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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 41 "Geometrical interpretation of der ivatives" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C. , Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 11.8.2004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Stationary points" }}{PARA 0 "" 0 "" {TEXT -1 42 "Take a look at the graph of the function " }{XPPEDIT 18 0 "f(x) = x^3-3*x;" "6#/-%\"fG6#%\"xG,&*$F'\"\"$\"\"\"*&F*F+F'F+!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 88 "Where are the points \+ on the graph at which the tangent line to the graph is horizontal?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(x^3-3*x,x=-2.2..2.2,y );" }}{PARA 13 "" 1 "" {GLPLOT2D 333 311 311 {PLOTDATA 2 "6%-%'CURVESG 6$7en7$$!3;+++++++A!#<$!3q-+++++[SF*7$$!3]LL$3EY?:#F*$!3RQs]b(R1^$F*7$ $!3$omm;_#4/@F*$!3R(=;A3sH+$F*7$$!35+]iH#oB1#F*$!3;9'=FW&*[e#F*7$$!3QL LePRk??F*$!3S;^2p7O)=#F*7$$!3A+]P21st>F*$!3)Ro;l&4hn F*$!3'G(GMf]$HP\"F*7$$!3wmmm3Hcz=F*$!327$$!3smm\"*e/9^;F*$ \"3u`Tk)Q_(>XFU7$$!3/++D%p#)3c\"F*$\"3u(yQYhSyz)FU7$$!3wmm\"\\)z`n9F*$ \"32,(eNuC?C\"F*7$$!3/++DLE\\u8F*$\"3bfS!Q=Xn_\"F*7$$!3OLLL_Syy7F*$\"3 'zp=s5r^u\"F*7$$!3ymm;#)Q[%>\"F*$\"3Yf0=/>F*7$$!32+++b,H/5F*$\"3s!3[ZqW***>F*7$$!3y+++I%*eC\"*FU$\" 3'o\\j`b!ox>F*7$$!3gom;\\Sn!H)FU$\"3$p>$=W/M<>F*7$$!3GLLL.!o!*H(FU$\"3 pz^G!Q_3!=F*7$$!3+LLL8w/fkFU$\"3mx1GktCo;F*7$$!3'3++D?/>[&FU$\"3q)fVs0 L)z9F*7$$!3vMLL8n'ph%FU$\"3+F*7$$\"3+ML$3F==:*FU$!3Kq<&\\#y -z>F*7$$\"3P+++n0I45F*$!3+2j&QpR(**>F*7$$\"3CmmTm*ey4\"F*$!3%\\>!pLPLq >F*7$$\"3E++v5!G/>\"F*$!3YME5ggI%)=F*7$$\"3[mmmr6$4G\"F*$!31;oi*Gh5u\" F*7$$\"3q***\\PfzcP\"F*$!3&>t8+$HdB:F*7$$\"3'QLL`eLpY\"F*$!39,'3W'\\6W 7F*7$$\"3qLL$=(RDg:F*$!3/!)*[0;9\\#))FU7$$\"3'pm;%=;!Gl\"F*$!36/Jm>weL WFU7$$\"3:+++%HVyt\"F*$\"3qKh\">*e%H\\$Fen7$$\"3.nm;^1JN=F*$\"3w-N4*\\ X1w'FU7$$\"3[++DM')*)y=F*$\"3SfQ]f$\\I'**FU7$$\"3#RLLth'[A>F*$\"317!*R xP'zL\"F*7$$\"35n;a)\\g*o>F*$\"3A:=v&G&RE_@F*$\"3\"QNrst*H7NF*7$$\"3;+++++++AF*$\"3q-+++++[ SF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fj]lFi]l-%+AXESLABELSG6$Q\"x6\"Q \"yF_^l-%%VIEWG6$;$!#AFh]l$\"#AFh]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 45 "A point on the graph of a function where the " }{TEXT 259 18 "derivative is zero" }{TEXT -1 13 " is called a " }{TEXT 259 16 "stationary point" }{TEXT -1 54 ".\nStationary points are points on the graph where the " }{TEXT 259 26 "tangent line is horizontal" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{TEXT 269 1 "x" }{TEXT -1 49 " for which the derivative is zero are called t he " }{TEXT 259 15 "critical values" }{TEXT -1 4 " of " }{TEXT 268 1 " x" }{TEXT -1 19 " for the function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "There are evidently two main steps inv olved in finding stationary points on the graph of a function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 20 "Find the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#- %$f~'G6#%\"xG" }{TEXT -1 18 " of the function. " }}{PARA 15 "" 0 "" {TEXT -1 19 "Solve the equation " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6#/- %$f~'G6#%\"xG\"\"!" }{TEXT -1 5 " for " }{TEXT 264 1 "x" }{TEXT -1 15 " to obtain the " }{TEXT 265 1 "x" }{TEXT -1 39 " coordinates of the s tationary points. " }}{PARA 0 "" 0 "" {TEXT -1 5 "\nThe " }{TEXT 266 1 "y" }{TEXT -1 74 " coordinates of the stationary points can be obtai ned by substituting the " }{TEXT 267 1 "x" }{TEXT -1 40 " values in th e formula for the function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 5 "Let " }{XPPEDIT 18 0 "f(x)=x^3-3*x" "6#/- %\"fG6#%\"xG,&*$F'\"\"$\"\"\"*&F*F+F'F+!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "`f '`(x) = 3*x^2-3; " "6#/-%$f~'G6#%\"xG,&*&\"\"$\"\"\"*$F'\"\"#F+F+F*!\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6 #/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "3 *x^2-3=0" "6#/,&*&\"\"$\"\"\"*$%\"xG\"\"#F'F'F&!\"\"\"\"!" }{TEXT -1 16 ", that is, when " }{XPPEDIT 18 0 "x^2=1" "6#/*$%\"xG\"\"#\"\"\"" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x = ``" "6#/%\"xG%!G" }{TEXT 272 1 " +" }{TEXT -1 37 " 1. These are the critical values of " }{TEXT 275 1 " x" }{TEXT -1 19 " for the function. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Si nce " }{XPPEDIT 18 0 "f(1)=-2" "6#/-%\"fG6#\"\"\",$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(-1)=2" "6#/-%\"fG6#,$\"\"\"!\"\"\"\"#" }{TEXT -1 48 ", the associated stationary points on the curve " } {XPPEDIT 18 0 "y=x^3-3*x" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"*&F(F)F'F)!\"\" " }{TEXT -1 4 " are" }{XPPEDIT 18 0 " ``(1,-2)" "6#-%!G6$\"\"\",$\"\"# !\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(-1,2)" "6#-%!G6$,$\"\"\" !\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The tangen t lines are horizontal at the points" }{XPPEDIT 18 0 " ``(-1,2)" "6#-% !G6$,$\"\"\"!\"\"\"\"#" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1,-2)" "6#-%!G6$\"\"\",$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 9 "The point" }{XPPEDIT 18 0 " ``( -1,2)" "6#-%!G6$,$\"\"\"!\"\"\"\"#" }{TEXT -1 20 " is an example of a \+ " }{TEXT 259 8 "relative" }{TEXT -1 4 " or " }{TEXT 259 19 "local maxi mum point" }{TEXT -1 33 ",\nThe value of the function when " } {XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 45 " is great er than values for nearby values of " }{TEXT 273 1 "x" }{TEXT -1 2 ". \n" }}{PARA 15 "" 0 "" {TEXT -1 9 "The point" }{XPPEDIT 18 0 " ``(1,-2 )" "6#-%!G6$\"\"\",$\"\"#!\"\"" }{TEXT -1 20 " is an example of a " } {TEXT 259 8 "relative" }{TEXT -1 4 " or " }{TEXT 259 19 "local minimum point" }{TEXT -1 33 ".\nThe value of the function when " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 42 " is less than values for n earby values of " }{TEXT 274 1 "x" }{TEXT -1 1 "." }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 379 380 380 {PLOTDATA 2 "6,-%'CURVESG6%7en 7$$!3;+++++++A!#<$!3q-+++++[SF*7$$!3]LL$3EY?:#F*$!3RQs]b(R1^$F*7$$!3$o mm;_#4/@F*$!3R(=;A3sH+$F*7$$!35+]iH#oB1#F*$!3;9'=FW&*[e#F*7$$!3QLLePRk ??F*$!3S;^2p7O)=#F*7$$!3A+]P21st>F*$!3)Ro;l&4hnF*$!3 'G(GMf]$HP\"F*7$$!3wmmm3Hcz=F*$!327$$!3smm\"*e/9^;F*$\"3u`T k)Q_(>XFU7$$!3/++D%p#)3c\"F*$\"3u(yQYhSyz)FU7$$!3wmm\"\\)z`n9F*$\"32,( eNuC?C\"F*7$$!3/++DLE\\u8F*$\"3bfS!Q=Xn_\"F*7$$!3OLLL_Syy7F*$\"3'zp=s5 r^u\"F*7$$!3ymm;#)Q[%>\"F*$\"3Yf0=/>F*7$$!32+++b,H/5F*$\"3s!3[ZqW***>F*7$$!3y+++I%*eC\"*FU$\"3'o\\j `b!ox>F*7$$!3gom;\\Sn!H)FU$\"3$p>$=W/M<>F*7$$!3GLLL.!o!*H(FU$\"3pz^G!Q _3!=F*7$$!3+LLL8w/fkFU$\"3mx1GktCo;F*7$$!3'3++D?/>[&FU$\"3q)fVs0L)z9F* 7$$!3vMLL8n'ph%FU$\"3+F*7$$\"3+ML$3F==:*FU$!3Kq<&\\#y-z>F*7 $$\"3P+++n0I45F*$!3+2j&QpR(**>F*7$$\"3CmmTm*ey4\"F*$!3%\\>!pLPLq>F*7$$ \"3E++v5!G/>\"F*$!3YME5ggI%)=F*7$$\"3[mmmr6$4G\"F*$!31;oi*Gh5u\"F*7$$ \"3q***\\PfzcP\"F*$!3&>t8+$HdB:F*7$$\"3'QLL`eLpY\"F*$!39,'3W'\\6W7F*7$ $\"3qLL$=(RDg:F*$!3/!)*[0;9\\#))FU7$$\"3'pm;%=;!Gl\"F*$!36/Jm>weLWFU7$ $\"3:+++%HVyt\"F*$\"3qKh\">*e%H\\$Fen7$$\"3.nm;^1JN=F*$\"3w-N4*\\X1w'F U7$$\"3[++DM')*)y=F*$\"3SfQ]f$\\I'**FU7$$\"3#RLLth'[A>F*$\"317!*RxP'zL \"F*7$$\"35n;a)\\g*o>F*$\"3A:=v&G&RE_@F*$\"3\"QNrst*H7NF*7$$\"3;+++++++AF*$\"3q-+++++[SF*-% 'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fj]lFi]l-%*THICKNESSG6#\"\"#-F$6&7$7$$ \"\"\"Fj]l$!\"#Fj]l7$$Fh]lFj]l$F^^lFj]l-%'SYMBOLG6#%'CIRCLEG-Fc]l6&Fe] lFj]lFj]lFj]l-%&STYLEG6#%&POINTG-F$6&Fa^l-F[_l6#%&CROSSGF^_lF`_l-F$6&F a^l-F[_l6#%(DIAMONDGF^_lF`_l-%%TEXTG6$7$$FUFh]l$\"#EFh]lQ;local~maximu m~point~(-1,2)6\"-F_`l6$7$$\"#=Fh]l$!#EFh]lQ;local~minimum~point~(1,-2 )Ff`l-F_`l6$7$$\"#CFh]l$!\"$Fh]lQ\"xFf`l-F_`l6$7$$!#:Ff^l$\"#ZFh]lQ\"y Ff`l-%+AXESLABELSG6%%!GFbbl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#CFh]lFbal ;$!#ZFh]lF\\bl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 236.000000 477.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "\nWe c an find the derivative using the operator " }{TEXT 0 1 "D" }{TEXT -1 7 " . . .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> x^3 -3*x;\nDf := x -> D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"F1*&F0F1F/F1!\"\"F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#%\"xG6\"6$%)operator G%&arrowGF(,&*&\"\"$\"\"\")9$\"\"#F/F/F.!\"\"F(F(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " . . . and then use " }{TEXT 0 5 "solve" }{TEXT -1 13 " to find the " }{TEXT 282 1 "x" } {TEXT -1 45 " coordinates of the stationary points.\n\n(The " }{TEXT 270 1 "y" }{TEXT -1 44 " coordinates can be obtained by getting the " }{TEXT 271 1 "x" }{TEXT -1 56 " coordinates in the form of a list usin g [ ] around the " }{TEXT 0 5 "solve" }{TEXT -1 25 " command, and then using " }{TEXT 0 3 "map" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "[solve(Df(x)=0,x)];\n map(f,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 145 " \nAlternatively, we can construct a sequence of stationary point s as follows, using the Maple format for coordinates as a list with tw o members.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xcrit := [so lve(Df(x)=0,x)]:\nop(map(x->[x,f(x)],xcrit));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$\"\"\"!\"#7$!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = x/2+2/x;" "6#/-%\"fG6#%\"xG,&*&F'\"\"\"\"\"#!\"\"F**&F+F*F'F,F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "` f '`(x) = 1/2-2/(x^2);" "6#/-%$f~'G6#%\"xG,&*&\"\"\"F*\"\"#!\"\"F**&F+ F**$F'F+F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 14 " \+ exactly when " }{XPPEDIT 18 0 "1/2-2/(x^2) = 0;" "6#/,&*&\"\"\"F&\"\"# !\"\"F&*&F'F&*$%\"xGF'F(F(\"\"!" }{TEXT -1 16 ", that is, when " } {XPPEDIT 18 0 "1/2 = 2/(x^2);" "6#/*&\"\"\"F%\"\"#!\"\"*&F&F%*$%\"xGF& F'" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x^2 = 4;" "6#/*$%\"xG\"\"#\"\"% " }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "x =``" "6#/%\"xG%!G" } {TEXT -1 1 " " }{TEXT 277 1 "+" }{TEXT -1 37 " 2. These are the critic al values of " }{TEXT 276 1 "x" }{TEXT -1 19 " for the function. " }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(2) = 2;" "6#/-% \"fG6#\"\"#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(-2) = -2;" "6#/-% \"fG6#,$\"\"#!\"\",$F(F)" }{TEXT -1 48 ", the associated stationary po ints on the curve " }{XPPEDIT 18 0 "y = x/2+2/x;" "6#/%\"yG,&*&%\"xG\" \"\"\"\"#!\"\"F(*&F)F(F'F*F(" }{TEXT -1 4 " are" }{XPPEDIT 18 0 "``(2, 2);" "6#-%!G6$\"\"#F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(2, -2);" "6#-%!G6$\"\"#,$F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "The point" }{XPPEDIT 18 0 "``(2,2)" "6#-%!G6$\"\"#F&" }{TEXT -1 31 " is a local minimum point while" }{XPPEDIT 18 0 " ``(-2,-2)" "6#-%!G6 $,$\"\"#!\"\",$F'F(" }{TEXT -1 27 " is a local maximum point. " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 427 343 343 {PLOTDATA 2 "6 ,-%'CURVESG6&7gn7$$!3;+++++++_!#<$!3o%Q:YQ:Y)HF*7$$!3oB+c)zam3&F*$!3'4 S2N68l$HF*7$$!3e*fS[uL!))\\F*$!3Q)[*Qpk(\\*GF*7$$!3c&3za0Cr([F*$!3W(=8 cwR'[GF*7$$!3?vBq%=zaw%F*$!31S#HZcCC!GF*7$$!3wOz]U\\OaYF*$!3)ej!*fj')o v#F*7$$!3cP1Q;yM^XF*$!3NM2ByT5:FF*7$$!3CT1E\\&zYW%F*$!3?Jo_rfJsEF*7$$! 3e2Y\"[2jVL%F*$!3JV[R[.hGEF*7$$!3A*>,nQ+WA%F*$!3KF?;l,k&e#F*7$$!3v'3We C!H6TF*$!3%f*R]7/6UDF*7$$!3A3+@sFm6SF*$!35JU&yxxV]#F*7$$!3o))4gLe]**QF *$!3)Qo+qTQEY#F*7$$!3_Bi#yO))oy$F*$!3u5BlwAe@CF*7$$!3/zK/s0OyOF*$!3Cu& )es2!HQ#F*7$$!3[>PKOp!)zNF*$!3')3AE:Gf[BF*7$$!3mK99JrhiMF*$!33+/$[N1*3 BF*7$$!3#H)*RV*>MjLF*$!31\\:QKwJwAF*7$$!3]xUl49'yC$F*$!397H.F0sRAF*7$$ !3%RQvZ_Tc9$F*$!35SU(z6@'3AF*7$$!30?I35,\\LIF*$!3EXpNQ;0w@F*7$$!374mzT apEHF*$!3Cr(H/I7n9#F*7$$!3nLh$*3kE:GF*$!3KX'GaaX!=@F*7$$!3K(y!Rs(RHr#F *$!3#o,zIbxO4#F*7$$!30+#G5nmDg#F*$!3YsZ^$[b(p?F*7$$!3c$\\2&Q+#z[#F*$!3 g?P3wV%y/#F*7$$!3WgP(y!)>\")Q#F*$!3-,<(\\#)Q:.#F*7$$!3'zmsY#GL!G#F*$!3 `IEfk8BmG#e1?F*7$$!3/@*z(p'R+1#F*$!3PB;PD \\(3+#F*7$$!3=G2C?fja>F*$!3/D95:k_+?F*7$$!32%zCs-.w$=F*$!3Z@;Wje<2?F*7 $$!3()zceY->29F*$!3nY&o\\0n[7#F*7$$!3`1; @?.`-8F*$!31qp(Q0Qn=#F*7$$!3oQPwS08$>\"F*$!3q-sb%yFGF#F*7$$!3UAV*)fA<' 3\"F*$!3[9(zpo9WQ#F*7$$!3Arg,A&o>u*!#=$!3[NfB'fr+a#F*7$$!3E0j+gA^j')Fh w$!3`&3\"Gd\"3y#*F*7$$!3!=#*)o![Sbl\"Fhw$!3*4jhQRUj@\"!#;7$$!3:-.C/3&)H6Fhw$!3wUDj tWzv$!31xwP%\\IWO#Fcz7$$!3y5:?TXD\\cF\\[l$!3$o C=1u9Ja$Fcz7$$!3uK6!f;TpB%F\\[l$!3odJr)*\\]AZFcz7$$!3Rb2g!zFY#GF\\[l$! 3_334P7*>3(Fcz7$$!3Pm0&H5r%=@F\\[l$!3sl$zW(*H=W*Fcz7$$!3qx.I:WJ79F\\[l $!3DS-'*ye=;9!#:7$$!3?$Gv92O#f5F\\[l$!3SCol#=1#))=Fh\\l7$$!3a))=]wsdhq !#?$!3pK\"p([PEKGFh\\l7$$!3JW4DQQzINFa]l$!3s=$oL>mWm&Fh\\l7$$!30++++++ S5!#D$!3*G#p2Bp2B>!\"*7gn7$$\"30++++++S5F\\^l$\"3*G#p2Bp2B>F_^l7$$\"35 e#**\\%z/UNFa]l$\"3Mm1wk'okk&Fh\\l7$$\"37;&)***[&3%3(Fa]l$\"3$[42C/lK# GFh\\l7$$\"3Mx(*\\.Bhi5F\\[l$\"3%pW&)H82A)=Fh\\l7$$\"3A.(**z0;oT\"F\\[ l$\"3'zzuzo'o69Fh\\l7$$\"3ka&**pcB_7#F\\[l$\"3/4>*)Gs$=T*Fcz7$$\"3U1%* *f2JO$GF\\[l$\"3SHb&=k)\\fqFcz7$$\"3G4\"**R4Y/D%F\\[l$\"3Q6j\"Ho8vq%Fc z7$$\"3%G\"))*>6hsm&F\\[l$\"3vJ'H8>v=`$Fcz7$$\"3a=#)*z9\"*3])F\\[l$\"3 SPvLj\\%pN#Fcz7$$\"3ci(*R=@XL6Fhw$\"33wD)G?)=qxeBxB\"Fcz7$$\"3=3SfbEm>@Fhw$\"3KC\\W7oWT&*F*7$$\"3aZ\"4#\\&f (GKFhw$\"3KVk9:twbjF*7$$\"3o]i(pD3_M%Fhw$\"3pk)*eX<.?[F*7$$\"3uJ1#*y1N caFhw$\"3Bw*f,4r#QRF*7$$\"3]AO>S>_'['Fhw$\"3UBRhxDk2MF*7$$\"3A!f$R6Y?` vFhw$\"3$yW\"RaHaDIF*7$$\"3'y#R&eNpjl)Fhw$\"3$R!>jCfDVFF*7$$\"352!))pB '*fv*Fhw$\"3*pTtF5@y`#F*7$$\"3%Q\"fbk(4()3\"F*$\"3%=\"\\GsxAF*7$$\"3p6!*zwT\\+8F*$\"3cT_ZCS7)=#F*7$ $\"3kwPdU;689F*$\"3(>8v?F*7$$ \"3W!GwS2$>?;F*$\"3yAr6=t^W?F*7$$\"3\\n&e#zGQP?F*7$$\" 3@<+1;!em$=F*$\"3\\phXcLE2?F*7$$\"3mAdu+'Q@&>F*$\"3#zK!)*=ne+?F*7$$\"3 ?;Yi&[eV0#F*$\"3QwG)[;>2+#F*7$$\"35!)pJ+*4l;#F*$\"3hf()4m')R1?F*7$$\"3 /\"R.'oXItAF*$\"3*pn-@\")Gk,#F*7$$\"3[mQY,Ot%Q#F*$\"3R/Kts&=n'>HF*$\"3NhiQvJ%[9#F*7$$\"3@czL'o@5.$F*$\"3w#G'*pNa`<#F*7$$ \"37z+iS.'*RJF*$\"3[Ff,4k`C$F*$\"3JByz>d%*QAF*7$$ \"331_<$)pRiLF*$\"3!=y9RD7gF#F*7$$\"3G?#)zUibnMF*$\"3QQl$H>`0J#F*7$$\" 3]F.C)>O)zNF*$\"3)fUT%zGg[BF*7$$\"3?4N&f_y:o$F*$\"3I\\[G+X.%Q#F*7$$\"3 s!Q9Qw4Gz$F*$\"3xJ$3pV=PU#F*7$$\"3i$R)=!ppu*QF*$\"3%=aH0H))=Y#F*7$$\"3 Xhijp%po+%F*$\"3U*ygK_xD]#F*7$$\"3txc]]x#Q6%F*$\"3TJps,\"zIa#F*7$$\"3. $R)>eJ!eA%F*$\"3ufQq!R%='e#F*7$$\"3ip$*R%y[OL%F*$\"307a%yDH$GEF*7$$\"3 h$zyO^ORW%F*$\"3m3vq!p>?n#F*7$$\"3O6n([+6Lb%F*$\"35&er$4j*er#F*7$$\"3% 4P#4.[\"Ql%F*$\"3C)=h$R#*GF*7$$\" 3%*pf(*H\\,(3&F*$\"3f5p#RMlm$HF*7$$\"3;+++++++_F*$\"3o%Q:YQ:Y)HF*-%'CO LOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`amF_am-%*THICKNESSG6#\"\"#-F$6&7$7$ $FdamF`amFiam7$$!\"#F`amF[bm-%'SYMBOLG6#%'CIRCLEG-Fi`m6&F[amF`amF`amF` am-%&STYLEG6#%&POINTG-F$6&Fgam-F^bm6#%&CROSSGFabmFcbm-F$6&Fgam-F^bm6#% (DIAMONDGFabmFcbm-%%TEXTG6$7$$!#K!\"\"$!#8FgcmQ " 0 "" {MPLTEXT 1 0 83 "f := x -> x/2+2/x;\nD(f);\nxcrit := [solve(D(f)(x)=0, x)];\nop(map(x->[x,f(x)],xcrit));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/9$F/F/*&F0F/ F1!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)ope ratorG%&arrowGF&,&#\"\"\"\"\"#F,*&F-F,9$!\"#!\"\"F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&xcritG7$\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$\"\"#F$7$!\"#F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = x^3-x^4 /4;" "6#/-%\"fG6#%\"xG,&*$F'\"\"$\"\"\"*&F'\"\"%F-!\"\"F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "`f '`(x) = 3*x^2-x^3;" "6#/-%$f~'G6#%\"xG,&*&\"\"$\"\"\"*$F'\"\"#F+F+*$F'F*!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 14 " exactly when " } {XPPEDIT 18 0 "3*x^2-x^3 = 0;" "6#/,&*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*$F) F&!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Factorin g the cubic gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*(3-x) = 0;" "6#/*&%\"xG\"\"#,&\"\"$\"\"\"F%!\"\"F)\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "which holds when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 33 "These are the critical values of \+ " }{TEXT 281 1 "x" }{TEXT -1 18 " for the function." }}{PARA 0 "" 0 " " {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(0) = 0;" "6#/-%\"fG6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(3) = 27-81/4;" "6#/-%\"fG6#\"\"$ ,&\"#F\"\"\"*&\"#\")F*\"\"%!\"\"F." }{XPPEDIT 18 0 "``=27/4" "6#/%!G*& \"#F\"\"\"\"\"%!\"\"" }{XPPEDIT 18 0 "``=6" "6#/%!G\"\"'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 48 ", t he associated stationary points on the curve " }{XPPEDIT 18 0 "y = x^3 -x^4/4;" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"*&F'\"\"%F+!\"\"F," }{TEXT -1 4 " are" }{XPPEDIT 18 0 "``(0,0);" "6#-%!G6$\"\"!F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(3,27/4);" "6#-%!G6$\"\"$*&\"#F\"\"\"\"\"%!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "By inspecting the graph of " }{XPPEDIT 18 0 "y = x^3-x^4/4" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"*&F' \"\"%F+!\"\"F," }{TEXT -1 12 " we see that" }{XPPEDIT 18 0 " ``(3,27/4 )" "6#-%!G6$\"\"$*&\"#F\"\"\"\"\"%!\"\"" }{TEXT -1 24 " is a maximum p oint, but" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 43 " \+ is neither a maximum nor a minimum point. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 31 ": The value of the function at " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 56 " is not only greater than the value at nearby values of " }{TEXT 283 2 "x," }{TEXT -1 65 " but is, \+ in fact, the maximum value obtained by the function for " }{TEXT 259 10 "all values" }{TEXT -1 4 " of " }{TEXT 284 1 "x" }{TEXT -1 1 "." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 345 363 363 {PLOTDATA 2 "6 ,-%'CURVESG6%7Y7$$!3++++++++:!#<$!3+++++]iSYF*7$$!3mmm;Ww&[O\"F*$!3'o8 P!Qj/5MF*7$$!3QL$eH+rsC\"F*$!3M[-b`'*RXDF*7$$!3amm\"p_K]6\"F*$!3WgR<%Q kFx\"F*7$$!3&fmmT)[<>)*!#=$!3/'\\pFbH\"z6F*7$$!3-LLeC7N%\\)F?$!3)=,e:W k0V(F?7$$!3Kmm\"H5qgE(F?$!3IY]jhk-LXF?7$$!3!****\\i9cU*fF?$!3mYDg#ojlZ #F?7$$!3!em;HLW*yYF?$!3)*=D$fVfT9\"F?7$$!3G++DT2&yO$F?$!3NttBWMeTT!#>7 $$!3)4LL$G)H#>?F?$!3sGEwG4f[')!#?7$$!3hSmmTzj8$)Fen$!3WME7)RHb'e!#@7$$ \"3gB++],#*e]Fen$\"3gcUos#Q$y7Fao7$$\"3E+++D(Q'[=F?$\"38S08#p\"oDgF[o7 $$\"3s.++&)RiUJF?$\"3yvA&zlR)fGFen7$$\"3?ML3nlo#*Q`,')*f\"F?7$$\"3_nmmEZh)*oF?$\"3)=UKSs))or #F?7$$\"3o,+vB')\\v#)F?$\"3;#*3Fp>([\\%F?7$$\"31mmmwTF%\\*F?$\"3C=[Z\" z&*o_'F?7$$\"3a+]P7k9$3\"F*$\"3eL`n]'3lE*F?7$$\"3K+]7<&y/@\"F*$\"3xa'* G%>=pB\"F*7$$\"31nmT5iLV8F*$\"3IDQ%=9=+h\"F*7$$\"3)pmTl/T`Y\"F*$\"3-_@ o')Gx$*>F*7$$\"3aLLeW(Rpf\"F*$\"37-tJ_5kYCF*7$$\"3nL$3n!QjLF*$\"3\"HK f9BtX#RF*7$$\"3=+++n/\"R6#F*$\"3#)*oe/s@TX%F*7$$\"3H++vi))zVAF*$\"3GC' elM@)f\\F*7$$\"3a+]PkDZpBF*$\"3UnQHB[zAaF*7$$\"32++D$y6!4DF*$\"3C^AIF*$\"3]#Q'yO5wZ nF*7$$\"3-n;/*3$)p9$F*$\"3Og1sMVJYmF*7$$\"3k+](yG@uF$F*$\"3U>$3-^&[fjF *7$$\"3\\mm;p%[\\S$F*$\"3;ka%z9VD(eF*7$$\"3U+]P4wXQNF*$\"3^&=6;PK?6&F* 7$$\"3AMLLzF/nOF*$\"3jh9f5Bl/TF*7$$\"3;MLe(oR&)z$F*$\"3%Q+z\"ewWgFF*7$ $\"3]n;/Nx%*GRF*$\"3#*=DP\")[Kx5F*7$$\"3AM3-hM')))RF*$\"3a\"z-O$\\+nh=%F*$!3j%=W/`SKT$F*7$$\"3o+]i%)*QvC%F*$!3;)f6cFtBu%F*7$ $\"3dLLL:%e*3VF*$!3Gm\\yf8bzhF*7$$\"3]mT&)QVWuVF*$!3DlPl%pFg$yF*7$$\"3 K+]Pi-$*RWF*$!3,^BMOE9E'*F*7$$\"3j+++Z'3E]%F*$!3?T/)\\v(*p9\"!#;7$$\"3 %4+D;.(GlXF*$!3VSyd/DmW8Fj[l7$$\"36+D\"e^VEj%F*$!3oXl\"H7#[s:Fj[l7$$\" 3;+++++++ZF*$!3\\++++D!p\"=Fj[l-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb]lF a]l-%*THICKNESSG6#\"\"#-F$6&7$7$Fa]lFa]l7$$\"\"$Fb]l$\"3+++++++]nF*-%' SYMBOLG6#%'CIRCLEG-F[]l6&F]]lFb]lFb]lFb]l-%&STYLEG6#%&POINTG-F$6&Fi]l- Fa^l6#%&CROSSGFd^lFf^l-F$6&Fi]l-Fa^l6#%(DIAMONDGFd^lFf^l-%%TEXTG6$7$F \\^l$\"#uF`]lQ7maximum~point~(3,27/4)6\"-Fe_l6$7$$!\"#Fb]l$\"\"\"Fb]lQ 7stationary~point~(0,0)F[`l-Fe_l6$7$$\"#ZF`]l$!#DF``lQ\"xF[`l-Fe_l6$7$ $!#:F``l$\"#tF`]lQ\"yF[`l-%+AXESLABELSG6%%!GFgal-%%FONTG6#%(DEFAULTG-% %VIEWG6$;F_`lFg`l;$!#BF`]l$\"#vF`]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 " We can use Maple to find the stationary point as follows .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> x^3-x^4/4; \nD(f);\nxcrit := [solve(D(f)(x)=0,x)];\nop(map(x->[x,f(x)],xcrit));\n evalf(evalf(%),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"F1*&#F1\"\"%F1*$)F/F4F1F1! \"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operator G%&arrowGF&,&*&\"\"$\"\"\")9$\"\"#F-F-*$)F/F,F-!\"\"F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xcritG7%\"\"$\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7$\"\"$#\"#F\"\"%7$\"\"!F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7$$\"\"$\"\"!$\"&+v'!\"%7$$F&F&F+F*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "f(x)=x^2 - 5*x + x*sqrt(x)" "6#/-%\"fG6#%\"xG,(*$F'\"\" #\"\"\"*&\"\"&F+F'F+!\"\"*&F'F+-%%sqrtG6#F'F+F+" }{XPPEDIT 18 0 "``=x^ 2-5*x+x^(3/2)" "6#/%!G,(*$%\"xG\"\"#\"\"\"*&\"\"&F)F'F)!\"\")F'*&\"\"$ F)F(F,F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "`f '`(x) = 2*x-5+3/2;" "6#/-%$f~'G6#%\"xG,(*&\"\"#\"\" \"F'F+F+\"\"&!\"\"*&\"\"$F+F*F-F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^( 3/2)" "6#)%\"xG*&\"\"$\"\"\"\"\"#!\"\"" }{XPPEDIT 18 0 "``=2*x-5+3*sqr t(x)/2" "6#/%!G,(*&\"\"#\"\"\"%\"xGF(F(\"\"&!\"\"*(\"\"$F(-%%sqrtG6#F) F(F'F+F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 14 " exactly w hen " }{XPPEDIT 18 0 "2*x-5+3*sqrt(x)/2=0" "6#/,(*&\"\"#\"\"\"%\"xGF'F '\"\"&!\"\"*(\"\"$F'-%%sqrtG6#F(F'F&F*F'\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "2*x-5+3*sq rt(x)/2 = 0" "6#/,(*&\"\"#\"\"\"%\"xGF'F'\"\"&!\"\"*(\"\"$F'-%%sqrtG6# F(F'F&F*F'\"\"!" }{TEXT -1 28 " can be written in the form " } {XPPEDIT 18 0 "2*x-5=-3*sqrt(x)/2" "6#/,&*&\"\"#\"\"\"%\"xGF'F'\"\"&! \"\",$*(\"\"$F'-%%sqrtG6#F(F'F&F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 47 "Squaring both sides of the last equation gives " } {XPPEDIT 18 0 "4*x^2-20*x+25=9*x/4" "6#/,(*&\"\"%\"\"\"*$%\"xG\"\"#F'F '*&\"#?F'F)F'!\"\"\"#DF'*(\"\"*F'F)F'F&F-" }{TEXT -1 9 " so that " } {XPPEDIT 18 0 "16*x^2-80*x+25=9*x" "6#/,(*&\"#;\"\"\"*$%\"xG\"\"#F'F'* &\"#!)F'F)F'!\"\"\"#DF'*&\"\"*F'F)F'" }{TEXT -1 4 " or " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "16*x^2-89*x+25=0" "6#/,(*&\"#;\" \"\"*$%\"xG\"\"#F'F'*&\"#*)F'F)F'!\"\"\"#DF'\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 30 "Factoring the quadratic gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(16*x-25)*(x-4)=0" "6#/*&,& *&\"#;\"\"\"%\"xGF(F(\"#D!\"\"F(,&F)F(\"\"%F+F(\"\"!" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 17 "which holds when " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "x=25/16" "6#/%\"xG*&\"#D\"\"\"\"#;!\" \"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=4" "6#/%\"xG \"\"%" }{TEXT -1 18 " is a solution of " }{XPPEDIT 18 0 "2*x-5 = 3*sqr t(x)/2" "6#/,&*&\"\"#\"\"\"%\"xGF'F'\"\"&!\"\"*(\"\"$F'-%%sqrtG6#F(F'F &F*" }{TEXT -1 13 " rather than " }{XPPEDIT 18 0 "2*x-5 = -3*sqrt(x)/2 " "6#/,&*&\"\"#\"\"\"%\"xGF'F'\"\"&!\"\",$*(\"\"$F'-%%sqrtG6#F(F'F&F*F *" }{TEXT -1 84 ", and was introduced with the squaring of this equati on. The only critical value of " }{TEXT 278 1 "x" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "x=25/16" "6#/%\"xG*&\"#D\"\"\"\"#;!\"\"" }{TEXT -1 21 " , and the associated " }{TEXT 279 1 "y" }{TEXT -1 10 " value is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(25/16)=625/256-125/ 16+125/64" "6#/-%\"fG6#*&\"#D\"\"\"\"#;!\"\",(*&\"$D'F)\"$c#F+F)*&\"$D \"F)F*F+F+*&F1F)\"#kF+F)" }{XPPEDIT 18 0 "``=(625-2000+500)/256" "6#/% !G*&,(\"$D'\"\"\"\"%+?!\"\"\"$+&F(F(\"$c#F*" }{XPPEDIT 18 0 "``=-875/2 56" "6#/%!G,$*&\"$v)\"\"\"\"$c#!\"\"F*" }{TEXT -1 1 " " }{TEXT 280 1 " ~" }{TEXT -1 10 " -3.4180. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 330 324 324 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3s +++]Z![:*!#>$!3Rnt#4%ef;U!#=7$$\"36++DT^.7F;7$$\"3%)***\\7> l15'F0$!3#[vmeF[;?#F;7$$\"3?,+DrGo\"*pF0$!3#*)R%f/mQACF;7$$\"3?++DYI%) zyF0$!3I4^so0_>EF;7$$\"3)4++]&\\U$z)F0$!3y)p!Q(4\")))z#F;7$$\"3U,+](R3 \")f*F0$!37*ozLG\"\\PHF;7$$\"3G++]O\"*R]5F;$!3Qgo+eR6sIF;7$$\"3/++](Rf 89\"F;$!3%o'Gd%)4t%=$F;7$$\"3F++]jk,H7F;$!3w/&[Ys+@F$F;7$$\"35+]7xuh38 F;$!3)HkO33+]P318O= F;$!3*[jaP!*[7K$F;7$$\"3E++D\")48E>F;$!3ccuEO6\\ZKF;7$$\"3`+]i=%z(3?F; $!3S@-%G$3jhJF;7$$\"3A++vBp#z4#F;$!33Z]prKl\\IF;7$$\"3/+]7Fh_!>#F;$!3( 3`fd\"R:7HF;7$$\"3/+](e+M6F#F;$!37SY:ct&\\x#F;7$$\"3=++DBF>eBF;$!3dJop #p`&3EF;7$$\"3E+++(*G8[CF;$!3Ab&4Y5OoT#F;7$$\"30++D5=7ODF;$!3Qd*R'G!p) 4AF;7$$\"3D+]73cD@EF;$!3G'[$fgwS\"*>F;7$$\"3D++vv@y:FF;$!3j)[(z1%Hzs\" F;7$$\"3(******4]=2!GF;$!3;%G!o&poCZ\"F;7$$\"3]++]dhS\"*GF;$!3M!)p6#3: -=\"F;7$$\"3E+]7`EetHF;$!39b'fJ6+0)*)F07$$\"3o++]oKUjIF;$!3A-,Z!eavq&F 07$$\"3!***\\7'Gcz9$F;$!3uDy&4As*[CF07$$\"3K+]iYwJOKF;$\"3y$3s4[TA9\"F 07$$\"3O++]FqqALF;$\"3O0,we\"\\d$[F07$$\"32+]7.([JT$F;$\"3WmBk*QM`*))F 07$$\"3i+++'ya-]$F;$\"3K'[@#o]7*H\"F;7$$\"3A++vOLL*e$F;$\"3Akb62:$ot\" F;7$$\"3U+]7sUnxOF;$\"3;yQer%*o*=#F;7$$\"3c+++'4SPJF;7$$\"3`+++i35NRF;$\"3y.Yk\\*3ch$F;7 $$\"3G+]7EP#Q-%F;$\"3H$>N1C;O9%F;7$$\"3*3+vy#Gu3TF;$\"3an&GH@*[mYF;7$$ \"3;+++++++UF;$\"3OjgSV'QuC&F;-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICK NESSG6#\"\"#-F$6&7#7$$\"3++++++]i:F;$!3++++](ozT$F;-%'SYMBOLG6#%'CIRCL EG-Fhz6&FjzF)F)F)-%&STYLEG6#%&POINTG-F$6&Fd[l-F[\\l6#%&CROSSGF^\\lF`\\ l-F$6&Fd[l-F[\\l6#%(DIAMONDGF^\\lF`\\l-%%TEXTG6$7$$Fa[lF)$!#UF][lQ?min imum~point~(25/16,-875/256)6\"-F_]l6$7$$\"#VF][l$!\"$F][lQ\"xFf]l-F_]l 6$7$$!#:!\"#$\"#dF][lQ\"yFf]l-%+AXESLABELSG6%%!GF[_l-%%FONTG6#%(DEFAUL TG-%%VIEWG6$;Fb^lFj]l;$!#`F][lFe^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 " We can use Maple to find the stationary point as follows.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f := x -> x^2-5*x+x*sqrt(x) ;\nD(f);\nxcrit := [solve(D(f)(x)=0,x)];\nop(map(x->[x,f(x)],xcrit)); \nevalf(evalf(%),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"F1*&\"\"&F1F/F1!\"\"*&F/F 1-%%sqrtG6#F/F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6 \"6$%)operatorG%&arrowGF&,**&\"\"#\"\"\"9$F-F-\"\"&!\"\"-%%sqrtG6#F.F- *&#F-F,F-*&F.F-F1F0F-F-F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xc ritG7##\"#D\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"#D\"#;#!$v)\"$ c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"&Dc\"!\"%$!&!=MF&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "Comparison of graphs of a function and it s derivative " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 127 "The examples in this section are adapted from \+ \"Exploring Calculus with Mathematica\", by Finch and Lehmann, Addison -Wesley.\nLet " }{XPPEDIT 18 0 "f(x)=2*x^3-21*x^2+60*x+3" "6#/-%\"fG6# %\"xG,**&\"\"#\"\"\"*$F'\"\"$F+F+*&\"#@F+*$F'F*F+!\"\"*&\"#gF+F'F+F+F- F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative of \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "`f '`(x) = 6*x^2-42*x+60;" "6#/-%$f~'G6#%\"xG,(*&\"\"' \"\"\"*$F'\"\"#F+F+*&\"#UF+F'F+!\"\"\"#gF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#% \"xG\"\"!" }{TEXT -1 14 " exactly when " }{XPPEDIT 18 0 "6*x^2-42*x+60 = 0;" "6#/,(*&\"\"'\"\"\"*$%\"xG\"\"#F'F'*&\"#UF'F)F'!\"\"\"#gF'\"\"! " }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x^2-7*x+10=0" "6#/,(*$%\"xG\"\" #\"\"\"*&\"\"(F(F&F(!\"\"\"#5F(\"\"!" }{TEXT -1 10 " so that " } {XPPEDIT 18 0 "(x-2)*(x-5)=0" "6#/*&,&%\"xG\"\"\"\"\"#!\"\"F',&F&F'\" \"&F)F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "The criti cal values of " }{TEXT 290 1 "x" }{TEXT -1 18 " for the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=5" "6 #/%\"xG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "f(2) = 55;" "6#/-%\"fG6#\"\"#\"#b" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(5) = 28;" "6#/-%\"fG6#\"\"&\"#G" }{TEXT -1 48 ", the associated stationary points on the curve " }{XPPEDIT 18 0 "y = 2*x^3 -21*x^2+60*x+3;" "6#/%\"yG,**&\"\"#\"\"\"*$%\"xG\"\"$F(F(*&\"#@F(*$F*F 'F(!\"\"*&\"#gF(F*F(F(F+F(" }{TEXT -1 4 " are" }{XPPEDIT 18 0 "``(2,55 );" "6#-%!G6$\"\"#\"#b" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(5, 28); " "6#-%!G6$\"\"&\"#G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 " By inspecting the graph we see that" }{XPPEDIT 18 0 " ``(2,55)" "6#-%! G6$\"\"#\"#b" }{TEXT -1 29 " is local maximum point while" }{XPPEDIT 18 0 " ``(5,28)" "6#-%!G6$\"\"&\"#G" }{TEXT -1 27 " is a local minimum point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "f := x -> 2*x^3-21*x^2+60*x+3;\nD(f);\nxcrit := [sol ve(D(f)(x)=0,x)];\nop(map(x->[x,f(x)],xcrit));\nevalf(evalf(%),5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,**&\"\"#\"\"\")9$\"\"$F/F/*&\"#@F/)F1F.F/!\"\"*&\"#gF/F1F/F/F2F/F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arr owGF&,(*&\"\"'\"\"\")9$\"\"#F-F-*&\"#UF-F/F-!\"\"\"#gF-F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xcritG7$\"\"&\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$\"\"&\"#G7$\"\"#\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$$\"\"&\"\"!$\"#GF&7$$\"\"#F&$\"#bF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The function " }{XPPEDIT 18 0 "f(x) = 2* x^3-21*x^2+60*x+3;" "6#/-%\"fG6#%\"xG,**&\"\"#\"\"\"*$F'\"\"$F+F+*&\"# @F+*$F'F*F+!\"\"*&\"#gF+F'F+F+F-F+" }{TEXT -1 3 " (" }{TEXT 260 3 "re d" }{TEXT -1 27 " graph) and its derivative " }{XPPEDIT 18 0 "`f '`(x) = 6*x^2-42*x+60;" "6#/-%$f~'G6#%\"xG,(*&\"\"'\"\"\"*$F'\"\"#F+F+*&\"# UF+F'F+!\"\"\"#gF+" }{TEXT -1 2 " (" }{TEXT 256 4 "blue" }{TEXT -1 49 " graph) are plotted together in the next picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "f := x -> 2* x^3-21*x^2+60*x+3;\nDf := x -> D(f)(x);\nplot([f(x),Df(x)],x=0..7,colo r=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$ %)operatorG%&arrowGF(,**&\"\"#\"\"\")9$\"\"$F/F/*&\"#@F/)F1F.F/!\"\"*& \"#gF/F1F/F/F2F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,(*&\"\"'\"\"\")9$\"\"#F/F/*&\"#UF/F1F/! \"\"\"#gF/F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$\"\"!F)$\"\"$F)7$$\"3+LLLeR+Hw!#>$\"315&)3Wo1cu! #<7$$\"3gmmm\"z+e_\"!#=$\"3/rOWz9In6!#;7$$\"3;+](oM'f*=#F6$\"3'yTbJOw^ ^\"F97$$\"3sLL3->R`GF6$\"3U)y\"3kFqX=F97$$\"3mmm;apSYVF6$\"38-T!fm\\v_ #F97$$\"3Onm;z'=$\\eF6$\"39J&3]V768$F97$$\"3!RL$3Ft3XtF6$\"3uDJ/w)\\Ll $F97$$\"3tmmTNj&=t)F6$\"3-N^T7[6rSF97$$\"33+](=`xn,\"F2$\"3c>Hm#zW)RWF 97$$\"3#omT&y/Gl6F2$\"3C,\"H!3(*fcZF97$$\"3++]PurI88F2$\"3s2M\\tg%3,&F 97$$\"3aLL$e#3dl9F2$\"36XfBj(=C@&F97$$\"3ymm\"Ht%o*f\"F2$\"3YK`uGE%HM& F97$$\"3K++]F_m]F2$\"3-INl([;7\\&F 97$$\"3;++]s2O[?F2$\"3u$p/It&F97$$\"3GLLLoD[lFF2$\"3ejDOs*GF97$$\"3'3++DE5!>[F2$\"3I1n35eHGGF 97$$\"3Mm;a)3rf&\\F2$\"3xeT8?ws,GF97$$\"3*4++vW0d5&F2$\"3kF2$\"3![K0&)=6n=&F97$$\"3'pmmmV,&elF2$\"3[p2M978VdF 97$$\"3<+](o(GP1nF2$\"3?_$Q:gHUT'F97$$\"3#3++]zQrx'F2$\"3QmCOS*>\\w'F9 7$$\"3g+]78Z!z%oF2$\"30M^iXQINrF97$$\"3I+DccB&R#pF2$\"3e0/^\"*4xbvF97$ $\"\"(F)$\"#!)F)-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7S7$F($\"#gF) 7$F4$\"3#R5Ne2KJP&F97$F@$\"35[i?tkU][F97$FE$\"3?qtr*fcyG%F97$FJ$\"3/)o 7)GLd[PF97$FO$\"3C\"))z'p^wQKF97$FT$\"3yO\"*\\HA4!z#F97$FY$\"3q*[?dHO) \\BF97$Fhn$\"3(pF97$F]o$\"3\"z!R=3`(*=:F97$Fbo$\"3#)y-#)Q7ML6 F97$Fgo$\"3e(**feo)=n\")F27$F\\p$\"3k\\dmxG.h[F27$Fap$\"3f)Qf$z+`;=F27 $Ffp$!3QZA*4>8Yc)F67$F[q$!3!*pmYT]*=1$F27$F`q$!3cj'R\"oKZ4aF27$Feq$!3% fcleQGX;(F27$Fjq$!3JO^fz%Gk$*)F27$F_r$!3Y\"oYb[!HE5F97$Fdr$!3]!\\c(ybo X6F97$Fir$!3T)3Rp!\\&RB\"F97$F^s$!3q>:`Sfh*H\"F97$Fcs$!3n7gbU88O8F97$F hs$!3W-\"es$G**\\8F97$F]t$!31:H^+s'e)=,8F97$Fgt$!3wZ7 5'>%*)Q7F97$F\\u$!3c!)R/5e+[6F97$Fau$!3,1rqxi*H.\"F97$Ffu$!33ORI$=V:(* )F27$F[v$!3-@mZ+_?!=(F27$F`v$!3xW&QfqflJ&F27$Fev$!3#*z!=Xcr71$F27$Fjv$ !3\"e2[)f8*)3yF67$F_w$\"31wqy/*R(p>F27$Fdw$\"3!>Q[riRN![F27$Fiw$\"3mg' *=nlH?!)F27$F^x$\"3Nxk\\euoT6F97$Fcx$\"3][9`FD$R_\"F97$Fhx$\"3\\yJ>wd& y\">F97$F]y$\"3S\\\"eKJcoM#F97$Fby$\"3m'e3#[oU)z#F97$Fgy$\"3C$z,L%yIOK F97$F\\z$\"3=`jDpE=lPF97$Faz$\"3AT_&GieEE%F97$Ffz$\"3[,eL:i\\=[F97$F`[ l$\"3;c()*3ez]P&F97$Fj[lFi\\l-F_\\l6&Fa\\lF(F(Fb\\l-%+AXESLABELSG6$Q\" x6\"Q!F_fl-%%VIEWG6$;F(Fj[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "The sign of the derivative can be determi ned from its graph.\n" }}{PARA 15 "" 0 "" {TEXT -1 28 "For example, th e derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 4 " is " }{TEXT 259 8 "negative" }{TEXT -1 6 " when " }{TEXT 300 1 "x " }{TEXT -1 60 " is between 2 and 5. Notice that the graph of the fun ction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is " } {TEXT 259 15 "going downwards" }{TEXT -1 19 " in this interval. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 15 "The deri vative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 4 " is " }{TEXT 259 8 "positive" }{TEXT -1 6 " when " }{TEXT 301 1 "x" } {TEXT -1 59 " is greater than 5. Notice that the graph of the functio n " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 259 13 "going upwards" }{TEXT -1 22 " in this interval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 9 " is also \+ " }{TEXT 259 8 "positive" }{TEXT -1 6 " when " }{TEXT 302 1 "x" } {TEXT -1 56 " is less than 2. Notice that the graph of the function \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 259 13 "going upwards" }{TEXT -1 19 " in this interval. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The sign of the de rivative determines whether the graph of the function goes up or down \+ as " }{TEXT 285 1 "x" }{TEXT -1 42 " increases.\n\nWhen the graph of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " " } {TEXT 259 5 "rises" }{TEXT -1 4 " as " }{TEXT 286 1 "x" }{TEXT -1 44 " increases, this means that any increase in " }{TEXT 287 1 "x" }{TEXT -1 29 " will produce an increase in " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 40 ". In this case we say that the function " } {TEXT 259 2 "f(" }{TEXT 298 1 "x" }{TEXT 259 15 ") is increasing" } {TEXT -1 2 ".\n" }}{PARA 256 "" 0 "" {TEXT 261 24 "When the derivative f '(" }{TEXT 291 1 "x" }{TEXT 261 18 ") of a function f(" }{TEXT 292 1 "x" }{TEXT 261 5 ") is " }{TEXT 260 8 "positive" }{TEXT 261 4 ", f( " }{TEXT 293 1 "x" }{TEXT 261 5 ") is " }{TEXT 260 10 "increasing" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 39 "_____ __________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Similarly, when the graph of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " " }{TEXT 259 5 "falls" }{TEXT -1 4 " as " }{TEXT 288 1 "x" }{TEXT -1 44 " increases, this means that any increase in " }{TEXT 289 1 "x" } {TEXT -1 28 " will produce a decrease in " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 40 ". In this case we say that the function " } {TEXT 259 2 "f(" }{TEXT 299 1 "x" }{TEXT 259 15 ") is decreasing" } {TEXT -1 2 ".\n" }}{PARA 256 "" 0 "" {TEXT 261 24 "When the derivative f '(" }{TEXT 294 1 "x" }{TEXT 261 18 ") of a function f(" }{TEXT 295 1 "x" }{TEXT 261 5 ") is " }{TEXT 260 8 "negative" }{TEXT 261 4 ", f( " }{TEXT 296 1 "x" }{TEXT 261 5 ") is " }{TEXT 260 10 "decreasing" } {TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 40 "_____ ___________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "At a " }{TEXT 259 16 "st ationary point" }{TEXT -1 37 " the graph, at least momentarily, is " } {TEXT 259 26 "neither rising nor falling" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "For another exampl e, we check how the derivative of the function " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x) = x^5+ 2*x^4-6*x^3+2*x-5;" "6#/-%\"fG6#%\"xG,,*$F'\"\"&\"\"\"*&\"\"#F+*$F'\" \"%F+F+*&\"\"'F+*$F'\"\"$F+!\"\"*&F-F+F'F+F+F*F4" }{TEXT -1 43 ", pro vides information about the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 29 " over the interval [-1.5, 2]." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> x^5+2*x^4-6*x^3+2*x-5;\nDf := x -> D(f)(x);\nplot([f(x),Df(x)],x=-1.5 ..2,y=-12..12,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,*$)9$\"\"&\"\"\"F1*&\"\"#F1)F /\"\"%F1F1*&\"\"'F1)F/\"\"$F1!\"\"*&F3F1F/F1F1F0F:F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#DfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**&\" \"&\"\"\")9$\"\"%F/F/*&\"\")F/)F1\"\"$F/F/*&\"#=F/)F1\"\"#F/!\"\"F:F/F (F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV ESG6$7\\o7$$!3++++++++:!#<$\"3+++++]7y9!#;7$$!3QL$3-)\\&=Y\"F*$\"38(Hr E)**zF8F-7$$!3vmmTg*4PU\"F*$\"391/$\\o,N=\"F-7$$!35]il#=?0R\"F*$\"3)[Z tKMPH1\"F-7$$!3ALe*[SItN\"F*$\"3[wBO+\\uq%*F*7$$!3\"*\\PfG]+?8F*$\"3*R `)zCJUC#)F*7$$!3em;H_'zEG\"F*$\"3?A%Hieh%QqF*7$$!3_mm;fo5X7F*$\"3$o9$o Unw0fF*7$$!3pm;/mS`27F*$\"3dx&)oa\"fV$[F*7$$!3')\\P%)\\)R,<\"F*$\"3=3Z 6,ElGQF*7$$!3DLekLcuK6F*$\"3Ydtv+!zH)GF*7$$!3sm\"HK=2M1\"F*$\"3m\"p2&) 3;hG\"F*7$$!3e**\\iSB6;**!#=$!3yT)[1,:)z:Fbo7$$!3%em\"H2wft\"*Fbo$!3Eq MJn*pfV\"F*7$$!3,+]7GTYL%)Fbo$!3LmIo14q-DF*7$$!3KKL$3(e9swFbo$!3oeZFbo$!3T2p:]d9 F_F*7$$!3Cm;HdV&[4%Fbo$!3QbY)z%4Gi`F*7$$!3VKL$3#o21LFbo$!3\"eLep3cWU&F *7$$!3qJLL$yyyj#Fbo$!3(o'ozdl.4aF*7$$!3[**\\i:cgg=Fbo$!3=.*3)e.IJ`F*7$ $!3gLLLeres6Fbo$!3U$p.\"Hz[C_F*7$$!3O\")*\\il=s<%!#>$!3QTp@x55$3&F*7$$ \"3NF+DJS)3,$Fds$!3$R)4h?W%*R\\F*7$$\"3[omT5:4^5Fbo$!3=u.])=#\\'z%F*7$ $\"3;o;a)[G)R+](=%[V8'Fbo$!3IG3Yi13)y%F*7$$\"3!G+vVt'zVoFbo$!3G\"RXuJLc' \\F*7$$\"36***\\78=:j(Fbo$!3\"\\w'p2AA._F*7$$\"3@kmmT3KR$)Fbo$!35%*)yj EV7W&F*7$$\"3K/+]780&4*Fbo$!3R-$o9#y=/dF*7$$\"3uJ$3FWb)z(*Fbo$!3)\\!\\ (oZW@$fF*7$$\"3]++vBF&G0\"F*$!3!fMEI#ybXhF*7$$\"3emT50pHB6F*$!3')yhtyM %\\G'F*7$$\"35+v=s8$p>\"F*$!3#eUS'[J>LjF*7$$\"3V$3_v%p#HB\"F*$!3(QPA`# 3#)3jF*7$$\"3umm\"H_A*o7F*$!3a&ztvl$4YiF*7$$\"3H$3FWb1mI\"F*$!3PKm*)zw LLhF*7$$\"3$)*\\Pfe!HW8F*$!3QEA!4q/e'fF*7$$\"3#RLL$))*yoT\"F*$!3Wm+v]X 4iaF*7$$\"3_L$eR666\\\"F*$!33%R\"oI5Q^YF*7$$\"3;nT5g&GZc\"F*$!3Q^N1+W) y[$F*7$$\"3!Q3-Q&>b)f\"F*$!3)zt]^Y>S\"GF*7$$\"3Y++]Z`PK;F*$!3,hvm/)QB/ #F*7$$\"3oLekt29r;F*$!3a0>;vSqF5F*7$$\"3\"pm\"z*>1*4%f1-xJ\"F*7$$\"3[LLL=2Dz%RP*Q) [yj#F*7$$\"3y;aQy&=i\"=F*$\"3A()\\(GOW1@%F*7$$\"33+vVQk=`=F*$\"3Yt.%z0 pg'fF*7$$\"3=++](Rp&))=F*$\"3\\CLjw:8HyF*7$$\"3I+DccB&R#>F*$\"3?&H!fV: %H))*F*7$$\"3Av=UnU'H%>F*$\"3X/T6)z[p5\"F-7$$\"39]7Gyh(>'>F*$\"3XC3%[! GlJ7F-7$$\"33D19*3))4)>F*$\"3KMQ(>-.EO\"F-7$$\"\"#\"\"!$\"#:Fc_l-%'COL OURG6&%$RGBG$\"*++++\"!\")$Fc_lFc_lF]`l-F$6$7U7$F($!3++++++v=SF-7$F4$! 3x6:.>!yGq$F-7$F>$!35vWUA5k>MF-7$FH$!3#*R***)e$4j4$F-7$FR$!3#)fP#Q\"p; qFF-7$Ffn$!39%)4Nkf;\\CF-7$F[o$!3-vZ!QIR\"e@F-7$F`o$!3Eh;%y3Il'=F-7$Ff o$!3Qm_Gu#)Gy:F-7$F[p$!3;>Nb_h928F-7$F`p$!3i'3MSGav/\"F-7$Fep$!3ej@.p_ Co$)F*7$Fjp$!3%=r?(G*fC@'F*7$F_q$!3G\")=()HAh\"H%F*7$Fdq$!3g`4q6Q!3o#F *7$Fiq$!3N&o%3XV#pU\"F*7$F^r$!3-0Wz60zn>Fbo7$Fcr$\"3Q$okcqM&[iFbo7$Fhr $\"3z%*H')y%H8L\"F*7$F]s$\"3MSB$HDa0u\"F*7$Fbs$\"3'o.y=jB!o>F*7$Fhs$\" 3w$Q:yq/R)>F*7$F]t$\"34Y$)>!RP5\"=F*7$Fbt$\"35#)4PMD&=]\"F*7$Fgt$\"3vb l:Gy\">.\"F*7$F\\u$\"3UO$=ys&RaUFbo7$Fau$!31b`%yk#R;Fbo7$Fby$\"3d`_s-&ybK#F*7$F\\z$\"3!**)[Fn\"yXB&F*7$Faz$\"3c4(o4ev 4x)F*7$Ffz$\"316Tg3H#>K\"F-7$F[[l$\"3E!3$*zyH]&=F-7$Fe[l$\"3__pk8adLCF -7$F_\\l$\"3&Ga7+U]4@$F-7$Fi\\l$\"3k(=SF-7$Fc]l$\"3w=mR " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "When the derivative of the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 " (" }{TEXT 256 4 "blue" } {TEXT -1 34 " graph) is negative, the graph of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 2 " (" }{TEXT 260 3 "red" }{TEXT -1 39 " graph) falls, and when the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6# -%$f~'G6#%\"xG" }{TEXT -1 27 " is positive, the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 8 " rises.\n" }}{PARA 0 "" 0 " " {TEXT -1 11 "We can use " }{TEXT 0 6 "fsolve" }{TEXT -1 13 " rather \+ than " }{TEXT 0 5 "solve" }{TEXT -1 43 " to find the stationary points numerically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "xcrit := [fsolve(Df(x)=0,x)]:\nop(map(x->[x,f(x) ],xcrit));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7$$!+%p3r%G!\"*$\"+h$o;@ (!\")7$$!+N#4+;$!#5$!+)[\")eU&F&7$$\"+H)=%QPF-$!+?OV>XF&7$$\"+NxE*=\"F &$!+k&GRL'F&" }}}{PARA 0 "" 0 "" {TEXT -1 55 "\nThe stationary point ( -2.847108694 , 72.11668361) has " }{TEXT 297 1 "x" }{TEXT -1 89 " coor dinate outside the interval considered above, but we can see it in a m odified plot. 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"" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 15 " minimum point" }{XPPEDIT 18 0 " ``(1/4, 3)" "6#-%!G6$*&\"\"\"F' \"\"%!\"\"\"\"$" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Tasks 2" }}{PARA 0 "" 0 "" {TEXT -1 62 "Ea ch of the following question involves a polynomial function." }}{PARA 15 "" 0 "" {TEXT -1 193 "Draw a graph of the function and its derivati ve in the same or separate plots.\n(Draw more than one plot if using o ne plot doesn't show the behaviour of the function and its derivative \+ clearly.)" }}{PARA 15 "" 0 "" {TEXT -1 152 "Find the coordinates of an y stationary points and say whether each stationary point is a local m aximum point, a local minimum point or neither of these." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = x^3 -6*x^2+11*x-6;" "6#/-%\"fG6#%\"xG,**$F'\"\"$\"\"\"*&\"\"'F+*$F'\"\"#F+ !\"\"*&\"#6F+F'F+F+F-F0" }}{PARA 0 "" 0 "" {TEXT -1 34 "______________ ____________________" }}{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 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x ) = x^5+4*x^4+x^3-10*x^2-4*x+8;" "6#/-%\"fG6#%\"xG,.*$F'\"\"&\"\"\"*& \"\"%F+*$F'F-F+F+*$F'\"\"$F+*&\"#5F+*$F'\"\"#F+!\"\"*&F-F+F'F+F5\"\")F +" }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 " " {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x) = 5*x^6-24*x^5-165*x^4+1120*x^ 3-765*x^2-5400*x+100;" "6#/-%\"fG6#%\"xG,0*&\"\"&\"\"\"*$F'\"\"'F+F+*& \"#CF+*$F'F*F+!\"\"*&\"$l\"F+*$F'\"\"%F+F1*&\"%?6F+*$F'\"\"$F+F+*&\"$l (F+*$F'\"\"#F+F1*&\"%+aF+F'F+F1\"$+\"F+" }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{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 34 "_________________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x) = 15*x^7-280*x^6+1722*x^5-1785*x^4-22925*x^3+99750*x^2-1575 00*x;" "6#/-%\"fG6#%\"xG,0*&\"#:\"\"\"*$F'\"\"(F+F+*&\"$!GF+*$F'\"\"'F +!\"\"*&\"%A " 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 34 "____________________ ______________" }}{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 37 "Code for max and min points examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "p1 := plot(x^3-3*x,x=-2.2.. 2.2,thickness=2):\np2 := plot([[[1,-2],[-1,2]]$3],style=point,symbol=[ circle,cross,diamond],color=black):\nt1 := plots[textplot]([[-1.8,2.6, `local maximum point (-1,2)`],\n [1.8,-2.6,`local minimum point (1, -2)`],[2.4,-.3,`x`],[-.15,4.7,`y`]]):\nplots[display]([p1,p2,t1],view= [-2.4..2.4,-4.7..4.7],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "p1 := plot(x/2+2/x,x=- 5.2..5.2,thickness=2,discont=true):\np2 := plot([[[2,2],[-2,-2]]$3],st yle=point,\n symbol=[circle,cross,diamond],color=black):\nt1 \+ := plots[textplot]([[-3.2,-1.3,`local maximum point (-2,-2)`],\n [3 .2,1.4,`local minimum point (2,2)`],[5.3,-.3,`x`],[-.3,5.7,`y`]]):\npl ots[display]([p1,p2,t1],view=[-5.2..5.3,-5.3..5.7],labels=[``,``]);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "p1 := plot(x^3-x^4/4,x=-1.5..4.7,thickness=2):\np2 := plot([[[0 ,0],[3,27/4]]$3],style=point,\n symbol=[circle,cross,diamond],col or=black):\nt1 := plots[textplot]([[3,7.4,`maximum point (3,27/4)`],[- 2,1,`stationary point (0,0)`],[4.7,-.25,`x`],[-.15,7.3,`y`]]):\nplots[ display]([p1,p2,t1],view=[-2..4.7,-2.3..7.5],labels=[``,``]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 315 "p1 := plot(x^2-5*x+x*sqrt(x),x=0..4.2,thickness=2):\np2 := plot([ [[25/16,-875/256]]$3],style=point,\n symbol=[circle,cross,diamond ],color=black):\nt1 := plots[textplot]([[2,-4.2,`minimum point (25/16, -875/256)`],[4.3,-.3,`x`],[-.15,5.7,`y`]]):\nplots[display]([p1,p2,t1] ,view=[-.15..4.3,-5.3..5.7],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }