{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "T imes" 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 {PARA 258 "" 0 "" {TEXT 256 47 "Calculus Exploration 4B: Newt on-Raphson Method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 473 "The following is a problem solver that uses the Newton- Raphson method to find the zeroes of a smooth (differentiable) functio n f(t). You should work through each example experimenting with differ ent starting points for the approximation process. Each exploration co ntains a math movie of the iteration process that you can step through using the VCR like controls at the top of the toolbar above. Note tha t these VCR controls appear only after you click on the plot image. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 208 "After exploring the first three examples below, use the last two sections t o create your own different examples. You should find two functions th at would be difficult for you to compute the zeroes of by hand." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 54 "Newton-Raphson estimation of zeroes of f(t): Example 1 " }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 39 "Click in the red area and pre ss [Enter]" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2788 "rest art:\nf:=t->sin(t): # enter here\ntleft:=-6: # enter here \ntri ght:=6: # enter here\nybot:=-2: # enter here \nytop:=2: \+ # enter here\nt0:=2: # enter start value of t here\nn:=10 : # enter number of iterations here\nm:=ytop: \ni:='i':t:='t ':b:='b':\nt[0]:=t0:\nfor i from 0 to n do\nb[i]:=f(t[i])-D(f)(t[i])*t [i]:\nL[i]:=t->evalf(D(f)(t[i])*t+b[i]):\nt[i+1]:=fsolve(L[i](t)=0,t): \nod:\nfor i from 0 to n do\nistr:=convert(i,name):\nListr:=` t`||(i +1)||` = solution L`||i||`(t) = 0`:\ntt[i]:=evalf(t[i],8):\ntstr:=conv ert(tt[i],name):\nftt[i]:=evalf(f(t[i]),8):\nftstr:=convert(ftt[i],nam e):\nTitle[i]:=`Newton-Raphson estimation of zeroes of a function f(t) `:\nitxtplot[i]:=plots[textplot]([0,m-.02*m,\n \+ ` i = `||istr],\n font=[HELVETICA,BOLD,12], \n color=blue,align=\{RIGHT\}):\nttxtplot[i]:= plots[textplot]([0,m-.12*m,\n ` t`||istr||` \+ = `||tstr],\n font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nfttxtplot[i]:=plots[ textplot]([0,m-.22*m,\n ` f(`||`t`||istr||`) = `||ftstr],\n font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nLitxtplot[i]:=plot s[textplot]([0,m-.32*m,\n Listr],\n \+ font=[HELVETICA,BOLD,12],\n \+ color=red,align=\{RIGHT\}):\npp[i]:=plot([t,L[i](t),t=tleft..tright] ,color=red,\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=``||Title[i]):\nppp[i]:=plot([[t[i],0],[t[i],f(t[i])]],linesty le=2,color=blue):\npppp[i]:=plot([[t[i+1],0],[t[i+1],f(t[i+1])]],lines tyle=2,color=red):\np[0]:=plot(f(t),t=tleft..tright,color=blue,\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[ i]):\np[i+1]:=plots[display]([pppp[i],ppp[i],pp[i],p[0],\n \+ itxtplot[i],ttxtplot[i],fttxtplot[i]],Litxtplot[i]):\nend: \nplotfcns:=plots[display](\n [seq(p[i],i=1..n+1)],\n \+ insequence=true,\n #scaling=constrained,\n \+ labels=[``,``],\n view=[tleft..tright,ybot..ytop]):\nplot fcns;\n'f'(t)=f(t),`f '`(t)=D(f)(t);\n`Estimate zeroes of `*'f'(t)*`us ing Newton-Raphson Method:`;``;\nL['i'](t)=`f '`(t['i'])*t+b['i'],` b `['i']='f'(t['i'])-`f '`(t['i'])*t['i'],'i'=0..'n';\n`Solve L`['i']('t ')=0,` or equivalently`,` t`['i'+1]=` t`['i']-'f'(` t`['i'])/'`f '`'(` t`['i']);\nfor i from 0 to n do\nprint(`___________________ `||`Itera tion `||i||` ____________________`):\nprint(` t`[i]=t[i],` L`[i](t)=L[ i](t));\nprint(`Solve L`[i](t)=0,` or equivalently`,` t`[i+1]=` t`[i]- 'f'(` t`[i])/'`f '`'(` t`[i]));\n#print(evalf(t[i]-f(t[i])/D(f)(t[i])) );\nprint(` t`[i+1]=t[i+1],` f`(` t`[i+1])=f(t[i+1]));\nod:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 54 "Newton-Raphson estimation of zeroes of f(t): Example 2 " }}{EXCHG {PARA 0 "" 0 "" {TEXT 259 39 "Click in the red area and pre ss [Enter]" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2823 "rest art:\nf:=t->t^2/2-ln(1+t): # enter here\ntleft:=0: # enter \+ here \ntright:=4: # enter here\nybot:=-3: # enter here \nytop:=3: # enter here\nt0:=.8: # ente r start value of t here\nn:=10: # enter number of iterat ions here\nm:=ytop: \ni:='i':t:='t':b:='b':\nt[0]:=t0:\nfor i from 0 \+ to n do\nb[i]:=f(t[i])-D(f)(t[i])*t[i]:\nL[i]:=t->evalf(D(f)(t[i])*t+b [i]):\nt[i+1]:=fsolve(L[i](t)=0,t):\nod:\nfor i from 0 to n do\nistr:= convert(i,name):\nListr:=` t`||(i+1)||` = solution L`||i||`(t) = 0`: \ntt[i]:=evalf(t[i],8):\ntstr:=convert(tt[i],name):\nftt[i]:=evalf(f(t [i]),8):\nftstr:=convert(ftt[i],name):\nTitle[i]:=`Newton-Raphson esti mation of zeroes of a function f(t)`:\nitxtplot[i]:=plots[textplot]([0 ,m-.02*m,\n ` i = `||istr],\n \+ font=[HELVETICA,BOLD,12],\n color=bl ue,align=\{RIGHT\}):\nttxtplot[i]:=plots[textplot]([0,m-.12*m,\n \+ ` t`||istr||` = `||tstr],\n \+ font=[HELVETICA,BOLD,12],\n color=blue,alig n=\{RIGHT\}):\nfttxtplot[i]:=plots[textplot]([0,m-.22*m,\n \+ ` f(`||`t`||istr||`) = `||ftstr],\n \+ font=[HELVETICA,BOLD,12],\n color=blue,al ign=\{RIGHT\}):\nLitxtplot[i]:=plots[textplot]([0,m-.32*m,\n \+ Listr],\n font=[HELVETICA ,BOLD,12],\n color=red,align=\{RIGHT\}):\npp[i ]:=plot([t,L[i](t),t=tleft..tright],color=red,\n titlefont= [HELVETICA,DEFAULT,14],\n title=``||Title[i]):\nppp[i]:=plo t([[t[i],0],[t[i],f(t[i])]],linestyle=2,color=blue):\npppp[i]:=plot([[ t[i+1],0],[t[i+1],f(t[i+1])]],linestyle=2,color=red):\np[0]:=plot(f(t) ,t=tleft..tright,color=blue,\n titlefont=[HELVETICA,DEFAULT ,14],\n title=``||Title[i]):\np[i+1]:=plots[display]([pppp[ i],ppp[i],pp[i],p[0],\n itxtplot[i],ttxtplot[i] ,fttxtplot[i]],Litxtplot[i]):\nend:\nplotfcns:=plots[display](\n \+ [seq(p[i],i=1..n+1)],\n insequence=true,\n \+ #scaling=constrained,\n labels=[``,``],\n view =[tleft..tright,ybot..ytop]):\nplotfcns;\n'f'(t)=f(t),`f '`(t)=D(f)(t) ;\n`Estimate zeroes of `*'f'(t)*`using Newton-Raphson Method:`;``;\nL[ 'i'](t)=`f '`(t['i'])*t+b['i'],` b`['i']='f'(t['i'])-`f '`(t['i'])*t[ 'i'],'i'=0..'n';\n`Solve L`['i']('t')=0,` or equivalently`,` t`['i'+1] =` t`['i']-'f'(` t`['i'])/'`f '`'(` t`['i']);\nfor i from 0 to n do\np rint(`___________________ `||`Iteration `||i||` ____________________`) :\nprint(` t`[i]=t[i],` L`[i](t)=L[i](t));\nprint(`Solve L`[i](t)=0,` \+ or equivalently`,` t`[i+1]=` t`[i]-'f'(` t`[i])/'`f '`'(` t`[i]));\n#p rint(evalf(t[i]-f(t[i])/D(f)(t[i])));\nprint(` t`[i+1]=t[i+1],` f`(` t `[i+1])=f(t[i+1]));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 54 "Newton-Raphson estimatio n of zeroes of f(t): Example 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT 260 39 "Click in the red area and press [Enter]" }{TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2809 "restart:\nf:=t->t^7-6*t^5+8*t^3: # enter h ere\ntleft:=-2.2: # enter here \ntright:=2.2: # enter here\n ybot:=-6: # enter here \nytop:=6: # enter here\nt0:=- 1.2: # enter start value of t here\nn:=10: # enter \+ number of iterations here\nm:=ytop: \ni:='i':t:='t':b:='b':\nt[0]:=t0 :\nfor i from 0 to n do\nb[i]:=f(t[i])-D(f)(t[i])*t[i]:\nL[i]:=t->eval f(D(f)(t[i])*t+b[i]):\nt[i+1]:=fsolve(L[i](t)=0,t):\nod:\nfor i from 0 to n do\nistr:=convert(i,name):\nListr:=` t`||(i+1)||` = solution L `||i||`(t) = 0`:\ntt[i]:=evalf(t[i],8):\ntstr:=convert(tt[i],name):\nf tt[i]:=evalf(f(t[i]),8):\nftstr:=convert(ftt[i],name):\nTitle[i]:=`New ton-Raphson estimation of zeroes of a function f(t)`:\nitxtplot[i]:=pl ots[textplot]([0,m-.02*m,\n ` i = `||istr], \n font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nttxtplot[i]:=plots[textplot]([0 ,m-.12*m,\n ` t`||istr||` = `||tstr],\n \+ font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nfttxtplot[i]:=plots[textplot]([0,m-.22 *m,\n ` f(`||`t`||istr||`) = `||ftstr],\n \+ font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nLitxtplot[i]:=plots[textplot]([0,m-. 32*m,\n Listr],\n \+ font=[HELVETICA,BOLD,12],\n color=red,align= \{RIGHT\}):\npp[i]:=plot([t,L[i](t),t=tleft..tright],color=red,\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[ i]):\nppp[i]:=plot([[t[i],0],[t[i],f(t[i])]],linestyle=2,color=blue): \npppp[i]:=plot([[t[i+1],0],[t[i+1],f(t[i+1])]],linestyle=2,color=red) :\np[0]:=plot(f(t),t=tleft..tright,color=blue,\n titlefont= [HELVETICA,DEFAULT,14],\n title=``||Title[i]):\np[i+1]:=plo ts[display]([pppp[i],ppp[i],pp[i],p[0],\n itxtp lot[i],ttxtplot[i],fttxtplot[i]],Litxtplot[i]):\nend:\nplotfcns:=plots [display](\n [seq(p[i],i=1..n+1)],\n insequence= true,\n #scaling=constrained,\n labels=[``,``], \n view=[tleft..tright,ybot..ytop]):\nplotfcns;\n'f'(t)=f(t ),`f '`(t)=D(f)(t);\n`Estimate zeroes of `*'f'(t)*`using Newton-Raphso n Method:`;``;\nL['i'](t)=`f '`(t['i'])*t+b['i'],` b`['i']='f'(t['i'] )-`f '`(t['i'])*t['i'],'i'=0..'n';\n`Solve L`['i']('t')=0,` or equival ently`,` t`['i'+1]=` t`['i']-'f'(` t`['i'])/'`f '`'(` t`['i']);\nfor i from 0 to n do\nprint(`___________________ `||`Iteration `||i||` ____ ________________`):\nprint(` t`[i]=t[i],` L`[i](t)=L[i](t));\nprint(`S olve L`[i](t)=0,` or equivalently`,` t`[i+1]=` t`[i]-'f'(` t`[i])/'`f \+ '`'(` t`[i]));\n#print(evalf(t[i]-f(t[i])/D(f)(t[i])));\nprint(` t`[i+ 1]=t[i+1],` f`(` t`[i+1])=f(t[i+1]));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 57 "Newton -Raphson estimation of zeroes of f(t): My Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT 261 39 "Click in the red area and press [Enter]" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2788 "restart:\nf:=t->sin(t): # enter here\ntleft:=-6: # enter here \ntright:=6: # ente r here\nybot:=-2: # enter here \nytop:=2: # enter here\nt 0:=2: # enter start value of t here\nn:=10: # enter \+ number of iterations here\nm:=ytop: \ni:='i':t:='t':b:='b':\nt[0]:=t0 :\nfor i from 0 to n do\nb[i]:=f(t[i])-D(f)(t[i])*t[i]:\nL[i]:=t->eval f(D(f)(t[i])*t+b[i]):\nt[i+1]:=fsolve(L[i](t)=0,t):\nod:\nfor i from 0 to n do\nistr:=convert(i,name):\nListr:=` t`||(i+1)||` = solution L `||i||`(t) = 0`:\ntt[i]:=evalf(t[i],8):\ntstr:=convert(tt[i],name):\nf tt[i]:=evalf(f(t[i]),8):\nftstr:=convert(ftt[i],name):\nTitle[i]:=`New ton-Raphson estimation of zeroes of a function f(t)`:\nitxtplot[i]:=pl ots[textplot]([0,m-.02*m,\n ` i = `||istr], \n font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nttxtplot[i]:=plots[textplot]([0 ,m-.12*m,\n ` t`||istr||` = `||tstr],\n \+ font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nfttxtplot[i]:=plots[textplot]([0,m-.22 *m,\n ` f(`||`t`||istr||`) = `||ftstr],\n \+ font=[HELVETICA,BOLD,12],\n \+ color=blue,align=\{RIGHT\}):\nLitxtplot[i]:=plots[textplot]([0,m-. 32*m,\n Listr],\n \+ font=[HELVETICA,BOLD,12],\n color=red,align= \{RIGHT\}):\npp[i]:=plot([t,L[i](t),t=tleft..tright],color=red,\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=``||Title[ i]):\nppp[i]:=plot([[t[i],0],[t[i],f(t[i])]],linestyle=2,color=blue): \npppp[i]:=plot([[t[i+1],0],[t[i+1],f(t[i+1])]],linestyle=2,color=red) :\np[0]:=plot(f(t),t=tleft..tright,color=blue,\n titlefont= [HELVETICA,DEFAULT,14],\n title=``||Title[i]):\np[i+1]:=plo ts[display]([pppp[i],ppp[i],pp[i],p[0],\n itxtp lot[i],ttxtplot[i],fttxtplot[i]],Litxtplot[i]):\nend:\nplotfcns:=plots [display](\n [seq(p[i],i=1..n+1)],\n insequence= true,\n #scaling=constrained,\n labels=[``,``], \n view=[tleft..tright,ybot..ytop]):\nplotfcns;\n'f'(t)=f(t ),`f '`(t)=D(f)(t);\n`Estimate zeroes of `*'f'(t)*`using Newton-Raphso n Method:`;``;\nL['i'](t)=`f '`(t['i'])*t+b['i'],` b`['i']='f'(t['i'] )-`f '`(t['i'])*t['i'],'i'=0..'n';\n`Solve L`['i']('t')=0,` or equival ently`,` t`['i'+1]=` t`['i']-'f'(` t`['i'])/'`f '`'(` t`['i']);\nfor i from 0 to n do\nprint(`___________________ `||`Iteration `||i||` ____ ________________`):\nprint(` t`[i]=t[i],` L`[i](t)=L[i](t));\nprint(`S olve L`[i](t)=0,` or equivalently`,` t`[i+1]=` t`[i]-'f'(` t`[i])/'`f \+ '`'(` t`[i]));\n#print(evalf(t[i]-f(t[i])/D(f)(t[i])));\nprint(` t`[i+ 1]=t[i+1],` f`(` t`[i+1])=f(t[i+1]));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 57 "Newton -Raphson estimation of zeroes of f(t): My Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 39 "Click in the red area and press [Enter]" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2823 "restart:\nf:=t->t^2/2-l n(1+t): # enter here\ntleft:=0: # enter here \ntright:=4: \+ # enter here\nybot:=-3: # enter here \nytop:=3: \+ # enter here\nt0:=.8: # enter start value of t \+ here\nn:=10: # enter number of iterations here\nm:=ytop: \ni:='i':t:='t':b:='b':\nt[0]:=t0:\nfor i from 0 to n do\nb[i]:=f(t[ i])-D(f)(t[i])*t[i]:\nL[i]:=t->evalf(D(f)(t[i])*t+b[i]):\nt[i+1]:=fsol ve(L[i](t)=0,t):\nod:\nfor i from 0 to n do\nistr:=convert(i,name):\nL istr:=` t`||(i+1)||` = solution L`||i||`(t) = 0`:\ntt[i]:=evalf(t[i] ,8):\ntstr:=convert(tt[i],name):\nftt[i]:=evalf(f(t[i]),8):\nftstr:=co nvert(ftt[i],name):\nTitle[i]:=`Newton-Raphson estimation of zeroes of a function f(t)`:\nitxtplot[i]:=plots[textplot]([0,m-.02*m,\n \+ ` i = `||istr],\n font=[HEL VETICA,BOLD,12],\n color=blue,align=\{RIGHT\}) :\nttxtplot[i]:=plots[textplot]([0,m-.12*m,\n \+ ` t`||istr||` = `||tstr],\n font=[HELVETICA, BOLD,12],\n color=blue,align=\{RIGHT\}):\nfttx tplot[i]:=plots[textplot]([0,m-.22*m,\n ` f( `||`t`||istr||`) = `||ftstr],\n font=[HELVETIC A,BOLD,12],\n color=blue,align=\{RIGHT\}):\nLi txtplot[i]:=plots[textplot]([0,m-.32*m,\n \+ Listr],\n font=[HELVETICA,BOLD,12],\n \+ color=red,align=\{RIGHT\}):\npp[i]:=plot([t,L[i](t), t=tleft..tright],color=red,\n titlefont=[HELVETICA,DEFAULT, 14],\n title=``||Title[i]):\nppp[i]:=plot([[t[i],0],[t[i],f (t[i])]],linestyle=2,color=blue):\npppp[i]:=plot([[t[i+1],0],[t[i+1],f (t[i+1])]],linestyle=2,color=red):\np[0]:=plot(f(t),t=tleft..tright,co lor=blue,\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=``||Title[i]):\np[i+1]:=plots[display]([pppp[i],ppp[i],pp[i],p[0 ],\n itxtplot[i],ttxtplot[i],fttxtplot[i]],Litx tplot[i]):\nend:\nplotfcns:=plots[display](\n [seq(p[i],i=1 ..n+1)],\n insequence=true,\n #scaling=constrai ned,\n labels=[``,``],\n view=[tleft..tright,ybo t..ytop]):\nplotfcns;\n'f'(t)=f(t),`f '`(t)=D(f)(t);\n`Estimate zeroes of `*'f'(t)*`using Newton-Raphson Method:`;``;\nL['i'](t)=`f '`(t['i' ])*t+b['i'],` b`['i']='f'(t['i'])-`f '`(t['i'])*t['i'],'i'=0..'n';\n` Solve L`['i']('t')=0,` or equivalently`,` t`['i'+1]=` t`['i']-'f'(` t` ['i'])/'`f '`'(` t`['i']);\nfor i from 0 to n do\nprint(`_____________ ______ `||`Iteration `||i||` ____________________`):\nprint(` t`[i]=t[ i],` L`[i](t)=L[i](t));\nprint(`Solve L`[i](t)=0,` or equivalently`,` \+ t`[i+1]=` t`[i]-'f'(` t`[i])/'`f '`'(` t`[i]));\n#print(evalf(t[i]-f(t [i])/D(f)(t[i])));\nprint(` t`[i+1]=t[i+1],` f`(` t`[i+1])=f(t[i+1])); \nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Filename: ExploreCalc04B.mws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Copyright 2005, All Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 44 "Permission is granted to use and modify for " }}{PARA 0 "" 0 "" {TEXT -1 37 "academic and non-commercial purposes." }}{PARA 0 "" 0 "" {TEXT -1 13 "Dr. John Pais" }}{PARA 0 "" 0 "" {TEXT -1 28 "Mathematics Department-MICDS" }}{PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@micds.o rg or pais@kinetigram.com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: http:/ /kinetigram.com/micds" }}{PARA 0 "" 0 "" {TEXT -1 37 "________________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }