3n%yERt>:%RF07$$\"31L$3xm\"zY 7F0$!3>-;$)[01O`F07$$\"37Leke**4!R\"F0$!3p))=n)R42,(F07$$\"3$)*\\(=Z-& [^\"F0$!32/%RG7k*Q')F07$$\"35L$ek(Re\\;F0$!3CF'zz\"y.b5F37$$\"3m****\\ -rx)y\"F0$!3gd?]1Qhl7F37$$\"3m**\\i?/&\\#>F0$!3_SZW&e'3![\"F37$$\"3Q* \\7y50n0#F0$!3U1N;0Ul!p\"F37$$\"3?**\\PCi*H?#F0$!3wO'p$p\"y7#>F37$$\"3 @mm;*HXWL#F0$!3)*e(>Dy(*)=@F37$$\"3/++vV_zuCF0$!3M6)o<;xg#F0$!3S\"R.R^qFY#F37$$\"3B++D,A,TFF0$!3ye(oQ*Gr#f#F37$$\"3[l\"z% 4r$=(GF0$!3e#H4n%34sEF37$$\"3#G$3xr>@SHF0$!3Is)oASNPp#F37$$\"3;+D1Moe3 IF0$!3)RP]Axm)*p#F37$$\"3%HeR(fdVvIF0$!3A_>$yM5%*o#F37$$\"3slmT&o%GUJF 0$!39sb@-P@hEF37$$\"3\\#ek`JpA@$F0$!3XVg290/6EF37$$\"3E*\\7`%RD#G$F0$! 3!fT%[l](z`#F37$$\"3mLLLy41F37$$\"3Wm\"zWvQ;p$F0$!3M)z#>:rP^:F37$$\"3I$e*[GOXaPF0$!3E4#[ IV#\\*H\"F37$$\"3;++]-&os\"QF0$!3fEBp4hT;5F37$$\"39L3xO9E*)QF0$!3%4o1t 8)y9lF07$$\"37m;/rVDhRF0$!3)*=bK\\sN3CF07$$\"3\"*)\\(o4dkDSF0$\"3]-g$f (z4t;F07$$\"3gKLL[q.!4%F0$\"3-s%eEy:.;'F07$$\"3;;HdX;peTF0$\"3IZtwRTOT 6F37$$\"3r*\\7GCYtA%F0$\"3&zhrkByur\"F37$$\"3O****\\_u0$H%F0$\"30T\"*[ lVu=BF37$$\"3#**\\(=i'o(eVF0$\"3[zH$fjA5(HF37$$\"3gD1k'\\wSR%F0$\"3=F$ >.yhLM$F37$$\"3S]P4JVQHWF0$\"3X$eQG\\Q9t$F37$$\"3?voal@pkWF0$\"3[t-X&= 8c8%F37$$\"3++++++++XF0$\"3++++++DcXF3-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F *Fj_l-%*THICKNESSG6#\"\"#-F$6&7$7$Fj_lFj_l7$$\"\"$F*$!#FF*-%'SYMBOLG6# %'CIRCLEG-Fd_l6&Ff_lF*F*F*-%&STYLEG6#%&POINTG-F$6&Fa`l-Fi`l6#%(DIAMOND GF\\alF^al-F$6&Fa`l-Fi`l6#%&CROSSGF\\alF^al-%%TEXTG6%7$$\"#XFi_lF(Q\"x 6\"F\\al-F]bl6%7$$F)Fi_lF+Q\"yFcblF\\al-F]bl6%7$Fd`l$!#KF*Q1min.~pt.~( 3,-27)FcblF\\al-%*AXESTICKSG6$\"\"'\"\")-%+AXESLABELSG6%%!GFgcl-%%FONT G6#%(DEFAULTG-%%VIEWG6$;F(F`blF[dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 47.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "It is instructive to plot the gr aph of the derivative " }{XPPEDIT 18 0 "`f '`(x) = 4*x^3-12*x^2;" "6#/ -%$f~'G6#%\"xG,&*&\"\"%\"\"\"*$F'\"\"$F+F+*&\"#7F+*$F'\"\"#F+!\"\"" } {TEXT -1 25 " along with the graph of " }{XPPEDIT 18 0 "f(x) = x^4-4*x ^3" "6#/-%\"fG6#%\"xG,&*$F'\"\"%\"\"\"*&F*F+*$F'\"\"$F+!\"\"" }{TEXT -1 89 " to see how it relates to the information gathered regarding th e sign of the derivative. 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" }}{PARA 0 "" 0 "" {TEXT -1 47 "Differentiating the derivative with respect to " }{TEXT 277 1 "x" }{TEXT -1 11 " gives the " }{TEXT 259 17 "second derivative " }{TEXT -1 28 " of the original function f(" }{TEXT 276 1 "x" }{TEXT -1 14 ") denoted by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "``(dy/dx)=d^2* y/(d*x^2)" "6#/-%!G6#*&%#dyG\"\"\"%#dxG!\"\"*(%\"dG\"\"#%\"yGF)*&F-F)* $%\"xGF.F)F+" }{XPPEDIT 18 0 "`` = `f ''`(x);" "6#/%!G-%%f~''G6#%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, if " } {XPPEDIT 18 0 "y=x^4-4*x^3" "6#/%\"yG,&*$%\"xG\"\"%\"\"\"*&F(F)*$F'\" \"$F)!\"\"" }{XPPEDIT 18 0 "``=f(x)" "6#/%!G-%\"fG6#%\"xG" }{TEXT -1 58 " is the function considered in the previous section, then " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = `f '`(x);" "6 #/*&%#dyG\"\"\"%#dxG!\"\"-%$f~'G6#%\"xG" }{XPPEDIT 18 0 "``=4*x^3-12*x ^2" "6#/%!G,&*&\"\"%\"\"\"*$%\"xG\"\"$F(F(*&\"#7F(*$F*\"\"#F(!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(x) = d^2*y/(d*x^2);" "6#/-%%f~ ''G6#%\"xG*(%\"dG\"\"#%\"yG\"\"\"*&F)F,*$F'F*F,!\"\"" }{XPPEDIT 18 0 " ``=12*x^2-24*x" "6#/%!G,&*&\"#7\"\"\"*$%\"xG\"\"#F(F(*&\"#CF(F*F(!\"\" " }{XPPEDIT 18 0 "``=12*x*(x-2)" "6#/%!G*(\"#7\"\"\"%\"xGF',&F(F'\"\"# !\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 270 10 "In general" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 14 " is positive, " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 4 " is " }{TEXT 259 10 "increasing" }{TEXT -1 63 ", that is, gradients of tangent line s increase as the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 63 " is followed in the direction from left to right. The gra ph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 60 " is \"co ncave upwards\" as suggested in the following picture." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{GLPLOT2D 391 217 217 {PLOTDATA 2 "6+-%'CURVESG 6%7S7$$!3++++++++:!#<$\"3+++++++]AF*7$$!3&*****\\P&3YV\"F*$\"3i))y'el, \"e?F*7$$!3!***\\iv$FX7$$!3')*****\\FPm(\\FX$\"3UU>pc=pwCFX7$$!3()*** ****4'*QS%FX$\"3%4_ff3I%R>FX7$$!3?++Dc>mPPFX$\"3'y&)>**o6qR\"FX7$$!3'3 +++&=$z9$FX$\"3bFWCK\\Z4**!#>7$$!3N***\\iX/4]#FX$\"3C06H*4BXD'Fgq7$$!3 C***\\(o8y%)=FX$\"3)47()z!3S_NFgq7$$!33****\\i:#>C\"FX$\"3#!#B7$$\"3i(***\\P>:mkFgq$\"3?(\\$)y376=%F\\s7$$\"3d***\\iv&QA7FX$ \"3S$)G3PpA%\\\"Fgq7$$\"3j++]PPBW=FX$\"3J@K`y!)>,MFgq7$$\"3%*)*****\\N m'[#FX$\"3'pf)*3h&\\$='Fgq7$$\"36****\\(yb^6$FX$\"3SRZR!e&>/(*Fgq7$$\" 3')***\\PMaKs$FX$\"3;C`#3HiiQ\"FX7$$\"3a****\\7TW)R%FX$\"3z!fy51JY$>FX 7$$\"3*y*****\\@80]FX$\"3-gj*QyM^]#FX7$$\"3_+++D6!Hl&FX$\"3)pi-H6Hb>$F X7$$\"3j)**\\P4w)RiFX$\"3Mu_`m`g$*QFX7$$\"3s,++vZf\")oFX$\"3OK2tkYjNZF X7$$\"3'z**\\P/-a[(FX$\"3IkwlvV7.cFX7$$\"3R++v=Yb;\")FX$\"3)o>:z)e%ye' FX7$$\"3s)****\\i@Ot)FX$\"3Jin')oYhFwFX7$$\"3g)**\\PfL'z$*FX$\"3+^.INE v(z)FX7$$\"3>+++!*>=+5F*$\"3Qg.78SO+5F*7$$\"3-++DE&4Q1\"F*$\"3T\\-932p J6F*7$$\"3=+]P%>5p7\"F*$\"3s+j='eE*p7F*7$$\"39+++bJ*[=\"F*$\"3Faew)yrR S\"F*7$$\"33++Dr\"[8D\"F*$\"3[>2pXA(ec\"F*7$$\"3++++Ijy58F*$\"3%*o[\"H !3;=~06\"-Fjz6&F\\[lFa[lFa[lFa[l-% %FONTG6$%*HELVETICAG\"#5-F\\^l6&7$F`[l$!\"&Fh\\lQ9graph~is~concave~upw ardsFb^l-Fjz6&F\\[l$\")#)eqkF_[l$\"))eqk\"F_[lFd_lFe^l-%+AXESLABELSG6% Q\"xFb^lQ!Fb^l-Ff^l6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#:Fh \\l$\"#:Fh\\lF]`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "No te that in the neighbourhood of the point of contact of a tangent line the graph is on or " }{TEXT 259 5 "above" }{TEXT -1 19 " that tangent line." }}{PARA 0 "" 0 "" {TEXT -1 150 "A car being driven (to the ri ght) along a road which has the form of the above curve, when viewed f rom a helicopter above the road, would need to be " }{TEXT 259 19 "ste ered to the left" }{TEXT -1 37 " with the driver's \"left hand down\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 14 " is n egative, " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 4 " \+ is " }{TEXT 259 10 "decreasing" }{TEXT -1 63 ", that is, gradients of \+ tangent lines decrease as the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 50 " is followed in the direction from left to ri ght. 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E ach schematic curve shows both positive and negative gradients when in reality the gradient may have only one sign in the corresponding inte rval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " A point on a curve where the concavity changes is called a " }{TEXT 259 19 "point of inflection" }{TEXT -1 37 ". A necessary condition for the point" }{XPPEDIT 18 0 "``(a,f(a))" "6#-%!G6$%\"aG-%\"fG6#F&" } {TEXT -1 37 " to be a point of inflection is that " }{XPPEDIT 18 0 "`f ''`(a) = 0;" "6#/-%%f~''G6#%\"aG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "In the following pic ture showing the graph of " }{XPPEDIT 18 0 "y=x^4-4*x^3" "6#/%\"yG,&*$ %\"xG\"\"%\"\"\"*&F(F)*$F'\"\"$F)!\"\"" }{TEXT -1 34 ", the sections o f the curve where " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" } {TEXT -1 61 " is positive, and the graph is concave upwards, are colou red " }{TEXT 260 3 "red" }{TEXT -1 26 ", while the section where " } {XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 62 " is negat ive, and the graph is concave downwards, is coloured " }{TEXT 256 4 "b lue" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 395 371 371 {PLOTDATA 2 "6/-%'CURVESG6%7S7$$!\"#\"\"!$\"#[F*7$$!3ymmm \"p0k&>!#<$\"3;^%z5qo-Y%!#;7$$!3FLL3F0$\"37%='\\tI0zTF37$$!3cmm;W p\"e(=F0$\"3s#zJ$4TGyQF37$$!3hmm;4m(G$=F0$\"3-Hj6BYb\"f$F37$$!3QLL3i.9 !z\"F0$\"3JG5$=0B;K$F37$$!3emmT!R=0v\"F0$\"31!=4v1eY3$F37$$!3)****\\P8 #\\4n?#F37$$!3mmm;>s %Ha\"F0$\"37A6'G&e2O?F37$$!3/+++N*4)*\\\"F0$\"3k_f`].[b=F37$$!3-+++Db \\c9F0$\"3AIT65n$fo\"F37$$!3*)*****\\1aZT\"F0$\"3y')oi6/GL:F37$$!3ommT ?)[oP\"F0$\"3]#p<\"elT.9F37$$!3ZLLL=exJ8F0$\"3Y3+lUXSf7F37$$!3SLLLtIf$ H\"F0$\"3u<_:j<*e9\"F37$$!3;++vju<\\7F0$\"3]sWr6u?B5F37$$!3aLLLB@')47F 0$\"3`'ziD%QVE#*F07$$!3'****\\P'psm6F0$\"3?;wtRK%e?)F07$$!35++D\"4_c7 \"F0$\"32gj$=hX2J(F07$$!3ULL$3x%z#3\"F0$\"3/\\vX0Lp_kF07$$!3MLL3s$QM/ \"F0$\"3IE\\$4GP'HdF07$$!3pmm;zr)4+\"F0$\"3JsDx:C\"e,&F07$$!3Iom;/K#*o &*!#=$\"3y+'z,j)3VVF07$$!3-,+]ih2&=*Fds$\"30*R&)\\eu8\"QF07$$!3snmmT3^ q()Fds$\"3a8zL/LF!H$F07$$!3q++++VAU$)Fds$\"3F.J1Qja1GF07$$!33-++v%HK#z Fds$\"3a4@6%\\1PQ#F07$$!3d,+]P/$y^(Fds$\"3#y\"*)[k+**=?F07$$!3y,++DRqn qFds$\"3yU^>/3sh;F07$$!3uMLLL_CjmFds$\"3OvL7Iq[!Q\"F07$$!3R+++]#*RJiFd s$\"3B@Y$f#)['=6F07$$!3enm;/E3SeFds$\"3(Fds7$$!3VNL$3(>t4]Fds$\"3A,:n')H8fcFds7$$!3?,+](ej*)e%Fd s$\"3+g'f3gZ*3VFds7$$!3=MLL$e&exTFds$\"3+\">KN&f!4A$Fds7$$!3#4++v$4\"p u$Fds$\"34)R47$$!3`++++Br+ @Fds$\"3rFds7$$\"3#pmmm@Xt=%Fds$!3$fU+2uw$HEFds7$$\"3QL LL3y_qXFds$!3_x-Bs?q#Q$Fds7$$\"3i******\\1!>+&Fds$!3o'**[2D`(zVFds7$$ \"3()******\\Z/NaFds$!3)Qk5.f\"R\\bFds7$$\"3'*******\\$fC&eFds$!3\"[!f ?Z)=]%oFds7$$\"3ELL$ez6:B'Fds$!33:iPwOJr\")Fds7$$\"3Smmm;=C#o'Fds$!3NM +lo1GT**Fds7$$\"3-mmmm#pS1(Fds$!3okkFlo+h6F07$$\"3]****\\i`A3vFds$!3Uc Z+Y;Ev8F07$$\"3slmmm(y8!zFds$!3'feo1(fT$e\"F07$$\"3V++]i.tK$)Fds$!3sM2 \\]))>K=F07$$\"39++](3zMu)Fds$!3i?;7$ok#*3#F07$$\"3#pmm;H_?<*Fds$!3#yz FroC(yBF07$$\"3emm;zihl&*Fds$!3svn)f\"3\"Qm#F07$$\"39LLL3#G,***Fds$!3O f>!QT3@*HF07$$\"3g'=5u\"p$F07$$\"3BLL$e\"*[H7\"F0$!3=g(faVhS2%F07$$\"3#*******pvxl6 F0$!37bX#[]e.\\%F07$$\"3z****\\_qn27F0$!3kKPx#zM$=\\F07$$\"3%)***\\i&p @[7F0$!3G0i*yf4;N&F07$$\"3#)****\\2'HKH\"F0$!3aQ=N'HXV&eF07$$\"3_mmmwa nL8F0$!3tCtp5L/DjF07$$\"3'******\\2goP\"F0$!3%pQSw,^o%oF07$$\"3CLLeR<* fT\"F0$!3-H;e!o#GOtF07$$\"3'******\\)Hxe9F0$!3-HNX`5s))yF07$$\"3Ymm\"H !o-*\\\"F0$!3'z3CK5mVU)F07$$\"3))***\\7k.6a\"F0$!3W(*4U/-')***)F07$$\" 3emmmT9C#e\"F0$!3QWmJcw.x&*F07$$\"3\"****\\i!*3`i\"F0$!3V82)Q)[c>5F37$ $\"3QLLL$*zym;F0$!3KnjphmU!3\"F37$$\"3GLL$3N1#4F0$!3YJg<&o,gY\"F37$$\"3/++v.Uac>F0$!3[*pqU>.0`\"F37$$F d[lF*$F3F*-F[[l6&F][lFizFizF^[lFa[l-F$6%7SFcjl7$$\"3'om;a)G\\a?F0$!35 \\Rz3I7(o\"F37$$\"3=Lek`o!>5#F0$!3o@'['fohi)G_:#F0$!39*3 #R$*=\"o%=F37$$\"3%omT&QU!*3AF0$!3]3Mz8'4/$>F37$$\"37LeRZXKiAF0$!3s<=l y]-7?F37$$\"3dm\"z>,_=J#F0$!3/z#)yxh)e3#F37$$\"3#**\\7G$[8jBF0$!35qq_d E7g@F37$$\"3gm\"z%*frhT#F0$!36G@/3E/MAF37$$\"3')*\\ilFQ!pCF0$!3mb&Q5fY VI#F37$$\"3ULL3_\"=M_#F0$!3!>Y'*pk.EP#F37$$\"3%omTg(fJrDF0$!3&)3N`!Gg) GCF37$$\"3&****\\7eP_i#F0$!3=,5'yYIt[#F37$$\"3#****\\Pf!QzEF0$!3G0WCbf FSDF37$$\"3-++v=ubJFF0$!3#[diwEF37$$\" 3\"**\\7.F!)eEF37$$\"3eLe*[.-d>$F0$!3%\\y'>-%=\\i#F37$$ \"3Um;/Egw[KF0$!3iLK7]!4fd#F37$$\"3zm\"z%*f%)QI$F0$!31xr\\4ZZ5DF37$$\" 3/+voza'=N$F0$!3LShYx*e2W#F37$$\"34n;zWho.MF0$!3Uya7_KQ^BF37$$\"3C++]i >AdMF0$!3*)>rcj(pGC#F37$$\"32+]i:jf4NF0$!3=47%41[*>@F37$$\"39+DJ&>r-c$ F0$!3AV^AmDU%)>F37$$\"3++]P4q`;OF0$!3bR`m`M&Q\"=F37$$\"3;LL$eM%4nOF0$! 31k$zn*enT;F37$$\"37++v$4v5s$F0$!3$Ql1Jc=rV\"F37$$\"3cm\"zWn*)*pPF0$!3 c;kHfdWK7F37$$\"3H++DJiYBQF0$!3%H*ft0\"Gt')*F07$$\"3-Lek.NytQF0$!3x+'> IXhqL(F07$$\"3S+Dc^&zj#RF0$!3)3867**3jX%F07$$\"3YLL3-=!y(RF0$!3GhR!\\? 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Note that in order to find the points of inflection along a curve " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 15 " the values of " } {XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 11 " for which " } {XPPEDIT 18 0 "`f ''`(x) = 0;" "6#/-%%f~''G6#%\"xG\"\"!" }{TEXT -1 17 " should be found." }}{PARA 0 "" 0 "" {TEXT -1 42 "One should then che ck that the derivative " }{TEXT 259 12 "changes sign" }{TEXT -1 4 " as " }{TEXT 278 1 "x" }{TEXT -1 19 " increases through " }{TEXT 279 1 "a " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "The point" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 14 " on the curve " } {XPPEDIT 18 0 "y=x^4" "6#/%\"yG*$%\"xG\"\"%" }{TEXT -1 61 " provides a n example of point at which the second derivative " }{XPPEDIT 18 0 "d^ 2*y/(d*x^2)=12*x^2" "6#/*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"* &\"#7F(*$F+F&F(" }{TEXT -1 71 " is zero, but such that the point is no t a point of inflection because " }{XPPEDIT 18 0 "d^2*y/(d*x^2)" "6#*( %\"dG\"\"#%\"yG\"\"\"*&F$F'*$%\"xGF%F'!\"\"" }{TEXT -1 41 " is poisiti ve on both sides of the value " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 373 291 291 {PLOTDATA 2 "6)-%'CURVESG6%7gn7$$!\"#\"\"!$\"#;F*7$$!3ymmm\"p0k&>! #<$\"3[SU\"zS$*\\Y\"!#;7$$!3MLLL$Q6G\">F0$\"3#o!3-@krQ8F37$$!31++v3-)[ (=F0$\"3?ic\\)3YcB\"F37$$!3bmm;M!\\p$=F0$\"3IWxDA_kQ6F37$$!3#)***\\7Y \"H%z\"F0$\"3=*>0pL1l.\"F37$$!3MLLL))Qj^VkB'F07$$!3OLL$3yO5]\"F0 $\"3;hNGf5^w]F07$$!3&*****\\nU)*=9F0$\"3O_bS(4[U0%F07$$!3SLL$3WDTL\"F0 $\"33ml.v3,oJF07$$!35++]d(Q&\\7F0$\"3A(>fpy/yV#F07$$!3gmmmc4`i6F0$\"3: '***!#=$ \"3%H4KxY.[)**F\\q7$$!3E++++0\"*H\"*F\\q$\"3=1VDE.5[pF\\q7$$!35++++83& H)F\\q$\"3'zxo8I#fMZF\\q7$$!3\\LLL3k(p`(F\\q$\"3(f(H5cJ#pA$F\\q7$$!3An mmmj^NmF\\q$\"3@VcX?xkQ>F\\q7$$!3)zmmmYh=(eF\\q$\"3I0&Q+p%y)=\"F\\q7$$ !3+,++v#\\N)\\F\\q$\"3Y()>'[U^\"oh!#>7$$!3commmCC(>%F\\q$\"3KAqNR?`.JF ]s7$$!39*****\\FRXL$F\\q$\"3%HaX7XbjB\"F]s7$$!3t*****\\#=/8DF\\q$\"3s$ 49>Y,%))R!#?7$$!3=mmm;a*el\"F\\q$\"3]5(ph82&=v!#@7$$!3komm;Wn(o)F]s$\" 3MQQS$Hzlp&!#A7$$!3IqLLL$eV(>F]t$\"39**yk57^>:!#G7$$\"3)Qjmm\"f`@')F]s $\"3/*[RQ&*z]_&Fit7$$\"3%z****\\nZ)H;F\\q$\"3o?mrb#zk0(Fct7$$\"3ckmm;$ y*eCF\\q$\"3sQAK^h5cOF]t7$$\"3f)******R^bJ$F\\q$\"3b)e=dMM%37F]s7$$\"3 'e*****\\5a`TF\\q$\"3yde')e;GwHF]s7$$\"3'o****\\7RV'\\F\\q$\"3=n!3#=Tf tgF]s7$$\"3Y'*****\\@fkeF\\q$\"3wb_s*y3H=\"F\\q7$$\"3_ILLL&4Nn'F\\q$\" 3sZ#R8?JM)>F\\q7$$\"3A*******\\,s`(F\\q$\"3FYLQh'3tA$F\\q7$$\"3%[mm;zM )>$)F\\q$\"3!R/6\"z+O\"z%F\\q7$$\"3M*******pfa<*F\\q$\"3+@JhUzz(3(F\\q 7$$\"39HLLeg`!)**F\\q$\"3?j#RbCrB#**F\\q7$$\"3w****\\#G2A3\"F0$\"3![') oVEX;P\"F07$$\"3;LLL$)G[k6F0$\"3:L&)4'G\"zQ=F07$$\"3#)****\\7yh]7F0$\" 3+R1q*[OiW#F07$$\"3xmmm')fdL8F0$\"3A!G$Qo^ziJF07$$\"3bmmm,FT=9F0$\"3!f 5A.&)>x/%F07$$\"3FLL$e#pa-:F0$\"3s_@9q6(p4&F07$$\"3!*******Rv&)z:F0$\" 3.^g@PcwHiF07$$\"3gmm;%)3;C;F0$\"3?y]+1*3&epF07$$\"3ILLLGUYo;F0$\"3KO, Ea9R\\xF07$$\"3\"*****\\n'*33+L*F07$$\"3ILLe*3k**y\"F0$\"3x/Q6+LaE5F37$$\"34++]sI@K=F0$\"37K #zW3[p7\"F37$$\"33+++S2ls=F0$\"3T$*)Q\"3#z(H7F37$$\"34++]2%)38>F0$\"33 M\\X6@\\R8F37$$\"3/++v.Uac>F0$\"3eCtzL$3aY\"F37$$\"\"#F*F+-%'COLOURG6& %$RGBG$\"*++++\"!\")$F*F*Ff^l-%*THICKNESSG6#F^^l-%%TEXTG6%7$$\"#B!\"\" $!\"&F`_lQ\"x6\"-F`^l6&Fb^lF*F*F*-F[_l6%7$$!#:F)$\"$v\"F`_lQ\"yFd_lFe_ l-F[_l6&7$$!#7F`_l$\"#5F*Q,f~''(x)~>~0Fd_l-F`^l6&Fb^l$\")#)eqkFe^l$\") )eqk\"Fe^lF[al-%%FONTG6$%*HELVETICAGFe`l-F[_l6&7$$\"#7F`_lFd`lFf`lFg`l F]al-%+AXESLABELSG6%%!GFial-F^al6#%(DEFAULTG-%%VIEWG6$;F(F^_l;Fa_lF\\` l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" "Curve 4" "Curve 5" }}{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Classification of turning points " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "A stationary p oint" }{XPPEDIT 18 0 " ``(c, f(c))" "6#-%!G6$%\"cG-%\"fG6#F&" }{TEXT -1 14 " on the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG " }{TEXT -1 14 " occurs where " }{XPPEDIT 18 0 "`f '`(c) = 0;" "6#/-%$ f~'G6#%\"cG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "A st ationary point" }{XPPEDIT 18 0 " ``(c, f(c))" "6#-%!G6$%\"cG-%\"fG6#F& " }{TEXT -1 88 " can be identified as a maximum point from the fact th at the derivative is positive for " }{TEXT 280 1 "x" }{TEXT -1 16 " ju st less than " }{TEXT 283 1 "c" }{TEXT -1 18 " and negative for " } {TEXT 281 1 "x" }{TEXT -1 19 " just greater than " }{TEXT 282 1 "c" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 259 21 "first derivative test" }{TEXT -1 22 " for a maximum point. " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 297 263 263 {PLOTDATA 2 " 6/-%'CURVESG6%7S7$$!\"#\"\"!$!\"%F*7$$!3MLLL$Q6G\">!#<$!3A!e4#)QZ)eOF0 7$$!3bmm;M!\\p$=F0$!3'*[e7a'***!#=$ !3mq\"))\\%))R#***Fbo7$$!3E++++0\"*H\"*Fbo$!3ND5!QdEbL)Fbo7$$!35++++83 &H)Fbo$!3yo4Oxt$3)oFbo7$$!3\\LLL3k(p`(Fbo$!3[Bt(zL,1o&Fbo7$$!3Anmmmj^N mFbo$!3%37I_u2IS%Fbo7$$!3)zmmmYh=(eFbo$!39$[s$3d(yW$Fbo7$$!3+,++v#\\N) \\Fbo$!3W.`jPjd$[#Fbo7$$!3commmCC(>%Fbo$!3'*4!*RKWoh6\"Fbo7$$!3t*****\\#=/8DFbo$!3CH$*>9#z`J'!#>7$$!3=m mm;a*el\"Fbo$!38sw$4j*)>u#Fbr7$$!3komm;Wn(o)Fbr$!37a/+x'ova(!#?7$$!3Iq LLL$eV(>F]s$!3gBHSG34)*Q!#B7$$\"3)Qjmm\"f`@')Fbr$!3[xsBc\")3LuF]s7$$\" 3%z****\\nZ)H;Fbo$!3)*)*GqVMScEFbr7$$\"3ckmm;$y*eCFbo$!3SBo$=Oul/'Fbr7 $$\"3f)******R^bJ$Fbo$!3+&>/'3\")G*4\"Fbo7$$\"3'e*****\\5a`TFbo$!3q1NS D.>D$)Fbo$!3$*4F1'4l>#pFbo7$$\"3M**** ***pfa<*Fbo$!3:3Cjqg!*=%)Fbo7$$\"39HLLeg`!)**Fbo$!3))y\"p6+56'**Fbo7$$ \"3w****\\#G2A3\"F0$!3EMgH-EF0$!3?')3\"\\D2*fOF07$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"*++++\"! \")$F*F*Fa[l-%*THICKNESSG6#Fiz-F$6%7$7$Fa[lFa[l7$Fa[lF+-F[[l6&F][lF*F* F*-%*LINESTYLEGFd[l-F$6&7#Fh[l-%'SYMBOLG6#%'CIRCLEGFj[l-%&STYLEG6#%&PO INTG-F$6&F`\\l-Fb\\l6#%(DIAMONDGFj[lFe\\l-F$6&F`\\l-Fb\\l6#%&CROSSGFj[ lFe\\l-%%TEXTG6&7$$Fbo!\"\"$!#9Fh]lQ+f~'(x)~>~06\"Fjz-%%FONTG6%%*HELVE TICAG%%BOLDG\"#5-Fd]l6&7$$\"#=Fh]lFi]lQ+f~'(x)~<~0F\\^lFjzF]^l-Fd]l6&7 $Fa[l$\"\"&Fh]lQ,f~''(c)~<~0F\\^lFj[lF]^l-Fd]l6&7$Fa[l$\"\"\"F*Q+f~'(c )~=~0F\\^lFj[lF]^l-Fd]l6&7$Fa[l$!#UFh]lQ\"cF\\^lFj[lF]^l-%+AXESLABELSG 6%Q\"xF\\^lQ!F\\^l-F^^l6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F( FhzFb`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv e 8" "Curve 9" "Curve 10" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Alternatively, a stationary point" }{XPPEDIT 18 0 " ``(c, f(c))" " 6#-%!G6$%\"cG-%\"fG6#F&" }{TEXT -1 34 " is a maximum point provided th at " }{XPPEDIT 18 0 "`f ''`(c);" "6#-%%f~''G6#%\"cG" }{TEXT -1 29 " is negative (in addition to " }{XPPEDIT 18 0 "`f '`(c) = 0;" "6#/-%$f~'G 6#%\"cG\"\"!" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 14 "The co ndition " }{XPPEDIT 18 0 "`f ''`(c) < 0;" "6#2-%%f~''G6#%\"cG\"\"!" } {TEXT -1 12 " means that " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#% \"xG" }{TEXT -1 41 " is negative in some interval containing " }{TEXT 289 1 "c" }{TEXT -1 19 ", and so the graph " }{XPPEDIT 18 0 "y=f(x)" " 6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 40 " is concave downwards in that int erval. " }}{PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 259 22 "s econd derivative test" }{TEXT -1 22 " for a maximum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "A stationary po int" }{XPPEDIT 18 0 " ``(c, f(c))" "6#-%!G6$%\"cG-%\"fG6#F&" }{TEXT -1 88 " can be identified as a minimum point from the fact that the de rivative is negative for " }{TEXT 284 1 "x" }{TEXT -1 16 " just less t han " }{TEXT 287 1 "c" }{TEXT -1 18 " and positive for " }{TEXT 285 1 "x" }{TEXT -1 19 " just greater than " }{TEXT 286 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 259 21 "first d erivative test" }{TEXT -1 22 " for a minimum point. " }}{PARA 256 "" 0 "" {GLPLOT2D 275 257 257 {PLOTDATA 2 "6/-%'CURVESG6%7S7$$!\"#\"\"!$ \"\"%F*7$$!3MLLL$Q6G\">!#<$\"3A!e4#)QZ)eOF07$$!3bmm;M!\\p$=F0$\"3'*[e7 a'***!#=$\"3mq\"))\\%))R#* **Fbo7$$!3E++++0\"*H\"*Fbo$\"3ND5!QdEbL)Fbo7$$!35++++83&H)Fbo$\"3yo4Ox t$3)oFbo7$$!3\\LLL3k(p`(Fbo$\"3[Bt(zL,1o&Fbo7$$!3Anmmmj^NmFbo$\"3%37I_ u2IS%Fbo7$$!3)zmmmYh=(eFbo$\"39$[s$3d(yW$Fbo7$$!3+,++v#\\N)\\Fbo$\"3W. `jPjd$[#Fbo7$$!3commmCC(>%Fbo$\"3'*4!*RKWoh6\"Fbo7$$!3t*****\\#=/8DFbo$\"3CH$*>9#z`J'!#>7$$!3=mmm;a*e l\"Fbo$\"38sw$4j*)>u#Fbr7$$!3komm;Wn(o)Fbr$\"37a/+x'ova(!#?7$$!3IqLLL$ eV(>F]s$\"3gBHSG34)*Q!#B7$$\"3)Qjmm\"f`@')Fbr$\"3[xsBc\")3LuF]s7$$\"3% z****\\nZ)H;Fbo$\"3)*)*GqVMScEFbr7$$\"3ckmm;$y*eCFbo$\"3SBo$=Oul/'Fbr7 $$\"3f)******R^bJ$Fbo$\"3+&>/'3\")G*4\"Fbo7$$\"3'e*****\\5a`TFbo$\"3q1 NSD.>D$)Fbo$\"3$*4F1'4l>#pFbo7$$ \"3M*******pfa<*Fbo$\"3:3Cjqg!*=%)Fbo7$$\"39HLLeg`!)**Fbo$\"3))y\"p6+5 6'**Fbo7$$\"3w****\\#G2A3\"F0$\"3EMgH-EF0$\"3?')3\"\\D2*fOF07$$\"\"#F*F+-%'COLOUR G6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-%*THICKNESSG6#Fiz-F$6%7$7$Fa[lFa[l7$F a[l$!3++++++++]Fbo-F[[l6&F][lF*F*F*-%*LINESTYLEGFd[l-F$6&7#Fh[l-%'SYMB OLG6#%'CIRCLEGF\\\\l-%&STYLEG6#%&POINTG-F$6&Fb\\l-Fd\\l6#%(DIAMONDGF\\ \\lFg\\l-F$6&Fb\\l-Fd\\l6#%&CROSSGF\\\\lFg\\l-%%TEXTG6&7$$Fbo!\"\"$\"# 9Fj]lQ+f~'(x)~<~06\"Fjz-%%FONTG6%%*HELVETICAG%%BOLDG\"#5-Ff]l6&7$$\"#= Fj]lF[^lQ+f~'(x)~>~0F^^lFjzF_^l-Ff]l6&7$Fa[l$\"\"&Fj]lQ,f~''(c)~>~0F^^ lF\\\\lF_^l-Ff]l6&7$Fa[l$\"\"\"F*Q+f~'(c)~=~0F^^lF\\\\lF_^l-Ff]l6&7$Fa [l$!\"(Fj]lQ\"cF^^lF\\\\lF_^l-%+AXESLABELSG6%Q\"xF^^lQ!F^^l-F`^l6#%(DE FAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F(FhzFd`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 48.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }} {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Alternatively, a statio nary point" }{XPPEDIT 18 0 " ``(c, f(c))" "6#-%!G6$%\"cG-%\"fG6#F&" } {TEXT -1 34 " is a minimum point provided that " }{XPPEDIT 18 0 "`f '' `(c);" "6#-%%f~''G6#%\"cG" }{TEXT -1 29 " is positive (in addition to \+ " }{XPPEDIT 18 0 "`f '`(c) = 0;" "6#/-%$f~'G6#%\"cG\"\"!" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 14 "The condition " }{XPPEDIT 18 0 " `f ''`(c);" "6#-%%f~''G6#%\"cG" }{TEXT -1 16 " > 0 means that " } {XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 41 " is posit ive in some interval containing " }{TEXT 288 1 "c" }{TEXT -1 19 ", and so the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 38 " is concave upwards in that interval. " }}{PARA 0 "" 0 " " {TEXT -1 12 "This is the " }{TEXT 259 22 "second derivative test" } {TEXT -1 22 " for a minimum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Cubic curves " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "f(x)=a*x^3+b*x^2+c*x+d" "6#/-%\"fG6#%\"xG,**&%\"aG\"\" \"*$F'\"\"$F+F+*&%\"bGF+*$F'\"\"#F+F+*&%\"cGF+F'F+F+%\"dGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a<>0" "6#0%\"aG\"\"!" }{TEXT -1 31 ", \+ be a general cubic function. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "`f '`(x) = 3*a*x^2+2*b*x+c;" "6#/-%$f~'G6#%\"xG,(*(\"\" $\"\"\"%\"aGF+F'\"\"#F+*(F-F+%\"bGF+F'F+F+%\"cGF+" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "`f ''`(x) = 6*a*x+2*b;" "6#/-%%f~''G6#%\"xG,&*(\"\" '\"\"\"%\"aGF+F'F+F+*&\"\"#F+%\"bGF+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "The graph of the second derivative is a straight lin e with non-zero gradient " }{XPPEDIT 18 0 "6*a" "6#*&\"\"'\"\"\"%\"aGF %" }{TEXT -1 24 ", and it intersects the " }{TEXT 290 1 "x" }{TEXT -1 12 " axis where " }{XPPEDIT 18 0 "6*a*x+2*b=0" "6#/,&*(\"\"'\"\"\"%\"a GF'%\"xGF'F'*&\"\"#F'%\"bGF'F'\"\"!" }{TEXT -1 17 ", that is, where " }{XPPEDIT 18 0 "x=-b/(3*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(! \"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "`f '`(-b/(3*a)) = 0;" "6#/-%$f~'G6#,$*&%\"bG\"\"\"*&\" \"$F*%\"aGF*!\"\"F.\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f ''`(x );" "6#-%%f~''G6#%\"xG" }{TEXT -1 17 " changes sign as " }{TEXT 291 1 "x" }{TEXT -1 29 " increases through the value " }{XPPEDIT 18 0 "x = - b/(3*a);" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F," }{TEXT -1 32 ", this means that the graph of " }{XPPEDIT 18 0 "y=a*x^3+b*x^2+c* x+d" "6#/%\"yG,**&%\"aG\"\"\"*$%\"xG\"\"$F(F(*&%\"bGF(*$F*\"\"#F(F(*&% \"cGF(F*F(F(%\"dGF(" }{TEXT -1 8 " has a " }{TEXT 259 19 "point of in flection" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "``(-b/(3*a),f(-b/(2*a))); " "6#-%!G6$,$*&%\"bG\"\"\"*&\"\"$F)%\"aGF)!\"\"F--%\"fG6#,$*&F(F)*&\" \"#F)F,F)F-F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "Now consider the derivative " }{XPPEDIT 18 0 "`f '`(x) = 3*a*x^2+2*b*x+c;" "6#/-%$f~'G6#%\"xG,(*(\"\"$\"\"\"% \"aGF+F'\"\"#F+*(F-F+%\"bGF+F'F+F+%\"cGF+" }{TEXT -1 10 ". Setting " } {XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 31 ", gives the quadratic equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "3*a*x^2+2*b*x+c=0" "6#/,(*(\"\"$\"\"\"%\"aGF'%\"xG\"\"# F'*(F*F'%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 13 " ------- (i)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "There are " }{TEXT 259 19 "three possibilities" }{TEXT -1 44 " for the stationa ry points on the graph of " }{XPPEDIT 18 0 "y=a*x^3+b*x^2+c*x+d" "6#/ %\"yG,**&%\"aG\"\"\"*$%\"xG\"\"$F(F(*&%\"bGF(*$F*\"\"#F(F(*&%\"cGF(F*F (F(%\"dGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 6 "Case I" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "If the discriminant " }{XPPEDIT 18 0 "4*b^2-12*a*c" "6#,&*&\"\" %\"\"\"*$%\"bG\"\"#F&F&*(\"#7F&%\"aGF&%\"cGF&!\"\"" }{TEXT -1 24 " is positive, that is, " }{XPPEDIT 18 0 "b^2>3*a*c" "6#2*(\"\"$\"\"\"%\"a GF&%\"cGF&*$%\"bG\"\"#" }{TEXT -1 15 ", then (i) has " }{TEXT 259 23 " two distinct real roots" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "They are " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=(-2 *b + sqrt(4*b^2-12*a*c))/(6*a)" "6#/%\"xG*&,&*&\"\"#\"\"\"%\"bGF)!\"\" -%%sqrtG6#,&*&\"\"%F)*$F*F(F)F)*(\"#7F)%\"aGF)%\"cGF)F+F)F)*&\"\"'F)F5 F)F+" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "x=(-2*b - sqrt(4*b^2-12* a*c))/(6*a)" "6#/%\"xG*&,&*&\"\"#\"\"\"%\"bGF)!\"\"-%%sqrtG6#,&*&\"\"% F)*$F*F(F)F)*(\"#7F)%\"aGF)%\"cGF)F+F+F)*&\"\"'F)F5F)F+" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (-b+sqrt(b^2-3*a*c))/(3*a)" "6#/%\" xG*&,&%\"bG!\"\"-%%sqrtG6#,&*$F'\"\"#\"\"\"*(\"\"$F/%\"aGF/%\"cGF/F(F/ F/*&F1F/F2F/F(" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "x = (-b-sqrt(b ^2-3*a*c))/(3*a)" "6#/%\"xG*&,&%\"bG!\"\"-%%sqrtG6#,&*$F'\"\"#\"\"\"*( \"\"$F/%\"aGF/%\"cGF/F(F(F/*&F1F/F2F/F(" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-b/(3*a)+p" "6#/%\"xG,&*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F,%\" pGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=-b/(3*a)-p" "6#/%\"xG,&*&% \"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F,%\"pGF," }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "p = sqrt(b^2-3*a*c)/(3*a) " "6#/%\"pG*&-%%sqrtG6#,&*$%\"bG\"\"#\"\"\"*(\"\"$F-%\"aGF-%\"cGF-!\" \"F-*&F/F-F0F-F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "Hence in this case the graph of " }{XPPEDIT 18 0 "y = a*x^3+b*x^2+c*x+d" " 6#/%\"yG,**&%\"aG\"\"\"*$%\"xG\"\"$F(F(*&%\"bGF(*$F*\"\"#F(F(*&%\"cGF( F*F(F(%\"dGF(" }{TEXT -1 69 " has two stationary points situated at a n equal horizontal distance " }{XPPEDIT 18 0 "abs(p)" "6#-%$absG6#%\"p G" }{TEXT -1 51 " to the left and right of the point of inflection. " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 292 1 "a" }{TEXT -1 1 " " } {TEXT 259 11 "is positive" }{TEXT -1 28 " then the second derivative \+ " }{XPPEDIT 18 0 "`f ''`(x) = 6*a*x+2*b;" "6#/-%%f~''G6#%\"xG,&*(\"\"' \"\"\"%\"aGF+F'F+F+*&\"\"#F+%\"bGF+F+" }{TEXT -1 17 " is negative for \+ " }{XPPEDIT 18 0 "x<-b/(3*a)" "6#2%\"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF( !\"\"F," }{TEXT -1 18 " and positive for " }{TEXT 296 1 "x" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "-b/(3*a);" "6#,$*&%\"bG\"\"\"*&\"\"$F&%\"aGF&! \"\"F*" }{TEXT -1 27 " ( The straight line graph " }{XPPEDIT 18 0 "y=6 *a*x+2*b" "6#/%\"yG,&*(\"\"'\"\"\"%\"aGF(%\"xGF(F(*&\"\"#F(%\"bGF(F(" }{TEXT -1 90 " slopes up from left to right. ), so that the graph of t he cubic is concave downwards for " }{XPPEDIT 18 0 "x<-b/(3*a)" "6#2% \"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F," }{TEXT -1 25 " and concave upwards for " }{TEXT 295 1 "x" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "-b/( 3*a);" "6#,$*&%\"bG\"\"\"*&\"\"$F&%\"aGF&!\"\"F*" }{TEXT -1 55 ". In p articular, this means that the cubic curve has a " }{TEXT 259 13 "maxi mum point" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = -b/(3*a)-p;" "6#/ %\"xG,&*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F,%\"pGF," }{TEXT -1 8 ", and a " }{TEXT 259 13 "minimum point" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = -b/(3*a)+p" "6#/%\"xG,&*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F,%\"p GF(" }{TEXT -1 2 ". 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AF*$\"3G#*\\cH8m)3%F*7$$\"3#**\\7V$e-]AF*$\"3-Q')=g)R4k%F*7$$\"3#)**** *********H#F*$\"3o(**********pE&F*-Fhz6&FjzF[[lFezFezF^[l-F$6%7$7$$!\" \"Ffz$Fa[lFfz7$F[[m$!\"'Ffz-Fhz6&FjzFfzFfzFfz-%*LINESTYLEGF`[l-F$6%7$F dz7$FezF_[mFa[mFc[m-F$6%7$7$$\"\"\"Ffz$!\"#Ffz7$F]\\mF_[mFa[mFc[m-F$6& 7%FjjlFdzF\\\\m-%'SYMBOLG6#%'CIRCLEGFa[m-%&STYLEG6#%&POINTG-F$6&Fd\\m- Ff\\m6#%(DIAMONDGFa[mFi\\m-F$6&Fd\\m-Ff\\m6#%&CROSSGFa[mFi\\m-%+AXESLA BELSG6%Q\"x6\"Q!F[^m-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG 6$;$!#BF\\[m$\"#BF\\[mF`^m" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-b/(3*a)-p" "6#,&*&%\"bG\"\"\"*&\"\"$F&%\"aGF&! \"\"F*%\"pGF*" }{TEXT -1 8 " " }{XPPEDIT 18 0 "-b/(3*a)" "6#,$* &%\"bG\"\"\"*&\"\"$F&%\"aGF&!\"\"F*" }{TEXT -1 8 " " }{XPPEDIT 18 0 "-b/(3*a)+p" "6#,&*&%\"bG\"\"\"*&\"\"$F&%\"aGF&!\"\"F*%\"pGF&" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 293 1 "a" }{TEXT -1 1 " " }{TEXT 259 11 "is ne gative" }{TEXT -1 28 " then the second derivative " }{XPPEDIT 18 0 "`f ''`(x) = 6*a*x+2*b;" "6#/-%%f~''G6#%\"xG,&*(\"\"'\"\"\"%\"aGF+F'F+F+* &\"\"#F+%\"bGF+F+" }{TEXT -1 17 " is positive for " }{XPPEDIT 18 0 "x< -b/(3*a)" "6#2%\"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F," }{TEXT -1 18 " and negative for " }{XPPEDIT 18 0 "x>-b/(3*a)" "6#2,$*&%\"bG\"\" \"*&\"\"$F'%\"aGF'!\"\"F+%\"xG" }{TEXT -1 27 " ( The straight line gra ph " }{XPPEDIT 18 0 "y=6*a*x+2*b" "6#/%\"yG,&*(\"\"'\"\"\"%\"aGF(%\"xG F(F(*&\"\"#F(%\"bGF(F(" }{TEXT -1 90 " slopes down from left to right. ), so that the graph of the cubic is concave upwards for " }{XPPEDIT 18 0 "x<-b/(3*a)" "6#2%\"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F," } {TEXT -1 27 " and concave downwards for " }{TEXT 294 1 "x" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "-b/(3*a);" "6#,$*&%\"bG\"\"\"*&\"\"$F&%\"aGF&! \"\"F*" }{TEXT -1 55 ". In particular, this means that the cubic curve has a " }{TEXT 259 13 "minimum point" }{TEXT -1 7 " where " } {XPPEDIT 18 0 "x = -b/(3*a)+p;" "6#/%\"xG,&*&%\"bG\"\"\"*&\"\"$F(%\"aG F(!\"\"F,%\"pGF(" }{TEXT -1 8 ", and a " }{TEXT 259 13 "maximum point " }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = -b/(3*a)-p;" "6#/%\"xG,&*& %\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F,%\"pGF," }{TEXT -1 18 ". Note that s ince " }{TEXT 297 1 "a" }{TEXT -1 15 " is negative, " }{XPPEDIT 18 0 "p = sqrt(b^2-3*a*c)/(3*a)" "6#/%\"pG*&-%%sqrtG6#,&*$%\"bG\"\"#\"\"\"* (\"\"$F-%\"aGF-%\"cGF-!\"\"F-*&F/F-F0F-F2" }{TEXT -1 22 " is negative, so that " }{XPPEDIT 18 0 "x = -b/(3*a)+p" "6#/%\"xG,&*&%\"bG\"\"\"*& \"\"$F(%\"aGF(!\"\"F,%\"pGF(" }{TEXT -1 19 " is to the left of " } {XPPEDIT 18 0 "x = -b/(3*a)-p" "6#/%\"xG,&*&%\"bG\"\"\"*&\"\"$F(%\"aGF (!\"\"F,%\"pGF," }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 302 7 "Case II" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "If the discriminant " }{XPPEDIT 18 0 "4*b^2-12*a*c" "6#,&*&\"\"%\" \"\"*$%\"bG\"\"#F&F&*(\"#7F&%\"aGF&%\"cGF&!\"\"" }{TEXT -1 20 " is ze ro, that is, " }{XPPEDIT 18 0 "b^2=3*a*c" "6#/*$%\"bG\"\"#*(\"\"$\"\" \"%\"aGF)%\"cGF)" }{TEXT -1 15 ", then (i) has " }{TEXT 259 22 "exactl y one real roots" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x=-b/(3*a)" "6#/%\"x G,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 65 "This means that point of inflection is also a stat ionary point ( " }{TEXT 259 32 "a stationary point of inflection" } {TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 87 "The two possiblities \+ for the graph of the cubic are shown in in the following picture. 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\"\"F," }{TEXT -1 63 " \+ " }{XPPEDIT 18 0 "x=-b/(3*a)" "6#/%\"xG,$*&%\"bG\"\"\"* &\"\"$F(%\"aGF(!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 303 8 "Case III" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "If the discriminant " }{XPPEDIT 18 0 "4*b^2-12*a* c" "6#,&*&\"\"%\"\"\"*$%\"bG\"\"#F&F&*(\"#7F&%\"aGF&%\"cGF&!\"\"" } {TEXT -1 24 " is negative, that is, " }{XPPEDIT 18 0 "b^2<3*a*c" "6#2 *$%\"bG\"\"#*(\"\"$\"\"\"%\"aGF)%\"cGF)" }{TEXT -1 15 ", then (i) has \+ " }{TEXT 259 13 "no real roots" }{TEXT -1 56 ". The roots can only be \+ expressed using complex numbers." }}{PARA 0 "" 0 "" {TEXT -1 60 "This \+ means that there are no stationary point on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 87 "The two possiblities for the graph of the cubic are \+ shown in in the following picture. 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$$\"3#p;aB[kR,'F*$!32DxV.6F=5F-7$$\"3g****\\&3=%egF*$!3sE(e406!y5F-7$$ \"3_m;HJpO4hF*$!3#[3O7]%[\\6F-7$$\"3'HLLj=O\\:'F*$!3Mrp3.@>;7F-7$$\"3V *\\(o;D_.iF*$!3[)\\J'R^F!H\"F-7$$\"3#**\\7V$e-]iF*$!3yjQu>V5k8F-7$$\"3 #)*************H'F*F+FjzFa[l-F$6%7$Fez7$Ffz$!#:Fhz-F[[l6&F][lFhzFhzFhz -%*LINESTYLEGFc[l-F$6%7$F]jm7$F^jmFeinFginFiin-F$6&7$FezF]jm-%'SYMBOLG 6#%'CIRCLEGFgin-%&STYLEG6#%&POINTG-F$6&Fajn-Fcjn6#%(DIAMONDGFginFfjn-F $6&Fajn-Fcjn6#%&CROSSGFginFfjn-%+AXESLABELSG6%Q\"x6\"Q!Fh[o-%%FONTG6#% (DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#j!\"\"$\"#jFh\\oF]\\o" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-b/(3*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"$F(%\"aGF(!\"\"F," } {TEXT -1 63 " \+ " }{XPPEDIT 18 0 "x=-b/(3*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"$F(% \"aGF(!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 63 "Every cubic curve is symmetrical about its point of inf lection " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "To see this let " }{XPPEDIT 18 0 "x[1]=-b/(3*a)" "6 #/&%\"xG6#\"\"\",$*&%\"bGF'*&\"\"$F'%\"aGF'!\"\"F." }{TEXT -1 8 " be t he " }{TEXT 299 1 "x" }{TEXT -1 44 " coordinate of the point of inflec tion. The " }{TEXT 298 1 "y" }{TEXT -1 43 " coordinate of the point of inflection is " }{XPPEDIT 18 0 "y[1]=``" "6#/&%\"yG6#\"\"\"%!G" } {XPPEDIT 18 0 "f(x[1]) = 2/27/(a^2)*b^3-c*b/(3*a)+d;" "6#/-%\"fG6#&%\" xG6#\"\"\",(**\"\"#F*\"#F!\"\"*$%\"aGF-F/%\"bG\"\"$F**(%\"cGF*F2F**&F3 F*F1F*F/F/%\"dGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "Tra nslating the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 32 " so that the point of inflection" }{XPPEDIT 18 0 " ``(x[1 ],y[1])" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 37 " moves to th e origin gives the graph:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y+y[1]=f(x+x[1])" "6#/,&%\"yG\"\"\"&F%6#F&F&-%\"fG6#,&% \"xGF&&F-6#F&F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = a*(x-b/(3*a))^3+ b*(x-b/(3*a))^2+c*(x-b/(3*a))-2*b^3/(27*a^2)+c*b/(3*a);" "6#/%\"yG,,*& %\"aG\"\"\"*$,&%\"xGF(*&%\"bGF(*&\"\"$F(F'F(!\"\"F0F/F(F(*&F-F(*$,&F+F (*&F-F(*&F/F(F'F(F0F0\"\"#F(F(*&%\"cGF(,&F+F(*&F-F(*&F/F(F'F(F0F0F(F(* (F6F(*$F-F/F(*&\"#FF(*$F'F6F(F0F0*(F8F(F-F(*&F/F(F'F(F0F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "This can be simplified to give: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=a*x^3+(c-b^2/(3 *a))*x" "6#/%\"yG,&*&%\"aG\"\"\"*$%\"xG\"\"$F(F(*&,&%\"cGF(*&%\"bG\"\" #*&F+F(F'F(!\"\"F3F(F*F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Because the expression " } {XPPEDIT 18 0 "a*x^3+(c-b^2/(3*a))*x" "6#,&*&%\"aG\"\"\"*$%\"xG\"\"$F& F&*&,&%\"cGF&*&%\"bG\"\"#*&F)F&F%F&!\"\"F1F&F(F&F&" }{TEXT -1 30 " on ly involves odd powers of " }{TEXT 300 1 "x" }{TEXT -1 18 ", it descri bes an " }{TEXT 259 12 "odd function" }{TEXT -1 2 " " }{XPPEDIT 18 0 "g(x)=a*x^3+(c-b^2/(3*a))*x" "6#/-%\"gG6#%\"xG,&*&%\"aG\"\"\"*$F'\"\"$ F+F+*&,&%\"cGF+*&%\"bG\"\"#*&F-F+F*F+!\"\"F5F+F'F+F+" }{TEXT -1 30 ", \+ which has the property that " }{XPPEDIT 18 0 "g(-x) = -g(x)" "6#/-%\"g G6#,$%\"xG!\"\",$-F%6#F(F)" }{TEXT -1 42 ". This means that the transl ated graph of " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" } {TEXT -1 59 " is symmetrical about the origin. Hence the original grap h " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 46 " is \+ symmetrical about the point of inflection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f := x -> a*x^3+b *x^2+c*x+d;\nx1 := -b/(3*a);\ny1 := f(x1);\ng := unapply(f(x+x1)-y1,x) ;\ng(x);\nexpand(simplify(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**&%\"aG\"\"\")9$\"\"$F/F/*&%\"b GF/)F1\"\"#F/F/*&%\"cGF/F1F/F/%\"dGF/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G,$*(\"\"$!\"\"%\"bG\"\"\"%\"aGF(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y1G,(**\"\"#\"\"\"\"#F!\"\"%\"aG!\"#%\"bG\"\" $F(**F.F*%\"cGF(F-F(F+F*F*%\"dGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,*&%\"aG\"\"\"),&9$F/*(\"\"$! \"\"%\"bGF/F.F5F5F4F/F/*&F6F/)F1\"\"#F/F/*&%\"cGF/F1F/F/**F9F/\"#FF5F. !\"#F6F4F5**F4F5F;F/F6F/F.F5F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,,*&%\"aG\"\"\"),&%\"xGF&*(\"\"$!\"\"%\"bGF&F%F,F,F+F&F&*&F-F&)F(\" \"#F&F&*&%\"cGF&F(F&F&**F0F&\"#FF,F%!\"#F-F+F,**F+F,F2F&F-F&F%F,F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"aG\"\"\")%\"xG\"\"$F&F&**F)!\" \"F%F+F(F&%\"bG\"\"#F+*&%\"cGF&F(F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "We show how to sketch the gr aph of " }{XPPEDIT 18 0 "y = x^3-6*x^2+9*x;" "6#/%\"yG,(*$%\"xG\"\"$ \"\"\"*&\"\"'F)*$F'\"\"#F)!\"\"*&\"\"*F)F'F)F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "f(x)=x^3-6*x^2+9*x" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\"\"\"* &\"\"'F+*$F'\"\"#F+!\"\"*&\"\"*F+F'F+F+" }{XPPEDIT 18 0 "`` = x*(x^2-6 *x+9);" "6#/%!G*&%\"xG\"\"\",(*$F&\"\"#F'*&\"\"'F'F&F'!\"\"\"\"*F'F'" }{XPPEDIT 18 0 "``=x*(x-3)^2" "6#/%!G*&%\"xG\"\"\"*$,&F&F'\"\"$!\"\"\" \"#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {XPPEDIT 18 0 "`f ''`(x) = 3*x^2-12*x+9;" "6#/-%%f~''G6#%\"xG,(*&\"\"$ \"\"\"*$F'\"\"#F+F+*&\"#7F+F'F+!\"\"\"\"*F+" }{XPPEDIT 18 0 "``=3*(x^2 -4*x+3)" "6#/%!G*&\"\"$\"\"\",(*$%\"xG\"\"#F'*&\"\"%F'F*F'!\"\"F&F'F' " }{XPPEDIT 18 0 "``=3*(x-1)*(x-3)" "6#/%!G*(\"\"$\"\"\",&%\"xGF'F'!\" \"F',&F)F'F&F*F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "and \+ " }{XPPEDIT 18 0 "`f ''`(x) = 6*x-12;" "6#/-%%f~''G6#%\"xG,&*&\"\"'\" \"\"F'F+F+\"#7!\"\"" }{XPPEDIT 18 0 "``=6*(x-2)" "6#/%!G*&\"\"'\"\"\", &%\"xGF'\"\"#!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "`f ' `(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 14 " is zero when " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 10 " and when " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "The fo llowing table indicates how the sign of the derivative changes as " } {TEXT 307 1 "x" }{TEXT -1 9 " varies. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 415 139 139 {PLOTDATA 2 "6>-%'CURVESG6$7$7$$!3+++++++ +N!#<$!3+++++++]7F*7$$\"3++++++++bF*F+-%'COLOURG6&%$RGBG\"\"!F4F4-F$6$ 7$7$F($\"3++++++++]!#=7$F.F9F0-F$6$7$7$F($\"\"\"F47$F.FAF0-F$6$7$F'F@F 0-F$6$7$7$$!3++++++++DF*F+7$FKFAF0-F$6$7$7$$!3++++++++]F;F+7$FRFAF0-F$ 6$7$7$F9F+7$F9FAF0-F$6$7$7$$\"3++++++++DF*F+7$FhnFAF0-F$6$7$7$$\"3++++ ++++NF*F+7$F_oFAF0-F$6$7$F-FCF0-F$6'7$7$$!\"#F4$!\"\"F47$F[p$!\"$F\\p7 %7$$!+w'4I>\"!\"*$!*1wbV$FdpF]p7$$!+C.*p5\"Fdp$!*%RUkYFdp-%&STYLEG6#%, PATCHNOGRIDGF0-%*THICKNESSG6#\"\"#-F$6'7$7$FAF^p7$$FcqF4$!#5F\\p7%7$$ \"+w'4I*=Fdp$!+ggdN$)F[rFhq7$$\"+C.*p!=Fdp$!+SRUk&*F[rF\\qF0F`q-F$6'7$ 7$$\"\"%F4F[p7$$\"\"&F4F^p7%7$$\"+C.*p![FdpFepF]s7$$\"+w'4I*[FdpFjpF\\ qF0F`q-%%TEXTG6&7$$F_pF4$\"#vFjoQ\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-Fh s6&7$$!#:F\\pF\\tQ&x~<~1F_tF0F`t-Fhs6&7$$F4F4F\\tQ\"1F_tF0F`t-Fhs6&7$$ \"#:F\\pF\\tQ*1~<~x~<~3F_tF0F`t-Fhs6&7$$\"\"$F4F\\tQ\"3F_tF0F`t-Fhs6&7 $$\"#XF\\pF\\tQ&x~>~3F_tF0F`t-Fhs6&7$F[t$FcqF\\pQ'f~'(x)F_tF0F`t-Fhs6& 7$F^uFevQ\"0F_tF0F`t-Fhs6&7$FiuFevFjvF0F`t-Fhs6&7$Fht$FjuF\\pQ\"+F_tF0 -Fat6$Fct\"#9-Fhs6&7$Fcu$F\\sF\\pQ\"_F_tF0Fcw-Fhs6&7$F_vFevFbwF0Fcw-%* AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F_tFex-%%VIEWG6$;$!#NF\\p$\"#bF\\ p;$!#8F\\p$\"#6F\\p" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(1)=4" "6#/-%\"fG6#\"\"\"\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " f(3)=0" "6#/-%\"fG6#\"\"$\"\"!" }{TEXT -1 33 ", we obtain the stationa ry points" }{XPPEDIT 18 0 "``(1,4)" "6#-%!G6$\"\"\"\"\"%" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(3,0)" "6#-%!G6$\"\"$\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 46 "Applying the first derivative test we see that" }{XPPEDIT 18 0 "``(1,4)" "6#-%!G6$\"\"\"\"\"%" }{TEXT -1 6 " is a " }{TEXT 259 13 "maximum point" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(3,0)" "6#-%!G6$\"\"$\"\"!" }{TEXT -1 6 " is a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "The second derivative " }{XPPEDIT 18 0 "` f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 14 " is zero when " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 79 "The following table indicates how the sign of the second \+ derivative changes as " }{TEXT 308 1 "x" }{TEXT -1 9 " varies. 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" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)=x*(x-3)^2" "6#/-%\"fG6#%\"xG* &F'\"\"\"*$,&F'F)\"\"$!\"\"\"\"#F)" }{TEXT -1 16 ", note that the " } {TEXT 309 1 "x" }{TEXT -1 1 " " }{TEXT 259 10 "intercepts" }{TEXT -1 26 " of the curve occur where " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 11 " and where " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "We can use the information ga thered to help in sketching the graph of " }{XPPEDIT 18 0 "y = x^3-6* x^2+9*x;" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"*&\"\"'F)*$F'\"\"#F)!\"\"*&\"\" *F)F'F)F)" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f (x) = x+1/x;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*&F)F)F'!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "`f '`(x) = \+ 1-1/(x^2);" "6#/-%$f~'G6#%\"xG,&\"\"\"F)*&F)F)*$F'\"\"#!\"\"F-" } {XPPEDIT 18 0 "`` = (x^2-1)/(x^2);" "6#/%!G*&,&*$%\"xG\"\"#\"\"\"F*!\" \"F**$F(F)F+" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`f ''`(x) = 2/(x^3 );" "6#/-%%f~''G6#%\"xG*&\"\"#\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The deri vative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 14 " i s zero when " }{XPPEDIT 18 0 "x = ``;" "6#/%\"xG%!G" }{TEXT -1 1 " " } {TEXT 317 1 "+" }{TEXT -1 4 " 1. " }}{PARA 0 "" 0 "" {TEXT -1 72 "The \+ following table indicates how the sign of the derivative changes as " }{TEXT 316 1 "x" }{TEXT -1 9 " varies. " }}{PARA 0 "" 0 "" {TEXT -1 13 "A column for " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 68 " is included in the table because the derivative is not defined for " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 47 " and so could chang e sign across this value of " }{TEXT 318 1 "x" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 588 107 107 {PLOTDATA 2 "6 E-%'CURVESG6$7$7$$!3++++++++N!#<$!3+++++++]7F*7$$\"3++++++++&)F*F+-%'C OLOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F.F9F0-F$6$7$7$F($ \"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF*F+7$FKFAF0-F$6$7$7 $$!3++++++++]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF0-F$6$7$7$$\"3++++++++DF* F+7$FhnFAF0-F$6$7$7$$\"3++++++++NF*F+7$F_oFAF0-F$6$7$7$$\"3++++++++bF* F+7$FfoFAF0-F$6$7$7$$\"3++++++++lF*F+7$F]pFAF0-F$6$7$F-FCF0-F$6'7$7$$! \"#F4$!\"\"F47$Fip$!\"$Fjp7%7$$!+w'4I>\"!\"*$!*1wbV$FbqF[q7$$!+C.*p5\" Fbq$!*%RUkYFbq-%&STYLEG6#%,PATCHNOGRIDGF0-%*THICKNESSG6#\"\"#-F$6'7$7$ FAF\\q7$$FarF4$!#5Fjp7%7$$\"+w'4I*=Fbq$!+ggdN$)FirFfr7$$\"+C.*p!=Fbq$! +SRUk&*FirFjqF0F^r-F$6'7$7$$\"\"%F4F\\q7$$\"\"&F4Fhr7%7$$\"+w'4I*[FbqF ^sF[t7$$\"+C.*p![FbqFcsFjqF0F^r-F$6'7$7$$\"\"(F4Fip7$$\"\")F4F\\q7%7$$ \"+C.*p!yFbqFcqF[u7$$\"+w'4I*yFbqFhqFjqF0F^r-%%TEXTG6&7$$F]qF4$\"#vFhp Q\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-Ffu6&7$$!#:FjpFjuQ'x~<~-1F]vF0F^v- Ffu6&7$$F4F4FjuQ#-1F]vF0F^v-Ffu6&7$$\"#:FjpFjuQ+-1~<~x~<~0F]vF0F^v-Ffu 6&7$$\"\"$F4FjuQ\"0F]vF0F^v-Ffu6&7$$\"#XFjpFjuQ*0~<~x~<~1F]vF0F^v-Ffu6 &7$$\"\"'F4FjuQ\"1F]vF0F^v-Ffu6&7$$F[vFjpFjuQ&x~>~1F]vF0F^v-Ffu6&7$Fiu $FarFjpQ'f~'(x)F]vF0F^v-Ffu6&7$F\\wF^yFiwF0F^v-Ffu6&7$FgwF^yQ#NDF]vF0F ^v-Ffu6&7$FcxF^yFiwF0F^v-Ffu6&7$Ffv$FhwFjpQ\"+F]vF0-F_v6$Fav\"#9-Ffu6& 7$Faw$FjsFjpQ\"_F]vF0F_z-Ffu6&7$F]xFezFfzF0F_z-Ffu6&7$FixF^yF^zF0F_z-% *AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F]vFd[l-%%VIEWG6$;$!#NFjp$\"#&)F jp;$!#8Fjp$\"#6Fjp" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur ve 32" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "f(-1) = -2;" "6#/-%\"fG6#,$\"\"\"!\"\",$\"\"#F)" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(1) = 2;" "6#/-%\"fG6#\"\"\"\"\"# " }{TEXT -1 33 ", we obtain the stationary points" }{XPPEDIT 18 0 "``( -1, -2);" "6#-%!G6$,$\"\"\"!\"\",$\"\"#F(" }{TEXT -1 4 " and" } {XPPEDIT 18 0 "``(1,2);" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 46 "Applying the first derivative test we see that" }{XPPEDIT 18 0 "``(-1, -2);" "6#-%!G6$,$\"\"\"!\"\",$\"\"#F(" } {TEXT -1 6 " is a " }{TEXT 259 13 "maximum point" }{TEXT -1 4 " and" } {XPPEDIT 18 0 "``(1, 2);" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 6 " is a " } {TEXT 259 13 "minimum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The second derivative " } {XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 31 " is never zero. However, since " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG " }{TEXT -1 20 " is not defined for " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\" \"!" }{TEXT -1 29 " there is a possibility that " }{XPPEDIT 18 0 "`f ' '`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 36 " could change sign across th e value " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 72 ". In fact there is such a sign change as the following table indicates. " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 344 125 125 {PLOTDATA 2 "6 7-%'CURVESG6$7$7$$!3++++++++N!#<$!3+++++++]7F*7$$\"3++++++++DF*F+-%'CO LOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F.F9F0-F$6$7$7$F($ \"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF*F+7$FKFAF0-F$6$7$7 $$!3++++++++]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF0-F$6$7$F-FCF0-F$6%7S7$$! 3\"4b_H`+'>\"*F;$!3#HjH_,%)z!))F;7$$!3S%oPVV)G,#*F;$!33w+L5Y\"*e%)F;7$ $!3%\\m=zL>#*G*F;$!3y5G4*)Qif\")F;7$$!37s;)>SMlS*F;$!3QAShhE-HyF;7$$!3 F-\"\\O'p$Ra*F;$!38$4`-bFP](F;7$$!3!e%o4G9X*p*F;$!3OQj(\\yh&)=(F;7$$!3 q1l^e@*)f)*F;$!3jD&4n'p&\\!pF;7$$!3rI%\\Ay)>/5F*$!3&el!)\\#)H4i'F;7$$! 3AsS_[%zY-\"F*$!3Qoz)f&yVQjF;7$$!3]X!H5E%pY5F*$!3y,Mt(Hu\"pgF;7$$!3a?? 2dw#42\"F*$!3YzVupK*f!eF;7$$!39Wv$p'*QN4\"F*$!3YAmUiW]'e&F;7$$!39jB,]A L?6F*$!3g%y#=$=3SN&F;7$$!35Jpzw2d[6F*$!3#H**)GE\"**o8&F;7$$!3S,%*oFp&p <\"F*$!3r[\"ya<$)Q%\\F;7$$!3!QG-$oGl.7F*$!3_f@dR#GHy%F;7$$!3%R:>g)>VO7 F*$!3V!)e;p))*)4YF;7$$!3Wgm0cA*\\E\"F*$!3-yJ>'))*QzWF;7$$!3'*4dXR!>!*H \"F*$!3?P%*HWNiYVF;7$$!3`2Ym?SwH8F*$!3[:0E^(olC%F;7$$!3+OcTt91k8F*$!3u N]18E-cTF;7$$!3#>xQe>krR\"F*$!3'>_n_c$y)3%F;7$$!30&\\g'Q#\\?V\"F*$!3#f TZTs,'QSF;7$$!3%[(oK7SHk9F*$!3+3F*ftL1,%F;7$$!3K5aUn!)=*\\\"F*$!3I&RBO \\0++%F;7$$!3bN41zWVN:F*$!3@$ow>\\s/,%F;7$$!3q#R&*4O')oc\"F*$!3+UC_&4) RPSF;7$$!3pdvU+Zk+;F*$!37G*QLd8]3%F;7$$!3OA(>='y^o;F*$!3gYJV7b\\TUF;7$$!3oj2BHkB+#yWF;7$$!3yH>RD2(\\w\"F*$!3[(ejlc!y;YF;7$$!3G-*R1=bj z\"F*$!39/\"y5'R(Hy%F;7$$!3#e;SyU+R#=F*$!3w'\\)4*>r$\\\\F;7$$!3/7/COd% H&=F*$!3YJHk'>%*y9&F;7$$!3%)3j]x(o\"z=F*$!3__Y&R2N*\\`F;7$$!3&)*o-7PW` !>F*$!3_3w(zgYid&F;7$$!33k0'G0L'H>F*$!3oIrMatt6eF;7$$!3/Bz/Kjf`>F*$!3- (f8M%z_sgF;7$$!3Mj\"p`i#=v>F*$!3)yg#y%3[mL'F;7$$!3w'R:Z!)oc*>F*$!3*)yx A%y()*=mF;7$$!3EI#)\\h3L9?F*$!3TL*Hv*)y-\"pF;7$$!3w)>V$*Gt*H?F*$!3!*y< <$[Cq=(F;7$$!3[M5G*>pg/#F*$!3lp$)z#))pQ^(F;7$$!3#3V!eM9we?F*$!3>H7dzo+ 9yF;7$$!3O#G_!pwZq?F*$!3&ov(o9j4T\")F;7$$!37V302/!*z?F*$!3'4P.El:+Y)F; 7$$!3MUOvY*R!)3#F*$!3F7=k\"*F;$!3=)=eB)f,#>%F;7$$\"3>@bqL%)G,#*F;$!3x`7H(Q&3TXF;7$$\"3/J P;P$>#*G*F;$!3+)ek&3hPS[F;7$$\"3FA64,W`1%*F;$!3+*\\\"4Ot(4<&F;7$$\"3r9 ^iip$Ra*F;$!31#z3vWsi\\&F;7$$\"3)\\dVpU^%*p*F;$!3ES$\\G@Q9\"eF;7$$\"31 +pCd@*)f)*F;$!35o>=JI/&4'F;7$$\"3#*=36#y)>/5F*$!3qLb)H(F;7$$\"3k*\\cn'*QN4\"F*$!3EZg!f`&\\8uF;7$$\"3f\" y@)\\AL?6F*$!3)Qzf_\"=*fk(F;7$$\"3yVufw2d[6F*$!3y@%pA(35jyF;7$$\"3)f*> [Fp&p<\"F*$!3=1n>Bo6c!)F;7$$\"35w#)3oGl.7F*$!3X/A@f<2<#)F;7$$\"3f[!)z& )>VO7F*$!33ZHvH65!R)F;7$$\"3O,-$eD#*\\E\"F*$!3c.G%G651_)F;7$$\"3![!QAR !>!*H\"F*$!3'\\7w[XwLl)F;7$$\"3(yfG/-k(H8F*$!3!e;T![7V`()F;7$$\"3?7fkrR\"F*$!39Q1JMk@6*)F;7$$\"3WafTQ# \\?V\"F*$!3QYQdv#)Rh*)F;7$$\"3;(=\"37SHk9F*$!3%H#3'QEm$*)*)F;7$$\"3w'G zr1)=*\\\"F*$!3[uKP1X*****)F;7$$\"3?T_\")yWVN:F*$!3)4no\"3v_*)*)F;7$$ \"3y-3vgj)oc\"F*$!33I>v/>gi*)F;7$$\"3*3#\\=+Zk+;F*$!3>BR2Fk)\\\"*)F;7$ $\"3^o;F*$!3!45e#)[/&e()F;7$$\" 3%*\\()**GkB+KQ)F;7$$\"3O'*eU!=bjz\"F*$!3*\\cP,/Eq@)F;7$$\" 3b&)HjF/!R#=F*$!3?k,B-)G10)F;7$$\"3Yv8/Od%H&=F*$!3Uw[![!e5_yF;7$$\"3') fbJx(o\"z=F*$!3wB2gF\\1]wF;7$$\"3MC7-rVM0>F*$!3wS^o$R`PU(F;7$$\"3ie()o _IjH>F*$!3M`_TZEE)=(F;7$$\"3[:o)=L'f`>F*$!38'3Z%e?ZFpF;7$$\"3A!*)=_i#= v>F*$!3YCm;<>NjmF;7$$\"3E0nd/)oc*>F*$!3W)[0y@75Q'F;7$$\"3l([r8'3L9?F*$ !3C)))zX5@(*3'F;7$$\"3%)3yA*Gt*H?F*$!353A+>b(H\"eF;7$$\"3i^!z\"*>pg/#F *$!3SV;W>,8'[&F;7$$\"3Wf2\\M9we?F*$!3yZ3sAJ*f=&F;7$$\"39Hg(*owZq?F*$!3 o\"Q_wo.*e[F;7$$\"3Ksw)pS+*z?F*$!3)*>ax\\V)*RXF;7$$\"3-XZqY*R!)3#F*$!3 'pOqZ)f,#>%F;F0F_^l-%%TEXTG6&7$$!\"$F4$\"#v!\"#Q\"x6\"F0-%%FONTG6$%*HE LVETICAG\"#5-F\\^m6&7$$!#:!\"\"Fa^mQ&x~<~0Fe^mF0Ff^m-F\\^m6&7$$F4F4Fa^ mQ\"0Fe^mF0Ff^m-F\\^m6&7$$\"#:F`_mFa^mQ&x~>~0Fe^mF0Ff^m-F\\^m6&7$F_^m$ Fb^lF`_mQ(f~''(x)Fe^mF0Ff^m-F\\^m6&7$Fe_mF``mQ#NDFe^mF0Ff^m-F\\^m6&7$F ^_m$\"\"%F`_mQ\"_Fe^mF0-Fg^m6$Fi^m\"#9-F\\^m6&7$Fj_m$\"\"$F`_mQ\"+Fe^m F0F\\am-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!Fe^mF\\bm-Fg^m6#%(DEFAU LTG-%%VIEWG6$;$!#NF`_m$\"#DF`_m;$!#8F`_m$\"#6F`_m" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "There \+ are no points of inflection on the graph of " }{XPPEDIT 18 0 "y=x+1/x " "6#/%\"yG,&%\"xG\"\"\"*&F'F'F&!\"\"F'" }{TEXT -1 21 ". There are al so no " }{TEXT 319 1 "x" }{TEXT -1 13 " intercepts. " }}{PARA 0 "" 0 " " {TEXT -1 3 "As " }{XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)o peratorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 8 " and as " } {XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\" ,$%)infinityG!\"\"F*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "1/x" "6#*& \"\"\"F$%\"xG!\"\"" }{TEXT -1 16 " tends to 0, so " }{XPPEDIT 18 0 "y= x+1/x" "6#/%\"yG,&%\"xG\"\"\"*&F'F'F&!\"\"F'" }{TEXT -1 1 " " }{TEXT 320 1 "~" }{TEXT -1 1 " " }{TEXT 321 1 "x" }{TEXT -1 6 " when " } {TEXT 322 1 "x" }{TEXT -1 46 " is large and positive or large and nega tive. " }}{PARA 0 "" 0 "" {TEXT -1 9 "The line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 37 " is as an asymptote for the graph of " } {XPPEDIT 18 0 "y=x+1/x" "6#/%\"yG,&%\"xG\"\"\"*&F'F'F&!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "We can use the information ga thered to help in sketching the graph of " }{XPPEDIT 18 0 "y = x+1/x; " "6#/%\"yG,&%\"xG\"\"\"*&F'F'F&!\"\"F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 364 364 364 {PLOTDATA 2 "6.-%'CURVES G6%7[o7$$!\"&\"\"!$!3;+++++++_!#<7$$!3)HL3#fy%Q*[F-$!3)*eP?y$F-$!3'\\rdk&\\ WYSF-7$$!3'***\\P.mcwOF-$!33UG'4(GF-$!3[B!)QV!y#>KF-7$$!3am\"zpE0mw#F-$!3Q;Ctr!f!GJF-7$$!3O; H-OCxqEF-$!3Q\"*>!)\\f>XIF-7$$!3YL3F\"y.uc#F-$!3AM)o\"*y#F-7$$!3YLe%**Q >cE#F-$!33s*>Kz**pq#F-7$$!35++0<;Lh@F-$!3G\"=5TM4Si#F-7$$!3?+D;xjIf?F- $!3%e(H$px1\\a#F-7$$!3!*\\7`6F-$!3MCKG7=kqCF-7$$!3A+vt0f)4&=F-$!3k ggvL&Q7R#F-7$$!3/nmJ9-]_#F-7$$!3)***\\_\"y()yW\"F-$!3'o\"> &=/\\&Q@F-7$$!3=o!4#F-7$$!3))\\ibLETZ7F-$!3%**zbF )>2\\?F-7$$!3%omT&*3Us9\"F-$!3d%)zf'p(*)=?F-7$$!3?]7G8GPU5F-$!3cM)H%pC s,?F-7$$!3>NLLiBr8%*!#=$!3H'*)p&49l.?F-7$$!3AI$3sND3Q)F^x$!3t`$H$QCGJ? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x ) = x/(1+x^2);" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F)F)*$F'\"\"#F)!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Using the quotient rule: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = (1*`.`*(1+x^2)-x*`.`*2*x)/(1+x^2);" " 6#/-%$f~'G6#%\"xG*&,&*(\"\"\"F+%\".GF+,&F+F+*$F'\"\"#F+F+F+**F'F+F,F+F /F+F'F+!\"\"F+,&F+F+*$F'F/F+F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= (1 -x^2)/(1+x^2)^2" "6#/%!G*&,&\"\"\"F'*$%\"xG\"\"#!\"\"F'*$,&F'F'*$F)F*F 'F*F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Using the quoti ent rule again: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` f ''`(x) = ((-2*x)*`.`*(1+x^2)^2-(1-x^2)*`.`*2*(1+x^2)*2*x)/((1+x^2)^4 );" "6#/-%%f~''G6#%\"xG*&,&*(,$*&\"\"#\"\"\"F'F.!\"\"F.%\".GF.,&F.F.*$ F'F-F.F-F.*.,&F.F.*$F'F-F/F.F0F.F-F.,&F.F.*$F'F-F.F.F-F.F'F.F/F.*$,&F. F.*$F'F-F.\"\"%F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=((-2*x)*(1+x^2)-4* x*(1-x^2))/(1+x^2)^3" "6#/%!G*&,&*&,$*&\"\"#\"\"\"%\"xGF+!\"\"F+,&F+F+ *$F,F*F+F+F+*(\"\"%F+F,F+,&F+F+*$F,F*F-F+F-F+*$,&F+F+*$F,F*F+\"\"$F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "( on dividing the top a nd bottom by" }{XPPEDIT 18 0 " ``(1+x^2)" "6#-%!G6#,&\"\"\"F'*$%\"xG\" \"#F'" }{TEXT -1 3 " ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(-2*x-2*x^3-4*x+4*x^3)/(1+x^2)^3" "6#/%!G*&,**&\"\"#\"\"\"%\" xGF)!\"\"*&F(F)*$F*\"\"$F)F+*&\"\"%F)F*F)F+*&F0F)*$F*F.F)F)F)*$,&F)F)* $F*F(F)F.F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(2*x^3-6*x)/(1+x^2)^3" " 6#/%!G*&,&*&\"\"#\"\"\"*$%\"xG\"\"$F)F)*&\"\"'F)F+F)!\"\"F)*$,&F)F)*$F +F(F)F,F/" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 7 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(x) = 2*x*(x^2- 3)/((1+x^2)^3);" "6#/-%%f~''G6#%\"xG**\"\"#\"\"\"F'F*,&*$F'F)F*\"\"$! \"\"F**$,&F*F**$F'F)F*F-F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "`f ''`(x) = (1-x^2)/((1+x^2)^2);" "6#/-%%f~''G6#%\"xG*&,&\"\"\"F**$ F'\"\"#!\"\"F**$,&F*F**$F'F,F*F,F-" }{TEXT -1 15 " is zero when " } {XPPEDIT 18 0 "x^2-1=0" "6#/,&*$%\"xG\"\"#\"\"\"F(!\"\"\"\"!" }{TEXT -1 15 ", that is when " }{XPPEDIT 18 0 "x = ``;" "6#/%\"xG%!G" }{TEXT -1 1 " " }{TEXT 310 1 "+" }{TEXT -1 4 " 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The following table indicates h ow the sign of the derivative changes as " }{TEXT 304 1 "x" }{TEXT -1 9 " varies. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 432 112 112 {PLOTDATA 2 "6>-%'CURVESG6$7$7$$!3++++++++N!#<$!3+++++++]7F*7$$\"3 ++++++++bF*F+-%'COLOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F .F9F0-F$6$7$7$F($\"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF*F +7$FKFAF0-F$6$7$7$$!3++++++++]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF0-F$6$7$ 7$$\"3++++++++DF*F+7$FhnFAF0-F$6$7$7$$\"3++++++++NF*F+7$F_oFAF0-F$6$7$ F-FCF0-F$6'7$7$$!\"#F4$!\"$!\"\"7$$F]pF4$!#5F]p7%7$$!+C.*p5\"!\"*$!+gg dN$)FapF^p7$$!+w'4I>\"Ffp$!+SRUk&*Fap-%&STYLEG6#%,PATCHNOGRIDGF0-%*THI CKNESSG6#\"\"#-F$6'7$7$FAF_p7$$FeqF4F[p7%7$$\"+C.*p!=Ffp$!*1wbV$FfpFjq 7$$\"+w'4I*=Ffp$!*%RUkYFfpF^qF0Fbq-F$6'7$7$$\"\"%F4F[p7$$\"\"&F4F`p7%7 $$\"+w'4I*[FfpFgpF]s7$$\"+C.*p![FfpF\\qF^qF0Fbq-%%TEXTG6&7$$F\\pF4$\"# vFjoQ\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-Fhs6&7$$!#:F]pF\\tQ'x~<~-1F_tF 0F`t-Fhs6&7$$F4F4F\\tQ#-1F_tF0F`t-Fhs6&7$$\"#:F]pF\\tQ+-1~<~x~<~1F_tF0 F`t-Fhs6&7$$\"\"$F4F\\tQ\"1F_tF0F`t-Fhs6&7$$\"#XF]pF\\tQ&x~>~1F_tF0F`t -Fhs6&7$F[t$FeqF]pQ'f~'(x)F_tF0F`t-Fhs6&7$F^uFevQ\"0F_tF0F`t-Fhs6&7$Fi uFevFjvF0F`t-Fhs6&7$Fht$F\\sF]pQ\"_F_tF0-Fat6$Fct\"#9-Fhs6&7$FcuFevQ\" +F_tF0Fcw-Fhs6&7$F_vFawFbwF0Fcw-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q !F_tFdx-%%VIEWG6$;$!#NF]p$\"#bF]p;$!#8F]p$\"#6F]p" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23 " "Curve 24" "Curve 25" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(-1) = -1/2; " "6#/-%\"fG6#,$\"\"\"!\"\",$*&F(F(\"\"#F)F)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "f(1) = 1/2;" "6#/-%\"fG6#\"\"\"*&F'F'\"\"#!\"\"" } {TEXT -1 33 ", we obtain the stationary points" }{XPPEDIT 18 0 "``(-1, -1/2);" "6#-%!G6$,$\"\"\"!\"\",$*&F'F'\"\"#F(F(" }{TEXT -1 4 " and" } {XPPEDIT 18 0 "``(1, 1/2);" "6#-%!G6$\"\"\"*&F&F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Applying the first derivative test we see that" }{XPPEDIT 18 0 "``(-1, -1/2);" "6#-%!G6$,$\"\"\"!\" \",$*&F'F'\"\"#F(F(" }{TEXT -1 6 " is a " }{TEXT 259 13 "minimum point " }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(1,1/2);" "6#-%!G6$\"\"\"*&F&F& \"\"#!\"\"" }{TEXT -1 6 " is a " }{TEXT 259 13 "maximum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The second derivative " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#% \"xG" }{TEXT -1 14 " is zero when " }{XPPEDIT 18 0 "2*x*(x^2-3)=0" "6# /*(\"\"#\"\"\"%\"xGF&,&*$F'F%F&\"\"$!\"\"F&\"\"!" }{TEXT -1 19 ". This occurs when " }{XPPEDIT 18 0 "x = -sqrt(3), x=0" "6$/%\"xG,$-%%sqrtG6 #\"\"$!\"\"/F$\"\"!" }{TEXT -1 10 " and when " }{XPPEDIT 18 0 "x=sqrt( 3)" "6#/%\"xG-%%sqrtG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "The following table indicates how the sign of the second \+ derivative " }{XPPEDIT 18 0 "`f ''`(x) = 2*x*(x^2-3)/((1+x^2)^3);" "6# /-%%f~''G6#%\"xG**\"\"#\"\"\"F'F*,&*$F'F)F*\"\"$!\"\"F**$,&F*F**$F'F)F *F-F." }{TEXT -1 12 " changes as " }{TEXT 305 1 "x" }{TEXT -1 15 " var ies, where " }{XPPEDIT 18 0 "r = sqrt(3);" "6#/%\"rG-%%sqrtG6#\"\"$" } {TEXT -1 2 ". 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They are" }{XPPEDIT 18 0 "``(-sqrt(3),-sqrt (3)/4), ``(0,0)" "6$-%!G6$,$-%%sqrtG6#\"\"$!\"\",$*&-F(6#F*\"\"\"\"\"% F+F+-F$6$\"\"!F4" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(sqrt(3),sqrt( 3)/4)" "6#-%!G6$-%%sqrtG6#\"\"$*&-F'6#F)\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "The only " }{TEXT 306 1 "x" }{TEXT -1 1 " " }{TEXT 259 9 "intercept" }{TEXT -1 36 " of the curve occurs a t the origin. " }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "x - > infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x/(1+x^2)" "6#*&%\"xG\"\"\",&F%F%*$F$ \"\"#F%!\"\"" }{TEXT -1 34 " tends to zero and, similarly, as " } {XPPEDIT 18 0 "x -> -infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\" ,$%)infinityG!\"\"F*F*F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "x/(1+x^2) " "6#*&%\"xG\"\"\",&F%F%*$F$\"\"#F%!\"\"" }{TEXT -1 36 " tends to zero . This means that the " }{TEXT 311 1 "x" }{TEXT -1 47 " axis is a hori zontal asymptote for the graph. " }}{PARA 0 "" 0 "" {TEXT -1 71 "We ca n use the information gathered to help in sketching the graph of " } {XPPEDIT 18 0 "y = x/(1+x^2);" "6#/%\"yG*&%\"xG\"\"\",&F'F'*$F&\"\"#F' !\"\"" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = 3* x^5-5*x^3;" "6#/-%\"fG6#%\"xG,&*&\"\"$\"\"\"*$F'\"\"&F+F+*&F-F+*$F'F*F +!\"\"" }{XPPEDIT 18 0 "`` = x^3*(3*x^2-5);" "6#/%!G*&%\"xG\"\"$,&*&F' \"\"\"*$F&\"\"#F*F*\"\"&!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "`f '`(x) = 15*x^4-15*x^2;" "6#/-%$ f~'G6#%\"xG,&*&\"#:\"\"\"*$F'\"\"%F+F+*&F*F+*$F'\"\"#F+!\"\"" } {XPPEDIT 18 0 "`` = 15*x^2*(x^2-1);" "6#/%!G*(\"#:\"\"\"*$%\"xG\"\"#F' ,&*$F)F*F'F'!\"\"F'" }{XPPEDIT 18 0 "`` = 15*x^2*(x-1)*(x+1);" "6#/%!G **\"#:\"\"\"*$%\"xG\"\"#F',&F)F'F'!\"\"F',&F)F'F'F'F'" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "and " }{XPPEDIT 18 0 "`f ''`(x) = 60* x^3-30*x;" "6#/-%%f~''G6#%\"xG,&*&\"#g\"\"\"*$F'\"\"$F+F+*&\"#IF+F'F+! \"\"" }{XPPEDIT 18 0 "`` = 30*x*(2*x^2-1);" "6#/%!G*(\"#I\"\"\"%\"xGF' ,&*&\"\"#F'*$F(F+F'F'F'!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivative " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 14 " is zero whe n " }{XPPEDIT 18 0 "x = -1,x = 0;" "6$/%\"xG,$\"\"\"!\"\"/F$\"\"!" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "The following table indicates ho w the sign of the derivative changes as " }{TEXT 312 1 "x" }{TEXT -1 9 " varies. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 511 106 106 {PLOTDATA 2 "6E-%'CURVESG6$7$7$$!3++++++++N!#<$!3+++++++]7F*7$$\"3 ++++++++&)F*F+-%'COLOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$ F.F9F0-F$6$7$7$F($\"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF* F+7$FKFAF0-F$6$7$7$$!3++++++++]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF0-F$6$7 $7$$\"3++++++++DF*F+7$FhnFAF0-F$6$7$7$$\"3++++++++NF*F+7$F_oFAF0-F$6$7 $7$$\"3++++++++bF*F+7$FfoFAF0-F$6$7$7$$\"3++++++++lF*F+7$F]pFAF0-F$6$7 $F-FCF0-F$6'7$7$$!\"#F4$!\"\"F47$Fip$!\"$Fjp7%7$$!+w'4I>\"!\"*$!*1wbV$ FbqF[q7$$!+C.*p5\"Fbq$!*%RUkYFbq-%&STYLEG6#%,PATCHNOGRIDGF0-%*THICKNES SG6#\"\"#-F$6'7$7$FAF\\q7$$FarF4$!#5Fjp7%7$$\"+w'4I*=Fbq$!+ggdN$)FirFf r7$$\"+C.*p!=Fbq$!+SRUk&*FirFjqF0F^r-F$6'7$7$$\"\"%F4F\\q7$$\"\"&F4Fhr 7%7$$\"+w'4I*[FbqF^sF[t7$$\"+C.*p![FbqFcsFjqF0F^r-F$6'7$7$$\"\"(F4Fip7 $$\"\")F4F\\q7%7$$\"+C.*p!yFbqFcqF[u7$$\"+w'4I*yFbqFhqFjqF0F^r-%%TEXTG 6&7$$F]qF4$\"#vFhpQ\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-Ffu6&7$$!#:FjpFj uQ'x~<~-1F]vF0F^v-Ffu6&7$$F4F4FjuQ#-1F]vF0F^v-Ffu6&7$$\"#:FjpFjuQ+-1~< ~x~<~0F]vF0F^v-Ffu6&7$$\"\"$F4FjuQ\"0F]vF0F^v-Ffu6&7$$\"#XFjpFjuQ*0~<~ x~<~1F]vF0F^v-Ffu6&7$$\"\"'F4FjuQ\"1F]vF0F^v-Ffu6&7$$F[vFjpFjuQ&x~>~1F ]vF0F^v-Ffu6&7$Fiu$FarFjpQ'f~'(x)F]vF0F^v-Ffu6&7$F\\wF^yFiwF0F^v-Ffu6& 7$FgwF^yFiwF0F^v-Ffu6&7$FcxF^yFiwF0F^v-Ffu6&7$Ffv$FhwFjpQ\"+F]vF0-F_v6 $Fav\"#9-Ffu6&7$Faw$FjsFjpQ\"_F]vF0F^z-Ffu6&7$F]xFdzFezF0F^z-Ffu6&7$Fi xF^yF]zF0F^z-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F]vFc[l-%%VIEWG6$; $!#NFjp$\"#&)Fjp;$!#8Fjp$\"#6Fjp" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv e 31" "Curve 32" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Sinc e " }{XPPEDIT 18 0 "f(-1) = 2,f(0) = 0;" "6$/-%\"fG6#,$\"\"\"!\"\"\"\" #/-F%6#\"\"!F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(1) = -1;" "6#/-% \"fG6#\"\"\",$F'!\"\"" }{TEXT -1 33 ", we obtain the stationary points " }{XPPEDIT 18 0 "``(-1,2),``(0,0);" "6$-%!G6$,$\"\"\"!\"\"\"\"#-F$6$ \"\"!F," }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(1, -2);" "6#-%!G6$\"\" \",$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "Applyi ng the first derivative test we see that" }{XPPEDIT 18 0 "``(-1, 2);" "6#-%!G6$,$\"\"\"!\"\"\"\"#" }{TEXT -1 6 " is a " }{TEXT 259 13 "maxim um point" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(1, 2);" "6#-%!G6$\"\" \"\"\"#" }{TEXT -1 6 " is a " }{TEXT 259 13 "minimum point" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " The second derivative " }{XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG " }{TEXT -1 14 " is zero when " }{XPPEDIT 18 0 "x = 0,x = 1/sqrt(2);" "6$/%\"xG\"\"!/F$*&\"\"\"F(-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x = -1/sqrt(2);" "6#/%\"xG,$*&\"\"\"F'-%%sqrtG6#\"\"#! \"\"F," }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "The following table indicates how the sign of the second derivative " }{XPPEDIT 18 0 "`f ''`(x) = 30*x*(2*x^2-1);" "6#/-%%f~''G6#%\"xG*(\"#I\"\"\"F'F*,&* &\"\"#F**$F'F-F*F*F*!\"\"F*" }{TEXT -1 12 " changes as " }{TEXT 313 1 "x" }{TEXT -1 15 " varies, where " }{XPPEDIT 18 0 "r = 1/sqrt(2);" "6# /%\"rG*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{XPPEDIT 18 0 "``=sqrt(2)/2" "6 #/%!G*&-%%sqrtG6#\"\"#\"\"\"F)!\"\"" }{TEXT -1 2 ". 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\"$F4$\"#v!\"#Q\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-Fdbn6&7$$!#:!\"\"Fib nQ'x~<~-rF]cnF0F^cn-Fdbn6&7$$F4F4FibnQ#-rF]cnF0F^cn-Fdbn6&7$$\"#:FhcnF ibnQ+-r~<~x~<~0F]cnF0F^cn-Fdbn6&7$$\"\"$F4FibnQ\"0F]cnF0F^cn-Fdbn6&7$$ \"#XFhcnFibnQ*0~<~x~<~rF]cnF0F^cn-Fdbn6&7$$\"\"'F4FibnQ\"rF]cnF0F^cn-F dbn6&7$$FjbnFhcnFibnQ&x~>~rF]cnF0F^cn-Fdbn6&7$Fgbn$F^`lFhcnQ(f~''(x)F] cnF0F^cn-Fdbn6&7$F]dnF_fnFjdnF0F^cn-Fdbn6&7$FhdnF_fnFjdnF0F^cn-Fdbn6&7 $FdenF_fnFjdnF0F^cn-Fdbn6&7$Ffcn$\"\"%FhcnQ\"_F]cnF0-F_cn6$Facn\"#9-Fd bn6&7$Fbdn$FidnFhcnQ\"+F]cnF0F`gn-Fdbn6&7$F^enF]gnF_gnF0F`gn-Fdbn6&7$F jenFfgnFggnF0F`gn-%+AXESLABELSG6%Q!F]cnFahn-F_cn6#%(DEFAULTG-%*AXESSTY LEG6#%%NONEG-%%VIEWG6$;$!#NFhcn$\"#&)Fhcn;$!#8Fhcn$\"#6Fhcn" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22 " "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "C urve 29" "Curve 30" "Curve 31" "Curve 32" }}{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "There are t hree points of inflection on the curve " }{XPPEDIT 18 0 "y = 3*x^5-5*x ^3;" "6#/%\"yG,&*&\"\"$\"\"\"*$%\"xG\"\"&F(F(*&F+F(*$F*F'F(!\"\"" } {TEXT -1 11 ". They are" }{XPPEDIT 18 0 "``(-sqrt(2)/2,7*sqrt(2)/8),` `(0,0);" "6$-%!G6$,$*&-%%sqrtG6#\"\"#\"\"\"F+!\"\"F-*(\"\"(F,-F)6#F+F, \"\")F--F$6$\"\"!F5" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(sqrt(2)/2, 7*sqrt(2)/8);" "6#-%!G6$*&-%%sqrtG6#\"\"#\"\"\"F*!\"\"*(\"\"(F+-F(6#F* F+\"\")F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)=x^3*(3*x^2-5)" "6#/-% \"fG6#%\"xG*&F'\"\"$,&*&F)\"\"\"*$F'\"\"#F,F,\"\"&!\"\"F," }{TEXT -1 16 ", note that the " }{TEXT 314 1 "x" }{TEXT -1 1 " " }{TEXT 259 10 " intercepts" }{TEXT -1 26 " of the curve occur where " }{XPPEDIT 18 0 " x=0" "6#/%\"xG\"\"!" }{TEXT -1 11 " and where " }{XPPEDIT 18 0 "x = `` ;" "6#/%\"xG%!G" }{TEXT 315 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt (5/3)" "6#-%%sqrtG6#*&\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "We can use the information gathered to help in \+ sketching the graph of " }{XPPEDIT 18 0 "y = 3*x^5-5*x^3;" "6#/%\"yG, &*&\"\"$\"\"\"*$%\"xG\"\"&F(F(*&F+F(*$F*F'F(!\"\"" }{TEXT -1 2 ". 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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "D(f)(x);\nnormal(%);\nD(D(f))(x);\nnormal(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xGF&F&*&\"\"%F&F'!\"#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&*$)%\"xG\"\"$F&F&F%! \"\"F&F*!\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"*&\"\") F%%\"xG!\"$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&*$)% \"xG\"\"$F&F&\"\"%F&F&F*!\"$F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "[solve(D(f)(x),x)]:\nremove (has,%,Complex(1));\nmap(_x->``(_x,f(_x)),%);\n[solve(D(D(f))(x),x)]: \nremove(has,%,Complex(1));\nmap(_x->``(_x,f(_x)),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#*$)\"\"##\"\"\"\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%!G6$*$)\"\"##\"\"\"\"\"$F+,$*&F,F+)F)#F)F,F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7#,$*$)\"\"##F'\"\"$\"\"\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7#-%!G6$,$*$)\"\"##F*\"\"$\"\"\"!\"\" \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 57 "__________________ _______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 57 "_________________________________ ________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "Dia grams for sign of derivative " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 741 "p1 := plot([[[-3,0],[6,0]], [[0,-.05],[0,.05]],\n [[3,-.05],[3,.05]]],color=black):\np2 := plo ttools[arrow]([-2,-.3],[-1,-1],\n 0,.15,.15,arrow,color=red,thi ckness=2):\np3 := plottools[arrow]([1,-.3],[2,-1],\n 0,.15,.15, arrow,color=red,thickness=2):\np4 := plottools[arrow]([4,-1],[5,-.3], \n 0,.15,.15,arrow,color=red,thickness=2):\nt1 := plots[textplo t]([[0,-.2,`0`],[3,-.2,`3`]],\n color=black,font=[HELVETICA,9]) :\nt2 := plots[textplot]([[-3.5,.3,`f '(x)`],[0,.3,`0`],[3,.3,`0`]],\n color=red,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([[ -1.5,.4,`_`],[1.5,.4,`_`],[4.5,.3,`+`]],\n color=red,font=[HELV ETICA,BOLD,14]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none,\n \+ view=[-3.5..6,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 583 "r1 := evalf(sqrt(2)):\np1 : = plot([r1*cos(t),r1*sin(t),t=-Pi/2+0.3..-0.3],\n color=re d,thickness=2):\np2 := plottools[arrow]([.6,-1.4],[1.4,-.6],\n \+ 0,.05,.05,arrow,color=navy,thickness=2):\np3 := plot([4+r1*cos(t),r1*s in(t),t=-Pi+0.3..-Pi/2-0.3],\n color=red,thickness=2):\np4 := plottools[arrow]([2.6,-.6],[3.4,-1.4],\n 0,.05,.05,arrow,co lor=navy,thickness=2):\nt1 := plots[textplot]([[.75,-.75,`f '(x) > 0`] ,\n [3.25,-.75,`f '(x) < 0`]],color=black,font=[HELVETICA,10]):\nplo ts[display]([p1,p2,p3,p4,t1],axes=none,\n view=[0..4,-1.5 ..-.3]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1001 "p1 := plot([[[-3.5,-1.25],[5.5,-1.25]],[[-3.5,.5],[ 5.5,.5]],\n [[-3.5,1],[5.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2 .5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[ [2.5,-1.25],[2.5,1]],\n [[3.5,-1.25],[3.5,1]],[[5.5,-1.25],[5.5,1] ]],color=black):\np2 := plottools[arrow]([-2,-.3],[-1,-1],\n 0, .15,.15,arrow,color=black,thickness=2):\np3 := plottools[arrow]([1,-.3 ],[2,-1],\n 0,.15,.15,arrow,color=black,thickness=2):\np4 := pl ottools[arrow]([4,-1],[5,-.3],\n 0,.15,.15,arrow,color=black,th ickness=2):\nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75,`x < 0`],\n [0,.75,`0`],[1.5,.75,`0 < x < 3`],[3,.75,`3`],\n [4.5,.75,`x > 3` ]],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2,`f '(x)`],[0,.2,`0`],[3,.2,`0`]],\n color=black,font=[HELVETICA,1 0]):\nt3 := plots[textplot]([[-1.5,.4,`_`],[1.5,.4,`_`],[4.5,.2,`+`]], \n color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3, p4,t1,t2,t3],axes=none,\n view=[-3.5..5.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "p 1 := plot(x^4-4*x^3,x=-2..4.5,thickness=2):\np2 := plot([[[0,0],[3,-27 ]]$3],style=point,\n symbol=[circle,diamond,cross],color=black):\nt 1 := plots[textplot]([[4.5,-2,`x`],[-.2,48,`y`],\n [3,-32,`min. \+ pt. (3,-27)`]],color=black):\nplots[display]([p1,p2,t1],tickmarks=[6,8 ],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "p1 := plo t([x^4-4*x^3,4*x^3-12*x^2],x=-2..4.5,y=-28..48,\n color=[red,blue],t hickness=2):\nt1 := plots[textplot]([[4.5,-2,`x`],[-.2,48,`y`]],color= black):\nplots[display]([p1,t1],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 44 "Diagrams for sign of the second d erivative " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 386 "p1 := plot(x^2,x=-1.5..1.5,color=red,thickness=2): \np2 := plot([[[-1.4,1.8],[-.6,.2]],[[-.4,0],[.4,0]],\n [[.6,.2],[ 1.4,1.8]]],color=COLOR(RGB,0,.7,0),thickness=2):\nt1 := plots[textplot ]([0,1,`f ''(x) > 0`],color=black,\n font=[HELVETICA,10]):\nt2 := \+ plots[textplot]([0,-.5,`graph is concave upwards`],color=brown,\n \+ font=[HELVETICA,10]):\nplots[display]([p1,p2,t1,t2],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "p1 := plot(-x^2,x=-1.5..1.5,color=blue,thickness=2):\np2 := plot( [[[-1.4,-1.8],[-.6,-.2]],[[-.4,0],[.4,0]],\n [[.6,-.2],[1.4,-1.8]] ],color=COLOR(RGB,0,.7,0),thickness=2):\nt1 := plots[textplot]([0,-1,` f ''(x) < 0`],color=black,\n font=[HELVETICA,10]):\nt2 := plots[te xtplot]([0,-2.5,`graph is concave downwards`],color=navy,\n font=[ HELVETICA,10]):\nplots[display]([p1,p2,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1017 "p1 := plot([[[-3.5,-1.25],[5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n [[-3. 5,1],[5.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[ [-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]], \n [[3.5,-1.25],[3.5,1]],[[5.5,-1.25],[5.5,1]]],color=black):\np2 \+ := plot([-1.5+.6*cos(t),-.3+.6*sin(t),t=-Pi+.2..-.2],\n co lor=black,thickness=2):\np3 := plot([1.5+.6*cos(t),-1+.6*sin(t),t=Pi-. 2..0.2],\n color=black,thickness=2):\np4 := plot([4.5+.6*c os(t),-.3+.6*sin(t),t=-Pi+.2..-.2],\n color=black,thicknes s=2):\nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75,`x < 0`],\n [0 ,.75,`0`],[1.5,.75,`0 < x < 2`],[3,.75,`2`],\n [4.5,.75,`x > 2`]],col or=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2,`f ''(x) `],[0,.2,`0`],[3,.2,`0`]],\n color=black,font=[HELVETICA,10]): \nt3 := plots[textplot]([[-1.5,.3,`+`],[1.5,.3,`_`],[4.5,.2,`+`]],\n \+ color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3,p4,t 1,t2,t3],axes=none,\n view=[-3.5..5.5,-1.3..1.1]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 556 "p1 : = plot(x^4-4*x^3,x=-2..0,color=red,thickness=2):\np2 := plot(x^4-4*x^3 ,x=0..2,color=blue,thickness=2):\np3 := plot(x^4-4*x^3,x=2..4.5,color= red,thickness=2):\np4 := plot([[[0,0],[2,-16]]$3],style=point,\n sy mbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[4.5 ,-2,`x`],[-.2,48,`y`]],color=black):\nt2 := plots[textplot]([[-2,10,`f ''(x) > 0`],\n [3.5,20,`f ''(x) > 0`]],color=brown,font=[HELVETICA,1 0]):\nt3 := plots[textplot]([[.9,-13,`f ''(x) < 0`]],color=navy,font=[ HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],labels=[``,``]) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "p1 := plot(x^4,x=-2..2,color=red,thickness=2):\nt1 : = plots[textplot]([[2.3,-.5,`x`],[-.15,17.5,`y`]],color=black):\nt2 := plots[textplot]([[-1.2,10,`f ''(x) > 0`],\n [1.2,10,`f ''(x) > 0`]], color=brown,font=[HELVETICA,10]):\nplots[display]([p1,t1,t2],labels=[` `,``],\n view=[-2..2.3,-.5..17.5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 55 "Code for classification of stationary point pictu res " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "p1 := plot(-x^2,x=-2..2,color=red,thickness=2):\np2 \+ := plot([[0,0],[0,-4]],color=black,linestyle=2):\np3 := plot([[[0,0]]$ 3],style=point,\n symbol=[circle,diamond,cross],color=black):\nt1 : = plots[textplot]([[-1.8,-1.4,`f '(x) > 0`],\n [1.8,-1.4,`f '(x) < 0` ]],color=red,font=[HELVETICA,BOLD,10]):\nt2 := plots[textplot]([[0,.5, `f ''(c) < 0`],[0,1,`f '(c) = 0`]],\n color=black,font=[HELVETICA, BOLD,10]):\nt3 := plots[textplot]([0,-4.2,`c`],\n color=black,font =[HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p3,t1,t2,t3],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "p1 := plot(x^2,x=-2..2,colo r=red,thickness=2):\np2 := plot([[0,0],[0,-.5]],color=black,linestyle= 2):\np3 := plot([[[0,0]]$3],style=point,\n symbol=[circle,diamond,c ross],color=black):\nt1 := plots[textplot]([[-1.8,1.4,`f '(x) < 0`],\n [1.8,1.4,`f '(x) > 0`]],color=red,font=[HELVETICA,BOLD,10]):\nt2 := \+ plots[textplot]([[0,.5,`f ''(c) > 0`],[0,1,`f '(c) = 0`]],\n color =black,font=[HELVETICA,BOLD,10]):\nt3 := plots[textplot]([0,-.7,`c`], \n color=black,font=[HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p 3,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "p1 := plot(-(x+5)^2,x=-6..-4,color =red,thickness=2):\np2 := plot((x-5)^2-1,x=4..6,color=red,thickness=2) :\np3 := plot(x^3-.5,x=-.8..0.8,color=red,thickness=2):\np4 := plot([[ [-5,0],[0,-.5],[5,-1]]$3],style=point,\n symbol=[circle,diamond,cr oss],color=black):\np5 := plot([[[-5,-1.3],[-5,0]],[[0,-1.3],[0,-.5]], \n [[5,-1.3],[5,-1]]],linestyle=2,color=black):\nt1 := plots[ textplot]([[-5,-1.35,`c`],[0,-1.35,`c`],[5,-1.35,`c`], [- 5,.15,`P`],[0,-.35,`Q`],[5,-.85,`R`]],font=[HELVETICA,10]):\nplots[dis play]([p1,p2,p3,p4,p5,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 41 "Code for cubic curve analysis pictures " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "p1 := plot(x*(x^2-3),x=-2.3..0,color=blue,thickness=2):\np2 := pl ot(x*(x^2-3),x=0..2.3,color=red,thickness=2):\np3 := plot([[[-1,2],[-1 ,-6]],[[0,0],[0,-6]],[[1,-2],[1,-6]]],color=black,linestyle=2):\np4 := plot([[[-1,2],[0,0],[1,-2]]$3],style=point,\n symbol=[circle,dia mond,cross],color=black):\nplots[display]([p1,p2,p3,p4],axes=none);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "p1 := plot(x*(3-x^2),x=-2.3 ..0,color=red,thickness=2):\np2 := plot(x*(3-x^2),x=0..2.3,color=blue, thickness=2):\np3 := plot([[[-1,-2],[-1,-6]],[[0,0],[0,-6]],[[1,2],[1, -6]]],color=black,linestyle=2):\np4 := plot([[[-1,-2],[0,0],[1,2]]$3], style=point,\n symbol=[circle,diamond,cross],color=black):\nplots [display]([p1,p2,p3,p4],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 429 "p1 := plot((x+4)^3,x=-6.3 ..-4,color=blue,thickness=2):\np2 := plot((x+4)^3,x=-4..-1.7,color=red ,thickness=2):\np3 := plot((4-x)^3,x=1.7..4,color=red,thickness=2):\np 4 := plot((4-x)^3,x=4..6.3,color=blue,thickness=2):\np5 := plot([[[-4, 0],[-4,-13]],[[4,0],[4,-13]]],color=black,linestyle=2):\np6 := plot([[ [-4,0],[4,0]]$3],style=point,\n symbol=[circle,diamond,cross], color=black):\nplots[display]([p1,p2,p3,p4,p5,p6],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "p1 := plot((x+4)^3+x+4,x=-6.3..-4,color=blue,thickness=2):\np2 := plot((x+4)^3+x+4,x=-4..-1.7,color=red,thickness=2):\np3 := plot((4-x) ^3+4-x,x=1.7..4,color=red,thickness=2):\np4 := plot((4-x)^3+4-x,x=4..6 .3,color=blue,thickness=2):\np5 := plot([[[-4,0],[-4,-15]],[[4,0],[4,- 15]]],color=black,linestyle=2):\np6 := plot([[[-4,0],[4,0]]$3],style=p oint,\n symbol=[circle,diamond,cross],color=black):\nplots[dis play]([p1,p2,p3,p4,p5,p6],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 30 "Code for example 1 pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1001 "p1 := plot ([[[-3.5,-1.25],[5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n [[-3.5,1],[5. 5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1. 25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]],\n [ [3.5,-1.25],[3.5,1]],[[5.5,-1.25],[5.5,1]]],color=black):\np2 := plott ools[arrow]([-2,-1],[-1,-.3],\n 0,.15,.15,arrow,color=black,thi ckness=2):\np3 := plottools[arrow]([1,-.3],[2,-1],\n 0,.15,.15, arrow,color=black,thickness=2):\np4 := plottools[arrow]([4,-1],[5,-.3] ,\n 0,.15,.15,arrow,color=black,thickness=2):\nt1 := plots[text plot]([[-3,.75,`x`],[-1.5,.75,`x < 1`],\n [0,.75,`1`],[1.5,.75,`1 < x < 3`],[3,.75,`3`],\n [4.5,.75,`x > 3`]],color=black,font=[HELVETIC A,10]):\nt2 := plots[textplot]([[-3,.2,`f '(x)`],[0,.2,`0`],[3,.2,`0`] ],\n color=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([ [-1.5,.3,`+`],[1.5,.4,`_`],[4.5,.2,`+`]],\n color=black,font=[H ELVETICA,14]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none,\n \+ view=[-3.5..5.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 808 "p1 := plot([[[-3.5,-1.25],[ 2.5,-1.25]],[[-3.5,.5],[2.5,.5]],\n [[-3.5,1],[2.5,1]],[[-3.5,-1.2 5],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n \+ [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]]],color=black):\np2 := plo t([-1.5+.6*cos(t),-1+.6*sin(t),t=Pi-.2..0.2],\n color=blac k,thickness=2):\np3 := plot([1.5+.6*cos(t),-.3+.6*sin(t),t=-Pi+.2..-.2 ],\n color=black,thickness=2):\nt1 := plots[textplot]([[-3 ,.75,`x`],[-1.5,.75,`x < 2`],\n [0,.75,`2`],[1.5,.75,`x > 2`]],colo r=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2,`f ''(x)` ],[0,.2,`0`]],\n color=black,font=[HELVETICA,10]):\nt3 := plots [textplot]([[-1.5,.4,`_`],[1.5,.3,`+`]],\n color=black,font=[HE LVETICA,14]):\nplots[display]([p1,p2,p3,t1,t2,t3],axes=none,\n vi ew=[-3.5..2.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 423 "f := x -> x*(x-3)^2:\np1 := plot(f (x),x=-.7..2,color=blue,thickness=2):\np2 := plot(f(x),x=2..4.7,color= red,thickness=2):\np3 := plot([[[1,4],[2,2],[3,0]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([ [4.7,-.8,`x`],[-.2,13.8,`y`],\n [1,5.5,` max. pt. (1,4)`],[3,-2.5,` \+ min. pt. (3,0)`],\n [2.7,3,` inflection. pt. (2,2)`]],color=black): \nplots[display]([p1,p2,p3,t1],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "Code for example 2 pictures " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1214 "p1 := plot([[[-3.5,-1.25],[8.5,-1.25]],[[-3.5,.5],[8.5,.5]],\n \+ [[-3.5,1],[8.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2 .5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[ 2.5,1]],\n [[3.5,-1.25],[3.5,1]],[[5.5,-1.25],[5.5,1]],\n [[6. 5,-1.25],[6.5,1]],[[8.5,-1.25],[8.5,1]]],color=black):\np2 := plottool s[arrow]([-2,-1],[-1,-.3],\n 0,.15,.15,arrow,color=black,thickn ess=2):\np3 := plottools[arrow]([1,-.3],[2,-1],\n 0,.15,.15,arr ow,color=black,thickness=2):\np4 := plottools[arrow]([4,-.3],[5,-1],\n 0,.15,.15,arrow,color=black,thickness=2):\np5 := plottools[arr ow]([7,-1],[8,-.3],\n 0,.15,.15,arrow,color=black,thickness=2): \nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75,`x < -1`],\n [0,. 75,`-1`],[1.5,.75,`-1 < x < 0`],[3,.75,`0`],\n [4.5,.75,`0 < x < \+ 1`],[6,.75,`1`],\n [7.5,.75,`x > 1`]],color=black,font=[HELVETICA ,10]):\nt2 := plots[textplot]([[-3,.2,`f '(x)`],[0,.2,`0`],[3,.2,`ND`] ,\n [6,.2,`0`]],color=black,font=[HELVETICA,10]):\nt3 := plots[te xtplot]([[-1.5,.3,`+`],[1.5,.4,`_`],[4.5,.4,`_`],\n [7.5,.2,`+`]] ,color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3,p4,p5,t1, t2,t3],axes=none,\n view=[-3.5..8.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 809 "p1 := \+ plot([[[-3.5,-1.25],[2.5,-1.25]],[[-3.5,.5],[2.5,.5]],\n [[-3.5,1] ,[2.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5 ,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]]],col or=black):\np2 := plot([-1.5+.6*cos(t),-1+.6*sin(t),t=Pi-.2..0.2],\n \+ color=black,thickness=2):\np3 := plot([1.5+.6*cos(t),-.3+.6 *sin(t),t=-Pi+.2..-.2],\n color=black,thickness=2):\nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75,`x < 0`],\n [0,.75,`0`],[1 .5,.75,`x > 0`]],color=black,font=[HELVETICA,10]):\nt2 := plots[textpl ot]([[-3,.2,`f ''(x)`],[0,.2,`ND`]],\n color=black,font=[HELVET ICA,10]):\nt3 := plots[textplot]([[-1.5,.4,`_`],[1.5,.3,`+`]],\n \+ color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3,t1,t2,t3 ],axes=none,\n view=[-3.5..2.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 476 "f := x -> x +1/x:\np1 := plot(f(x),x=-5..-0.13,color=blue,thickness=2):\np2 := plo t(f(x),x=0.13..5,color=red,thickness=2):\np3 := plot(x,x=-5..5,color=b lack,linestyle=2):\np4 := plot([[[-1,-2],[1,2]]$3],style=point,\n s ymbol=[circle,diamond,cross],color=black):\nt1 := plots[textplot]([[5. 3,-.4,`x`],[-.2,8.9,`y`],\n [2,4.7,` min. pt. (1,2)`],\n [-2,- 4.7,` max. pt. (-1,-2)`]],color=black):\nplots[display]([p1,p2,p3,p4,t 1],labels=[``,``],\n view=[-5..5.5,-8..8.9]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "Code for example 3 pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1004 "p1 := plot([[[-3.5,-1.25],[5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n [[-3.5,1],[5.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[ -2.5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25] ,[2.5,1]],\n [[3.5,-1.25],[3.5,1]],[[5.5,-1.25],[5.5,1]]],color=bl ack):\np2 := plottools[arrow]([-2,-.3],[-1,-1],\n 0,.15,.15,arr ow,color=black,thickness=2):\np3 := plottools[arrow]([1,-1],[2,-.3],\n 0,.15,.15,arrow,color=black,thickness=2):\np4 := plottools[arr ow]([4,-.3],[5,-1],\n 0,.15,.15,arrow,color=black,thickness=2): \nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75,`x < -1`],\n [0,.75 ,`-1`],[1.5,.75,`-1 < x < 1`],[3,.75,`1`],\n [4.5,.75,`x > 1`]],color =black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2,`f '(x)`], [0,.2,`0`],[3,.2,`0`]],\n color=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[-1.5,.4,`_`],[1.5,.2,`+`],[4.5,.4,`_`]],\n \+ color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3,p4,t1,t2 ,t3],axes=none,\n view=[-3.5..5.5,-1.3..1.1]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1227 "p1 := pl ot([[[-3.5,-1.25],[8.5,-1.25]],[[-3.5,.5],[8.5,.5]],\n [[-3.5,1],[ 8.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,- 1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]],\n \+ [[3.5,-1.25],[3.5,1]],[[5.5,-1.25],[5.5,1]],\n [[6.5,-1.25],[6.5, 1]],[[8.5,-1.25],[8.5,1]]],color=black):\np2 := plot([-1.5+.6*cos(t),- 1+.6*sin(t),t=Pi-.2..0.2],\n color=black,thickness=2):\np3 := plot([1.5+.6*cos(t),-.3+.6*sin(t),t=-Pi+.2..-.2],\n co lor=black,thickness=2):\np4 := plot([4.5+.6*cos(t),-1+.6*sin(t),t=Pi-. 2..0.2],\n color=black,thickness=2):\np5 := plot([7.5+.6*c os(t),-.3+.6*sin(t),t=-Pi+.2..-.2],\n color=black,thicknes s=2):\nt1 := plots[textplot]([[-3,.75,`x`],[-1.5,.75,`x < -r`],\n [ 0,.75,`-r`],[1.5,.75,`-r < x < 0`],[3,.75,`0`],\n [4.5,.75,`0 < x < r`],[6,.75,`r`],\n [7.5,.75,`x > r`]],color=black,font=[HELVETICA, 10]):\nt2 := plots[textplot]([[-3,.2,`f ''(x)`],[0,.2,`0`],[3,.2,`0`], [6,.2,`0`]],\n color=black,font=[HELVETICA,10]):\nt3 := plots[t extplot]([[-1.5,.4,`_`],[1.5,.3,`+`],\n [4.5,.4,`_`],[7.5,.3,`+`]], color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3,p4,p5,t1,t 2,t3],axes=none,\n view=[-3.5..8.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 715 "cc := ev alf(sqrt(3)):\nf := x -> x/(1+x^2):\np1 := plot(f(x),x=-6..-cc,color=b lue,thickness=2):\np2 := plot(f(x),x=-cc..0,color=red,thickness=2):\np 3 := plot(f(x),x=0..cc,color=blue,thickness=2):\np4 := plot(f(x),x=cc. .6,color=red,thickness=2):\np5 := plot([[[-cc,-cc/4],[-1,-1/2],[0,0],[ 1,1/2],[cc,cc/4]]$3],style=point,\n symbol=[circle,diamond,cross],c olor=black):\nt1 := plots[textplot]([[6.3,-.08,`x`],[-.2,.65,`y`],\n \+ [1.7,.65,`max. pt. (1,1/2)`],[-1.7,-.65,` min. pt. (-1/-1/2)`],\n [ -1.4,.1,`inflection. pt. (0,0)`],\n [3.5,.44,`inflection. pt. (r,r/4 )`],\n [-3.5,-.44,`inflection. pt. (-r,-r/4)`]],color=black):\nplots [display]([p1,p2,p3,p4,p5,t1],ytickmarks=3,labels=[``,``],\n view=[ -6..6.3,-.65..0.65]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "Code for example 4 pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1213 "p1 := plot([[[-3 .5,-1.25],[8.5,-1.25]],[[-3.5,.5],[8.5,.5]],\n [[-3.5,1],[8.5,1]], [[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[- .5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]],\n [[3.5,- 1.25],[3.5,1]],[[5.5,-1.25],[5.5,1]],\n [[6.5,-1.25],[6.5,1]],[[8. 5,-1.25],[8.5,1]]],color=black):\np2 := plottools[arrow]([-2,-1],[-1,- .3],\n 0,.15,.15,arrow,color=black,thickness=2):\np3 := plottoo ls[arrow]([1,-.3],[2,-1],\n 0,.15,.15,arrow,color=black,thickne ss=2):\np4 := plottools[arrow]([4,-.3],[5,-1],\n 0,.15,.15,arro w,color=black,thickness=2):\np5 := plottools[arrow]([7,-1],[8,-.3],\n \+ 0,.15,.15,arrow,color=black,thickness=2):\nt1 := plots[textplot ]([[-3,.75,`x`],[-1.5,.75,`x < -1`],\n [0,.75,`-1`],[1.5,.75,`-1 \+ < x < 0`],[3,.75,`0`],\n [4.5,.75,`0 < x < 1`],[6,.75,`1`],\n \+ [7.5,.75,`x > 1`]],color=black,font=[HELVETICA,10]):\nt2 := plots[te xtplot]([[-3,.2,`f '(x)`],[0,.2,`0`],[3,.2,`0`],\n [6,.2,`0`]],co lor=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[-1.5,.3,`+`], [1.5,.4,`_`],[4.5,.4,`_`],\n [7.5,.2,`+`]],color=black,font=[HELV ETICA,14]):\nplots[display]([p1,p2,p3,p4,p5,t1,t2,t3],axes=none,\n \+ view=[-3.5..8.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1227 "p1 := plot([[[-3.5,-1.25], [8.5,-1.25]],[[-3.5,.5],[8.5,.5]],\n [[-3.5,1],[8.5,1]],[[-3.5,-1. 25],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n \+ [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]],\n [[3.5,-1.25],[3.5 ,1]],[[5.5,-1.25],[5.5,1]],\n [[6.5,-1.25],[6.5,1]],[[8.5,-1.25],[ 8.5,1]]],color=black):\np2 := plot([-1.5+.6*cos(t),-1+.6*sin(t),t=Pi-. 2..0.2],\n color=black,thickness=2):\np3 := plot([1.5+.6*c os(t),-.3+.6*sin(t),t=-Pi+.2..-.2],\n color=black,thicknes s=2):\np4 := plot([4.5+.6*cos(t),-1+.6*sin(t),t=Pi-.2..0.2],\n \+ color=black,thickness=2):\np5 := plot([7.5+.6*cos(t),-.3+.6*sin(t ),t=-Pi+.2..-.2],\n color=black,thickness=2):\nt1 := plots [textplot]([[-3,.75,`x`],[-1.5,.75,`x < -r`],\n [0,.75,`-r`],[1.5,. 75,`-r < x < 0`],[3,.75,`0`],\n [4.5,.75,`0 < x < r`],[6,.75,`r`], \n [7.5,.75,`x > r`]],color=black,font=[HELVETICA,10]):\nt2 := plot s[textplot]([[-3,.2,`f ''(x)`],[0,.2,`0`],[3,.2,`0`],[6,.2,`0`]],\n \+ color=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[-1.5,. 4,`_`],[1.5,.3,`+`],\n [4.5,.4,`_`],[7.5,.3,`+`]],color=black,font= [HELVETICA,14]):\nplots[display]([p1,p2,p3,p4,p5,t1,t2,t3],axes=none, \n view=[-3.5..8.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 570 "x1 := evalf(sqrt(2)/2): \ny1 := evalf(7*sqrt(2)/8):\nf := x -> 3*x^5-5*x^3:\np1 := plot(f(x),x =-1.45..-x1,color=blue,thickness=2):\np2 := plot(f(x),x=-x1..0,color=r ed,thickness=2):\np3 := plot(f(x),x=0..x1,color=blue,thickness=2):\np4 := plot(f(x),x=x1..1.45,color=red,thickness=2):\np5 := plot([[[-1,2], [-x1,y1],[0,0],[1,-2],[x1,-y1]]$3],style=point,\n symbol=[circle,di amond,cross],color=black):\nt1 := plots[textplot]([[1.45,-.2,`x`],[-.2 ,4.1,`y`],\n [-1,2.5,` max. pt. (-1,2)`],[1,-2.5,` min. pt. (1,-2)`] ],color=black):\nplots[display]([p1,p2,p3,p4,p5,t1],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK " 4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }