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{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 36 "Taylor polynomials and Taylor ser ies" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can ada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{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 13 "Introduction " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Introductor y question " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 294 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 58 "(a) Find the equation of the tangent line to the graph of " }{XPPEDIT 18 0 "y=sqrt(x)" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 " ``(4,2)" "6#-%!G6$\"\"%\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "(b) Use the answer from part ( a) to estimate " }{XPPEDIT 18 0 "sqrt(4.2)" "6#-%%sqrtG6#-%&FloatG6$\" #U!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 295 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 20 ", the derivative is " }{XPPEDIT 18 0 "`f '`(x) = 1/(2*sqrt(x));" "6#/-%$f~' G6#%\"xG*&\"\"\"F)*&\"\"#F)-%%sqrtG6#F'F)!\"\"" }{TEXT -1 41 ", so the gradient of the tangent line is " }{XPPEDIT 18 0 "`f '`(4) = 1/4;" "6 #/-%$f~'G6#\"\"%*&\"\"\"F)F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Using the point-slope form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-y[1]=m*(x-x[1])" "6#/,&%\"yG\"\"\"&F%6#F&! \"\"*&%\"mGF&,&%\"xGF&&F-6#F&F)F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "for the equation of the straight line with gradient m and passing through the point" }{XPPEDIT 18 0 "``(x[1],y[1])" "6#-%!G6$&% \"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 35 ", we see that the tangent line i s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-2=1/4" "6#/, &%\"yG\"\"\"\"\"#!\"\"*&F&F&\"\"%F(" }{XPPEDIT 18 0 "``(x-4)" "6#-%!G6 #,&%\"xG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=x/4+1" "6 #/%\"yG,&*&%\"xG\"\"\"\"\"%!\"\"F(F(F(" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 54 "It is therefore reasonable to use the linear function " } {XPPEDIT 18 0 "g(x) = x/4+1;" "6#/-%\"gG6#%\"xG,&*&F'\"\"\"\"\"%!\"\"F *F*F*" }{TEXT -1 29 " to approximate the function " }{XPPEDIT 18 0 "f( x);" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " }{TEXT 297 1 "x" }{TEXT -1 12 " is near 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 47 "Comparing the values of the two functions when " } {XPPEDIT 18 0 "x=4.2" "6#/%\"xG-%&FloatG6$\"#U!\"\"" }{TEXT -1 11 ", w e have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(4.2) = \+ 2.05;" "6#/-%\"gG6#-%&FloatG6$\"#U!\"\"-F(6$\"$0#!\"#" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(4.2);" "6#-%\"fG6#-%&FloatG6$\"#U!\"\"" } {TEXT -1 1 " " }{TEXT 298 1 "~" }{TEXT -1 14 " 2.049390153. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "g := x -> x/4+1: 'g(x)'=g(x);\nf := x -> sqrt(x): 'f(x)'=f(x);\nxx := 4 .2:\n'g'(xx)=g(xx);\n'f'(xx)=f(xx);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*&\"\"%!\"\"F'\"\"\"F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#$\"#U!\"\"$\"++++]?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"#U!\"\"$\"+`,R\\?!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 93 "Comments concerning the introductory qu estion and extension to a second degree approximation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 261 12 "tangent line" }{TEXT -1 14 " to the curve " } {XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 13 " at th e point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "y = f(a)+`f '`(a)*(x-a);" "6#/%\"yG, &-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F)F*,&%\"xGF*F)!\"\"F*F*" }{TEXT -1 2 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 421 330 330 {PLOTDATA 2 "60-%'CURVESG6%7S7$$\"3))**************H!#=$\"3qmDN$y$ed5F *7$$\"3w*****\\x[BR$F*$\"3ADw#Ha\"f$R\"F*7$$\"3')***\\iMHPt$F*$\"3M>DE j`p&o\"F*7$$\"3y****\\-vksF*7$$\"3( ****\\iq%[D6Fdq$\"372[F!=mjW(F*7$$\"3#***\\7la!4;\"Fdq$\"3&RBL(f!eQk(F *7$$\"3)****\\(Q:6*>\"Fdq$\"3\"eOBpBAX%yF*7$$\"3%***\\i6pzQ7Fdq$\"3rwc ^7\"\\'Q!)F*7$$\"3%***\\PXJMt7Fdq$\"3]ln!3\"y=&>)F*7$$\"3*****\\U-a1J \"Fdq$\"3YP(\\w4f1N)F*7$$\"3-+++8)*>\\8Fdq$\"3EI^/%R_e\\)F*7$$\"3/++DZ $4pQ\"Fdq$\"33vtd5W:A')F*7$$\"3****\\ig_RB9Fdq$\"3NuAbvi.H()F*7$$\"3%* ***\\nk1RY\"Fdq$\"3;/V/'z3&H))F*7$$\"3()******GzI+:Fdq$\"385!y+*F*7$$\"3:++]'o&*Gh\"Fdq$\"3\"\\'*G'3:(f-*F*7$$\"3'***\\iAT7\\;Fdq$ \"3!zeX$y,-J!*F*7$$\"3-+]7xK*po\"Fdq$\"35R'3SOPz,*F*7$$\"3))****\\(H4,!=Fdq$\"3-;_b4psl))F*7$$\"35++v:dGQ=Fdq$\"3q67@%H%*ex)F*7$$\"3=+ ]i;h9w=Fdq$\"394/$3jrwm)F*7$$\"3++++$*e$4\">Fdq$\"31w^4(G::b)F*7$$\"3' ****\\F!*33&>Fdq$\"3GL,^gT'))R)F*7$$\"3'******zzrk)>Fdq$\"3y#>#*y5b[C) F*7$$\"38+]i#)e\\C?Fdq$\"3SQ7Wv[si!)F*7$$\"3'***\\P$y*)31#Fdq$\"3cPr[T \"f8(yF*7$$\"33+++++++@Fdq$\"3k%)fO1S\\ZwF*-%'COLOURG6&%$RGBG$\"*++++ \"!\")$\"\"!F`[lF_[l-%*THICKNESSG6#\"\"#-F$6%7S7$F($\"3#)************* *>F*7$F.$\"3m******\\echAF*7$F3$\"3!*****\\(*G:*[#F*7$F8$\"3m*****\\L) 4XFF*7$F=$\"3`*****\\MSF+$F*7$FB$\"3Q****\\Fy:fKF*7$FG$\"3o****\\d'*)o \\$F*7$FL$\"3:++](>ZIu$F*7$FQ$\"39++]xOi(*RF*7$FV$\"3%*****\\FPQ^UF*7$ Fen$\"3:+++IrS7XF*7$Fjn$\"3\")*****\\o;Bu%F*7$F_o$\"3b*******QS6+&F*7$ Fdo$\"3-******\\o-h_F*7$Fio$\"3k+++5cZ6bF*7$F^p$\"3^****\\xq!*QdF*7$Fc p$\"31+++!4X$4gF*7$Fhp$\"3s)*****f:WQiF*7$F]q$\"3f)***\\<_$\\]'F*7$Fbq $\"3l******fs#3u'F*7$Fhq$\"3O****\\<#Q'**pF*7$F]r$\"3k****\\_u3YsF*7$F br$\"3/*****\\PJK](F*7$Fgr$\"3\\****\\n(p$RxF*7$F\\s$\"3))*****\\#p2%* zF*7$Fas$\"3C****\\xgke#)F*7$Ffs$\"3;****\\-V&*)[)F*7$F[t$\"3f*****\\ \\$pP()F*7$F`t$\"3!)******>am%**)F*7$Fet$\"3')*****\\JigC*F*7$Fjt$\"3% *****\\PEdRF\"Fdq7$ F`y$\"3!*****\\o#R0I\"Fdq7$Fey$\"3#)*****>`9VK\"Fdq7$Fjy$\"3%****\\<#R m\\8Fdq7$F_z$\"3!****\\A_ERP\"Fdq7$Fdz$\"3!**************R\"Fdq-Fiz6&F [[lF_[lF_[lF\\[lFa[l-F$6&7#7$$\"\"\"F`[l$\"3ImmmmmmmmF*-%'SYMBOLG6#%'C IRCLEG-Fiz6&F[[lF`[lF`[lF`[l-%&STYLEG6#%&POINTG-F$6&F_el-Ffel6#%(DIAMO NDGFielF[fl-F$6&F_el-Ffel6#%&CROSSGFielF[fl-F$6%7$7$F(F_[l7$FdzF_[l-%* LINESTYLEG6#FbelFiel-F$6%7$7$FaelF_[lF`el-F_glFc[lFiel-%%TEXTG6%7$$\" \"*!\"\"$\"#v!\"#Q)(a,f(a))6\"Fiel-Fggl6%7$Fael$!\"%F_hlQ&x~=~aFahlFie l-Fggl6%7$$\"#8F\\hl$\"#6F\\hlQ7y~=~f(a)+f~'(a)(x~-~a)FahlF[el-Fggl6%7 $$\"#>F\\hl$\"#wF_hlQ)y~=~f(x)FahlFhz-%+AXESLABELSG6%Q\"xFahlQ\"yFahl- %%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$\"\"$F\\hl$\"#@F \\hl;Fehl$\"#7F\\hl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The corresponding function:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = f(a)+`f '`(a)* (x-a);" "6#/-%\"gG6#%\"xG,&-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F,F-,&F'F-F,! \"\"F-F-" }{TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 21 " can be regarded as a " }{TEXT 261 20 "linear approximation" }{TEXT -1 17 " to the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 15 " for values of " }{TEXT 299 1 "x" }{TEXT -1 6 " near " } {TEXT 300 1 "a" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The de rivative of the function " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" } {TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `g '`(x) = `f '`(a);" "6#/-%$g~'G6#%\"xG-%$f~'G6#%\"aG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 71 "that is, it is the constant given by the gradient of the tangent line. " }}{PARA 0 "" 0 "" {TEXT -1 13 "Th e function " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 33 " h as the crucial properties that " }{TEXT 261 39 "its value and derivati ve match those of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#% \"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 10 ", namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " g(a) = f(a)" "6#/-%\"gG6#%\"aG-%\"fG6#F'" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`g '`(a) = `f '`(a);" "6#/-%$g~'G6#%\"aG-%$f~'G6#F'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 301 7 "_______" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Consider the " }{TEXT 261 13 "second degree" }{TEXT -1 22 " polynomial function: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x) = f(a)+`f '`(a)*(x-a)+ 1/2;" "6#/-%\"qG6#%\"xG,(-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F,F-,&F'F-F,!\" \"F-F-*&F-F-\"\"#F3F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(a)*(x-a) ^2;" "6#*&-%%f~''G6#%\"aG\"\"\"*$,&%\"xGF(F'!\"\"\"\"#F(" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative of \+ " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`q '`(x) = `f '`(a)+`f ''`( a)*(x-a);" "6#/-%$q~'G6#%\"xG,&-%$f~'G6#%\"aG\"\"\"*&-%%f~''G6#F,F-,&F 'F-F,!\"\"F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "and the second derivative is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`q ''`(x) = `f ''`(a);" "6#/-%%q~''G6#%\"xG-%%f~''G6#%\"aG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "This means that " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG " }{TEXT -1 25 " has the properties that " }{TEXT 261 28 "its value, f irst derivative " }{TEXT 302 3 "and" }{TEXT 261 33 " second derivative match those of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\" xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 10 ", namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(a ) = f(a);" "6#/-%\"qG6#%\"aG-%\"fG6#F'" }{TEXT -1 2 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`q '`(a) = `f '`(a);" "6#/-%$q~' G6#%\"aG-%$f~'G6#F'" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 4 "a nd " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`q ''`(a) = `f ''`(a);" "6#/-%%q~''G6#%\"aG-%%f~''G6#F'" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 303 8 "________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "This woul d seem to suggest that it would do a better job of approximating the f unction " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 6 " near \+ " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 29 " than the quadrat ic function " }{XPPEDIT 18 0 "q(x);" "6#-%\"qG6#%\"xG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 440 309 309 {PLOTDATA 2 "64-%'CURVESG6%7S7$$\"3))*********** ***H!#=$\"3qmDN$y$ed5F*7$$\"3w*****\\x[BR$F*$\"3ADw#Ha\"f$R\"F*7$$\"3' )***\\iMHPt$F*$\"3M>DEj`p&o\"F*7$$\"3y****\\-vksF*7$$\"3(****\\iq%[D6Fdq$\"372[F!=mjW(F*7$$\"3#***\\7la!4; 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" }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "f(4) = 2,`f '`(4) \+ = 1/4;" "6$/-%\"fG6#\"\"%\"\"#/-%$f~'G6#F'*&\"\"\"F.F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f ''`(4) = -1/32;" "6#/-%%f~''G6#\"\"%,$ *&\"\"\"F*\"#K!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 " The corresponding second degree approximation is: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x)=2+1/4" "6#/-%\"qG6#%\"xG,&\"\"# \"\"\"*&F*F*\"\"%!\"\"F*" }{XPPEDIT 18 0 " ``(x-4)-1/64" "6#,&-%!G6#,& %\"xG\"\"\"\"\"%!\"\"F)*&F)F)\"#kF+F+" }{XPPEDIT 18 0 "``(x-4)^2" "6#* $-%!G6#,&%\"xG\"\"\"\"\"%!\"\"\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 10 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "q(x)=3/4+3/8" "6#/-%\"qG6#%\"xG,&*&\"\"$\"\"\"\"\"%!\" \"F+*&F*F+\"\")F-F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x-1/64" "6#,&%\"x G\"\"\"*&F%F%\"#k!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2" "6#*$% \"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "plot1 := plot([sqrt(x),3/4+3*x/8-x ^2/64],x=-1..8,color=[red,blue],thickness=2):\nplot2 := plot([[[4,2]]$ 3],color=black,style=point,symbol=[circle,diamond,cross]):\nplot3 := p lot([[4,0],[4,2]],color=black,linestyle=2):\nplots[display]([plot1,plo t2,plot3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 290 290 {PLOTDATA 2 "6*- %'CURVESG6%7T7$$\"3A++++D#on)!#A$\"3Fu+)HKY\\J*!#?7$$\"3#********)zgec F-$\"3&yi%pf>PAv!#>7$$\"32+++vZ/B6F3$\"3c9H0p!Q(f5!#=7$$\"3&******zs7u B#F3$\"3af$o!e$)z&\\\"F97$$\"35+++!o!y^LF3$\"35aR&=!pyI=F97$$\"3;+++! \\_#4yF3$\"3C7@CzR]%z#F97$$\"3&*******HCnE7F9$\"3SB@\"3P)Q-NF97$$\"37+ ++0Kw\\@F9$\"3+x95)*QbOYF97$$\"3))******zR&G2$F9$\"3S*o6W\"HLVbF97$$\" 35+++!ex@)\\F9$\"3U7Va2UXeqF97$$\"3!)******fzP&)oF9$\"3M?`Jhx\"yH)F97$ $\"3Y+++![`I%))F9$\"3hr&=E5^PS*F97$$\"3%******R^Pn0\"!#<$\"3Jt\"H)*Qxz -\"Fdo7$$\"3-+++#Hb3D\"Fdo$\"3QgKb@kT=6Fdo7$$\"35+++Q,xX9Fdo$\"3FPU_?= S-7Fdo7$$\"33+++3ngL;Fdo$\"3iX-&)[i7y7Fdo7$$\"32+++3.=/=Fdo$\"3Cm&Q=z( >V8Fdo7$$\"33+++=)3q+#Fdo$\"3*>\"Hc'Q*o;9Fdo7$$\"3/+++q6$)y@Fdo$\"37S4 eI#e;Fdo7$$\"3(*******)e lX$HFdo$\"3]8M`!QdIr\"Fdo7$$\"3\")*****4`Bu7$Fdo$\"3S%\\(>)H_%o/8u>#>Fdo7$$\"3))*****ps:n'QFdo$\"3 AY)=_j'Rm>Fdo7$$\"3s*****47qK0%Fdo$\"3RZi!R[tK,#Fdo7$$\"3O+++l!**fC%Fd o$\"3![wa]@#eg?Fdo7$$\"3r*****ftYXV%Fdo$\"3eSnr]j$e5#Fdo7$$\"3R+++.j(p h%Fdo$\"3m>wJN]r[@Fdo7$$\"3n*****RBL&>[Fdo$\"3KQFRMNM&>#Fdo7$$\"3E+++X 'R:+&Fdo$\"3a?*)3EATOAFdo7$$\"3u*****zLqe>&Fdo$\"3Oqj0#=X%zAFdo7$$\"3- +++GG'>P&Fdo$\"3Rx\"fz^\\xJ#Fdo7$$\"3e*****>VyWc&Fdo$\"35>7ZsY\"*eBFdo 7$$\"3#)*****Hh?cu&Fdo$\"3%)z!Q1Tk yhV#Fdo7$$\"35+++)['3?hFdo$\"3)oyfc&3)QZ#Fdo7$$\"3-+++y+*QJ'Fdo$\"3#e$ Q\"f_XF^#Fdo7$$\"37+++qfa+lFdo$\"3inV2Goh\\DFdo7$$\"33+++z&G9p'Fdo$\"3 OW(4odzne#Fdo7$$\"31+++$eI2)oFdo$\"3H8E5sY6BEFdo7$$\"3U+++l%zY0(Fdo$\" 3UtP`dY1cEFdo7$$\"3G+++9X/asFdo$\"3gueCYLL$p#Fdo7$$\"3++++!**eBV(Fdo$ \"3tj]\\9`BEFFdo7$$\"3w*****HTzCi(Fdo$\"3!z7B,\"Q)3w#Fdo7$$\"3@+++<*[W !yFdo$\"3!)4?7zVk$z#Fdo7$$\"\")\"\"!$\"3H!>YZ7F%GGFdo-%'COLOURG6&%$RGB G$\"*++++\"!\")$F^[lF^[lFh[l-%*THICKNESSG6#\"\"#-F$6%7S7$$!\"\"F^[l$\" 3)**********\\Pf$F97$$!3k+++DhDQ!)F9$\"3ybEBj_p%Q%F97$$!3g***\\(oKNJjF 9$\"3CE)fd43J1&F97$$!3W****\\([i2K&H$*F97$$\"3Q)**\\i&zP&)oF9$\"3U5VB1Tz+5Fdo7$$\"3 F-++vM0V))F9$\"35$)4j=eRp5Fdo7$$\"3#****\\P^Pn0\"Fdo$\"3zZO?@#G)G6Fdo7 $$\"3G++]#Hb3D\"Fdo$\"3cOT*3DBY>\"Fdo7$$\"3%)****\\P,xX9Fdo$\"3AW'[^i. &f7Fdo7$$\"3E++]2ngL;Fdo$\"3=AR*G`/4K\"Fdo7$$\"3&)**\\73.=/=Fdo$\"3YV$ o,?2dP\"Fdo7$$\"3/++]<)3q+#Fdo$\"3*pq8z[*oR9Fdo7$Faq$\"39Wu%R/&)G\\\"F do7$$\"3%***\\789qyBFdo$\"3)*H=(fL.Ob\"Fdo7$$\"3u*****\\W?cb#Fdo$\"3Mc 'z:t2jg\"Fdo7$$\"3U+]7j'G(\\FFdo$\"3'**>MI]2Im\"Fdo7$$\"3X+]P*elX$HFdo $\"3Y-:/rY!fr\"Fdo7$$\"3s++DJNUFJFdo$\"3'=iY)o\"f*pFdo7$$\"3K+](os:n'QFdo$\"3C**3Ke8Sm>Fdo7$$ \"3u***\\77qK0%Fdo$\"3w$4f6>tK,#Fdo7$$\"3Y*****\\1**fC%Fdo$\"3c)zP4@a0 1#Fdo7$$\"3t***\\itYXV%Fdo$\"3Ow9y,io0@Fdo7$$\"3'***\\7.j(ph%Fdo$\"3;Z G0niH[@Fdo7$$\"3l***\\PBL&>[Fdo$\"3#eU=f,*Q%>#Fdo7$$\"3P*****\\kR:+&Fd o$\"3&=e/.w6ZB#Fdo7$$\"3e++]P.(e>&Fdo$\"3[W#[-=AmF#Fdo7$$\"3e**\\7GG'> P&Fdo$\"3p+Qv]+e8BFdo7$$\"3_++]K%yWc&Fdo$\"3)o.g6(f(GN#Fdo7$$\"3S**\\7 81iXdFdo$\"3)>kG))y#z)Q#Fdo7$$\"3n**\\i&Qm\\$fFdo$\"3'H$GQ8,CDCFdo7$$ \"31++](['3?hFdo$\"3y[7'>*3zfCFdo7$$\"3e**\\7y+*QJ'Fdo$\"3wT!ob!\\\"[ \\#Fdo7$$\"3,,++qfa+lFdo$\"3x*>ORePu_#Fdo7$$\"31++vy&G9p'Fdo$\"3AY#)Rg HnfDFdo7$$\"3_+]7$eI2)oFdo$\"3mH,'\\)o^!f#Fdo7$$\"3a*****\\YzY0(Fdo$\" 3%oU&4]>(yh#Fdo7$$\"3Q***\\P^WSD(Fdo$\"3['e9%R51[EFdo7$$\"3*3+++**eBV( Fdo$\"3)o%)3QC5Sn#Fdo7$$\"3L**\\78%zCi(Fdo$\"3ohphBIe+FFdo7$$\"3v**\\( o\"*[W!yFdo$\"3/&Qw5he\\s#Fdo7$F\\[l$\"3+++++++]FFdo-Fb[l6&Fd[lFh[lFh[ lFe[lFi[l-F$6&7#7$$\"\"%F^[l$F\\\\lF^[l-%'SYMBOLG6#%'CIRCLEG-Fb[l6&Fd[ lF^[lF^[lF^[l-%&STYLEG6#%&POINTG-F$6&Fe[m-F[\\m6#%(DIAMONDGF^\\mF`\\m- F$6&Fe[m-F[\\m6#%&CROSSGF^\\mF`\\m-F$6%7$7$Fg[mFh[lFf[mF^\\m-%*LINESTY LEGF[\\l-%+AXESLABELSG6%Q\"x6\"Q!Fh]m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fa \\lF\\[lF]^m" 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" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Comparing the values of the two functions when " }{XPPEDIT 18 0 "x=4.2" "6#/%\"xG-%&FloatG6$\"#U!\"\" " }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "q(4.2) = 2.049375;" "6#/-%\"qG6#-%&FloatG6$\"#U!\"\"-F( 6$\"(v$\\?!\"'" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 6 "while \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(4.2);" "6#-%\"f G6#-%&FloatG6$\"#U!\"\"" }{TEXT -1 1 " " }{TEXT 304 1 "~" }{TEXT -1 14 " 2.049390153. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 114 "q := x -> 3/4+3*x/8-x^2/64: 'q(x)'=q(x);\nf := x -> sqrt(x): 'f(x)'=f(x);\nxx := 4.2:\n'q'(xx)=q(xx);\n'f'(xx)=f( xx);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,(#\"\"$\"\"% \"\"\"*&#F*\"\")F,F'F,F,*&#F,\"#kF,*$)F'\"\"#F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"qG6#$\"#U!\"\"$\"++]P\\?!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#$\"#U!\"\"$\"+`,R\\?!\"*" }}}{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 35 "Linear approximation of a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Given a function " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 22 ", the equation o f the " }{TEXT 261 12 "tangent line" }{TEXT -1 14 " at the point " } {XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" }{TEXT -1 17 " t o the graph of " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = f(a)+`f '`(a)*(x-a);" "6#/%\"yG,&-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F)F* ,&%\"xGF*F)!\"\"F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "T he function " }{XPPEDIT 18 0 "g(x) = f(a)+`f '`(a)*(x-a);" "6#/-%\"gG6 #%\"xG,&-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F,F-,&F'F-F,!\"\"F-F-" }{TEXT -1 12 " provides a " }{TEXT 261 20 "linear approximation" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 6 " when " }{TEXT 277 1 "x" }{TEXT -1 13 " is close to " } {TEXT 278 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 39 "We find the ta ngent line to the graph " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6 #%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x) = sin*x;" "6#/-%\" fG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 24 ", at the point given by " } {XPPEDIT 18 0 "x = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }}{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 ) = cos*x;" "6#/-%$f~'G6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 10 ", so t hat " }{XPPEDIT 18 0 "`f '`(Pi/3) = cos(Pi/3)" "6#/-%$f~'G6#*&%#PiG\" \"\"\"\"$!\"\"-%$cosG6#*&F(F)F*F+" }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\" \"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The \+ tangent line to the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$s inG\"\"\"%\"xGF'" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(Pi/3 ,sqrt(3)/2))" "6#-%!G6$*&%#PiG\"\"\"\"\"$!\"\"*&-%%sqrtG6#F)F(\"\"#F* " }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(3)/2+1/2" "6#/%\"yG,&*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F+*&F+ F+F,F-F+" }{XPPEDIT 18 0 "``(x-Pi/3)" "6#-%!G6#,&%\"xG\"\"\"*&%#PiGF( \"\"$!\"\"F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = sqrt(3)/2+x/2 -Pi/6;" "6#/%\"yG,(*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F+*&%\"xGF+F,F-F+* &%#PiGF+\"\"'F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "A li near approximation for " }{XPPEDIT 18 0 "f(x) = sin*x;" "6#/-%\"fG6#% \"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 5 " for " }{TEXT 305 1 "x" }{TEXT -1 6 " near " }{XPPEDIT 18 0 "Pi/3" "6#*&%#PiG\"\"\"\"\"$!\"\"" } {TEXT -1 14 " is given by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = sqrt(3)/2+x/2-Pi/6; " "6#/-%\"gG6#%\"xG,(*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F.*&F'F.F/F0F.*& %#PiGF.\"\"'F0F0" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "f := x -> sin(x):\n'f(x)'=f (x);\na := Pi/3;\nf(a) + D(f)(a)*(x-a):\ng := unapply(%,x):\n'g(x)'=g( x);\nplot([f(x),g(x)],x=0..2,thickness=2,legend=[`f(x)`,`g(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aG,$*&\"\"$!\"\"%#PiG\"\"\"F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(*&\"\"#!\"\"\"\"$#\"\"\"F*F.*&F* F+F'F.F.*&\"\"'F+%#PiGF.F+" }}{PARA 13 "" 1 "" {GLPLOT2D 477 324 324 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3!)y==T, 0eVF-7$$\"3'pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLLLe0$=C\"!#=$\"3I#\\^x@ T'Q7F87$$\"3ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm;zjf)4#F8$\"3ctIO$>EK3# F87$$\"3=LL$e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p****\\i'y]!HF8$\"3)z'**Hz%) QkGF87$$\"3,LL$ezs$HLF8$\"3%f8@S`IT% F87$$\"3i******\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3()******\\Z/NaF8$\"3Uh8% *pkQr^F87$$\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF87$$\"3ELL$ez6:B'F8$\"3% o#4^'[pf$eF87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F87$$\"3-mmmm#pS1(F8$\"3w N76%>Z5\\'F87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C#oF87$$\"3slmmm(y8!zF8$ \"3NBP6&4.X5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S(F87$$\"39++](3zMu)F8$\" 3;afM1NDrwF87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF87$$\"3emm;zihl&*F8$\"3O `uvYv9s\")F87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%)F87$$\"3*F87$$\"3 z****\\_qn27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p@[7Fas$\"3uZ3fv(3U[*F87 $$\"3#)****\\2'HKH\"Fas$\"3t$**F87$$\"3Ymm\"H!o-*\\ \"Fas$\"3qbf1InDu**F87$$\"3))***\\7k.6a\"Fas$\"3_Q!)*f/#f&***F87$$\"3e mmmT9C#e\"Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i!*3`i\"Fas$\"3Iw,Vsb9&)* *F87$$\"3QLLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3GLL$3N1#4Fas$\"3jyU@b\\q4%*F87$$\"3/++v.Uac>Fas$ \"37dI([,t^E*F87$$\"\"#F)$\"35/I&F87$FZ$\"3;IZ>!*)Qz^&F87$Fin$\"3hl!Gg=I&4d F87$F^o$\"3i(Rhog;_#fF87$Fco$\"3'))Rhol)yThF87$Fho$\"3M)Rho&f\\]jF87$F ]p$\"3bl!y(z@-SlF87$Fbp$\"3WIZ>!>(QlnF87$Fgp$\"3%>t%>:4IcpF87$F\\q$\"3 -(R6J'*y$yrF87$Faq$\"37IZ>lc&\\P(F87$Ffq$\"3-)R6JYJ1f(F87$F[r$\"3K(R6c #e+'z(F87$F`r$\"3rIZpFCH5!)F87$Fer$\"3)*HZW@W22#)F87$Fjr$\"3Qk!GgQI$>% )F87$F_s$\"3mj!y(zY!)R')F87$Fes$\"3S)R61?G<$))F87$Fjs$\"3&R1G5'3,R!*F8 7$F_t$\"3Y(Rh=8aJD*F87$Fdt$\"3y'RhVa^EY*F87$Fit$\"3-(R6J1^`m*F87$F^u$ \"3#pRh$>VT!*)*F87$Fcu$\"3$HZ>lOk#45Fas7$Fhu$\"3)*Rholm&3.\"Fas7$F]v$ \"3I1y(z\\A/0\"Fas7$Fbv$\"3))Rho?J\"=2\"Fas7$Fgv$\"3AtWkH+%>4\"Fas7$F \\w$\"3$)R6\")[%yH6\"Fas7$Faw$\"3'HZ>!\\taL6Fas7$Ffw$\"3iR6J\"3\"3b6Fa s7$F[x$\"3o1G&[i?e<\"Fas7$F`x$\"3i1Gg.)Hq>\"Fas7$Fex$\"3@tWkfL1=7Fas7$ Fjx$\"3()Rh=85RP7Fas7$F_y$\"3Mt%p_oU&f7Fas7$Fdy$\"3k1G&[Sb$z7Fas7$Fiy$ \"3pR6J'*)z/I\"Fas7$F^z$\"3pR61I()p?8Fas7$Fcz$\"3nRh=GmUU8Fas-Fhz6&Fjz F(F[[lF(-F_[l6#%%g(x)G-%+AXESLABELSG6$Q\"x6\"Q!Fael-%*THICKNESSG6#Fdz- %%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "g(x)" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " Comparing the values of the two functions when " }{XPPEDIT 18 0 "x = 1 ;" "6#/%\"xG\"\"\"" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(1);" "6#-%\"gG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 15 " 0.8424266282, " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(1);" "6#-%\"fG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 306 1 "~" }{TEXT -1 15 " 0.8414709848. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xx := 1.; \nevalf(g(xx));\nf(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\" \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#GmUU)!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+[)4ZT)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 39 "We find the tangent line to the graph " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\" fG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x) = 1/(1-x);" "6# /-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }{TEXT -1 24 ", at the point \+ given by " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x) = (1-x)^(-1);" "6#/-%\"fG6#%\"xG), &\"\"\"F*F'!\"\",$F*F+" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "`f \+ '`(x) = (1-x)^(-2);" "6#/-%$f~'G6#%\"xG),&\"\"\"F*F'!\"\",$\"\"#F+" } {XPPEDIT 18 0 "``=1/(1-x)^2" "6#/%!G*&\"\"\"F&*$,&F&F&%\"xG!\"\"\"\"#F *" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '`(0) = 1;" "6#/-%$f~'G6#\" \"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The tangent line to the curve " }{XPPEDIT 18 0 "y = 1/(1-x);" "6#/%\"yG*&\"\"\"F& ,&F&F&%\"xG!\"\"F)" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(0, 1);" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1+x;" "6#/%\"yG,&\"\"\"F&%\"xGF&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "A linear approximation \+ for " }{XPPEDIT 18 0 "f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F) F)F'!\"\"F+" }{TEXT -1 5 " for " }{TEXT 310 1 "x" }{TEXT -1 21 " near \+ 0 is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g( x) = 1+x;" "6#/-%\"gG6#%\"xG,&\"\"\"F)F'F)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "f := x -> 1/(1-x):\n'f(x)'=f(x);\na := 0;\nf(a) + D(f)(a)*(x-a):\ng := \+ unapply(%,x):\n'g(x)'=g(x);\nplot([f(x),g(x)],x=-1.2..1,y=0..3.4,thick ness=2,legend=[`f(x)`,`g(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"aG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&\"\" \"F)F'F)" }}{PARA 13 "" 1 "" {GLPLOT2D 395 465 465 {PLOTDATA 2 "6'-%'C URVESG6%7ep7$$!3%**************>\"!#<$\"3IXXXXXXXX!#=7$$!3'******4EY?: \"F*$\"3W!=)*)e-uYYF-7$$!3\"*******o>K56F*$\"3/4.6L\"*f\\F-7$$!3a+++$)R a\"p*F-$\"3Qnq0[>Ky]F-7$$!33+++&H-dD*F-$\"3?o#=\"HoE$>&F-7$$!3%******4 Z8W!))F-$\"3Tr]l50!zJ&F-7$$!3G+++D**oP$)F-$\"3gwCX'e\\KX&F-7$$!3/+++nJ YsyF-$\"3Q%)=@p'*>&f&F-7$$!3'******>E?RR(F-$\"37j'[MCN\"\\dF-7$$!3S+++ 6%>C(pF-$\"3!ehBf!>\">*eF-7$$!3w*****\\G4z\\'F-$\"3)>$zJ(4u81'F-7$$!39 +++v2X@gF-$\"39oyv4?jTiF-7$$!3;+++:ZHibF-$\"3fsY$Qd(yDkF-7$$!3o*****\\ -P`9&F-$\"33C#3(*G#p-mF-7$$!3y*****>+M&\\YF-$\"3&GfXzpbh#oF-7$$!3>+++2 Q_HUF-$\"3%o#4,g?kFqF-7$$!3G+++-@&4u$F-$\"3g@['*)3;vF(F-7$$!3'******pN $[3LF-$\"3Y\"y7?@/S^(F-7$$!39+++-m*R$GF-$\"3]$[?i30=z(F-7$$!3)******R+ t@Q#F-$\"3m69Rho7w!)F-7$$!3-+++![U2\">F-$\"33Y&))\\A#y&R)F-7$$!3)***** ***3AyZ\"F-$\"3aLj3*F-7$$ !3K++++`:e_!#>$\"3iYEj0:X+&*F-7$$!3(********zPe.\"Fcs$\"3W=swz\"yu*)*F -7$$\"3?++++2QCNFcs$\"3h1'=%48`O5F*7$$\"3g++++F`N#)Fcs$\"3-MW,Bku*3\"F *7$$\"3#*******zvW%G\"F-$\"37\"zo;?ut9\"F*7$$\"3:+++?lQIF*7$$\"3`+++gel/aF-$\"3-K_n%e:h<#F* 7$$\"3)*******pzRyeF-$\"3b,_.T5CECF*7$$\"3s******HzmMjF-$\"3=Ad#Rwl#GF F*7$$\"3*)******f)p7!oF-$\"3w96'**eSi7$F*7$$\"3O+++!43SE(F-$\"39#=Wvc \")\\l$F*7$$\"3')******pk@*o(F-$\"3\"*)Q[)pk`FVF*7$$\"3W+++]Kbw\")F-$ \"3SC+Um&>T[&F*7$$\"3G+++qJ\\%R)F-$\"39esWFFcGiF*7$$\"38+++!4LCh)F-$\" 38'*=wn'fo?(F*7$$\"3\\*******y<'G()F-$\"3#p1VcMba'yF*7$$\"3P+++&\\-[%) )F-$\"3#))o`;RGll)F*7$$\"39++++s)4'*)F-$\"3Ut$)Q10_C'*F*7$$\"3_******* *=Fc[l7$$\"3%*******Ri)>_*F-$\"3#pF.z)**)>4#Fc[l7$$\"3O++++ @'=b*F-$\"3kIo4()fXJAFc[l7$$\"3m******fzt\"e*F-$\"3I')3Y+d%3R#Fc[l7$$ \"3U+++!*en'f*F-$\"3Q,(=7a&RzCFc[l7$$\"34+++?Qh6'*F-$\"3E\\')=3pvuDFc[ l7$$\"3u******\\;X=Zxn#Fc[l7$$\"3^+++!o*[T'*F-$\"3=S5(Q) *>$*y#Fc[l7$$\"3;+++5wUc'*F-$\"3\"RZydw%f5HFc[l7$$\"3#)******RbOr'*F-$ \"3[5$fy_%*G/$Fc[l7$$\"3[******pMI'o*F-$\"3wy6GnUz(=$Fc[l7$$\"3C++++9C ,(*F-$\"3yX_k!)R=ZLFc[l7$$\"3*)******H$zhr*F-$\"3+RRde:NBNFc[l7$$\"3b* *****fs6J(*F-$\"3&>/G=J$4>PFc[l7$$\"3K+++!>bgu*F-$\"3Qn9k1N'y$RFc[l7$$ \"3'*******>J*4w*F-$\"3'Qb1e(*zR=%Fc[l7$$\"3i******\\5$fx*F-$\"3i]O>u> \"HY%Fc[l7$$\"3Q+++!)*o3z*F-$\"3Q&y@4S\"p\"y%Fc[l7$$\"3/+++5p!e!)*F-$ \"3a)Hxj\"Q^\\^Fc[l7$$\"3q******R[u?)*F-$\"3ei?un*R'ybFc[l7$$\"3Y+++qF oN)*F-$\"3cT'=d0*y&3'Fc[l7$$\"37++++2i])*F-$\"3c\"\\!HhzO%p'Fc[l7$$\"3 y******H'eb')*F-$\"3!R3cOi'=QuFc[l7$$\"3a+++gl\\!))*F-$\"33YJh^*fzO)Fc [l7$$\"3?+++!\\Ma*)*F-$\"3wqN%=!GQj&*Fc[l7$$\"3&)******>CP5**F-$\"3^7% [N*zs:6!#:7$$\"3^******\\.JD**F-$\"3O)3eAft)Q8Fdel7$$\"3F+++!G[-%**F-$ \"3AHEK!*>ft;Fdel7$$\"3#*******4i=b**F-$\"3.Do4()fXJAFdel7$$\"3e****** RT7q**F-$\"3['>X1)R=ZLFdel7$$\"3M+++q?1&)**F-$\"3yS1HhzO%p'Fdel7$%*und efinedGF_gl-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FhglFggl-%'LEGENDG6#%%f(x )G-F$6%7S7$F($!3a**************>F-7$$!3GLL$3EY?:\"F*$!3!GLL$3EY?:F-7$$ !3cm;zo>K56F*$!3elm\"zo>K5\"F-7$$!3;LLeQ')Rj5F*$!3n;LLeQ')RjFcs7$$!3KL L3qU;;5F*$!33KLL3qU;;Fcs7$$!33mm\"H)Ra\"p*F-$\"3CRL$3\"F-7$$!3ZK LeC**oP$)F-$\"3anmTv+Ji;F-7$$!3'))**\\i;jC(yF-$\"38,+vLo`F@F-7$$!3]lmm h-#RR(F-$\"3^MLLQ(zgg#F-7$$!3iKL$3T>C(pF-$\"3Rnm;*e!eFIF-7$$!3k)****\\ G4z\\'F-$\"3N,++:24-NF-7$$!3.*****\\x]9-'F-$\"3)4++]A\\&yRF-7$$!31**** *\\r%HibF-$\"3%4++]G0xV%F-7$$!3)HL$eCqLX^F-$\"3/nmTvHma[F-7$$!3Ilmm,S` \\YF-$\"3qMLL)*fY]`F-7$$!3;lmm1Q_HUF-$\"3%[LLL>w/x&F-7$$!36***\\75_4u$ F-$\"3!4+]()*y/fiF-7$$!3/mmmcL[3LF-$\"3'RLLLk;:p'F-7$$!3T)**\\7g'*R$GF -$\"3e,+v)R.g;(F-7$$!3u(**\\P+t@Q#F-$\"3E-+D'*p#yh(F-7$$!3&\\mm\"zCu5> F-$\"31NL$3_d#*3)F-7$$!3Alm\"H4AyZ\"F-$\"3yML32zF*7$$\"3[/+DTi)>_*F-$\"3W+]7C')>_>F*7$$\"\"\"Fhgl$\"\" #Fhgl-Fagl6&FcglFgglFdglFggl-Fjgl6#%%g(x)G-%+AXESLABELSG6$Q\"x6\"Q\"yF \\hm-%*THICKNESSG6#Fbgm-%%VIEWG6$;$!#7FfglF_gm;Fggl$\"#MFfgl" 1 2 0 1 10 2 2 9 1 4 2 1.000000 8.000000 74.000000 0 1 "f(x)" "g(x)" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Comparing the values of the two functions when " }{XPPEDIT 18 0 "x = 1/10;" "6#/%\"xG*&\"\"\"F&\"#5! \"\"" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(1/10) = 1.1;" "6#/-%\"gG6#*&\"\"\"F(\"#5!\"\"-%&Float G6$\"#6F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(1/10) = 1/(1-1/10); " "6#/-%\"fG6#*&\"\"\"F(\"#5!\"\"*&F(F(,&F(F(*&F(F(F)F*F*F*" }{TEXT -1 1 " " }{TEXT 311 1 "~" }{TEXT -1 14 " 1.111111111. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "xx := \+ 0.1;\nevalf(g(xx));\nevalf(f(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#xxG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#6!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+66666!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 39 "We find the tangent line to the graph " }{XPPEDIT 18 0 "y = f(x); " "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x) \+ = exp(x*ln(x+1));" "6#/-%\"fG6#%\"xG-%$expG6#*&F'\"\"\"-%#lnG6#,&F'F,F ,F,F," }{XPPEDIT 18 0 "`` = (x+1)^x;" "6#/%!G),&%\"xG\"\"\"F(F(F'" } {TEXT -1 24 ", at the point given by " }{XPPEDIT 18 0 "x=1" "6#/%\"xG \"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative \+ of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = exp(x*ln(x +1));" "6#/-%$f~'G6#%\"xG-%$expG6#*&F'\"\"\"-%#lnG6#,&F'F,F,F,F," } {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(x*ln(x+1),x);" "6#-%%DiffG6$*&%\"x G\"\"\"-%#lnG6#,&F'F(F(F(F(F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(x*ln(x+1))*(ln(x+1)+x/(x+1)); " "6#/%!G*&-%$expG6#*&%\"xG\"\"\"-%#lnG6#,&F*F+F+F+F+F+,&-F-6#,&F*F+F+ F+F+*&F*F+,&F*F+F+F+!\"\"F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(x+1)^x*(ln(x+1)+x/(x+1)" "6#/%!G*&) ,&%\"xG\"\"\"F)F)F(F),&-%#lnG6#,&F(F)F)F)F)*&F(F),&F(F)F)F)!\"\"F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "`f '`(1) = 2*(ln(2)+1/2);" "6#/-%$f~'G6#\"\"\"*&\"\"#F',&-%#lnG6#F) F'*&F'F'F)!\"\"F'F'" }{XPPEDIT 18 0 "``=2*ln(2)+1" "6#/%!G,&*&\"\"#\" \"\"-%#lnG6#F'F(F(F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The tangent line to the curve " }{XPPEDIT 18 0 "y = (x+1)^x;" "6#/ %\"yG),&%\"xG\"\"\"F(F(F'" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1,2);" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 2+(2*ln(2)+1)*(x-1);" "6#/%\"y G,&\"\"#\"\"\"*&,&*&F&F'-%#lnG6#F&F'F'F'F'F',&%\"xGF'F'!\"\"F'F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=(2*ln(2)+1)*x+1-2*ln(2)" "6#/% \"yG,(*&,&*&\"\"#\"\"\"-%#lnG6#F)F*F*F*F*F*%\"xGF*F*F*F**&F)F*-F,6#F)F *!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "A linear appro ximation for " }{XPPEDIT 18 0 "f(x) = (x+1)^x;" "6#/-%\"fG6#%\"xG),&F' \"\"\"F*F*F'" }{TEXT -1 5 " for " }{TEXT 308 1 "x" }{TEXT -1 21 " near 1 is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g (x) = (2*ln(2)+1)*x+1-2*ln(2);" "6#/-%\"gG6#%\"xG,(*&,&*&\"\"#\"\"\"-% #lnG6#F,F-F-F-F-F-F'F-F-F-F-*&F,F--F/6#F,F-!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "f := x -> exp(x*ln(x+1)):\n'f(x)'=f(x);\n'D(f)(x)'=D(f)(x);\na : = 1;\nf(a) + D(f)(a)*(x-a):\ng := unapply(%,x):\n'g(x)'=g(x);\nplot([f (x),g(x)],x=-0.5..2,y=-1.1..5,thickness=2,legend=[`f(x)`,`g(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expG6#*&F'\"\"\"-%#ln G6#,&F'F,F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/--%\"DG6#%\"fG6#% \"xG*&,&-%#lnG6#,&F*\"\"\"F1F1F1*&F*F1F0!\"\"F1F1-%$expG6#*&F*F1F-F1F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&\"\"#\"\"\"*&,&*&F)F*-%#lnG6#F)F*F*F*F* F*,&F'F*F*!\"\"F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 525 414 414 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$!3++++++++]!#=$\"3:&4tBc8UT\"!#<7$$!3k LL$e9r]X%F*$\"3Y;XP+vX+8F-7$$!3]m;aj9$4)RF*$\"3SJ:&R%='RA\"F-7$$!31LL3 -=rZMF*$\"33))G+Rv\"p:\"F-7$$!3pKLe9w&4\"HF*$\"3_Fs_yML06F-7$$!3qm;/EX vwBF*$\"3#*4P5='**R5F-7$$!3E+](=n^'o 8F*$\"3+I&3/o[.-\"F-7$$!31HL3_+%GQ)!#>$\"3a;&=!4jO25F-7$$!3s(**\\PMsh4 $FT$\"3z@ugcU(4+\"F-7$$\"37ULL3_\"=M#FT$\"3<6G&RBU0+\"F-7$$\"3)GnmTg(f JrFT$\"3xlLj')[#\\+\"F-7$$\"3k++]7eP_7F*$\"3Cv#=?.()[,\"F-7$$\"3Q++]Pf !Qz\"F*$\"3pj[:/#Q+.\"F-7$$\"3@++](=ubJ#F*$\"3?E4)[g5%\\5F-7$$\"37n;zW (*Q*y#F*$\"3sV4`gs.r5F-7$$\"3#QLL3F-GN$F*$\"3T_Rl@zz,6F-7$$\"3=MLL$e'3 IQF*$\"3\"\\&=m6lBK6F-7$$\"3?+]7.7$>q]T!Q\"F-7$$\"3yL$e*[. -dpF*$\"3[@$)[nD(RW\"F-7$$\"3WmmTg-m([(F*$\"3B;bs!))p'>:F-7$$\"3&zm\"z %*f%)Q!)F*$\"36B*y\"F-7$$\"3C+++D'>Ad*F*$\"3E!4l(yvz,>F-7$$\"32 +]i:jf45F-$\"3F:^#*[45B?F-7$$\"39+DJ&>r-1\"F-$\"37Cs$y[8?:#F-7$$\"3++] P4q`;6F-$\"3w@z\")RWz4BF-7$$\"3;LL$eM%4n6F-$\"3;V@-.c/mCF-7$$\"37++v$4 v5A\"F-$\"390e'>^'f\\EF-7$$\"3cm\"zWn*)*p7F-$\"3,>Z!)o#[B$GF-7$$\"3H++ DJiYB8F-$\"3%4&eBy;*=0$F-7$$\"3CLek.Nyt8F-$\"3U*o'ef?DzKF-7$$\"3=+Dc^& zjU\"F-$\"3)HY\\\\%F-7$$\"3* pmT&Qz]O;F-$\"3%oo]Zz>n)[F-7$$\"3iLekG=4*o\"F-$\"3Y'e)p(z9nJ&F-7$$\"3F ++]i4TPKU=F-$\"3m$*\\;A*=>&oF-7$$\"3/+DJqJ8&*=F-$\"3WG9D2\">w\\(F-7$$\"3 <++]7nS?>F-$\"3K&zoW!*>9$yF-7$$\"3G+voa-oX>F-$\"3%yq%o**R[#=)F-7$$\"39 ]PMF,%G(>F-$\"3+Mr\"4t*3!e)F-7$$\"\"#\"\"!$\"3y,++++++!*F--%'COLOURG6& %$RGBG$\"#5!\"\"$Fa[lFa[lF[\\l-%'LEGENDG6#%%f(x)G-F$6%7S7$F($!3?O)z;aT %z:F-7$F/$!3S#*))[waS\\9F-7$F4$!349`Y\"zhiL\"F-7$F9$!3E'oFV))>!47F-7$F >$!3+7QBNX$43\"F-7$FC$!3(RJeW`zX`*F*7$FH$!3U#e+*R%3FN)F*7$FM$!3tK:8y$ \\*GrF*7$FR$!3s_X%e+OL'eF*7$FX$!3g/Npr9y,YF*7$Fgn$!3;Uebov6/LF*7$F\\o$ !3M.!Rj\\M6;#F*7$Fao$!3cF@F=F1W()FT7$Ffo$\"3#4B2S#Q0wTFT7$F[p$\"3C;r=: !)pi;F*7$F`p$\"3[(4jxShLz#F*7$Fep$\"3wS1kT&Hy8%F*7$Fjp$\"3)eP5_SqnF&F* 7$F_q$\"3(3^!*))QH;g'F*7$Fdq$\"3;JSq8GiG5F-7$Fcr$\"3w\"yd7agk:\"F-7$Fhr$\"3')p;x\"[bQF\"F-7$F]s$ \"3l([@Yy\"[+9F-7$Fbs$\"3%pO3c#4,K:F-7$Fgs$\"3a\"3\"\\Hu]Y;F-7$F\\t$\" 37P'=/zm,x\"F-7$Fat$\"3#\\(RL5!>z*=F-7$Fft$\"3;kM'Qj**G-#F-7$F[u$\"3Zu @^8[#Q9#F-7$F`u$\"3CQLL3;4yAF-7$Feu$\"33+rB&HO()R#F-7$Fju$\"3nw'*f\\-b FDF-7$F_v$\"3()H-xP[FWEF-7$Fdv$\"3CyXkVc)=x#F-7$Fiv$\"3#zoHqYd>*GF-7$F ^w$\"3\">U!f>rY6O%H'Q%F--Fe[l6&Fg[lF[\\lFh[lF[ \\l-F]\\l6#%%g(x)G-%+AXESLABELSG6$Q\"x6\"Q\"yF_fl-%*THICKNESSG6#F`[l-% %VIEWG6$;$!\"&Fj[lF_[l;$!#6Fj[l$\"\"&Fa[l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "g(x)" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "Comparing the values of the two functions when " } {XPPEDIT 18 0 "x = 9/10;" "6#/%\"xG*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g( 9/10);" "6#-%\"gG6#*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 312 1 "~" }{TEXT -1 14 " 1.761370564, " }}{PARA 0 "" 0 "" {TEXT -1 6 "whil e " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(9/10);" "6#-% \"fG6#*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 309 1 "~" }{TEXT -1 14 " 1.781879128. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "xx := 0.9;\nevalf(g(xx));\nevalf(f( xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+k0Ph " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Quadratic \+ approximation of a function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given a function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 15 ", the function " }{XPPEDIT 18 0 "q(x) = `f `(a)+`f '`(a)*(x-a)+`f \"`(a)/2;" "6#/-%\"qG6#%\"xG,(- %#f~G6#%\"aG\"\"\"*&-%$f~'G6#F,F-,&F'F-F,!\"\"F-F-*&-%$f~\"G6#F,F-\"\" #F3F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x-a)^2" "6#*$,&%\"xG\"\"\"%\"a G!\"\"\"\"#" }{TEXT -1 13 " provides a " }{TEXT 261 23 "quadratic app roximation" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x);" " 6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " }{TEXT 275 1 "x" }{TEXT -1 13 " \+ is close to " }{TEXT 276 1 "a" }{TEXT -1 2 ". " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 7 " is the" }{TEXT 261 27 " degree 2 Taylo r polynomial" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#% \"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "q(a) = f(a);" "6#/-%\"qG6#%\"aG-%\"fG6 #F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`q '`(a) = `f '`(a);" "6#/-%$q~' G6#%\"aG-%$f~'G6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`q \"`(a) = ` f \"`(a);" "6#/-%$q~\"G6#%\"aG-%$f~\"G6#F'" }{TEXT -1 94 ", that is, t he value, first derivative and second derivatives of the two functions match when " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 2 ".\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 43 "We find the degree 2 Taylor polynomial for " }{XPPEDIT 18 0 "f(x) = sin*x;" "6#/- %\"fG6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x \+ = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "g(x) = sin*x;" "6#/-%\"g G6#%\"xG*&%$sinG\"\"\"F'F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`f '`(x) \+ = cos*x;" "6#/-%$f~'G6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "`f ''`(x) = -sin*x;" "6#/-%%f~''G6#%\"xG,$*&%$sinG\"\" \"F'F+!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } {XPPEDIT 18 0 "g(Pi/3) = sin(Pi/3);" "6#/-%\"gG6#*&%#PiG\"\"\"\"\"$!\" \"-%$sinG6#*&F(F)F*F+" }{XPPEDIT 18 0 "``=sqrt(3)/2" "6#/%!G*&-%%sqrtG 6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`g '`(Pi/3) \+ = cos(Pi/3);" "6#/-%$g~'G6#*&%#PiG\"\"\"\"\"$!\"\"-%$cosG6#*&F(F)F*F+ " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "`f ''`(Pi/3) = -sin(Pi/3);" "6#/-%%f~''G6#*&%#P iG\"\"\"\"\"$!\"\",$-%$sinG6#*&F(F)F*F+F+" }{XPPEDIT 18 0 "``=-sqrt(3) /2" "6#/%!G,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "The degree 2 Taylor polynomial for " } {XPPEDIT 18 0 "f(x) = sin*x;" "6#/-%\"fG6#%\"xG*&%$sinG\"\"\"F'F*" } {TEXT -1 4 " at " }{XPPEDIT 18 0 "x = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\" \"$!\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "q(x)=sqrt(3)/2+1/2" "6#/- %\"qG6#%\"xG,&*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F.*&F.F.F/F0F." } {XPPEDIT 18 0 "``(x-Pi/3)-sqrt(3)/4;" "6#,&-%!G6#,&%\"xG\"\"\"*&%#PiGF )\"\"$!\"\"F-F)*&-%%sqrtG6#F,F)\"\"%F-F-" }{XPPEDIT 18 0 "``(x-Pi/3)^2 " "6#*$-%!G6#,&%\"xG\"\"\"*&%#PiGF)\"\"$!\"\"F-\"\"#" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 40 " provides a quadratic approximation fo r " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " } {TEXT 279 1 "x" }{TEXT -1 13 " is close to " }{XPPEDIT 18 0 "Pi/3" "6# *&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "f := x -> sin(x):\n' f(x)'=f(x);\na := Pi/3;\nf(a) + D(f)(a)*(x-a)+(D@@2)(f)(a)*(x-a)^2/2: \nq := unapply(%,x):\n'q(x)'=q(x);\nplot([f(x),q(x)],x=0..2,color=[red ,blue],thickness=2,legend=[`f(x)`,`q(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,$*&\"\"$!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,**&\"\"#!\"\"\"\"$#\"\"\"F*F.*&F*F+F'F.F.*&\"\"'F +%#PiGF.F+*(\"\"%F+F,F-,&F'F.*&F,F+F2F.F+F*F+" }}{PARA 13 "" 1 "" {GLPLOT2D 446 326 326 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"39 LLLL3VfV!#>$\"3!)y==T,0eVF-7$$\"3'pmm;H[D:)F-$\"3;VE!pV?N9)F-7$$\"3LLL Le0$=C\"!#=$\"3I#\\^x@T'Q7F87$$\"3ILLL3RBr;F8$\"3=:p?3^Yj;F87$$\"3Ymm; zjf)4#F8$\"3ctIO$>EK3#F87$$\"3=LL$e4;[\\#F8$\"3UFv?6l,pCF87$$\"3p**** \\i'y]!HF8$\"3)z'**Hz%)QkGF87$$\"3,LL$ezs$HLF8$\"3%f8@S`IT%F87$$\"3i******\\1!>+&F8$\"3M:FtwK#fz%F87$$\"3( )******\\Z/NaF8$\"3Uh8%*pkQr^F87$$\"3'*******\\$fC&eF8$\"3#)HQ5YS/CbF8 7$$\"3ELL$ez6:B'F8$\"3%o#4^'[pf$eF87$$\"3Smmm;=C#o'F8$\"3ux&[t$3$f>'F8 7$$\"3-mmmm#pS1(F8$\"3wN76%>Z5\\'F87$$\"3]****\\i`A3vF8$\"3\"f)3()*p.C #oF87$$\"3slmmm(y8!zF8$\"3NBP6&4.X5(F87$$\"3V++]i.tK$)F8$\"3!>,MN2j8S( F87$$\"39++](3zMu)F8$\"3;afM1NDrwF87$$\"3#pmm;H_?<*F8$\"3f%f\"=mQ0RzF8 7$$\"3emm;zihl&*F8$\"3O`uvYv9s\")F87$$\"39LLL3#G,***F8$\"3)y9#)p*>P4%) F87$$\"3*F87$$\"3z****\\_qn27Fas$\"3#yNZUKMzM*F87$$\"3%)***\\i&p @[7Fas$\"3uZ3fv(3U[*F87$$\"3#)****\\2'HKH\"Fas$\"3t $**F87$$\"3Ymm\"H!o-*\\\"Fas$\"3qbf1InDu**F87$$\"3))***\\7k.6a\"Fas$\" 3_Q!)*f/#f&***F87$$\"3emmmT9C#e\"Fas$\"3]Ds(Q0X$****F87$$\"3\"****\\i! *3`i\"Fas$\"3Iw,Vsb9&)**F87$$\"3QLLL$*zym;Fas$\"3q\"4lgOjR&**F87$$\"3G LL$3N1#4Fas$\"3jyU@b\\q4% *F87$$\"3/++v.Uac>Fas$\"37dI([,t^E*F87$$\"\"#F)$\"35(F-7$F1$!3Y1JGZOZg?F-7$F6$\"3qtIo:\"\\5c$F-7$F <$\"3KI!\\)3nng!*F-7$FA$\"3)4IhA\"=cP9F87$FF$\"3\"=)HUo[>;>F87$FK$\"37 M'>`gkuR#F87$FP$\"33lAl$eh)zGF87$FU$\"3Gmb9hd>XLF87$FZ$\"36,I*o*oo2QF8 7$Fin$\"3#G)efoJZ,UF87$F^o$\"3n`26o\"o&HYF87$Fco$\"3-q:Kbj?V]F87$Fho$ \"3)>2.2!*\\kU&F87$F]p$\"31$*QP\")*)RhdF87$Fbp$\"33.R\\>3\\VhF87$Fgp$ \"3_Wh%>a2MX'F87$F\\q$\"3q@gt='G!)z'F87$Faq$\"3)>#)yNFA))3(F87$Ffq$\"3 ?ZQ]P)oCR(F87$F[r$\"3'4acaiMmm(F87$F`r$\"3!p%>$\\#>7PzF87$Fer$\"3g$>^$ ))H]r\")F87$Fjr$\"3)HeRY$oF4%)F87$F_s$\"3z@x_\\AtR')F87$Fes$\"3(3JKVQN m#))F87$Fjs$\"3\"3NJjUjT,*F87$F_t$\"3o5!*\\fsE#>*F87$Fdt$\"3Ef=f!)[8^$ *F87$Fit$\"3#pF)>@eP!\\*F87$F^u$\"3b8[waSIG'*F87$Fcu$\"3y*)y3l=FP(*F87 $Fhu$\"3')>z5*zzz$)*F87$F]v$\"32$os)p!)G:**F87$Fbv$\"3_5*)G3Aj%)**F87$ Fgv$\"3kRRFG1a.5Fas7$F\\w$\"3z6F]LzM25Fas7$Faw$\"3DCg4_ze45Fas7$Ffw$\" 3g/&e-si.,\"Fas7$F[x$\"3WH:(eP!f45Fas7$F`x$\"3c$*=8S&fs+\"Fas7$Fex$\"3 Gbw)=04M+\"Fas7$Fjx$\"3UlyX>q>&)**F87$F_y$\"31#ery$[C8**F87$Fdy$\"33MU (4=*[M)*F87$Fiy$\"3l4\">U$3aN(*F87$F^z$\"3&*)H'z)=cji*F87$Fcz$\"3%[vWq @OK\\*F8-Fhz6&FjzF(F(F[[l-F_[l6#%%q(x)G-%*THICKNESSG6#Fdz-%+AXESLABELS G6$Q\"x6\"Q!Fdel-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "q(x)" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "Comparing the values of the two functions when " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 11 ", we have: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(1);" "6#-%\"qG6#\" \"\"" }{TEXT -1 1 " " }{TEXT 319 1 "~" }{TEXT -1 15 " 0.8414620453, " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(1);" "6#-%\"fG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 318 1 "~" }{TEXT -1 15 " 0.8414709848. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xx := 1.;\nevalf(q(xx));\nf(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+`/i9%)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[)4ZT)!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 43 "We find the degree 2 Taylor polynomial for " } {XPPEDIT 18 0 "f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\" \"F+" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 " f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" } {XPPEDIT 18 0 "``=(1-x)^(-1)" "6#/%!G),&\"\"\"F'%\"xG!\"\",$F'F)" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "`f '`(x) = (1-x)^(-2);" "6#/-%$f~'G6#% \"xG),&\"\"\"F*F'!\"\",$\"\"#F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "` f \"`(x) = 2*(1-x)^(-3);" "6#/-%$f~\"G6#%\"xG*&\"\"#\"\"\"),&F*F*F'!\" \",$\"\"$F-F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "f(0) = 1;" "6#/-%\"fG6#\"\"!\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "`f '`(0) = 1;" "6#/-%$f~'G6#\"\"!\"\"\"" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "`f \"`(0) = 2;" "6#/-%$f~\"G6#\"\"!\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "The degree 2 Taylor pol ynomial for " }{XPPEDIT 18 0 "f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\" \"F),&F)F)F'!\"\"F+" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 0;" "6#/% \"xG\"\"!" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "q(x) = 1+x+x^2;" "6#/-% \"qG6#%\"xG,(\"\"\"F)F'F)*$F'\"\"#F)" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"x G" }{TEXT -1 40 " provides a quadratic approximation for " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " }{TEXT 280 1 "x" } {TEXT -1 13 " is close to " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f := x -> 1/(1-x):\n'f(x)'=f(x);\na := 0;\nf(a) + D( f)(a)*(x-a)+(D@@2)(f)(a)*(x-a)^2/2:\nq := unapply(%,x):\n'q(x)'=q(x); \nplot([f(x),q(x)],x=-1.2..0.9,y=0..4,color=[red,blue],thickness=2,leg end=[`f(x)`,`q(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"x G*&\"\"\"F),&F)F)F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,(\"\"\"F)F'F)*$) F'\"\"#F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 441 493 493 {PLOTDATA 2 "6'- %'CURVESG6%7en7$$!3%**************>\"!#<$\"3IXXXXXXXX!#=7$$!3-++DwfAa6 F*$\"3%*>/RP&Q?k%F-7$$!3*)*\\PHC)R96F*$\"3]L&pE,y%HZF-7$$!3%***\\P\"z2 'p5F*$\"3DW\"H_PL=$[F-7$$!3%)**\\iR/_C5F*$\"3c')p8j8WR\\F-7$$!3s**\\(= !QZ'z*F-$\"3?aYFfYS^]F-7$$!3')**\\P*4V/Q*F-$\"3C\")z/Z2%)f^F-7$$!33** \\P/um\\*)F-$\"3)Ht$3vu8x_F-7$$!3^**\\Pk&eT])F-$\"38(='=71>/aF-7$$!3!* )*\\(oZy+1)F-$\"3qfC()*\\uq`&F-7$$!3i****\\AvG.wF-$\"3EVVBnqv!o&F-7$$! 3G)**\\7!e%4?(F-$\"37r1p0Qj8eF-7$$!3=)***\\2jF -7$$!33**\\P9E\"pX&F-$\"3;*3wCH(fpkF-7$$!3C****\\#4YO)\\F-$\"3o+A5.I%R n'F-7$$!3!))*****pss#e%F-$\"3U<&RC%yUdoF-7$$!3.**\\PpLO;TF-$\"3qUZDlu( R3(F-7$$!3$))****\\H_Nq$F-$\"3mIB:CzP(H(F-7$$!3A)*\\P>Jj]KF-$\"33Wz>0# 4oa(F-7$$!3e)*\\7epM>GF-$\"3w\"R&=v$42!yF-7$$!3C)**\\P4X$pBF-$\"35`,<# R-X3)F-7$$!3)o*\\(o!H5c>F-$\"3%*3Hz[$HRO)F-7$$!3U)**\\7Ql.^\"F-$\"3[R' )e9:#yo)F-7$$!3G**\\Pk$pt/\"F-$\"3%*G%*H<(G>0*F-7$$!3;%**\\iq*HVk!#>$ \"3!RXB%eIn%R*F-7$$!3m')**\\PQO!4#Fhs$\"3q0*4lwV_z*F-7$$\"30<++][k1CFh s$\"3/&zA`#*fY-\"F*7$$\"322+]7041oFhs$\"3=q5w%\\JI2\"F*7$$\"3o,]iS!yi5 \"F-$\"36g<#pk)QC6F*7$$\"3q,+vy3\"*y:F-$\"3IF&3W&[\\(=\"F*7$$\"3I+++0D f.?F-$\"3)*4;4\"eh0D\"F*7$$\"3/.+](yIqX#F-$\"3P*H=$yytD8F*7$$\"3q,]ilK \"z'GF-$\"3^0g&z=9@S\"F*7$$\"3(Q++DM;rJ$F-$\"3ggHYJ+O'\\\"F*7$$\"3!*** \\iI9yRPF-$\"3;DuE;$)Q(f\"F*7$$\"31-]7L#)e\"=%F-$\"3gQ_]t?o=0l&=F*7$$\"3!3+Dc^Vd1&F-$\"33v88!zZm-#F*7$$\"3`.++ IRF,bF-$\"3%z)Hg,:&GA#F*7$$\"3_,+v$omm%fF-$\"3'=&ebt_5nCF*7$$\"3c-]ig8 P)Q'F-$\"3oD_N_R$)oFF*7$$\"3?.++&3_Uz'F-$\"3G-!Qi'oR>JF*7$$\"3#G+]()>P %fsF-$\"35KKT[b))[OF*7$$\"3'H+vVvquY(F-$\"37jIgD\" )F-$\"3)3t\")eTcmJ&F*7$$\"3P.++&Q;9L)F-$\"3?)f8U\"p5$*fF*7$$\"3!\\+v$R TrV&)F-$\"3[<3%Hp$ymoF*7$$\"3-a7`/cyd')F-$\"3gf7QqXP]uF*7$$\"37.vopq&= x)F-$\"3\"4>x?wuB9)F*7$$\"3nFcE-G*)G))F-$\"3^*HiV'z#*Q&)F*7$$\"3C_P%[` Gf)))F-$\"3#\\#[\")\\[3w*)F*7$$\"3mv=UnU'H%*)F-$\"3Qz3#>7=/Y*F*7$$\"3A +++++++!*F-$\"3=+++++++5!#;-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj]lF i]l-%'LEGENDG6#%%f(x)G-F$6%7S7$F($\"3**************R7F*7$F/$\"3[l-+G;, y6F*7$F4$\"3=(oC4?&[F6F*7$F9$\"3W0^*f.`W2\"F*7$F>$\"3..We\"p@^-\"F*7$F C$\"3j@t?$4;1!)*F-7$FH$\"3q0iou\"G)=%*F-7$FM$\"37M#=0E()*f!*F-7$FR$\"3 ;m#GWs7zs)F-7$FW$\"3NhLYG!3kV)F-7$Ffn$\"3K=/[#f5x<)F-7$F[o$\"3!***R+Ui T%)zF-7$F`o$\"3wj)*R4>b0yF-7$Feo$\"373R:.uBnwF-7$Fjo$\"33<^QC%)3tvF-7$ F_p$\"3A+vr8p(3_(F-7$Fdp$\"3U^!H]uE+](F-7$Fip$\"38l,7`;T)F-7$Fgr$\"3t%[`8&4`E%)F-7$F\\s$\"34ZyBx\\v<() F-7$Fas$\"39I\"oT*)GB1*F-7$Ffs$\"3-i&z/9'=(R*F-7$F\\t$\"3UHJAPKL&z*F-7 $Fat$\"3R.MWUckC5F*7$Fft$\"3UM9$>>$ps5F*7$F[u$\"36\">%4:j'G7\"F*7$F`u$ \"3=^`]$o?G=\"F*7$Feu$\"3y@4wzI]S7F*7$Fju$\"3CHsl\"3tgI\"F*7$F_v$\"3sy Rb\"fS!p8F*7$Fdv$\"3L1oaUUuT9F*7$Fiv$\"3yv4b%zPQ^\"F*7$F^w$\"3)\\fE[i: If\"F*7$Fcw$\"3yr%*Ry0?u;F*7$Fhw$\"3enMAD>>jF*7$Fgx$\"3mP]ZA+&p/#F*7$F\\y$\"3.p_WAQ/T@F*7$Fay$\"3 E`4I/!QHD#F*7$F[z$\"3e'GG^4%ocBF*7$Fez$\"37?TWOF6rCF*7$F_[l$\"3Wk5*o#> K%e#F*7$F]]l$\"3'*************4FF*-Fc]l6&Fe]lFi]lFi]lFf]l-F\\^l6#%%q(x )G-%+AXESLABELSG6$Q\"x6\"Q\"yF^hl-%*THICKNESSG6#\"\"#-%%VIEWG6$;$!#7! \"\"$\"\"*Fjhl;Fi]l$\"\"%Fj]l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 29.000000 65.000000 0 1 "f(x)" "q(x)" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Comparing the values of the two functions when " } {XPPEDIT 18 0 "x = 1/10;" "6#/%\"xG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 11 " , we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(1/10 ) = 1.11;" "6#/-%\"qG6#*&\"\"\"F(\"#5!\"\"-%&FloatG6$\"$6\"!\"#" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "while " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(1/10) = 1/(1-1/10);" "6#/-%\"fG6 #*&\"\"\"F(\"#5!\"\"*&F(F(,&F(F(*&F(F(F)F*F*F*" }{TEXT -1 1 " " } {TEXT 317 1 "~" }{TEXT -1 14 " 1.111111111. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "xx := 0.1;\nevalf (q(xx));\nevalf(f(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\" \"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$6\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+66666!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 43 "We find the de gree 2 Taylor polynomial for " }{XPPEDIT 18 0 "f(x) = exp(x*ln(x+1)); " "6#/-%\"fG6#%\"xG-%$expG6#*&F'\"\"\"-%#lnG6#,&F'F,F,F,F," }{XPPEDIT 18 0 "`` = (x+1)^x;" "6#/%!G),&%\"xG\"\"\"F(F(F'" }{TEXT -1 5 " for " }{TEXT 313 1 "x" }{TEXT -1 6 " near " }{XPPEDIT 18 0 "1;" "6#\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = exp(x*ln(x+1));" "6#/-%$f~'G 6#%\"xG-%$expG6#*&F'\"\"\"-%#lnG6#,&F'F,F,F,F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(x*ln(x+1),x);" "6#-%%DiffG6$*&%\"xG\"\"\"-%#lnG6#, &F'F(F(F(F(F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(x*ln(x+1))*(ln(x+1)+x/(x+1));" "6#/%!G*&-%$exp G6#*&%\"xG\"\"\"-%#lnG6#,&F*F+F+F+F+F+,&-F-6#,&F*F+F+F+F+*&F*F+,&F*F+F +F+!\"\"F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(x+1)^x*(ln(x+1)+x/(x+1)" "6#/%!G*&),&%\"xG\"\"\"F)F )F(F),&-%#lnG6#,&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 8 "so tha t " }{XPPEDIT 18 0 "`f '`(1) = 2*(ln(2)+1/2);" "6#/-%$f~'G6#\"\"\"*&\" \"#F',&-%#lnG6#F)F'*&F'F'F)!\"\"F'F'" }{XPPEDIT 18 0 "``=2*ln(2)+1" "6 #/%!G,&*&\"\"#\"\"\"-%#lnG6#F'F(F(F(F(" }{TEXT -1 2 ". " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(x) = exp(x*ln(x+1))*(ln(x +1)+x/(x+1))^2+exp(x*ln(x+1))*(1/(x+1)+1/((x+1)^2));" "6#/-%%f~''G6#% \"xG,&*&-%$expG6#*&F'\"\"\"-%#lnG6#,&F'F.F.F.F.F.*$,&-F06#,&F'F.F.F.F. *&F'F.,&F'F.F.F.!\"\"F.\"\"#F.F.*&-F+6#*&F'F.-F06#,&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 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(x+1)^x*((ln(x+1)+x/(x+1 ))^2+1/(x+1)+1/((x+1)^2))" "6#/%!G*&),&%\"xG\"\"\"F)F)F(F),(*$,&-%#lnG 6#,&F(F)F)F)F)*&F(F),&F(F)F)F)!\"\"F)\"\"#F)*&F)F),&F(F)F)F)F3F)*&F)F) *$,&F(F)F)F)F4F3F)F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "`f ''`(1) \+ = 2*((ln(2)+1/2)^2+1/2+1/4);" "6#/-%%f~''G6#\"\"\"*&\"\"#F',(*$,&-%#ln G6#F)F'*&F'F'F)!\"\"F'F)F'*&F'F'F)F1F'*&F'F'\"\"%F1F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*(ln(2)^2+ln(2)+1)" "6#/%!G*&\"\"#\"\"\",(*$-%# lnG6#F&F&F'-F+6#F&F'F'F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The tangent line to the curve " }{XPPEDIT 18 0 "y = (x+1)^x;" " 6#/%\"yG),&%\"xG\"\"\"F(F(F'" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1,2);" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 2+(2*ln(2)+1)*(x-1);" "6#/% \"yG,&\"\"#\"\"\"*&,&*&F&F'-%#lnG6#F&F'F'F'F'F',&%\"xGF'F'!\"\"F'F'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=(2*ln(2)+1)*x+1-2*ln(2)" "6#/% \"yG,(*&,&*&\"\"#\"\"\"-%#lnG6#F)F*F*F*F*F*%\"xGF*F*F*F**&F)F*-F,6#F)F *!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "A linear appro ximation for " }{XPPEDIT 18 0 "f(x) = (x+1)^x;" "6#/-%\"fG6#%\"xG),&F' \"\"\"F*F*F'" }{TEXT -1 5 " for " }{TEXT 314 1 "x" }{TEXT -1 21 " near 1 is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g (x) = (2*ln(2)+1)*x+1-2*ln(2);" "6#/-%\"gG6#%\"xG,(*&,&*&\"\"#\"\"\"-% #lnG6#F,F-F-F-F-F-F'F-F-F-F-*&F,F--F/6#F,F-!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The 2nd d egree approximation for " }{XPPEDIT 18 0 "f(x) = (x+1)^x;" "6#/-%\"fG6 #%\"xG),&F'\"\"\"F*F*F'" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 1;" "6 #/%\"xG\"\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x)=2+(2*ln(2)+1)*(x-1)+(ln(2)^2+ln(2)+1)*(x-1)^2" "6 #/-%\"qG6#%\"xG,(\"\"#\"\"\"*&,&*&F)F*-%#lnG6#F)F*F*F*F*F*,&F'F*F*!\" \"F*F**&,(*$-F/6#F)F)F*-F/6#F)F*F*F*F**$,&F'F*F*F2F)F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "f := x -> exp(x*ln(x+1)):\n'f(x)'=f(x);\na := 1;\nf( a) + D(f)(a)*(x-a)+(D@@2)(f)(a)*(x-a)^2/2:\nmap(simplify,%):\nq := una pply(%,x):\n'q(x)'=q(x);\nplot([f(x),q(x)],x=-.5..2,y=-1..5,color=[red ,blue],thickness=2,legend=[`f(x)`,`q(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expG6#*&F'\"\"\"-%#lnG6#,&F'F,F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,(\"\"#\"\"\"*&,&*&F)F*-%#lnG6#F)F*F*F*F* F*,&F'F*F*!\"\"F*F**&,(F*F**$)F.F)F*F*F.F*F*)F1F)F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 478 398 398 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$!3++++++++] !#=$\"3:&4tBc8UT\"!#<7$$!3kLL$e9r]X%F*$\"3Y;XP+vX+8F-7$$!3]m;aj9$4)RF* $\"3SJ:&R%='RA\"F-7$$!31LL3-=rZMF*$\"33))G+Rv\"p:\"F-7$$!3pKLe9w&4\"HF *$\"3_Fs_yML06F-7$$!3qm;/EXvwBF*$\"3#*4P5='**R5F-7$$!3E+](=n^'o8F*$\"3+I&3/o[.-\"F-7$$!31HL3_+%GQ)!#>$\"3a ;&=!4jO25F-7$$!3s(**\\PMsh4$FT$\"3z@ugcU(4+\"F-7$$\"37ULL3_\"=M#FT$\"3 <6G&RBU0+\"F-7$$\"3)GnmTg(fJrFT$\"3xlLj')[#\\+\"F-7$$\"3k++]7eP_7F*$\" 3Cv#=?.()[,\"F-7$$\"3Q++]Pf!Qz\"F*$\"3pj[:/#Q+.\"F-7$$\"3@++](=ubJ#F*$ \"3?E4)[g5%\\5F-7$$\"37n;zW(*Q*y#F*$\"3sV4`gs.r5F-7$$\"3#QLL3F-GN$F*$ \"3T_Rl@zz,6F-7$$\"3=MLL$e'3IQF*$\"3\"\\&=m6lBK6F-7$$\"3?+]7.7$>q]T!Q\"F-7$$\"3yL$e*[.-dpF*$\"3[@$)[nD(RW\"F-7$$\"3WmmTg-m([(F *$\"3B;bs!))p'>:F-7$$\"3&zm\"z%*f%)Q!)F*$\"36B*y\"F-7$$\"3C+++D '>Ad*F*$\"3E!4l(yvz,>F-7$$\"32+]i:jf45F-$\"3F:^#*[45B?F-7$$\"39+DJ&>r- 1\"F-$\"37Cs$y[8?:#F-7$$\"3++]P4q`;6F-$\"3w@z\")RWz4BF-7$$\"3;LL$eM%4n 6F-$\"3;V@-.c/mCF-7$$\"37++v$4v5A\"F-$\"390e'>^'f\\EF-7$$\"3cm\"zWn*)* p7F-$\"3,>Z!)o#[B$GF-7$$\"3H++DJiYB8F-$\"3%4&eBy;*=0$F-7$$\"3CLek.Nyt8 F-$\"3U*o'ef?DzKF-7$$\"3=+Dc^&zjU\"F-$\"3)HY\\\\%F-7$$\"3*pmT&Qz]O;F-$\"3%oo]Zz>n)[F-7$$\"3iLekG=4 *o\"F-$\"3Y'e)p(z9nJ&F-7$$\"3F++]i4TPKU=F-$\"3m$*\\;A*=>&oF-7$$\"3/+DJqJ 8&*=F-$\"3WG9D2\">w\\(F-7$$\"3<++]7nS?>F-$\"3K&zoW!*>9$yF-7$$\"3G+voa- oX>F-$\"3%yq%o**R[#=)F-7$$\"39]PMF,%G(>F-$\"3+Mr\"4t*3!e)F-7$$\"\"#\" \"!$\"3y,++++++!*F--%'COLOURG6&%$RGBG$\"*++++\"!\")$Fa[lFa[lF[\\l-%'LE GENDG6#%%f(x)G-F$6%7S7$F($\"3b%*f*e*)e6J$F-7$F/$\"3_tAbs@J#4$F-7$F4$\" 3t\"RW#GsR7HF-7$F9$\"38=T^:%R$\"36RqIm5IUDF-7$FC$\"3uu9t64:wB F-7$FH$\"3wJMyw5>LAF-7$FM$\"3_,JO/;S'4#F-7$FR$\"3gV_7u(fp'>F-7$FX$\"3+ pz5uI9)\\0:F-7$F[p$\"3/!*GB7%*y\\9F-7$F`p$\"3V<)z<#RX49F- 7$Fep$\"3at*ocO$>u8F-7$Fjp$\"3-&)>a,&>^N\"F-7$F_q$\"3-H,7Qt^*=ww`\"F-7$Fbs$\"3^Opy1 .h:;F-7$Fgs$\"3/$\\Bfb/Up\"F-7$F\\t$\"3otS]&))H.z\"F-7$Fat$\"3GPg\"Gh' *=!>F-7$Fft$\"3Z:k7*z*4B?F-7$F[u$\"3_Df\\q1s^@F-7$F`u$\"3lX_!***4h2BF- 7$Feu$\"3%R0HQMC%fCF-7$Fju$\"3[v-(Q@$yLEF-7$F_v$\"3h=-ms\"=F!GF-7$Fdv$ \"3CQ+V2.J**HF-7$Fiv$\"3!z]v%y+k&>$F-7$F^w$\"3f3`%ffEET$F-7$Fcw$\"3K0p c]oROOF-7$Fhw$\"35.#>c4!)H)QF-7$F]x$\"3Z.mqC%zB8%F-7$Fbx$\"3Uwf'z!G^*R %F-7$Fgx$\"3qOxLoY]wYF-7$F\\y$\"33I%e+\"*G;%\\F-7$Fay$\"3)[=,UgtzD&F-7 $Ffy$\"3#*3*[+.6Ab&F-7$F[z$\"3%>*zt-vnxeF-7$Fez$\"3-zb$fLY0?'F-7$F_[l$ \"3qP!)fbX*)flF--Fe[l6&Fg[lF[\\lF[\\lFh[l-F]\\l6#%%q(x)G-%+AXESLABELSG 6$Q\"x6\"Q\"yF_fl-%*THICKNESSG6#F`[l-%%VIEWG6$;$!\"&!\"\"F_[l;$FjflFa[ l$\"\"&Fa[l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 " f(x)" "q(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Comparing the values of the two functions when " } {XPPEDIT 18 0 "x = 9/10;" "6#/%\"xG*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 11 ", we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q( 9/10);" "6#-%\"qG6#*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 316 1 "~" }{TEXT -1 14 " 1.783106566, " }}{PARA 0 "" 0 "" {TEXT -1 6 "whil e " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(9/10);" "6#-% \"fG6#*&\"\"*\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 315 1 "~" }{TEXT -1 14 " 1.781879128. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "xx := 0.9;\nevalf(q(xx));\nevalf(f( xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ml5$y\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G\"z=y\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Taylor \+ polynomials, Taylor series and Maclaurin series .. " }{TEXT 0 26 "tayl or,convert(..,polynom)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 12 "The general " } {TEXT 261 17 "Taylor polynomial" }{TEXT -1 10 " of order " }{TEXT 265 1 "n" }{TEXT -1 16 " for a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" } {TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " p[n](x) = f(a)+`f '`(a)*(x-a)+1/2!" "6#/-&%\"pG6#%\"nG6#%\"xG,(-%\"fG6 #%\"aG\"\"\"*&-%$f~'G6#F/F0,&F*F0F/!\"\"F0F0*&F0F0-%*factorialG6#\"\"# F6F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(a)*(x-a)^2+1/3!;" "6#,&*&- %$f~'G6#%\"aG\"\"\"*$,&%\"xGF)F(!\"\"\"\"#F)F)*&F)F)-%*factorialG6#\" \"$F-F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '''`(a)*(x-a)^3+` . . . `+ 1/n!;" "6#,(*&-%&f~'''G6#%\"aG\"\"\"*$,&%\"xGF)F(!\"\"\"\"$F)F)%(~.~.~ .~GF)*&F)F)-%*factorialG6#%\"nGF-F)" }{TEXT -1 2 " f" }{XPPEDIT 18 0 " `@@`(``,n)" "6#-%#@@G6$%!G%\"nG" }{XPPEDIT 18 0 "``(a)*(x-a)^n;" "6#*& -%!G6#%\"aG\"\"\"),&%\"xGF(F'!\"\"%\"nGF(" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 52 "____________________________ ________________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "where f" }{XPPEDIT 18 0 "`@@`(``,n)*``(a);" "6#*&-%#@@G6$%!G%\"nG\" \"\"-F'6#%\"aGF)" }{TEXT -1 13 " denotes the " }{TEXT 266 1 "n" } {TEXT -1 18 " th derivative of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "\nThe value of the Taylor poly nomial " }{XPPEDIT 18 0 "p[n](x);" "6#-&%\"pG6#%\"nG6#%\"xG" }{TEXT -1 4 " at " }{TEXT 259 1 "a" }{TEXT -1 37 " and the value of its deriv atives at " }{TEXT 260 1 "a" }{TEXT -1 11 " up to the " }{TEXT 267 1 " n" }{TEXT -1 42 " th derivative coincide with the value of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 31 " and its derivati ves up to the " }{TEXT 268 1 "n" }{TEXT -1 152 " th derivative. Becaus e of this, we would expect the Taylor polynomials of successively high er order to provide progressively better approximations for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 26 " in the neighbour hood of " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The infin ite series:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a)+` f '`(a)*(x-a)+1/2!;" "6#,(-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F'F(,&%\"xGF(F '!\"\"F(F(*&F(F(-%*factorialG6#\"\"#F/F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x-a)^2+1/3!;" "6#,&*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#F'*&F'F'-%*fac torialG6#\"\"$F)F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '''`(a)*(x-a)^3 +` . . . `+1/n!;" "6#,(*&-%&f~'''G6#%\"aG\"\"\"*$,&%\"xGF)F(!\"\"\"\"$ F)F)%(~.~.~.~GF)*&F)F)-%*factorialG6#%\"nGF-F)" }{TEXT -1 2 " f" } {XPPEDIT 18 0 "`@@`(``,n)" "6#-%#@@G6$%!G%\"nG" }{XPPEDIT 18 0 "``(a)* (x-a)^n+` . . . `;" "6#,&*&-%!G6#%\"aG\"\"\"),&%\"xGF)F(!\"\"%\"nGF)F) %(~.~.~.~GF)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " } {XPPEDIT 18 0 "Sum(1/n!,n = 0 .. infinity);" "6#-%$SumG6$*&\"\"\"F'-%* factorialG6#%\"nG!\"\"/F+;\"\"!%)infinityG" }{TEXT -1 2 " f" } {XPPEDIT 18 0 "`@@`(``,n)*``(a)*(x-a)^n" "6#*(-%#@@G6$%!G%\"nG\"\"\"-F '6#%\"aGF)),&%\"xGF)F,!\"\"F(F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 281 20 "____________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 14 "is called the " }{TEXT 261 13 "Taylor ser ies" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 4 " at " }{TEXT 270 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 4 ": f" }{XPPEDIT 18 0 "`@@`(``,n)*``(a) ;" "6#*&-%#@@G6$%!G%\"nG\"\"\"-F'6#%\"aGF)" }{TEXT -1 21 " is understo od to be " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 78 "It turns out that, under suitable conditi ons, the Taylor series of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 43 " converges to the value of the function at " } {TEXT 283 1 "x" }{TEXT -1 16 ", at least when " }{TEXT 269 1 "x" } {TEXT -1 26 " is sufficiently close to " }{TEXT 271 1 "a" }{TEXT -1 56 ". The convergence usually becomes more rapid as we take " }{TEXT 272 1 "x" }{TEXT -1 25 " progressively closer to " }{TEXT 273 1 "a" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The Taylor series of a function " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 20 " can be found using " }{TEXT 0 6 "serie s" }{TEXT -1 4 " or " }{TEXT 0 6 "taylor" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 83 "Only a finite number of terms are constructed as d etermined by the global variable " }{TEXT 0 5 "Order" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "The default value Order is 6 which mean s that terms as far as " }{XPPEDIT 18 0 "(x-a)^5;" "6#*$,&%\"xG\"\"\"% \"aG!\"\"\"\"&" }{TEXT -1 11 " are given." }}{PARA 0 "" 0 "" {TEXT -1 39 "The order term function O, denotes the " }{TEXT 261 18 "rest of th e series" }{TEXT -1 45 ", beyond the order specified by the variable \+ " }{TEXT 0 5 "Order" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Order := 6;\nx := 'x':\ntaylor(sin(x),x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1,&% \"xG\"\"\"!\"#F&-%$sinG6#\"\"#\"\"!-%$cosGF*F&,$F(#!\"\"F+F+,$F-#F1\" \"'\"\"$,$F(#F&\"#C\"\"%,$F-#F&\"$?\"\"\"&-%\"OG6#F&F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "More formally, the t erm " }{XPPEDIT 18 0 "O((x-a)^6);" "6#-%\"OG6#*$,&%\"xG\"\"\"%\"aG!\" \"\"\"'" }{TEXT -1 24 ", has the property that " }{XPPEDIT 18 0 "Limit (O((x-a)^6)/((x-a)^5),x = a) = 0;" "6#/-%&LimitG6$*&-%\"OG6#*$,&%\"xG \"\"\"%\"aG!\"\"\"\"'F.*$,&F-F.F/F0\"\"&F0/F-F/\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Limit(O((x-a)^6)/((x-a)^5),x=a);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%&LimitG6$*&-%\"OG6#*$),&%\"xG\"\"\"%\"aG!\"\"\"\"'F .F.F,!\"&/F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "See the help page: " }{HYPERLNK 17 "Order" 2 "Order" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Maple handles the " } {TEXT 274 1 "n" }{TEXT -1 17 " th derivative f " }{XPPEDIT 18 0 "`@@`( ``,n)*``(a);" "6#*&-%#@@G6$%!G%\"nG\"\"\"-F'6#%\"aGF)" }{TEXT -1 22 " \+ using the expression " }{XPPEDIT 18 0 "`@@`(D,n)(f)(a);" "6#---%#@@G6$ %\"DG%\"nG6#%\"fG6#%\"aG" }{TEXT -1 39 ", where D is the differential \+ operator." }}{PARA 0 "" 0 "" {TEXT -1 84 "The order of the Taylor poly nomial can be specified by means of a 3rd parameter for " }{TEXT 0 6 " taylor" }{TEXT -1 4 " or " }{TEXT 0 6 "series" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f := 'f': x := 'x': a := 'a':\ntaylor(f(x),x=a,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/,&%\"xG\"\"\"%\"aG!\"\"-%\"fG6#F'\"\"!--%\"DG6#F *F+F&,$*&#F&\"\"#F&---%#@@G6$F/F4F0F+F&F&F4,$*&#F&\"\"'F&---F86$F/\"\" $F0F+F&F&FB,$*&#F&\"#CF&---F86$F/\"\"%F0F+F&F&FK-%\"OG6#F&\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The value of this polynomial and its derivatives up to order 4 match the value \+ and derivatives of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "taylor(f(x),x=a,5):\nconvert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%\"fG6#%\"aG\"\"\"*&--%\"DG6#F%F&F(,&%\"xGF( F'!\"\"F(F(*&#F(\"\"#F(*&---%#@@G6$F,F3F-F&F()F.F3F(F(F(*&#F(\"\"'F(*& ---F86$F,\"\"$F-F&F()F.FCF(F(F(*&#F(\"#CF(*&---F86$F,\"\"%F-F&F()F.FMF (F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 427 "p := x -> f(a)+D(f)(a)*(x-a)+1/2*`@@`(D,2)(f)(a)*(x- a)^2+1/6*`@@`(D,3)(f)(a)*(x-a)^3+\n 1/24*`@@`(D,4)(f)(a)*(x-a)^4:\n' p(x)'=p(x);\nEval('p(x)',x=a)=eval('p(x)',x=a);\nEval(Diff('p(x)',x),x =a)=eval(diff(p(x),x),x=a);\nEval(Diff('p(x)',x$2),x=a)=eval(diff(p(x) ,x$2),x=a);\nEval(Diff('p(x)',x$3),x=a)=eval(diff(p(x),x$3),x=a);\nEva l(Diff('p(x)',x$4),x=a)=eval(diff(p(x),x$4),x=a);\nEval(Diff('p(x)',x$ 5),x=a)=eval(diff(p(x),x$5),x=a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"pG6#%\"xG,,-%\"fG6#%\"aG\"\"\"*&--%\"DG6#F*F+F-,&F'F-F,!\"\"F-F-*& #F-\"\"#F-*&---%#@@G6$F1F7F2F+F-)F3F7F-F-F-*&#F-\"\"'F-*&---F<6$F1\"\" $F2F+F-)F3FGF-F-F-*&#F-\"#CF-*&---F<6$F1\"\"%F2F+F-)F3FQF-F-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%EvalG6$-%\"pG6#%\"xG/F*%\"aG-%\"fG 6#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%EvalG6$-%%DiffG6$-%\"pG6#% \"xGF-/F-%\"aG--%\"DG6#%\"fG6#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %%EvalG6$-%%DiffG6$-%\"pG6#%\"xG-%\"$G6$F-\"\"#/F-%\"aG---%#@@G6$%\"DG F16#%\"fG6#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%EvalG6$-%%DiffG6$ -%\"pG6#%\"xG-%\"$G6$F-\"\"$/F-%\"aG---%#@@G6$%\"DGF16#%\"fG6#F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%EvalG6$-%%DiffG6$-%\"pG6#%\"xG-%\" $G6$F-\"\"%/F-%\"aG---%#@@G6$%\"DGF16#%\"fG6#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%EvalG6$-%%DiffG6$-%\"pG6#%\"xG-%\"$G6$F-\"\"&/F-%\" aG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "When the expansion i s about " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 35 ", the r esulting series is called a " }{TEXT 261 9 "Maclaurin" }{TEXT -1 8 " s eries." }}{PARA 0 "" 0 "" {TEXT -1 24 "The Maclaurin series of " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(0)+`f '`(0)*x+1/2!;" "6#, (-%\"fG6#\"\"!\"\"\"*&-%$f~'G6#F'F(%\"xGF(F(*&F(F(-%*factorialG6#\"\"# !\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+1/3!;" "6#,&*$%\"xG\"\"# \"\"\"*&F'F'-%*factorialG6#\"\"$!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f '''`(0)*x^3+` . . . `+1/n!;" "6#,(*&-%&f~'''G6#\"\"!\"\"\"*$%\"x G\"\"$F)F)%(~.~.~.~GF)*&F)F)-%*factorialG6#%\"nG!\"\"F)" }{TEXT -1 2 " f" }{XPPEDIT 18 0 "`@@`(``,n)" "6#-%#@@G6$%!G%\"nG" }{XPPEDIT 18 0 "` `(0)*x^n+` . . . ` = ``;" "6#/,&*&-%!G6#\"\"!\"\"\")%\"xG%\"nGF*F*%(~. ~.~.~GF*F'" }{XPPEDIT 18 0 "Sum(1/n!,n = 0 .. infinity);" "6#-%$SumG6$ *&\"\"\"F'-%*factorialG6#%\"nG!\"\"/F+;\"\"!%)infinityG" }{TEXT -1 2 " f" }{XPPEDIT 18 0 "`@@`(``,n)*``(0)*x^n;" "6#*(-%#@@G6$%!G%\"nG\"\"\" -F'6#\"\"!F))%\"xGF(F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 284 51 "_________________________________________________ __" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 8 "where f" }{XPPEDIT 18 0 "`@@`(``,n)*``(0);" "6#*&-%#@@G 6$%!G%\"nG\"\"\"-F'6#\"\"!F)" }{TEXT -1 21 " is understood to be " } {XPPEDIT 18 0 "f(0);" "6#-%\"fG6#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "When using " } {TEXT 0 6 "taylor" }{TEXT -1 28 " to expand a function about " } {XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 67 " to obtain a Macla urin series, the second parameter can simply be \"" }{TEXT 264 1 "x" } {TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The corresponding Taylor polynomial of degree \"Order min us 1\" can be obtained by the scheme " }{TEXT 0 19 "convert(..,polynom )" }{TEXT -1 60 ", which converts a series data structure, as construc ted by " }{TEXT 0 6 "taylor" }{TEXT -1 4 " or " }{TEXT 0 6 "series" } {TEXT -1 56 ", to the polynomial obtained by deleting the order term. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "srs := taylor(exp(x),x,7);\npol := convert(srs,polynom);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$srsG+3%\"xG\"\"\"\"\"!F'F'#F'\"\"#F *#F'\"\"'\"\"$#F'\"#C\"\"%#F'\"$?\"\"\"&#F'\"$?(F,-%\"OG6#F'\"\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polG,0\"\"\"F&%\"xGF&*$)F'\"\"#F&#F &F**$)F'\"\"$F&#F&\"\"'*$)F'\"\"%F&#F&\"#C*$)F'\"\"&F&#F&\"$?\"*$)F'F0 F&#F&\"$?(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The type of each of the variables " }{TEXT 0 3 "srs" }{TEXT -1 5 " and " }{TEXT 0 3 "pol" }{TEXT -1 22 " can be checked using " } {TEXT 0 4 "type" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "type(srs,series);\ntype(pol, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Converting expressions to functions in Maple .. " }{TEXT 0 7 "unapply" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple command " }{TEXT 0 7 "unapply" } {TEXT -1 54 " can be used to convert an expression into a function." } }{PARA 0 "" 0 "" {TEXT -1 51 "The following command defines the functi on f where " }{XPPEDIT 18 0 "f(x) = sqrt(x)+2;" "6#/-%\"fG6#%\"xG,&-%% sqrtG6#F'\"\"\"\"\"#F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 58 "It is essentially equivalent to using the arrow notation: " }{TEXT 262 19 "f := x -> sqrt(x)+2" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := unapply (sqrt(x)+2,x);\nf(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Here is another use of " }{TEXT 0 7 "unapply" } {TEXT -1 67 " to convert a derivative obtained as an expression into a function." }}{PARA 0 "" 0 "" {TEXT -1 77 "The end result is exactly t he same as that obtained by applying the operator " }{TEXT 0 1 "D" } {TEXT -1 17 " to the function " }{TEXT 262 19 "f := x -> sqrt(x)+2" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := x -> sqrt(x)+2;\nDiff(f(x),x);\nvalue(%);\n df := unapply(%,x);\nD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%%sqrtG6#9$\"\"\"\"\"#F1F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*$-%%sqrtG6#%\"xG\"\"\"F, \"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$-%%sqrtG6#% \"xGF%!\"\"#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.*$-%%sqrtG6#9$F.!\"\"#F.\"\"#F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arro wGF&,$*&\"\"\"F,-%%sqrtG6#9$!\"\"#F,\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 49 ": A p rocedure similar to, but much simpler than, " }{TEXT 0 7 "unapply" } {TEXT -1 75 " for functions of one variable, can easily be constructed .by making use of " }{TEXT 0 4 "subs" }{TEXT -1 34 " to substitute for the expression " }{TEXT 264 3 "_FX" }{TEXT -1 18 " and the variable \+ " }{TEXT 264 2 "_X" }{TEXT -1 17 " in the template " }{TEXT 264 8 "_X \+ ->_FX" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simple_unapply := (fx,x) -> subs(\{'_FX'= fx, '_X'=x\},_X->_FX);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/simple_ unapplyGf*6$%#fxG%\"xG6\"6$%)operatorG%&arrowGF)-%%subsG6$<$/.%$_FXG9$ /.%#_XG9%f*6#F7F)F*F)F3F)F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := simple_unapply(sqrt(x )+2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,&*$-%%sqrtG6#9$\"\"\"F2\"\"#F2F(F(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Diff(f(x), x);\nvalue(%);\nsimple_unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%%DiffG6$,&*$-%%sqrtG6#%\"xG\"\"\"F,\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$-%%sqrtG6#%\"xGF%!\"\"#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&,$*&\"\" \"F,*$-%%sqrtG6#9$F,!\"\"#F,\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Examples of Taylor polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 " Example 1" }}{PARA 0 "" 0 "" {TEXT -1 35 "The degree 1 Taylor polynomi al for " }{XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F '" }{TEXT -1 13 ", centred at " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" } {TEXT -1 67 " and considered in an earlier section, can be obtained as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 107 "f := x -> sqrt(x):\n'f(x)'=f(x);\ntaylor(f(x),x=4, 2);\nconvert(%,polynom):\np1 := unapply(%,x):\n'p1(x)'=p1(x);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"xG\"\"\"\"\"%!\"\"\"\"#\"\"!#F& F'F&-%\"OG6#F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p1G6#%\"xG,&\" \"\"F)*&\"\"%!\"\"F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 35 "The degree 2 Taylor polynomial for " }{XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 13 ", cen tred at " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 30 ", can be \+ obtained as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> sqrt(x):\n'f(x)'=f(x);\ntaylor (f(x),x=4,3);\nconvert(%,polynom):\np2 := unapply(%,x):\n'p2(x)'=p2(x) ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++,&%\"xG\"\"\"\"\"%!\"\"\"\"#\"\" !#F&F'F&#F(\"#kF)-%\"OG6#F&\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %#p2G6#%\"xG,(\"\"\"F)*&\"\"%!\"\"F'F)F)*&\"#kF,,&F'F)F+F,\"\"#F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can ob tain the degree 3 Taylor polynomial in a similar way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f := x - > sqrt(x):\n'f(x)'=f(x);\ntaylor(f(x),x=4,4);\nconvert(%,polynom):\np3 := unapply(%,x):\n'p3(x)'=p3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"fG6#%\"xG*$F'#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-,&% \"xG\"\"\"\"\"%!\"\"\"\"#\"\"!#F&F'F&#F(\"#kF)#F&\"$7&\"\"$-%\"OG6#F&F '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p3G6#%\"xG,*\"\"\"F)*&\"\"%! \"\"F'F)F)*&\"#kF,,&F'F)F+F,\"\"#F,*&\"$7&F,F/\"\"$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The following pictu re shows Taylor polynomial approximations of degree 1 to 3 along with \+ the graph of " }{XPPEDIT 18 0 "y= sqrt(x)" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "plot([f(x),p1(x),p2(x),p3(x)],x=-3..15,color=[b lack,green,blue,magenta],\n thickness=2,legend=[`f(x)`,`p1(x)`,`p2(x )`,`p3(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 710 394 394 {PLOTDATA 2 "6 )-%'CURVESG6%7T7$$\"3O*****\\M=0L)!#?$\"3emU\"*Gt;F\"*!#>7$$\"3%)***** *o?bA?F-$\"3hw/qrX;A9!#=7$$\"39+++/B07KF-$\"3gxgRP*>Az\"F37$$\"3l***** z`_:S%F-$\"3o'\\Eat()z4#F37$$\"3q+++0Ib!y'F-$\"3fb%Q7]\\Rg#F37$$\"3!** ****\\Z`&f\"*F-$\"3B+*y6Uvk-$F37$$\"3#******4WbGF3$\"3awqqDAd4`F37$$\"3)) ******4fvqPF3$\"3aeV(45[19'F37$$\"3a+++I9VGdF3$\"3w7LI)=S'ovF37$$\"35+ 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"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 171 "It is also possible to construct an animation illustrate s how the Taylor polynomials of successively higher degree provide suc cessively better approximating functions for " }{XPPEDIT 18 0 "f(x) = \+ sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 90 "[Click on the graph and use the animation controls in the context bar. In particular, the " }{TEXT 264 3 "->|" }{TEXT -1 126 " button can be used to advance one frame at a time in order to show each successive Taylor polynomial along with the graph of " } {XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 2 " ]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> sqrt(x): 'f(x)'=f(x);\nclr := [green,blue, magenta,brown,green,coral]:\nframes := [seq(plot([f(x),convert(taylor( f(x),x=4,i),polynom)],x=-5..15,\n y=-3..5,color=[red,clr[i-1]],thick ness=2),i=2..7)]:\nplots[display](frames,insequence=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"\"#" }}{PARA 13 " " 1 "" {GLPLOT2D 515 345 345 {PLOTDATA 2 "6#-%(ANIMATEG6(7&-%'CURVESG6 %7J7$$\"3/+++++l+>!#?$\"31S\"=4[W'fV!#>7$$\"3)*******\\n'=5\"!#=$\"3d! >5*GyV>LF57$$\"3#*********ps%=#F5$\"3#>$fpJj5uYF57$$\"3G+++]senKF5$\"3 &o%yT_8G;dF57$$\"3!)*******\\Z/N%F5$\"3'[Z'G>Az&f'F57$$\"3\\*******\\$ fC&)F5$\"34*p$f@D(GB*F57$$\"3%******fz6:B\"!#<$\"3#G88;,N(46FO7$$\"3'* *****p\"=C#o\"FO$\"3ski0^E,(H\"FO7$$\"3#)*****pEpS1#FO$\"34@ER1poO9FO7 $$\"3@+++j`A3DFO$\"3i:LF#yPPe\"FO7$$\"3>+++n(y8!HFO$\"3%>p<%3MM.FO7$$ \"3/+++#H_?<%FO$\"3d@<&>KV; GFO7$$\"3k+++qZvO$)FO$\"3)zTZ`*QM()GFO7$$\"3%4+++v+'o()FO$\"31]>*HK#=h HFO7$$\"3W*******R<*f\"*FO$\"3f[%\\TaNl-$FO7$$\"3c+++])Hxe*FO$\"31\")> \"p&fS'4$FO7$$\"3I******H!o-***FO$\"37E5/E&Q2;$FO7$$\"3&******4k.6/\"! #;$\"3\"pgL$oPhEKFO7$$\"3%******>WTA3\"Fht$\"3U!3`G#Qu*G$FO7$$\"32+++1 *3`7\"Fht$\"3Ew(>XUiXN$FO7$$\"3)******H*zym6Fht$\"3#f+pU&y#eT$FO7$$\"3 \"******4N1#47Fht$\"3W->$)[WOxMFO7$$\"31+++jMF^7Fht$\"3ODP7'RMt`$FO7$$ \"3/+++q(G**G\"Fht$\"3m4][Pyb\"f$FO7$$\"3$******R6KUL\"Fht$\"3W^T[:Ur_ OFO7$$\"3/+++`v&QP\"Fht$\"3D:/@[%elq$FO7$$\"3.+++Ol5;9Fht$\"3OBd*)*4>J w$FO7$$\"33+++/Uac9Fht$\"3IB&[+%)pk\"QFO7$$\"#:\"\"!$\"3-M Le*)>VBeF57$$!3#)****\\(y$pZ7FO$\"3Y++DJbw!)oF57$$!3JELLLyaE\")F5$\"3U ommTIOozF57$$!3qhmm;>s%H%F5$\"3eML$3_>j#*)F57$$\"3L.0+++l+>F.$\"38++]i ^Z+5FO7$$\"39.+++vW]VF5$\"33++](=h(36FO7$$\"3r,+++NfC&)F5$\"3/++]P[687 FO7$$\"3qLL$ez6:B\"FO$\"3VL$e*[z(yI\"FO7$$\"3/nmm;=C#o\"FO$\"3wmm;a/c? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f := x -> sin(x):\n'f(x)'=f(x);\nt aylor(f(x),x=Pi/3,2);\nconvert(%,polynom):\np1 := unapply(%,x):\n'p1(x )'=p1(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"xG\"\"\"*&\"\"$!\"\"%#PiGF&F) ,$*&\"\"#F)F(#F&F-F&\"\"!F.F&-%\"OG6#F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p1G6#%\"xG,(*&\"\"#!\"\"\"\"$#\"\"\"F*F.*&F*F+F'F.F .*&\"\"'F+%#PiGF.F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 35 "The degree 2 Taylor polynomial for " }{XPPEDIT 18 0 "f( x) = sin(x);" "6#/-%\"fG6#%\"xG-%$sinG6#F'" }{TEXT -1 13 ", centred at " }{XPPEDIT 18 0 "x = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" } {TEXT -1 29 ", can be obtained as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f := x -> sin(x): \n'f(x)'=f(x);\ntaylor(f(x),x=Pi/3,3);\nconvert(%,polynom):\np2 := una pply(%,x):\n'p2(x)'=p2(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6 #%\"xG-%$sinGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++,&%\"xG\"\"\"*&\" \"$!\"\"%#PiGF&F),$*&\"\"#F)F(#F&F-F&\"\"!F.F&,$*&\"\"%F)F(F.F)F--%\"O G6#F&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p2G6#%\"xG,**&\"\"#!\" \"\"\"$#\"\"\"F*F.*&F*F+F'F.F.*&\"\"'F+%#PiGF.F+*(\"\"%F+F,F-,&F'F.*&F ,F+F2F.F+F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The degree 3 Taylor polynomial can be obtained in a simil ar way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f := x -> sin(x):\n'f(x)'=f(x);\ntaylor(f(x),x=Pi/3, 4);\nconvert(%,polynom):\np3 := unapply(%,x):\n'p3(x)'=p3(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#+-,&%\"xG\"\"\"*&\"\"$!\"\"%#PiGF&F),$*&\"\"#F)F (#F&F-F&\"\"!F.F&,$*&\"\"%F)F(F.F)F-#F)\"#7F(-%\"OG6#F&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p3G6#%\"xG,,*&\"\"#!\"\"\"\"$#\"\"\"F*F.*&F *F+F'F.F.*&\"\"'F+%#PiGF.F+*(\"\"%F+F,F-,&F'F.*&F,F+F2F.F+F*F+*&\"#7F+ F5F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The following picture shows Taylor polynomial approximations of d egree 1 to 3 along with the graph of " }{XPPEDIT 18 0 "y = sin(x);" " 6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> sin(x): 'f(x) '=f(x);\nplot([f(x),p1(x),p2(x),p3(x)],x=-1..3.5,color=[black,green,bl ue,magenta],\n thickness=2,legend=[`f(x)`,`p1(x)`,`p2(x)`,`p3(x) `]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }} {PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6)-%'CURVESG6%7S7$ $!\"\"\"\"!$!30l*y![)4ZT)!#=7$$!3K++]i!G\">!*F-$!3wT51`\\9XyF-7$$!3!)* *\\PMmnl\")F-$!3gA)[u9**zG(F-7$$!3s***\\PC\")e?(F-$!3aZ=$=8n#)f'F-7$$! 3&****\\iqB(RiF-$!3M`oDYfjUeF-7$$!3]+](o9e\"y_F-$!3_vc2jgZO]F-7$$!3/+] P%yjmQ%F-$!3AgP?.\\KZUF-7$$!3c+]P4IdjMF-$!3[OjDzot%R$F-7$$!3!***\\P47 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"p3(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following code makes use of the Maple procedure " }{TEXT 0 3 "seq" }{TEXT -1 59 " to construct the Taylor polynomials and draw th eir graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 157 "f := x -> sin(x): 'f(x)'=f(x);\nplot([f(x),seq(con vert(taylor(f(x),x=Pi/3,i),polynom),i=2..4)], x=-1..3.5,color=[b lack,green,blue,magenta],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 456 353 353 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$!\"\"\"\"!$!30l*y![)4ZT)!#=7$$!3K+ +]i!G\">!*F-$!3wT51`\\9XyF-7$$!3!)**\\PMmnl\")F-$!3gA)[u9**zG(F-7$$!3s ***\\PC\")e?(F-$!3aZ=$=8n#)f'F-7$$!3&****\\iqB(RiF-$!3M`oDYfjUeF-7$$!3 ]+](o9e\"y_F-$!3_vc2jgZO]F-7$$!3/+]P%yjmQ%F-$!3AgP?.\\KZUF-7$$!3c+]P4I djMF-$!3[OjDzot%R$F-7$$!3!***\\P47\"*3DF-$!3CX`a!zsE[#F-7$$!3#3+v=-6tb 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"" 0 "" {TEXT -1 149 "The following ani mation illustrates how the Taylor polynomials of successively higher d egree provide successively better approximating functions for " } {XPPEDIT 18 0 "f(x) = sin(x);" "6#/-%\"fG6#%\"xG-%$sinG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 90 "[Click on the graph and use th e animation controls in the context bar. In particular, the " }{TEXT 264 3 "->|" }{TEXT -1 126 " button can be used to advance one frame at a time in order to show each successive Taylor polynomial along with \+ the graph of " }{XPPEDIT 18 0 "f(x) = sin(x);" "6#/-%\"fG6#%\"xG-%$sin G6#F'" }{TEXT -1 2 " ]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "f := x -> sin(x): 'f(x)'=f(x);\ncl r := [green,blue,magenta,brown,COLOR(RGB,.9,0,.5),coral,COLOR(RGB,.5,0 ,1),cyan]:\nframes := [seq(plot([f(x),convert(taylor(f(x),x=Pi/3,i),po lynom)],x=-4..7,\ny=-2..3,color=[red,clr[i-1]],thickness=2),i=2..9)]: \nplots[display](frames,insequence=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 469 347 347 {PLOTDATA 2 "6#-%(ANIMATEG6*7&-%'CURVESG6%7\\q7$$!\"%\"\"! $\"3-#GzI&\\-ov!#=7$$!3imm;/8BgP!#<$\"3WX(R]\\m#*z&F17$$!3#)*\\iSd?fl$ F5$\"3:U)>#>$)\\>\\F17$$!3YL$eR%)4;b$F5$\"3%*fYqN4D')RF17$$!34+vV=:IMM 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$\"3')RT***fP=$)*F17$FS$\"3!>5.\"3rzXlF17$FX$\"3)>FnKZ7ur$F17$Fgn$\"3- $)4)RLJfE\"F17$Fao$!3w!\\WJGz!\\CF17$F[p$!3knWR)RW.O&F17$F`p$!3$F17$Fju$!3'=1+(e%Q]9\"F17 $Fev$\"3+kb*zDG;H\"F17$F_w$\"3AM;(3\"G5*Q$F17$Fiw$\"3\"*R9@]FK0bF17$Fc x$\"3:XOvcgPNsF1Fgx7$Fby$\"3*3&>DuzAC&*F17$Fgy$\"3f]'oIZ'R9)*F17$F\\z$ \"3w$H'y#=i4(**F17$F`[l$\"3AqzZe3')*)**F17$Fj[l$\"3[1$p'zi/i)*F17$F_\\ l$\"3[Mfn0huK'*F17$Fd\\l$\"3u?OGZ4@'H*F17$Fi\\l$\"3N/\\&>Q$eA#)F17$F^] l$\"3YR*=5nbrm'F17$Fh]l$\"3)Q[\"ova1)y%F17$Fb^l$\"3;TEWQ<&ps#F17$F\\_l $\"38Apumt]CGFQ7$Ff_l$!3MgX3V`rP>F17$F``l$!3\"=Q5&\\nT4UF17$Fj`l$!3Q;@ <`-J\"3'F17$F_al$!3$\\3IWk<$HyF17$Fdal$!3H4)y#>AC=\"*F17$F^bl$!3o!puB3 )Q/5F57$Fhbl$!3dVub()o*H.\"F57$Fbcl$!3+]t3+nY]5F57$F\\dl$!3UzQ*yRPo0\" F57$Ffdl$!3-/o)>&z;^5F57$F[el$!3y!z\\Y\"4dM5F57$F`el$!3&Qcuj?]t+\"F57$ Feel$!3[R'yV0*>*>*F17$Fjel$!3Hs@8zf:EzF17$Fdfl$!3cQ)RFodQR'F17$F^gl$!3 I3t`)yr\\9%F17$Fhgl$!3%>+`u`(4W:F17$Fbhl$\"3\"[jt$*G8O8#F17$Fghl$\"3?* GWdj8%fVF17$F\\il$\"3Wi5R<&oa)pF17$Fail$\"3WAyA)4hg.\"F57$Ffil$\"3>+Pw $eSYW\"F5-F[jl6&F]jlFajlF^jlF^jlFbjlF^dmFddm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 35 "The degree 1 Taylor polynomial for " }{XPPEDIT 18 0 "f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }{TEXT -1 13 ", centred at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 66 " and considered in an earlier section, can be obtained as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> 1/(1-x):\n'f(x)'=f(x);\ntaylor(f(x),x,2);\nconvert(%,po lynom):\np1 := unapply(%,x):\n'p1(x)'=p1(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"xG\"\"\"\"\"!F%F%-%\"OG6#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p1G6#%\"xG,&\"\"\"F)F'F)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The degree 2 Taylor pol ynomial for " }{XPPEDIT 18 0 "f(x) = 1/(1-x);" "6#/-%\"fG6#%\"xG*&\"\" \"F),&F)F)F'!\"\"F+" }{TEXT -1 13 ", centred at " }{XPPEDIT 18 0 "x = \+ 0;" "6#/%\"xG\"\"!" }{TEXT -1 30 ", can be obtained as follows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> 1/(1-x):\n'f(x)'=f(x);\ntaylor(f(x),x,3);\nconvert(%,po lynom):\np2 := unapply(%,x):\n'p2(x)'=p2(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\"\"!F%F%F%\"\"#-%\"OG6#F%\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p2G6#%\"xG,(\"\"\"F)F'F)*$)F'\"\"# F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 " The degree 3 Taylor polynomial can be obtained in a similar way." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> 1/(1-x):\n'f(x)'=f(x);\ntaylor(f(x),x,4);\nconvert(%,po lynom):\np3 := unapply(%,x):\n'p3(x)'=p3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"xG\"\"\"\"\"!F%F%F%\"\"#F%\"\"$-%\"OG6#F%\"\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p3G6#%\"xG,*\"\"\"F)F'F)*$)F' \"\"#F)F)*$)F'\"\"$F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 104 "The following picture shows Taylor polynomial app roximations of degree 1 to 3, along with the graph of " }{XPPEDIT 18 0 "y = 1/(1-x);" "6#/%\"yG*&\"\"\"F&,&F&F&%\"xG!\"\"F)" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "plot([f(x),p1(x),p2(x),p3(x)],x=-0.7..0.7,color=[black,green, blue,magenta],\n thickness=2,legend=[`f(x)`,`p1(x)`,`p2(x)`,` p3(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 585 518 518 {PLOTDATA 2 "6)-%' CURVESG6%7U7$$!3a**************p!#=$\"3yqk;KHkF*$\"3%)Q8'3eym3'F*7$$!3bmm;4'=28'F*$\" 3a*[/W$=N*>'F*7$$!3`mm;ki8IeF*$\"3Ej(\\k,lqJ'F*7$$!3+LLeMD)4`&F*$\"3A] =1tEuQkF*7$$!3Umm\"HtGOD&F*$\"3UDIZ%4$F *$\"37Pf^\"4f$yvF*7$$!3f*****\\XyK!HF*$\"3W]Y4y$o*\\xF*7$$!3$om;HuTzj# F*$\"3^KVl!)3o7zF*7$$!3ULLLG2VABF*$\"31*=*zw>G:\")F*7$$!37LLL8::b?F*$ \"3m3Jp*))3_H)F*7$$!3o++DYACW()F*7$$!39++DY()3n6F*$\"3iX:I-a)[&*)F*7$$!3;++](QYcz)! #>$\"3eiZeoUa\">*F*7$$!3%\\LL$eRj&z&F^r$\"39.h%R!f=_%*F*7$$!3eHL$e/'oS IF^r$\"3;%3F$GV!\\q*F*7$$!3+ulmmTD5p!#@$\"3agU6J5!#<7$$\"3e!***\\ioY/dF^r$\"3%*Q]H@c\\g5Ffs7$$\"3gKL L3TU1')F^r$\"3e!)[33)oT4\"Ffs7$$\"31*******)HWg6F*$\"3:h/25%y78\"Ffs7$ 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V$e^8e>z6#Ffs7$F`[l$\"3%**************=#Ffs-Fe[l6&Fg[lFeelFeelFfel-Fj[ l6#%&p2(x)G-F$6%7S7$F($\"3?,+++++qWF*7$F.$\"3AQvTV?c'y%F*7$F3$\"3%z/mv Hgm/&F*7$F8$\"3n@#)>*HyNK&F*7$F=$\"33soQ/3A(e&F*7$FB$\"330IG6*oh$eF*7$ FG$\"3cXY\\?*)RcgF*7$FL$\"3O&*z]y.6viF*7$FQ$\"3)e#*eiR=G\\'F*7$FV$\"3N -ewd))y-nF*7$Fen$\"3oI7zG#yI\"pF*7$Fjn$\"3Em/$)=b&[4(F*7$F_o$\"3=)\\&) )p!QrH(F*7$Fdo$\"3SQ?5$QV$*\\(F*7$Fio$\"3u&fyS.1\\p(F*7$F^p$\"3=^^S-XO uyF*7$Fcp$\"37e3_MJn\"4)F*7$Fhp$\"3;q\"R/!4T!G)F*7$F]q$\"3Mn\"z,OHp])F *7$Fbq$\"3\"*o0[v(o]r)F*7$Fgq$\"3MZ.8(*RA`*)F*7$F\\r$\"3]'[Lf9%*4>*F*7 $Fbr$\"3W\"*)Q+Ez?X*F*7$Fgr$\"30I#e@.'*[q*F*7$F\\s$\"3YS[p(3I\"Ffs7$F[v$\"3!4o\")>'>v^8Ffs7$F`v$\"3n< wJr\"Ffs7$F^x$\"3Q1n`t PH'y\"Ffs7$Fcx$\"3!*))HK&y`_'=Ffs7$Fhx$\"3]W!Quo%*y%>Ffs7$F]y$\"3b?o^K 1xF?Ffs7$Fby$\"3!H![)\\(H6C@Ffs7$Fgy$\"3[Y1,b=w9AFfs7$F\\z$\"3Rq,\"e(4 A;BFfs7$Ffz$\"3bQ`o*e=\"=CFfs7$F`[l$\"3#************H`#Ffs-Fe[l6&Fg[lF felFeelFfel-Fj[l6#%&p3(x)G-%+AXESLABELSG6$Q\"x6\"Q!Ffim-%*THICKNESSG6# \"\"#-%%VIEWG6$;$!\"(!\"\"$\"\"(Fbjm%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "p1(x)" "p2(x)" "p3(x)" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following code makes use of the Maple proce dure " }{TEXT 0 3 "seq" }{TEXT -1 59 " to construct the Taylor polynom ials and draw their graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "f := x -> 1/(1-x): 'f(x)'=f(x);\np lot([f(x),seq(convert(taylor(f(x),x=0,i),polynom),i=2..4)],x=-.7..0.7, color=[black,green,blue,magenta],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&F)F)F'!\"\"F+" }}{PARA 13 "" 1 "" {GLPLOT2D 486 423 423 {PLOTDATA 2 "6)-%'CURVESG6$7U7$$!3a******** ******p!#=$\"3yqk;KHkF*$\"3%)Q8'3eym3'F*7$$!3bmm;4'=28'F*$\"3a*[/W$=N*>'F*7$$!3`mm;ki8 IeF*$\"3Ej(\\k,lqJ'F*7$$!3+LLeMD)4`&F*$\"3A]=1tEuQkF*7$$!3Umm\"HtGOD&F *$\"3UDIZ%4$F*$\"37Pf^\"4f$yvF*7$$!3f** 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> (x+1)^x:\n'f(x)'=f(x);\ntaylor((x+1)^ x,x=1,2);\nconvert(%,polynom):\np1 := unapply(%,x):\n'p1(x)'=p1(x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG),&\"\"\"F*F'F*F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+),&%\"xG\"\"\"F&!\"\"\"\"#\"\"!,&*&F( F&-%#lnG6#F(F&F&F&F&F&-%\"OG6#F&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%#p1G6#%\"xG,&\"\"#\"\"\"*&,&*&F)F*-%#lnG6#F)F*F*F*F*F*,&F'F*F*!\"\" F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " The degree 2 Taylor polynomial for " }{XPPEDIT 18 0 "f(x) = (x+1)^x;" "6#/-%\"fG6#%\"xG),&F'\"\"\"F*F*F'" }{TEXT -1 13 ", centred at " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 29 ", can be obtaine d as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> (x+1)^x:\n'f(x)'=f(x);\ntaylor((x+1)^ x,x=1,3);\nconvert(%,polynom):\np2 := unapply(%,x):\n'p2(x)'=p2(x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG),&\"\"\"F*F'F*F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#++,&%\"xG\"\"\"F&!\"\"\"\"#\"\"!,&*&F( F&-%#lnG6#F(F&F&F&F&F&,(F&F&*$)F,F(F&F&F,F&F(-%\"OG6#F&\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p2G6#%\"xG,(\"\"#\"\"\"*&,&*&F)F*-%#lnG6 #F)F*F*F*F*F*,&F'F*F*!\"\"F*F**&,(F*F**$)F.F)F*F*F.F*F*)F1F)F*F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The degre e 3 Taylor polynomial can be obtained in a similar way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> (x+1)^x:\n'f(x)'=f(x);\ntaylor((x+1)^x,x=1,4);\nconvert(%,polynom) :\np3 := unapply(%,x):\n'p3(x)'=p3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG),&\"\"\"F*F'F*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-,&%\"xG\"\"\"F&!\"\"\"\"#\"\"!,&*&F(F&-%#lnG6#F(F&F&F&F&F&,(F&F&* $)F,F(F&F&F,F&F(,*#F&\"\"%F&F,F&*&#F&F(F&F0F&F&*&#F&\"\"$F&*$)F,F9F&F& F&F9-%\"OG6#F&F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p3G6#%\"xG,*\" \"#\"\"\"*&,&*&F)F*-%#lnG6#F)F*F*F*F*F*,&F'F*F*!\"\"F*F**&,(F*F**$)F.F )F*F*F.F*F*)F1F)F*F**&,*#F*\"\"%F*F.F**&#F*F)F*F5F*F**&#F*\"\"$F**$)F. F@F*F*F*F*)F1F@F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 103 "The following picture shows Taylor polynomial approxim ations of degree 1 to 3 along with the graph of " }{XPPEDIT 18 0 "y = (x+1)^x;" "6#/%\"yG),&%\"xG\"\"\"F(F(F'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "f := x -> (x+1)^x: 'f(x)'=f(x);\nplot([f(x),p1(x),p2(x),p3(x)],x=-0.5..3.5 ,y=-1..8,\n color=[black,green,blue,magenta],thickness=2,\n \+ legend=[`f(x)`,`p1(x)`,`p2(x)`,`p3(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG),&\"\"\"F*F'F*F'" }}{PARA 13 "" 1 "" {GLPLOT2D 589 416 416 {PLOTDATA 2 "6)-%'CURVESG6%7W7$$!3++++++++]!#=$ \"3:&4tBc8UT\"!#<7$$!3PLLLLQ6GTF*$\"3)z4Y7K3eC\"F-7$$!3immmT.\\pLF*$\" 3E'f'R(G'\\[6F-7$$!3LLLL$))Qj^#F*$\"3iJ!p3Hlc2\"F-7$$!3ULLL$=Kvl\"F*$ \"3vjW:7X\\I5F-7$$!3gqmm;C2G!)!#>$\"37)*4Yb5u15F-7$$!3phLLL3yO5!#?$\"3 d&osa2,++\"F-7$$\"3t$*****\\Kd,\")FE$\"3PfDsr6L15F-7$$\"3-mmm\"fX(e;F* $\"39)[mW\"QyD5F-7$$\"3.*****\\U7Y]#F*$\"3].\"*e9ydd5F-7$$\"3'QLLLV!pu LF*$\"3G_v[)*\\5.6F-7$$\"3xmmm;c0TTF*$\"3Y!oA>9\"Ha6F-7$$\"3B*******H, Q+&F*$\"3*)ftqu*)3D7F-7$$\"3u*******\\*3qeF*$\"3*4(=Sg-U68F-7$$\"3!*** *****p=\\q'F*$\"3q@&y6@P1T\"F-7$$\"3_mmm\"fBIY(F*$\"3v?yFDH)f^\"F-7$$ \"3yKLLLO[k$)F*$\"3'3l)*4HkEm\"F-7$$\"3.KLLL&Q\"G\"*F*$\"31KV<&o^w!=F- 7$$\"3!*****\\s]k,5F-$\"3]6IL::$R+#F-7$$\"39LLL`dF!3\"F-$\"3Q^&oj=oi?# F-7$$\"33++]sgam6F-$\"3'e1f*fEFkCF-7$$\"3/++]F-$\"3%4I)[Rl LjxF-7$$\"3o****\\7RV'*>F-$\"3!)))3&)*)paV*)F-7$$\"3k*****\\@fk3#F-$\" 31-e0!G@,0\"!#;7$$\"3/LLL`4Nn@F-$\"32yj&RQ-n@\"Feu7$$\"3#*******\\,s`A F-$\"3S[#))pr4\"G9Feu7$$\"3[mm;zM)>L#F-$\"3q'4sb#)>bl\"Feu7$$\"3$***** **pfaFeu7$$\"3#HLLeg`!)\\#F-$\"3)34WD94IG#Feu7$$\"3 w****\\#G2Ae#F-$\"3SsC\"y4Q&Feu7$$\"3FLL$e#pa-IF-$\"3oK`Y>P# \\V'Feu7$$\"3!*******Rv&)zIF-$\"3=tF/*\\0!)f(Feu7$$\"3ILLLGUYoJF-$\"3_ I^^>RQ7#*Feu7$$\"3\"*****\\n'*33KF-$\"3g5*omy%*[+\"!#:7$$\"3_mmm1^rZKF -$\"3O,ZQNUl'4\"F\\z7$$\"33LLe*3k**G$F-$\"3jt:Ck>M/7F\\z7$$\"34++]sI@K LF-$\"3eBl\"H/$GB8F\\z7$$\"33+++S2lsLF-$\"39Z\")pU)*z[9F\\z7$$\"34++]2 %)38MF-$\"3i'4.1E^pe\"F\\z7$$\"3/++v.UacMF-$\"37\"z>Rp>5v\"F\\z7$$\"3+ +++++++NF-$\"38C(oI;`I$>F\\z-%'COLOURG6&%$RGBG\"\"!Fd\\lFd\\l-%'LEGEND G6#%%f(x)G-F$6%7S7$F($!3?O)z;aT%z:F-7$F/$!3oDVPPQQr8F-7$F4$!3m+mLTRN!> \"F-7$F9$!3%Q'R;**)ow')*F*7$F>$!3Ry>m8LI=yF*7$FC$!3k*Ga]+y'ydF*7$FI$!3 'Q\">w`Uo()QF*7$FO$!3XL9$\\vp'H>F*7$FT$\"3yBLxsS;J&*FK7$FY$\"3N0%p`*)) z8@F*7$Fhn$\"3/n'*QS61!>%F*7$F]o$\"3Z5m$f0M)=gF*7$Fbo$\"3uoa::#*fx!)F* 7$Fgo$\"3,'y\"[95F-7$F\\p$\"39SyV2mp87F-7$Fap$\"37(**\\-v-YR\"F-7$F fp$\"3://n^wr4;F-7$F[q$\"3Ch:%)*=[>z\"F-7$F`q$\"3P$QIsiDR+#F-7$Feq$\"3 G%)R^x:c\">#F-7$Fjq$\"3)[TtO&zU(R#F-7$F_r$\"3W^V$pUhMf#F-7$Fdr$\"3i^.- \"z,!)z#F-7$Fir$\"3ItD%epLe)HF-7$F^s$\"3e\"G-/yN%)=$F-7$Fcs$\"3?(G\")f S#)))R$F-7$Fhs$\"3GJO>7o2#e$F-7$F]t$\"3g>xn*yJ*zPF-7$Fbt$\"3)3EU:MNV)R F-7$Fgt$\"3mV%*=RVI%=%F-7$F\\u$\"3WztiEYyxVF-7$Fau$\"3#=CT$)\\6Ef%F-7$ Fgu$\"3%=E(Q(*Hk&y%F-7$F\\v$\"3]$QnVKX<*\\F-7$Fav$\"3?q-WlY]y^F-7$Ffv$ \"3=E#R[&>o#Q&F-7$F[w$\"3K,aDsozubF-7$F`w$\"3+vDN;BhvdF-7$Few$\"3M4*y' =h%>(fF-7$Fjw$\"3E)yWWT*[xhF-7$F_x$\"3c1vmP?XvjF-7$Fdx$\"3mTzG7u*yd'F- 7$Fix$\"3YbS*pVm'ynF-7$F^y$\"3i(oM'>B:jpF-7$Fcy$\"3yG?OgRfurF-7$F^z$\" 3]gEW%))4PO(F-7$Fhz$\"3GXo$Q!pMlvF-7$Fb[l$\"3AF5qf#R$exF-7$F\\\\l$\"3+ F(*z-ftlzF--Fa\\l6&Fc\\l$Fd\\lFd\\l$\"*++++\"!\")Fafl-Ff\\l6#%&p1(x)G- F$6%7S7$F($\"3b%*f*e*)e6J$F-7$F/$\"3Eo+,6/?nHF-7$F4$\"3u\"G$Qm,\"[p#F- 7$F9$\"3qLrdmxO=CF-7$F>$\"3gP>I\"4]?<#F-7$FC$\"3[*yp4jP(e>F-7$FI$\"37V #y&*>T$*y\"F-7$FO$\"3gE@ME\"3Fk\"F-7$FT$\"3\\(>c!\\m%=_\"F-7$FY$\"37X6 5t8F-7$F]o$\"3%[qM\"G,-[8F-7$Fbo$\"3Q8edLSL]8F -7$Fgo$\"338R;HX@&Q\"F-7$F\\p$\"3E.'G&plp\\9F-7$Fap$\"398XZx5]M:F-7$Ff p$\"3Czq\"Hefym\"F-7$F[q$\"3<[q*okq%3=F-7$F`q$\"3'*=`c4:$R+#F-7$Feq$\" 3f@)QWoob?#F-7$Fjq$\"3wbD'RR=xX#F-7$F_r$\"3p&e![]x*ys#F-7$Fdr$\"3e1.ir i2TIF-7$Fir$\"3K!)**=LQ!oN$F-7$F^s$\"3%z\\bi$GbFPF-7$Fcs$\"3%QY9ztOe9% F-7$Fhs$\"3]$fmN\"Feu7$Fjw$\"3c1w/\"4%)QG\"Feu7$F_x$\"3A<5`%y4$o8Feu7$Fdx$\"3 ))4$3iSTxX\"Feu7$Fix$\"3%=3j3`A&\\:Feu7$F^y$\"31iHm4GdO;Feu7$Fcy$\"3Dy '=cuP&RFeu7$Fb[l$\"3mtiRw. _T?Feu7$F\\\\l$\"3s9)G=rt]:#Feu-Fa\\l6&Fc\\lFaflFaflFbfl-Ff\\l6#%&p2(x )G-F$6%7S7$F($!3QJ$oH.!Qd5F-7$F/$!3oKX!)3a&)HoF*7$F4$!3)3y$o&=EQ)RF*7$ F9$!3[yrkQL['>\"F*7$F>$\"3F)*fD1sV97F*7$FC$\"35ZjH!)Ri;F-7$F[q$\"3ob(z'4Gh2=F-7$F`q$\"3q1zK::$R+#F-7$Feq$\"3 ej:o%GQi?#F-7$Fjq$\"3um@G/zpjCF-7$F_r$\"3s]vS&f2yu#F-7$Fdr$\"3_!)Qc#e# [*3$F-7$Fir$\"3G'ocZAo![MF-7$F^s$\"3%z!=b_9W()QF-7$Fcs$\"3aB#)yeDf1WF- 7$Fhs$\"3e\"4?%*Q\"o9\\F-7$F]t$\"3<1gp*e4\"Feu7$Fav$\"3xt@r.=P47Feu7$Ffv$\"3 lUV&f+SPM\"Feu7$F[w$\"3OFeu7$F_x$\"3!GD$p`%Hi;#Feu7$Fdx$\"3S #[0!4,irBFeu7$Fix$\"3g)o'RG\"*)*)e#Feu7$F^y$\"3>'Gu#yj8,GFeu7$Fcy$\"3W 'yvcYr$fIFeu7$F^z$\"3_ZU\\HqT/LFeu7$Fhz$\"3u^Z#*H.y!e$Feu7$Fb[l$\"3/V) *=(f3.'QFeu7$F\\\\l$\"3;\"*[:QaaxTFeu-Fa\\l6&Fc\\lFbflFaflFbfl-Ff\\l6# %&p3(x)G-%+AXESLABELSG6$Q\"x6\"Q\"yFbjm-%*THICKNESSG6#\"\"#-%%VIEWG6$; $!\"&!\"\"$\"#NF^[n;$F^[nFd\\l$\"\")Fd\\l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "p1(x)" "p2(x)" "p3(x)" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following code makes use of the Maple proce dure " }{TEXT 0 3 "seq" }{TEXT -1 59 " to construct the Taylor polynom ials and draw their graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "f := x -> (x+1)^x: 'f(x)'=f(x);\np lot([f(x),seq(convert(taylor(f(x),x=Pi/3,i),polynom),i=2..4)], \+ x=-0.5..3.5,y=-1..8,color=[black,green,blue,magenta],thickness=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG),&\"\"\"F*F'F*F'" }} {PARA 13 "" 1 "" {GLPLOT2D 449 311 311 {PLOTDATA 2 "6)-%'CURVESG6$7W7$ $!3++++++++]!#=$\"3:&4tBc8UT\"!#<7$$!3PLLLLQ6GTF*$\"3)z4Y7K3eC\"F-7$$! 3immmT.\\pLF*$\"3E'f'R(G'\\[6F-7$$!3LLLL$))Qj^#F*$\"3iJ!p3Hlc2\"F-7$$! 3ULLL$=Kvl\"F*$\"3vjW:7X\\I5F-7$$!3gqmm;C2G!)!#>$\"37)*4Yb5u15F-7$$!3p hLLL3yO5!#?$\"3d&osa2,++\"F-7$$\"3t$*****\\Kd,\")FE$\"3PfDsr6L15F-7$$ \"3-mmm\"fX(e;F*$\"39)[mW\"QyD5F-7$$\"3.*****\\U7Y]#F*$\"3].\"*e9ydd5F -7$$\"3'QLLLV!puLF*$\"3G_v[)*\\5.6F-7$$\"3xmmm;c0TTF*$\"3Y!oA>9\"Ha6F- 7$$\"3B*******H,Q+&F*$\"3*)ftqu*)3D7F-7$$\"3u*******\\*3qeF*$\"3*4(=Sg -U68F-7$$\"3!********p=\\q'F*$\"3q@&y6@P1T\"F-7$$\"3_mmm\"fBIY(F*$\"3v ?yFDH)f^\"F-7$$\"3yKLLLO[k$)F*$\"3'3l)*4HkEm\"F-7$$\"3.KLLL&Q\"G\"*F*$ \"31KV<&o^w!=F-7$$\"3!*****\\s]k,5F-$\"3]6IL::$R+#F-7$$\"39LLL`dF!3\"F -$\"3Q^&oj=oi?#F-7$$\"33++]sgam6F-$\"3'e1f*fEFkCF-7$$\"3/++]F-$\"3%4I)[RlLjxF-7$$\"3o****\\7RV'*>F-$\"3!)))3&)*)paV*)F-7$$\"3k *****\\@fk3#F-$\"31-e0!G@,0\"!#;7$$\"3/LLL`4Nn@F-$\"32yj&RQ-n@\"Feu7$$ \"3#*******\\,s`AF-$\"3S[#))pr4\"G9Feu7$$\"3[mm;zM)>L#F-$\"3q'4sb#)>bl \"Feu7$$\"3$*******pfaFeu7$$\"3#HLLeg`!)\\#F-$\"3)3 4WD94IG#Feu7$$\"3w****\\#G2Ae#F-$\"3SsC\"y4Q&Feu7$$\"3FLL$e# pa-IF-$\"3oK`Y>P#\\V'Feu7$$\"3!*******Rv&)zIF-$\"3=tF/*\\0!)f(Feu7$$\" 3ILLLGUYoJF-$\"3_I^^>RQ7#*Feu7$$\"3\"*****\\n'*33KF-$\"3g5*omy%*[+\"!# :7$$\"3_mmm1^rZKF-$\"3O,ZQNUl'4\"F\\z7$$\"33LLe*3k**G$F-$\"3jt:Ck>M/7F \\z7$$\"34++]sI@KLF-$\"3eBl\"H/$GB8F\\z7$$\"33+++S2lsLF-$\"39Z\")pU)*z [9F\\z7$$\"34++]2%)38MF-$\"3i'4.1E^pe\"F\\z7$$\"3/++v.UacMF-$\"37\"z>R p>5v\"F\\z7$$\"3++++++++NF-$\"38C(oI;`I$>F\\z-%'COLOURG6&%$RGBG\"\"!Fd \\lFd\\l-F$6$7S7$F($!3e`Wv(\\dd!>F-7$F/$!3U$z>aJI!z;F-7$F4$!3-jSP1iv\" [\"F-7$F9$!3=_r\\Q3!*f7F-7$F>$!3Xw.Bp[dO5F-7$FC$!3n=97$H.J9)F*7$FI$!3+ **Rm*[EC3'F*7$FO$!3euDu=]r[RF*7$FT$!3lL)4KcF?u\"F*7$FY$\"3mE#pmF-7$Feq$\"3aH/vsgi. 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" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Firstly, " }{XPPEDIT 18 0 "p[1](0) = (1 = 0);" "6#/-&%\"pG6#\"\"\"6#\"\"!/F(F*" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "f(0) = exp(0);" "6#/-%\"fG6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%! G\"\"\"" }{TEXT -1 20 ", so the values of " }{XPPEDIT 18 0 "p[1](x)" "6#-&%\"pG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x) = \+ exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 12 " agree when " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "p[1]*`'`(x) = 1;" "6#/*&&%\"pG6#\" \"\"F(-%\"'G6#%\"xGF(F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '`(x) \+ = exp(x);" "6#/-%$f~'G6#%\"xG-%$expG6#F'" }{TEXT -1 6 ", so " } {XPPEDIT 18 0 "p[1]*`'`(0) = 1;" "6#/*&&%\"pG6#\"\"\"F(-%\"'G6#\"\"!F( F(" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "`f '`(0) = exp(0);" "6#/-%$ f~'G6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 8 ". Hence " }{TEXT 257 15 "the derivatives" }{TEXT -1 5 " of " } {XPPEDIT 18 0 "p[1](x)" "6#-&%\"pG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" } {TEXT -1 6 " also " }{TEXT 257 5 "agree" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "p[2](x)(x) = 1+x+x^2/2;" "6#/--&%\"pG6#\"\"#6#%\"xG6#F+,(\"\"\"F.F+ F.*&F+F)F)!\"\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then \+ " }{XPPEDIT 18 0 "p[2]*`'`(x) = 1+x;" "6#/*&&%\"pG6#\"\"#\"\"\"-%\"'G6 #%\"xGF),&F)F)F-F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p[2]*`''`(x)=1 " "6#/*&&%\"pG6#\"\"#\"\"\"-%#''G6#%\"xGF)F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 29 "The second degree polynomial " }{XPPEDIT 18 0 "p[2](x) = 1+x+x^2/2;" "6#/-&%\"pG6#\"\"#6#%\"xG,(\"\"\"F,F*F,*&F *F(F(!\"\"F," }{TEXT 261 38 " has the same value, first derivative " } {TEXT 285 10 "and second" }{TEXT 261 11 " derivative" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" } {TEXT -1 7 ", when " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Firstly, " }{XPPEDIT 18 0 "p[2](0) = 1+0+0;" "6#/-&%\"pG6#\"\"#6#\"\"!,(\"\"\"F,F*F,F*F," }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "f(0) = \+ exp(0);" "6#/-%\"fG6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\" \"\"" }{TEXT -1 19 ", so the values of " }{XPPEDIT 18 0 "p[2](x);" "6# -&%\"pG6#\"\"#6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x) = exp( x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 12 " agree when " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "p[2]*`'`(x) = 1+x;" "6#/*&&%\"pG6# \"\"#\"\"\"-%\"'G6#%\"xGF),&F)F)F-F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '`(x) = exp(x);" "6#/-%$f~'G6#%\"xG-%$expG6#F'" }{TEXT -1 5 " , so " }{XPPEDIT 18 0 "p[2]*`'`(0) = 1+0;" "6#/*&&%\"pG6#\"\"#\"\"\"-% \"'G6#\"\"!F),&F)F)F-F)" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "`f '`(0) = exp(0);" "6#/-%$f~'G6#\"\"! -%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 8 ". Hence " }{TEXT 257 21 "the derivatives agree" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 8 "Lastly, " }{XPPEDIT 18 0 "p[2]*`''`(x)=1" "6#/*&&% \"pG6#\"\"#\"\"\"-%#''G6#%\"xGF)F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f ''`(x) = exp(x);" "6#/-%%f~''G6#%\"xG-%$expG6#F'" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "p[2]*`''`(0) = 1;" "6#/*&&%\"pG6#\"\"#\"\"\"-%#' 'G6#\"\"!F)F)" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "`f ''`(0) = exp( 0);" "6#/-%%f~''G6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\" \"" }{TEXT -1 8 ". Hence " }{TEXT 330 22 "the second derivatives" } {TEXT -1 8 " agree. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 45 "The following picture compares the graphs of " } {XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p[2](x) = 1+x+x^2/2;" "6#/-&%\"pG6#\"\"#6 #%\"xG,(\"\"\"F,F*F,*&F*F(F(!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "f := x -> exp(x): 'f(x)'=f(x);\np2 := x -> 1+x+x^2/2: 'p2(x)'=p2(x);\nplot([f(x ),p2(x)],x=-2..2,y=-0.2..4,color=[red,blue],thickness=2,legend=[`f(x)` ,`p2(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expGF& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p2G6#%\"xG,(\"\"\"F)F'F)*&#F) \"\"#F)*$)F'F,F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 471 445 445 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"#\"\"!$\"3-FhOKGN`8!#=7$$!3MLLL$Q6G \">!#<$\"3mL4@0mkw9F-7$$!3bmm;M!\\p$=F1$\"3q(zuc7FIf\"F-7$$!3MLLL))Qj^ ;Te'f>CF-7$$!3SLL$3WDTL\"F1$\"3>&e(=NU)Qj#F-7$$!35++]d( Q&\\7F1$\"3OD#)Hb(pj'GF-7$$!3gmmmc4`i6F1$\"3DqR_Tw$p7$F-7$$!3KLLLQW*e3 \"F1$\"3=$*H3()o*fP$F-7$$!3w++++()>'***F-$\"3'*[&ew4$>!o$F-7$$!3E++++0 \"*H\"*F-$\"31/%os_?K,%F-7$$!35++++83&H)F-$\"3e#z1k=QEO%F-7$$!3\\LLL3k (p`(F-$\"3%*3#z8OJiq%F-7$$!3Anmmmj^NmF-$\"3'Hf#)HA*=]^F-7$$!3)zmmmYh=( eF-$\"3o![.*pF*)ebF-7$$!3+,++v#\\N)\\F-$\"3KmY(yn#HvgF-7$$!3commmCC(>% F-$\"372'*[w-GslF-7$$!39*****\\FRXL$F-$\"397`(G1\\W;(F-7$$!3t*****\\#= /8DF-$\"3smIfnu&yx(F-7$$!3=mmm;a*el\"F-$\"3))R?QF)RRZ)F-7$$!3komm;Wn(o )!#>$\"3Kv$eR)3!z;*F-7$$!3IqLLL$eV(>!#?$\"3A!4+V*eF!)**F-7$$\"3)Qjmm\" f`@')Fjr$\"3I4?O`5/!4\"F17$$\"3%z****\\nZ)H;F-$\"3'Rk81w=q<\"F17$$\"3c kmm;$y*eCF-$\"3e$Ryr\"*o(y7F17$$\"3f)******R^bJ$F-$\"3g:?^hH8$R\"F17$$ \"3'e*****\\5a`TF-$\"3RF\")*=32\\^\"F17$$\"3'o****\\7RV'\\F-$\"3@9IJdA &Gk\"F17$$\"3Y'*****\\@fkeF-$\"3O(f&Qv@h(z\"F17$$\"3_ILLL&4Nn'F-$\"31M Zg0t1\\>F17$$\"3A*******\\,s`(F-$\"3Rf[`T-*[7#F17$$\"3%[mm;zM)>$)F-$\" 3i$[nV+syH#F17$$\"3M*******pfa<*F-$\"3B(Qnc1SJ]#F17$$\"39HLLeg`!)**F-$ \"3)QXji7'*Hr#F17$$\"3w****\\#G2A3\"F1$\"3K'Go$pk=^HF17$$\"3;LLL$)G[k6 F1$\"3MP8ivaE/KF17$$\"3#)****\\7yh]7F1$\"3%o$fA,+]#\\$F17$$\"3xmmm')fd L8F1$\"3+9#RDb)e%z$F17$$\"3bmmm,FT=9F1$\"3))**e)[!)e08%F17$$\"3FLL$e#p a-:F1$\"3YrqL[\"=J\\%F17$$\"3!*******Rv&)z:F1$\"3WGb\\BUEa[F17$$\"3ILL LGUYo;F1$\"3uo!4=y:SI&F17$$\"3_mmm1^rZF1$\"3;#Gn\"Gt(Rx'F17$$\"\"#F*$\" 3S]1$*)4c!*Q(F1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fb[l-%'LEGENDG6#%% f(x)G-F$6%7S7$F($\"\"\"F*7$F/$\"3unXr2J7m\"*F-7$F5$\"3FzD'*G%=C])F-7$F :$\"3=GaX5vwCyF-7$F?$\"3%Q&fgQn8;sF-7$FD$\"3Kbp>%fGOo'F-7$FI$\"3K[\"R( y#*=biF-7$FN$\"3eeD1#3Rx(eF-7$FS$\"3SZ.1^!*>ebF-7$FX$\"3?>KZdzM6`F-7$F gn$\"3<4\\Pf:3K^F-7$F\\o$\"3\"R*Hos#*)o.&F-7$Fao$\"3c%3%\\A2++]F-7$Ffo $\"3'=^+py_y.&F-7$F[p$\"3I%[!o)QP`9&F-7$F`p$\"3!)G`lgUK.`F-7$Fep$\"3?$ R[f]()fc&F-7$Fjp$\"3gt&>vQw?&eF-7$F_q$\"3I]w\"Q*)Q#eiF-7$Fdq$\"3'e$G` \\(*f$o'F-7$Fiq$\"3Rwe#QL=9A(F-7$F^r$\"3,n*4d8FF!yF-7$Fcr$\"3#y@!)[1/7 [)F-7$Fhr$\"3)[L=$**3 \"F17$Fis$\"3?9&=Zymi<\"F17$F^t$\"3o]eZ.28w7F17$Fct$\"3s4-VX&>lQ\"F17$ Fht$\"3&\\;F17$Fbu$\"3]2#H/UE%e!)pRt\"3y@F17$Ffv$\"3M? ;`t7\\QBF17$F[w$\"3#>z\"*e54h\\#F17$F`w$\"3m;![Oe$znEF17$Few$\"3r*[@h2 $\\UGF17$Fjw$\"3.U6(*o-kKIF17$F_x$\"3t`OxU%)yAKF17$Fdx$\"3gK8z(**fVU$F 17$Fix$\"3rL*)*zbq8j$F17$F^y$\"3ADuMjC$y#QF17$Fcy$\"371t%*o1NgSF17$Fhy $\"3u0@q`\"p\\F%F17$F]z$\"3^X*>SW:2^%F17$Fbz$\"3(HWb\\.UIu%F17$Fgz$\" \"&F*-F\\[l6&F^[lFb[lFb[lF_[l-Fd[l6#%&p2(x)G-%+AXESLABELSG6$Q\"x6\"Q\" yFfel-%*THICKNESSG6#Fhz-%%VIEWG6$;F(Fgz;$F)!\"\"$\"\"%F*" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "p2(x)" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "p[3](x) = \+ 1+x+x^2/2+x^3/6;" "6#/-&%\"pG6#\"\"$6#%\"xG,*\"\"\"F,F*F,*&F*\"\"#F.! \"\"F,*&F*F(\"\"'F/F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 " The third degree polymomial " }{XPPEDIT 18 0 "p[3](x) = 1+x+x^2/2+x^3/ 6;" "6#/-&%\"pG6#\"\"$6#%\"xG,*\"\"\"F,F*F,*&F*\"\"#F.!\"\"F,*&F*F(\" \"'F/F," }{TEXT -1 1 " " }{TEXT 261 56 "has the same value, first deri vative, second derivative " }{TEXT 326 9 "and third" }{TEXT 261 11 " d erivative" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\" fG6#%\"xG-%$expG6#F'" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "x = 0" "6# /%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Firstly, " }{XPPEDIT 18 0 "p[3](0) = 1+0+0+0;" " 6#/-&%\"pG6#\"\"$6#\"\"!,*\"\"\"F,F*F,F*F,F*F," }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "f(0) = exp(0);" "6#/-%\"fG6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" } {TEXT -1 19 ", so the values of " }{XPPEDIT 18 0 "p[3](x);" "6#-&%\"pG 6#\"\"$6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x) = exp(x);" "6 #/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 12 " agree when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 " Then " }{XPPEDIT 18 0 "p[3]*`'`(x) = 1+x+x^2/2;" "6#/*&&%\"pG6#\"\"$ \"\"\"-%\"'G6#%\"xGF),(F)F)F-F)*&F-\"\"#F0!\"\"F)" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "`f '`(x) = exp(x);" "6#/-%$f~'G6#%\"xG-%$expG6#F'" } {TEXT -1 5 ", so " }{XPPEDIT 18 0 "p[3]*`'`(0) = 1+0+0;" "6#/*&&%\"pG6 #\"\"$\"\"\"-%\"'G6#\"\"!F),(F)F)F-F)F-F)" }{XPPEDIT 18 0 "``=0" "6#/% !G\"\"!" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "`f '`(0) = exp(0);" "6 #/-%$f~'G6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" } {TEXT -1 8 ". Hence " }{TEXT 257 21 "the derivatives agree" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "p[3]*`''` (x) = 1+x;" "6#/*&&%\"pG6#\"\"$\"\"\"-%#''G6#%\"xGF),&F)F)F-F)" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "`f ''`(x) = exp(x);" "6#/-%%f~''G6# %\"xG-%$expG6#F'" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "p[3]*`''`(0) = 1 +0;" "6#/*&&%\"pG6#\"\"$\"\"\"-%#''G6#\"\"!F),&F)F)F-F)" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "`f ''`(0 ) = exp(0);" "6#/-%%f~''G6#\"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6# /%!G\"\"\"" }{TEXT -1 12 ". Hence the " }{TEXT 330 21 "2nd derivatives agree" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "Lastly, " } {XPPEDIT 18 0 "p[3]*`'''`(x) = 1;" "6#/*&&%\"pG6#\"\"$\"\"\"-%$'''G6#% \"xGF)F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '''`(x) = exp(x);" "6 #/-%&f~'''G6#%\"xG-%$expG6#F'" }{TEXT -1 5 ", so " }{XPPEDIT 18 0 "p[3 ]*`'''`(0) = 1;" "6#/*&&%\"pG6#\"\"$\"\"\"-%$'''G6#\"\"!F)F)" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "`f '''`(0) = exp(0);" "6#/-%&f~'''G6# \"\"!-%$expG6#F'" }{XPPEDIT 18 0 "``=1" "6#/%!G\"\"\"" }{TEXT -1 12 ". Hence the " }{TEXT 329 21 "3rd derivatives agree" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 45 "The following picture compares the graphs of " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "p[3](x) = 1+x+x^2/2+x^3/6;" "6#/-&% \"pG6#\"\"$6#%\"xG,*\"\"\"F,F*F,*&F*\"\"#F.!\"\"F,*&F*F(\"\"'F/F," } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f := x -> exp(x): 'f(x)'=f(x);\np3 := x -> 1+x+ x^2/2+x^3/6: 'p3(x)'=p3(x);\nplot([f(x),p3(x)],x=-2..2,y=-0.2..4,color =[red,brown],thickness=2,legend=[`f(x)`,`p3(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$expGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p3G6#%\"xG,*\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F )*$)F'\"\"$F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT2D 608 472 472 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"#\"\"!$\"3-FhOKGN`8!#=7$$!3MLLL$Q6G\">!#<$\"3 mL4@0mkw9F-7$$!3bmm;M!\\p$=F1$\"3q(zuc7FIf\"F-7$$!3MLLL))Qj^;Te'f>CF-7$$!3SLL$3WDTL\"F1$\"3>&e(=NU)Qj#F-7$$!35++]d(Q&\\7F1$\"3 OD#)Hb(pj'GF-7$$!3gmmmc4`i6F1$\"3DqR_Tw$p7$F-7$$!3KLLLQW*e3\"F1$\"3=$* H3()o*fP$F-7$$!3w++++()>'***F-$\"3'*[&ew4$>!o$F-7$$!3E++++0\"*H\"*F-$ \"31/%os_?K,%F-7$$!35++++83&H)F-$\"3e#z1k=QEO%F-7$$!3\\LLL3k(p`(F-$\"3 %*3#z8OJiq%F-7$$!3Anmmmj^NmF-$\"3'Hf#)HA*=]^F-7$$!3)zmmmYh=(eF-$\"3o![ .*pF*)ebF-7$$!3+,++v#\\N)\\F-$\"3KmY(yn#HvgF-7$$!3commmCC(>%F-$\"372'* [w-GslF-7$$!39*****\\FRXL$F-$\"397`(G1\\W;(F-7$$!3t*****\\#=/8DF-$\"3s mIfnu&yx(F-7$$!3=mmm;a*el\"F-$\"3))R?QF)RRZ)F-7$$!3komm;Wn(o)!#>$\"3Kv $eR)3!z;*F-7$$!3IqLLL$eV(>!#?$\"3A!4+V*eF!)**F-7$$\"3)Qjmm\"f`@')Fjr$ \"3I4?O`5/!4\"F17$$\"3%z****\\nZ)H;F-$\"3'Rk81w=q<\"F17$$\"3ckmm;$y*eC F-$\"3e$Ryr\"*o(y7F17$$\"3f)******R^bJ$F-$\"3g:?^hH8$R\"F17$$\"3'e**** *\\5a`TF-$\"3RF\")*=32\\^\"F17$$\"3'o****\\7RV'\\F-$\"3@9IJdA&Gk\"F17$ $\"3Y'*****\\@fkeF-$\"3O(f&Qv@h(z\"F17$$\"3_ILLL&4Nn'F-$\"31MZg0t1\\>F 17$$\"3A*******\\,s`(F-$\"3Rf[`T-*[7#F17$$\"3%[mm;zM)>$)F-$\"3i$[nV+sy H#F17$$\"3M*******pfa<*F-$\"3B(Qnc1SJ]#F17$$\"39HLLeg`!)**F-$\"3)QXji7 '*Hr#F17$$\"3w****\\#G2A3\"F1$\"3K'Go$pk=^HF17$$\"3;LLL$)G[k6F1$\"3MP8 ivaE/KF17$$\"3#)****\\7yh]7F1$\"3%o$fA,+]#\\$F17$$\"3xmmm')fdL8F1$\"3+ 9#RDb)e%z$F17$$\"3bmmm,FT=9F1$\"3))**e)[!)e08%F17$$\"3FLL$e#pa-:F1$\"3 YrqL[\"=J\\%F17$$\"3!*******Rv&)z:F1$\"3WGb\\BUEa[F17$$\"3ILLLGUYo;F1$ \"3uo!4=y:SI&F17$$\"3_mmm1^rZF1$\"3;#Gn\"Gt(Rx'F17$$\"\"#F*$\"3S]1$*)4c !*Q(F1-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fb[l-%'LEGENDG6#%%f(x)G-F$6 %7S7$F($!3fKLLLLLLLF-7$F/$!3Y*HjBy^$)\\#F-7$F5$!3^@f&z2F&G=F-7$F:$!3H% 42vVmD8\"F-7$F?$!3Y)\\Xm8GB([Fjr7$FD$\"3q0c5#R1F1\"Fjr7$FI$\"31z+[1V<& ='Fjr7$FN$\"3#Qj[qqDe6\"F-7$FS$\"3gr0$yiO0g\"F-7$FX$\"3)fl!o^BT@Z\\!G!HF-7$Fao$\"34!)[U$)RBNLF-7$Ffo$ \"3gG43FfZpPF-7$F[p$\"3)QoDxH_S>%F-7$F`p$\"3'zH(\\*G[(*e%F-7$Fep$\"3u$ [9+M]!z]F-7$Fjp$\"3K+@iXAl9bF-7$F_q$\"3'4\\qG9b>0'F-7$Fdq$\"3W\"*RK/GO glF-7$Fiq$\"3_5Pw#yvix(F-7$Fcr$\"3EcGW^mjt%)F-7$F hr$\"3[[*HZbxy;*F-7$F^s$\"3%3w$H%*eF!)**F-7$Fds$\"3'GIs4rQ+4\"F17$Fis$ \"3]Wzat$))p<\"F17$F^t$\"3Ai,Np(3'y7F17$Fct$\"3k>gZATf#R\"F17$Fht$\"3= _&H$pjb8:F17$F]u$\"3IlDKrz/S;F17$Fbu$\"3!)fk>rO/#z\"F17$Fgu$\"3#Q-!oOX cR>F17$F\\v$\"3=l[%Gk,n0JF17$Fjw$\" 3k!3%F17$F^y$\"3#*379%**R][%F17$Fcy$\"3zYGI9\\XM[F17$Fhy$\"3i=$4O9/Z ;&F17$F]z$\"366.Y1,%e`&F17$Fbz$\"3bqz$>R'**4fF17$Fgz$\"3.LLLLLLLjF1-F \\[l6&F^[l$\")#)eqkFa[l$\"))eqk\"Fa[lFael-Fd[l6#%&p3(x)G-%+AXESLABELSG 6$Q\"x6\"Q\"yFjel-%*THICKNESSG6#Fhz-%%VIEWG6$;F(Fgz;$F)!\"\"$\"\"%F*" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x)" "p3(x) " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "General Ta ylor polynomials for " }{XPPEDIT 18 0 "exp(x);" "6#-%$expG6#%\"xG" } {TEXT -1 59 " and the notion of convergence of the Maclaurin series fo r " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 60 "The polynomials considered in the last section have the form" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p[n](x) = 1+x+x^2/2 !+x^3/3!+x^4/4!+` . . . `+x^n/n!;" "6#/-&%\"pG6#%\"nG6#%\"xG,0\"\"\"F, F*F,*&F*\"\"#-%*factorialG6#F.!\"\"F,*&F*\"\"$-F06#F4F2F,*&F*\"\"%-F06 #F8F2F,%(~.~.~.~GF,*&)F*F(F,-F06#F(F2F," }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum(x^i/i!,i = 0 .. n);" " 6#/%!G-%$SumG6$*&)%\"xG%\"iG\"\"\"-%*factorialG6#F+!\"\"/F+;\"\"!%\"nG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "p[n](x),p[n]*`'`(x),` . . . `,p[n]^``(n)*``(x)" "6&-&%\"pG6#%\"nG6# %\"xG*&&F%6#F'\"\"\"-%\"'G6#F)F-%(~.~.~.~G*&)&F%6#F'-%!G6#F'F--F76#F)F -" }{TEXT -1 26 " all have the value1 when " }{XPPEDIT 18 0 "x = 0" "6 #/%\"xG\"\"!" }{TEXT -1 67 ", and so agree with the values of the corr esponding derivatives of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "If we pick a specific \+ value of " }{TEXT 291 1 "x" }{TEXT -1 49 " different from 0, the value s of successive sums " }{XPPEDIT 18 0 "p[n](x), p[n+1](x),` . . . `" " 6%-&%\"pG6#%\"nG6#%\"xG-&F%6#,&F'\"\"\"F.F.6#F)%(~.~.~.~G" }{TEXT -1 18 " approach a limit." }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, wh en " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 11 ", we obtain " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[0](1) = 1" "6#/-& %\"pG6#\"\"!6#\"\"\"F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[1](1) = 1 + 1" "6#/-&%\"pG6#\"\"\"6#F(,&F(F(F(F( " }{TEXT -1 6 " = 2 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[2](1) = 1 + 1 + 1/2" "6#/-&%\"pG6#\"\"#6#\"\"\",(F*F*F*F**&F*F*F( !\"\"F*" }{TEXT -1 8 " = 2.5 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "p[3](1) = 1 + 1 + 1/2 + 1/6" "6#/-&%\"pG6#\"\"$6#\"\"\" ,*F*F*F*F**&F*F*\"\"#!\"\"F**&F*F*\"\"'F.F*" }{TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 15 " 2.666666667 \n " }{XPPEDIT 18 0 "p[4](1) = 1 \+ + 1 + 1/2 + 1/6 + 1/24" "6#/-&%\"pG6#\"\"%6#\"\"\",,F*F*F*F**&F*F*\"\" #!\"\"F**&F*F*\"\"'F.F**&F*F*\"#CF.F*" }{TEXT -1 1 " " }{TEXT 289 1 "~ " }{TEXT -1 13 " 2.708333333 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "p[5](1) = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120" "6#/-&%\"pG 6#\"\"&6#\"\"\",.F*F*F*F**&F*F*\"\"#!\"\"F**&F*F*\"\"'F.F**&F*F*\"#CF. F**&F*F*\"$?\"F.F*" }{TEXT -1 1 " " }{TEXT 288 1 "~" }{TEXT -1 13 " 2. 716666667 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[6](1) = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/600" "6#/-&%\"pG6#\"\"'6#\"\"\", 0F*F*F*F**&F*F*\"\"#!\"\"F**&F*F*F(F.F**&F*F*\"#CF.F**&F*F*\"$?\"F.F** &F*F*\"$+'F.F*" }{TEXT -1 1 " " }{TEXT 287 1 "~" }{TEXT -1 13 " 2.7180 55556 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The values of these successive sums approach, or " }{TEXT 261 8 "conv erge" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "exp(1)^1" "6#*$-%$expG6#\"\" \"F'" }{TEXT -1 1 " " }{TEXT 286 1 "~" }{TEXT -1 13 " 2.718281828." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "p[n](x)=Sum(x^i/i!,i=0..n)" "6#/-&%\"pG6#%\"nG6#%\"xG-% $SumG6$*&)F*%\"iG\"\"\"-%*factorialG6#F0!\"\"/F0;\"\"!F(" }{TEXT -1 30 " can be calculated as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "x := 'x': i := 'i':\nSum(x^i /i!,i=0..5);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*& )%\"xG%\"iG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\"\"F$%\"xGF$*$)F%\"\"#F$#F$F(*$)F%\"\"$F$#F$ \"\"'*$)F%\"\"%F$#F$\"#C*$)F%\"\"&F$#F$\"$?\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We can calculate successi ve approximations for " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for n from 0 to 12 do\n print(Sum(1/i!,i=0..n) =evalf(Sum(1/i!,i=0..n)));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"!F0$F(F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\" iG!\"\"/F,;\"\"!F($\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG 6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"!\"\"#$\"+++++D!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\" iG!\"\"/F,;\"\"!\"\"$$\"+nmmmE!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"!\"\"%$\"+LLL3F! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factoria lG6#%\"iG!\"\"/F,;\"\"!\"\"&$\"+nmm;F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"! \"\"'$\"+cb0=F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\" \"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"!\"\"($\"+oRD=F!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F, ;\"\"!\"\")$\"+q(y#=F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6 $*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"!\"\"*$\"+E:G=F!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\" iG!\"\"/F,;\"\"!\"#5$\"+,=G=F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$SumG6$*&\"\"\"F(-%*factorialG6#%\"iG!\"\"/F,;\"\"!\"#6$\"+E=G=F!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6 #%\"iG!\"\"/F,;\"\"!\"#7$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "Taking the sum of 13 terms gives " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 22 " correct to 1 0 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "The following procedure provides another way to obtain finite s ums:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p[n](x)=1+x+x^2/2!+x^3/3!+x^4/4!" "6#/-&%\"pG6#%\" nG6#%\"xG,,\"\"\"F,F*F,*&F*\"\"#-%*factorialG6#F.!\"\"F,*&F*\"\"$-F06# F4F2F,*&F*\"\"%-F06#F8F2F," }{TEXT -1 13 " + . . . . + " }{XPPEDIT 18 0 "x^n/n!" "6#*&)%\"xG%\"nG\"\"\"-%*factorialG6#F&!\"\"" }{TEXT -1 6 " = " }{XPPEDIT 18 0 "Sum(x^i/i!,i=0..n)" "6#-%$SumG6$*&)%\"xG%\"iG \"\"\"-%*factorialG6#F)!\"\"/F);\"\"!%\"nG" }{TEXT -1 5 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "E := proc(x::algebraic,n::nonnegint)\nlocal i,term,sum;\nsum := 1 ;\nterm := 1;\n for i from 1 to n do\n term := term*x/i;\n \+ sum := sum + term;\n end do;\n return sum;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "E(x ,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4\"\"\"F$%\"xGF$*&#F$\"\"#F$) F%F(F$F$*&#F$\"\"'F$)F%\"\"$F$F$*&#F$\"#CF$)F%\"\"%F$F$*&#F$\"$?\"F$)F %\"\"&F$F$*&#F$\"$?(F$)F%F,F$F$*&#F$\"%S]F$)F%\"\"(F$F$*&#F$\"&?.%F$)F %\"\")F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "evalf(E(1,12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "taylor" }{TEXT -1 52 " can al so be used to obtain such Taylor polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sx := taylor (exp(x),x=0,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sxG+1%\"xG\"\"\" \"\"!F'F'#F'\"\"#F*#F'\"\"'\"\"$#F'\"#C\"\"%#F'\"$?\"\"\"&-%\"OG6#F'F, " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The \+ order term " }{XPPEDIT 18 0 "O(x^5)" "6#-%\"OG6#*$%\"xG\"\"&" }{TEXT -1 19 " can be removed by " }{TEXT 0 19 "convert(..,polynom)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{TEXT 0 7 "unapply" } {TEXT -1 62 " can be used to set up a function to evaluate the polynom ial. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "convert(sx,polynom);\np5 := unapply(%,x):\n'p5(x)'=p5 (x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\"\"F$%\"xGF$*&#F$\"\"#F$* $)F%F(F$F$F$*&#F$\"\"'F$*$)F%\"\"$F$F$F$*&#F$\"#CF$*$)F%\"\"%F$F$F$*&# F$\"$?\"F$*$)F%\"\"&F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#p5G6 #%\"xG,.\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F )F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The following p icture compares the graph of " }{XPPEDIT 18 0 "p[5](x) = 1+x+x^2/2+x^3 /6+x^4/24+x^5/120;" "6#/-&%\"pG6#\"\"&6#%\"xG,.\"\"\"F,F*F,*&F*\"\"#F. !\"\"F,*&F*\"\"$\"\"'F/F,*&F*\"\"%\"#CF/F,*&F*F(\"$?\"F/F," }{TEXT -1 19 " with the graph of " }{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#% \"xG-%$expG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "f := x -> exp(x): 'f(x)'=f( x);\nplot([f(x),p5(x)],x=-3..3,color=[red,blue],thickness=2,legend=[`f (x)`,`p5(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$ex pGF&" }}{PARA 13 "" 1 "" {GLPLOT2D 457 445 445 {PLOTDATA 2 "6'-%'CURVE SG6%7V7$$!\"$\"\"!$\"3W%R'yOoqy\\!#>7$$!3!******\\2<#pG!#<$\"3M-!*)o\\ LVn&F-7$$!3#)***\\7bBav#F1$\"3aWlPp$3#ejF-7$$!36++]K3XFEF1$\"3S4#R.nVi A(F-7$$!3%)****\\F)H')\\#F1$\"3SKY*=ja(>#)F-7$$!3#****\\i3@/P#F1$\"3(y I7;6PTM*F-7$$!3;++Dr^b^AF1$\"3#Hb!4\"QaB0\"!#=7$$!3$****\\7Sw%G@F1$\"3 it^1A\\=!>\"FM7$$!3*****\\7;)=,?F1$\"3[tt(eyXFM7$$!3!******\\!)H%*\\\"F1$\"3]/HHTUdKAFM7$$!3/+++ vl[p8F1$\"3pJ?;zXPUDFM7$$!3\"******\\>iUC\"F1$\"3'Q$)HwIM:)GFM7$$!3-++ DhkaI6F1$\"3G9X)[zn&GKFM7$$!3s******\\XF`**FM$\"3K!*[tlR-'p$FM7$$!3u** *****>#z2))FM$\"3rMf7&HVk\"F 17$$\"3A)****\\d6.B'FM$\"3crX*RHrX'=F17$$\"3s****\\(o3lW(FM$\"32%Q9)Qh q0@F17$$\"35*****\\A))oz)FM$\"3G]rQ0'\\,T#F17$$\"3e******Hk-,5F1$\"3p0 3!pQt5s#F17$$\"36+++D-eI6F1$\"39!y6')>`u4$F17$$\"3u***\\(=_(zC\"F1$\"3 'zx1k#HG$[$F17$$\"3M+++b*=jP\"F1$\"3A=?'Gt'HgRF17$$\"3g***\\(3/3(\\\"F 1$\"3[;db_MioWF17$$\"33++vB4JB;F1$\"37&=I/W[)p]F17$$\"3u*****\\KCnu\"F 1$\"3+K(zxI$yNdF17$$\"3s***\\(=n#f(=F1$\"3[\"3/m)['o_'F17$$\"3P+++!)RO +?F1$\"3%zK&3dgu\"R(F17$$\"30++]_!>w7#F1$\"3y#f*)zGb[R)F17$$\"3O++v)Q? 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$\"3y)frqX^*\\:Fby7$F][l$\"3lczFs9EP;Fby7$Fb[l$\"3Dy&)[eX)ft\"Fby7$Fg[ l$\"3')************R=Fby-F\\\\l6&F^\\lFb\\lFb\\lF_\\l-Fd\\l6#%&p5(x)G- %+AXESLABELSG6$Q\"x6\"Q!F\\gl-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(Fg[l%(DE FAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "f(x) " "p5(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The following picture shows the graph of some Taylor polynomial s for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 31 " draw n together with the graph " }{XPPEDIT 18 0 "y = exp(x)" "6#/%\"yG-%$ex pG6#%\"xG" }{TEXT -1 16 " (shown in red)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "i := 'i': n := 'n':\n plot([exp(x),seq(sum(x^i/i!,i=0..n),n=1..5)],x=-3..3,thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 335 353 353 {PLOTDATA 2 "6+-%'CURVESG6$7V7$ $!\"$\"\"!$\"1%R'yOoqy\\!#<7$$!1+++vq@pG!#:$\"1-!*)o\\LVn&F-7$$!1++D^N UbFF1$\"1YlPp$3#ejF-7$$!1++]K3XFEF1$\"13#R.nViA(F-7$$!1++]F)H')\\#F1$ 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1f]jV(H=S\"FM7$F\\p$\"1S'ev\"Ht(3#FM7$Fap$\"1Mh1!*H*on#FM7$Ffp$\"1UpNG ipcLFM7$F[q$\"1p],$\\hA$RFM7$F`q$\"1`I*y!=\\AYFM7$Feq$\"1L!pjf1,F&FM7$ Fjq$\"18%[\"yh`SgFM7$F_r$\"1r5+1Tk^oFM7$Fdr$\"1$RH\")o!4*z(FM7$Fir$\"1 ^/`E12y()FM7$F^s$\"1O0BN%G/(**FM7$Fds$\"1Sh+Rd/Q6F17$Fis$\"1&)4H_qzw7F 17$F^t$\"1mkR\"=M_W\"F17$Fct$\"1!eZ(>V]T;F17$Fht$\"1$H3Q%=Ud=F17$F]u$ \"1FO>)3A24#F17$Fbu$\"1&>bgDt+Q#F17$Fgu$\"1XaS&zL#pEF17$F\\v$\"1f6\"e^ R0,$F17$Fav$\"1dA*f#yj]LF17$Ffv$\"1Xx9Q@'zv$F17$F[w$\"15teKr#p<%F17$F` w$\"1!*>#4e@Ql%F17$Few$\"17\"H5$*p/;&F17$Fjw$\"18wPh)Rdt&F17$F_x$\"1LN 1KM:NjF17$Fdx$\"1uZcV\\?'*pF17$Fix$\"1Z3B\\yz,xF17$F^y$\"1J2'RI0eR)F17 $Fdy$\"1*Qc>mUqC*F17$Fiy$\"1emn&)\\215Fby7$Fcz$\"161/oXZ)4\"Fby7$F][l$ \"1ySyA*\\D>\"Fby7$Fg[l$\"#8F*-F\\\\l6&F^\\lF*F*F_\\l-F$6$7U7$F($\"1++ ++++v8F17$F/$\"1h_#>\"43M6F17$F5$\"1b^c#z:*e&*FM7$F:$\"1&oc&yofpyFM7$F ?$\"1Zy#[(R(4Z'FM7$FD$\"1WRDNSoY`FM7$FI$\"1Yy()FM7$F^s$\"1)ei_VG/(**FM7$ Fds$\"1o;Z$Rd!Q6F17$Fis$\"1&G+***e%pF\"F17$F^t$\"1hxG\"R0gW\"F17$Fct$ \"1\"eX4O`Sk\"F17$Fht$\"1jC$z$**pj=F17$F]u$\"1BnyvN`.@F17$Fbu$\"1i%Q$Q _-0CF17$Fgu$\"1&p3tzr5r#F17$F\\v$\"19s)yb:'yIF17$Fav$\"105mxbq^MF17$Ff v$\"1T(**fPqu!RF17$F[w$\"1/z/^rA'Q%F17$F`w$\"1)zuQ+`J%\\F17$Few$\"1@0% z'*R$[bF17$Fjw$\"1UQ#4(Gu^iF17$F_x$\"1\"Q%*y`0B+(F17$Fdx$\"1$4#\\\\3-] yF17$Fix$\"14FV0-%px)F17$F^y$\"1$*41$\\'*)4(*F17$Fdy$\"1RR=#*z;)3\"Fby 7$Fiy$\"1j[l)3!)G?\"Fby7$Fcz$\"1\\/hz,>O8Fby7$Fhz$\"11;w&4=US\"Fby7$F] [l$\"1.T/S!*4v9Fby7$Fb[l$\"1eb*pf[Xb\"Fby7$Fg[l$\"1+++++]P;Fby-F\\\\l6 &F^\\lF_\\lF*F_\\l-F$6$7U7$F($!1+++++++lFM7$F/$!1))R'Qc'pj[FM7$F5$!1`0 W73?xOFM7$F:$!13Tv`gQlDFM7$F?$!1zyO0.xW;FM7$FD$!1m?CNf#*)*))F-7$FI$!1D /1wN+cIF-7$FO$\"1BN\\7#3D2#F-7$FT$\"1Uk\"\\tPqi'F-7$FY$\"157$G)p^i5FM7 $Fhn$\"10Li6OgP9FM7$F]o$\"1fxMo.;_FF17$F\\v$\"102GV'3S4$F17$Fav$\"1\"[gveJpZ$F17$Ffv$ \"16mL!eC'[RF17$F[w$\"1yb[U\\*)[WF17$F`w$\"1#[nd(z3P]F17$Few$\"1#)Git. %Qo&F17$Fjw$\"1ykVP'R`W'F17$F_x$\"1<9BY\\@psF17$Fdx$\"1&zZs%)RL@)F17$F ix$\"1.g2eddh#*F17$F^y$\"1Gl]b>FL5Fby7$Fdy$\"12k,1!))*p6Fby7$Fiy$\"1#4 =T'y118Fby7$Fcz$\"1Er#\\(Q&oY\"Fby7$Fhz$\"1*frqX^*\\:Fby7$F][l$\"1dzFs 9EP;Fby7$Fb[l$\"1y&)[eX)ft\"Fby7$Fg[l$\"1++++++S=Fby-F\\\\l6&F^\\lF*F_ \\lF_\\l-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fg[ l%(DEFAULTG" 1 2 0 1 10 2 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The infinite power series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+x+x^2/2!+x^3/3!+x^4/4!+x^ 5/5!+` . . . `" "6#,0\"\"\"F$%\"xGF$*&F%\"\"#-%*factorialG6#F'!\"\"F$* &F%\"\"$-F)6#F-F+F$*&F%\"\"%-F)6#F1F+F$*&F%\"\"&-F)6#F5F+F$%(~.~.~.~GF $" }{TEXT -1 12 " ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Sum(x^i/i!,i = 0 .. infinity);" "6#/%!G-%$SumG6$*& )%\"xG%\"iG\"\"\"-%*factorialG6#F+!\"\"/F+;\"\"!%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "is defined to be the limit of \+ the sequence of Taylor polynomials " }{XPPEDIT 18 0 "Sum(x^i/i!,i=0..n )" "6#-%$SumG6$*&)%\"xG%\"iG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!%\"nG " }{TEXT -1 4 " as " }{TEXT 292 1 "n" }{TEXT -1 19 " tends to infinity ." }}{PARA 0 "" 0 "" {TEXT -1 22 "The series (i) is the " }{TEXT 261 16 "Maclaurin series" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "exp(x)" "6#- %$expG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "It turn s out that, for any value of " }{TEXT 293 1 "x" }{TEXT -1 14 ", this s eries " }{TEXT 261 9 "converges" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "ex p(x)" "6#-%$expG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 32 "This means that we can identify " }{XPPEDIT 18 0 "exp(x)" "6#-%$ex pG6#%\"xG" }{TEXT -1 36 " with the infinite series and write:" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)=Sum(x^i/i!,i = 0 .. infinity)" "6#/-%$expG6#%\"xG-%$SumG6$*&)F'%\"iG\"\"\"-%*factori alG6#F-!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Maple \"knows\" this result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum(x^i/i!,i=0..infin ity);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)%\"xG% \"iG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 14 "More examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }{TEXT 331 46 " .. the Taylor polynomial of degree 3 for cos(" }{TEXT 335 1 "x" } {TEXT 336 5 ") at " }{TEXT 333 1 "x" }{TEXT 334 3 " = " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 332 4 "/4 " }}{PARA 0 "" 0 "" {TEXT 322 8 "Qu estion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the T aylor polynomial of degree 3 for the function " }{XPPEDIT 18 0 "f(x) = cos*x;" "6#/-%\"fG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 4 " at " } {XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 69 "(b) Use your answer for part (a) to obtain an approximate value for " }{XPPEDIT 18 0 "cos(3/4);" "6#- %$cosG6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }{TEXT 327 0 "" }} {PARA 0 "" 0 "" {TEXT 328 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[f(x) = cos*x, ` `, f(Pi/4) = sqrt(2)/2], [`f '`(x) = - sin*x, ` `, `f '`(Pi/4) = -sqrt(2)/2], [`f ''`(x) = -cos*x, ` `, `f \+ ''`(Pi/4) = -sqrt(2)/2], [`f '''`(x) = sin*x, ` `, `f '''`(Pi/4) = sq rt(2)/2]]);" "6#-%'matrixG6#7&7%/-%\"fG6#%\"xG*&%$cosG\"\"\"F,F/%#~~G/ -F*6#*&%#PiGF/\"\"%!\"\"*&-%%sqrtG6#\"\"#F/F " 0 "" {MPLTEXT 1 0 76 "taylor(cos (x),x=Pi/4,4):\nconvert(%,polynom):\np := unapply(%,x):\n'p(x)'=p(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,**&\"\"#!\"\"F*#\" \"\"F*F-*(F*F+F*F,,&F'F-*&\"\"%F+%#PiGF-F+F-F+*(F1F+F*F,F/F*F+*(\"#7F+ F*F,F/\"\"$F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "3/4" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 323 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 18 ", we would expect " }{XPPEDIT 18 0 "p(Pi/4)" "6#-%\"pG6#* &%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 33 " to provide an approximation for " }{XPPEDIT 18 0 "cos(3/4)" "6#-%$cosG6#*&\"\"$\"\"\"\"\"%!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(Pi/4) = sqrt(2)/2-sqrt(2)/2;" "6#/-% \"pG6#*&%#PiG\"\"\"\"\"%!\"\",&*&-%%sqrtG6#\"\"#F)F1F+F)*&-F/6#F1F)F1F +F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(3/4-Pi/4)-sqrt(2)/4;" "6#,&-%! G6#,&*&\"\"$\"\"\"\"\"%!\"\"F**&%#PiGF*F+F,F,F**&-%%sqrtG6#\"\"#F*F+F, F," }{XPPEDIT 18 0 "``(3/4-Pi/4)^2+sqrt(2)/12;" "6#,&*$-%!G6#,&*&\"\"$ \"\"\"\"\"%!\"\"F+*&%#PiGF+F,F-F-\"\"#F+*&-%%sqrtG6#F0F+\"#7F-F+" } {XPPEDIT 18 0 "``(3/4-Pi/4)^3;" "6#*$-%!G6#,&*&\"\"$\"\"\"\"\"%!\"\"F* *&%#PiGF*F+F,F,F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sqrt(2)/2-sqrt(2)* (3-Pi)/8 - sqrt(2)*(3-Pi)^2/64+sqrt(2)*(3-Pi)^3/768" "6#/%!G,**&-%%sqr tG6#\"\"#\"\"\"F*!\"\"F+*(-F(6#F*F+,&\"\"$F+%#PiGF,F+\"\")F,F,*(-F(6#F *F+*$,&F1F+F2F,F*F+\"#kF,F,*(-F(6#F*F+*$,&F1F+F2F,F1F+\"$o(F,F+" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=s qrt(2)*(1/2-(3-Pi)/8-(3-Pi)^2/64+(3-Pi)^3/768)" "6#/%!G*&-%%sqrtG6#\" \"#\"\"\",**&F*F*F)!\"\"F**&,&\"\"$F*%#PiGF-F*\"\")F-F-*&,&F0F*F1F-F) \"#kF-F-*&,&F0F*F1F-F0\"$o(F-F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 324 1 "~" } {TEXT -1 15 " 0.7316888223. " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" } {TEXT -1 28 ": A more accurate value for " }{XPPEDIT 18 0 "cos(3/4)" " 6#-%$cosG6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 17 " is 0.7316888689." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "evalf(evalf[15](p(3/4)));\nevalf(evalf[15](sqrt(2)*(1/2-(3-Pi)/ 8-(3-Pi)^2/64+(3-Pi)^3/768)));\nevalf(evalf[15](cos(3/4)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+B#))oJ(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+B#))oJ(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*o))oJ(!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }{TEXT 337 49 " .. the Taylor polynomial of degree 3 for arctan(" } {TEXT 338 1 "x" }{TEXT 339 5 ") at " }{TEXT 340 1 "x" }{TEXT 341 5 " = 1 " }}{PARA 0 "" 0 "" {TEXT 320 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the Taylor polynomial of degree \+ 3 for the function " }{XPPEDIT 18 0 "f(x) = arctan*x;" "6#/-%\"fG6#%\" xG*&%'arctanG\"\"\"F'F*" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 1;" "6 #/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "(b) Use your answer for part (a) to obtain an approximate value for " } {XPPEDIT 18 0 "arctan(1.1);" "6#-%'arctanG6#-%&FloatG6$\"#6!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[f(x) = arctan(x), ` `, f(1) = Pi/4], [`f '`(x) = 1/(1+x^2), ` `, `f '`(1 ) = 1/2], [`f ''`(x) = (-2*x)/((1+x^2)^3), ` `, `f ''`(1) = -1/2], [` f '''`(x) = (6*x^2-2)/((1+x^2)^3), ` `, `f '''`(1) = 1/2]]);" "6#-%'m atrixG6#7&7%/-%\"fG6#%\"xG-%'arctanG6#F,%#~~G/-F*6#\"\"\"*&%#PiGF4\"\" %!\"\"7%/-%$f~'G6#F,*&F4F4,&F4F4*$F,\"\"#F4F8F0/-F<6#F4*&F4F4FAF87%/-% %f~''G6#F,*&,$*&FAF4F,F4F8F4*$,&F4F4*$F,FAF4\"\"$F8F0/-FI6#F4,$*&F4F4F AF8F87%/-%&f~'''G6#F,*&,&*&\"\"'F4*$F,FAF4F4FAF8F4*$,&F4F4*$F,FAF4FQF8 F0/-FZ6#F4*&F4F4FAF8" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The Tay lor polynomial of degree 3 for " }{XPPEDIT 18 0 "f(x) = arctan*x;" "6# /-%\"fG6#%\"xG*&%'arctanG\"\"\"F'F*" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = f(1)+`f '`(1)*(x-1)+`f \"`(1)*(x -1)^2/2!+`f '''`(1)*(x-1)^3/3!;" "6#/-%\"pG6#%\"xG,*-%\"fG6#\"\"\"F,*& -%$f~'G6#F,F,,&F'F,F,!\"\"F,F,*(-%$f~\"G6#F,F,*$,&F'F,F,F2\"\"#F,-%*fa ctorialG6#F9F2F,*(-%&f~'''G6#F,F,*$,&F'F,F,F2\"\"$F,-F;6#FCF2F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = Pi/4+1/2;" "6#/-%\"pG6#% \"xG,&*&%#PiG\"\"\"\"\"%!\"\"F+*&F+F+\"\"#F-F+" }{XPPEDIT 18 0 "``(x-1 )-1/4;" "6#,&-%!G6#,&%\"xG\"\"\"F)!\"\"F)*&F)F)\"\"%F*F*" }{XPPEDIT 18 0 "``(x-1)^2+1/12;" "6#,&*$-%!G6#,&%\"xG\"\"\"F*!\"\"\"\"#F**&F*F* \"#7F+F*" }{XPPEDIT 18 0 "``(x-1)^3;" "6#*$-%!G6#,&%\"xG\"\"\"F)!\"\" \"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "taylor(arctan(x),x=1,4):\nconvert(% ,polynom):\np := unapply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,,*&\"\"%!\"\"%#PiG\"\"\"F-#F-\"\"#F+*&F/ F+F'F-F-*&F*F+,&F-F+F'F-F/F+*&\"#7F+F2\"\"$F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "1.1;" "6#-%&FloatG6$\"#6!\"\"" } {TEXT -1 32 " is close to 1, we would expect " }{XPPEDIT 18 0 "p(1.1); " "6#-%\"pG6#-%&FloatG6$\"#6!\"\"" }{TEXT -1 33 " to provide an approx imation for " }{XPPEDIT 18 0 "arctan*1.1;" "6#*&%'arctanG\"\"\"-%&Floa tG6$\"#6!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(1) = Pi/4+1/2;" "6# /-%\"pG6#\"\"\",&*&%#PiGF'\"\"%!\"\"F'*&F'F'\"\"#F,F'" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(1-1.1)-1/4;" "6#,&-%!G6#,&\"\"\"F(-%&FloatG6$\"#6 !\"\"F-F(*&F(F(\"\"%F-F-" }{XPPEDIT 18 0 "``(1.1-1)^2+1/12;" "6#,&*$-% !G6#,&-%&FloatG6$\"#6!\"\"\"\"\"F.F-\"\"#F.*&F.F.\"#7F-F." }{XPPEDIT 18 0 "``(1.1-1)^3;" "6#*$-%!G6#,&-%&FloatG6$\"#6!\"\"\"\"\"F-F,\"\"$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= Pi/4 +1/20 -1/400 + 1/12000" "6#/%!G,**&%#PiG\"\"\"\"\"%!\"\"F(*&F(F( \"#?F*F(*&F(F(\"$+%F*F**&F(F(\"&+?\"F*F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 325 1 " ~" }{TEXT -1 15 " 0.8329814967. " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note " }{TEXT -1 28 ": A more accurate value for " }{XPPEDIT 18 0 "arctan(1 .1);" "6#-%'arctanG6#-%&FloatG6$\"#6!\"\"" }{TEXT -1 17 " is 0.8329812 667." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "evalf(evalf[15](p(11/10)));\nevalf(evalf[15](Pi/4 +1 /20 -1/400 + 1/12000));\nevalf(evalf[15](arctan(1.1)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+n\\\")H$)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+n\\\")H$)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nE\")H$) !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 57 "(a) Find the \+ Taylor polynomials of degree 1, 2 and 3 for " }{XPPEDIT 18 0 "ln(x)" " 6#-%#lnG6#%\"xG" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=1" "6#/%\"xG \"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 83 "(b) Construct \+ graphs and/or an animation to illustrate some Taylor polynomials for \+ " }{XPPEDIT 18 0 "f(x) = ln(x);" "6#/-%\"fG6#%\"xG-%#lnG6#F'" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 84 "(c) Compare the value of each of Taylor \+ polynomials found in (a) with the value of " }{XPPEDIT 18 0 "f(x) = l n(x);" "6#/-%\"fG6#%\"xG-%#lnG6#F'" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=1.1" "6#/%\"xG-%&FloatG6$\"#6!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "__ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 57 "(a) Find the Taylor polynomials of degree 1, 2 and 3 for " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" } {TEXT -1 7 " about " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 83 "(b) Construct graphs and/or an a nimation to illustrate some Taylor polynomials for " }{XPPEDIT 18 0 "f (x) = arctan(x);" "6#/-%\"fG6#%\"xG-%'arctanG6#F'" }{TEXT -1 7 " about " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 84 "(c) Compare the value of each of Taylor polynomials \+ found in (a) with the value of " }{XPPEDIT 18 0 "f(x) = arctan(x);" " 6#/-%\"fG6#%\"xG-%'arctanG6#F'" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 ".9. " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________________ ___________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6# %\"xG" }{TEXT -1 25 " of degree 11 centred at " }{XPPEDIT 18 0 "x=0" " 6#/%\"xG\"\"!" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "f(x)= exp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 76 "(b) Use the Taylor polynomial found in (a) to find an a pproximate value for " }{XPPEDIT 18 0 "exp(1.3)" "6#-%$expG6#-%&FloatG 6$\"#8!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "(c) Increa se the degree of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\" pG6#%\"xG" }{TEXT -1 85 " until the the value of p(1.3) agrees with th e value of Maple's exponential function " }{TEXT 264 6 "exp(x)" } {TEXT -1 27 " to 10 digits. What is the " }{TEXT 261 14 "minimum degre e" }{TEXT -1 74 " of the Taylor polynomial needed for the two values t o agree to 10 digits?" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 30 ": To ensure that the value of " }{XPPEDIT 18 0 "exp(1.3)" "6#-%$ex pG6#-%&FloatG6$\"#8!\"\"" }{TEXT -1 126 " is correct to 10 digits, per form the evaluation using 15 digit arithmetic, and then round the resu lt to 10 digits using . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf[15](exp(1.3));\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0C>wm'HpO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+omHpO!\"*" }}}{PARA 0 "" 0 "" {TEXT 261 2 "OR" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(evalf[15](exp(1.3))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+omHpO!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " . . . assuming that Mapl e is currently running with the default setting of " }{TEXT 262 9 "Dig its=10" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "Also evaluate y our polynomial in a similar way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "(d) Find a Taylor polynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 30 " of minimum degree centred at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 24 " such that t he value of " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=1.3" "6#/%\"xG-%&FloatG6$\"#8!\"\"" }{TEXT -1 50 " agrees with that of Maple's exponential function " }{TEXT 264 6 "exp (x)" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=1.3" "6#/%\"xG-%&FloatG6$\"# 8!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "______________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________ ________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 35 "Use a T aylor polynomial centred at " }{XPPEDIT 18 0 "x=Pi/6" "6#/%\"xG*&%#PiG \"\"\"\"\"'!\"\"" }{TEXT -1 14 " to calculate " }{XPPEDIT 18 0 "sin(29 ^o)" "6#-%$sinG6#)\"#H%\"oG" }{TEXT -1 22 " correct to 10 digits." }} {PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "29 ^o = 29/180*Pi" "6#/)\"#H%\"oG*(F%\"\"\"\"$!=!\"\"%#PiGF(" }{TEXT -1 9 " radians." }}{PARA 0 "" 0 "" {TEXT -1 39 "_________________________ ______________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }} {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 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 596 "f := x -> x-x^2/3+sin(2*(x-1))/6: \ng := unapply(f(1)+D(f)(1)*(x-1),x):\nplt1 := plot([f(x),g(x)],x=0.3. .2.1,y=0..1,color=[red,blue],thickness=2):\nplt2 := plot([[[1,2/3]]$3] ,color=black,style=point,\n symbol=[circle,diamond,cross]): \nplt3 := plot([[[0.3,0],[2.1,0]],[[1,0],[1,2/3]]],color=black,linesty le=[1,2]):\nt1 := plots[textplot]([[.9,.75,`(a,f(a))`],[1,-.04,`x = a` ]],color=black):\nt2 := plots[textplot]([1.3,1.1,`y = f(a)+f '(a)(x - \+ a)`],color=blue):\nt3 := plots[textplot]([1.9,.76,`y = f(x)`],color=re d):\nplots[display]([plt1,plt2,plt3,t1,t2,t3],axes=none,view=[.3..2.1, -.04..1.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 779 "f := x -> x-x^2/3+sin(2*(x-1))/6:\ng := unapply (f(1)+D(f)(1)*(x-1)+D(D(f))(1)/2*(x-1)^2,x):\nplt1 := plot([f(x),g(x)] ,x=0.3..2.1,y=0..1,color=[red,blue],thickness=2):\nplt2 := plot([[[1,2 /3]]$3],color=black,style=point,\n symbol=[circle,diamond,c ross]):\nplt3 := plot([[[0.3,0],[2.1,0]],[[1,0],[1,2/3]]],color=black, linestyle=[1,2]):\nt1 := plots[textplot]([[.92,.73,`(a,f(a))`],[1,-.04 ,`x = a`]],color=black):\nt2 := plots[textplot]([[1.35,1.05,`y = f(a) \+ + f '(a)(x - a) + f ''(a)(x - a)`],\n [1.51,1.07,`1`], [1.51,1.06,`_`],[1.51,1.,`2`]],color=blue):\nt3 := plots[textplot]([1. 81,1.08,`2`],color=blue,font=[HELVETICA,8]):\nt4 := plots[textplot]([1 .97,.76,`y = f(x)`],color=red):\nplots[display]([plt1,plt2,plt3,t1,t2, t3,t4],axes=none,view=[.3..2.1,-.04..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }