{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We look for an approximate solution to the equation " }{XPPEDIT 18 0 "x^3 - 5*x + 3 = 0" "6#/,(* $%\"xG\"\"$\"\"\"*&\"\"&F(F&F(!\"\"F'F(\"\"!" }{TEXT -1 57 ", using Ne wton's method. To begin, we define a function " }{XPPEDIT 18 0 "f " " 6#%\"fG" }{TEXT -1 98 " to be the left hand side of our equation -- th at is, the expression we want to set equal to zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> x^3 - 5*x + 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\" \"$\"\"\"F1*&\"\"&F1F/F1!\"\"F0F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(x),x=-1..2.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$!\"\"\"\"!$\"\"(F*7$$!3Umm ;/'*4P#*!#=$\"3,(yC9V./$o!#<7$$!3UL$e*[SIt&)F0$\"3E/x8a4]cmF37$$!3%pm; H_'zEyF0$\"3Qc!*RQ&QRV'F37$$!3)om;/mS`2(F0$\"3'e$*f\\ivM='F37$$!3]K$ek LcuK'F0$\"3`@jxtwR5fF37$$!3>n;HK=2McF0$\"3q'Hic*[>QcF37$$!3e**\\iSB6; \\F0$\"3$e%Q+yFCR`F37$$!3%em\"H2wftTF0$\"3m6[Q`#*49]F37$$!3,+]7GTYLMF0 $\"3%p5w\"ogDwYF37$$!3KKL$3(e9sEF0$\"3j#*oUUG*pJ%F37$$!3-mmTNjd,?F0$\" 3=;/xO#pF*RF37$$!3Y)***\\iQnY7F0$\"39j$pgN*R@OF37$$!3u'****\\(or')[!#> $\"3e[@%*[\">UC$F37$$\"3W2++D'Q!=CFho$\"3WJh!o?7\"zGF37$$\"3aPL3FkX^!* Fho$\"3m$[$=a(o\"[DF37$$\"3cnm;zJ#Rp\"F0$\"39qr[7*)*y:#F37$$\"3Iomm;77 iBF0$\"3NO\"=D7>@$=F37$$\"3^+]P%Q%RRJF0$\"330Q)=/W7Y\"F37$$\"3SmmmTGTF QF0$\"3A`dv!phB9\"F37$$\"3'=+vV8yAe%F0$\"3Wv?2tFw]!)F07$$\"3t-]7.%)3,` F0$\"37:h0q`C%)\\F07$$\"3[omT5:4^gF0$\"35O'oWZ#>g>F07$$\"3;o;a)[G)RnF0 $!3Ue+](=%[V86F3$!33/e -vw!o=\"F37$$\"3G+vVt'zV=\"F3$!37`fk3O]g7F37$$\"3#***\\78=:j7F3$!3!o=B &GSL+8F37$$\"3Umm;%3KRL\"F3$!3q*Gg%y_4'H\"F37$$\"3V++DJ^]49F3$!3QG1:m^ DZ7F37$$\"3=L3FWb)zZ\"F3$!3sB[N.^Nh6F37$$\"3]++vBF&Gb\"F3$!3lAQDZ(*y>5 F37$$\"3emT50pHB;F3$!3!y]SCqk%*Q)F07$$\"35+v=s8$pp\"F3$!3qFhO*oQ@)fF07 $$\"3umm\"H_A*oF3$\"3i%y+AUK.f%F07$$\"3_L$eR666*>F3$\"31*=u9p6DQ*F 07$$\"3;nT5g&GZ1#F3$\"3Uo\\`Q)4&y9F37$$\"3Y++]Z`PK@F3$\"3iMEEsk3M?F37$ $\"3\"pm\"z*>1*4AF3$\"3D'>#eqc&Hu#F37$$\"3?+D1f%yXC#F3$\"39<(4eH\"f&3$ F37$$\"3[LLL=2DzAF3$\"3b*4)f]qTWMF37$$\"3+C]9d%o%F37$ $\"3I+DccB&RU#F3$\"3oFU>J%zA7&F37$$\"39]7Gyh(>Y#F3$\"3\"=aU(e5'Hh&F37$ $\"3++++++++DF3$\"3+++++++DhF3-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fd\\l-%+AX ESLABELSG6$Q\"x6\"Q!Fi\\l-%%VIEWG6$;F($\"#DF)%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "It looks like there's a root between x=1.5 and x=2. We'll use x=2 as an initial guess, and apply Newton's method to improve our guess." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The iteration rule for Newton's method says " } {XPPEDIT 18 0 "x[n+1]" "6#&%\"xG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 6 " - (f(" } {XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 5 ")/f'(" }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 2 "))" }{TEXT -1 25 ". In thi s case, we have " }{XPPEDIT 18 0 "f(x) = x^3 - 5*x + 3" "6#/-%\"fG6#% \"xG,(*$F'\"\"$\"\"\"*&\"\"&F+F'F+!\"\"F*F+" }{TEXT -1 8 " and f'(" } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "3*x^2 - 5" "6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F&F&\"\"&!\"\"" }{TEXT -1 31 ", so h ere's the iteration rule:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "nextx := x -> evalf(x - (x^3 - 5*x + 3)/(3*x^2 - 5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nextxGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%& evalfG6#,&9$\"\"\"*&,(*$)F0\"\"$F1F1*&\"\"&F1F0F1!\"\"F6F1F1,&*$)F0\" \"#F1F6F8F9F9F9F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "We set o ur initial guess, and begin iterating." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x0 := 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\" #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x1 := nextx(x0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+dG9d=!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }