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\"\"\"\"#CF(*.,(%\"xGF(%#x0G!\"\"%\"hGF(F(,&F,F(F-F.F(,(F,F(F-F.F/F.F(
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#\"\"\"\"#7F(*&%\"hG!\"\"-%\"yG6#,&%#x0GF(*&\"\"#F(F+F(F,F(F(F(*&#F3\"
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{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+BOcOa!\"*$!#LF%" }}}{EXCHG {PARA 
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" {MPLTEXT 1 0 200 "lagrange:=proc(xn)local n,pol,i;   n:=nops(xn);   \+
pol:=0;   for i to n do\npol:=pol+mul(x-xn[k],k=1..i-1)*mul(x-xn[k],k=
i+1..n)*y(xn[i])\n/mul(xn[i]-xn[k],k=1..i-1)/mul(xn[i]-xn[k],k=i+1..n)
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\"\"\"\"\"#F(**,(%\"xGF(%#x0G!\"\"%\"hGF.F(,(F,F(F-F.*&F)F(F/F(F.F(-%
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{MPLTEXT 1 0 29 "rule:=eval(diff(pol,x),x=x0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ruleG,(*&#\"\"$\"\"#\"\"\"*&%\"hG!\"\"-%\"yG6#%#x0GF
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;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*--%\"DG6#%\"yG6#%#x0G\"\"\"*&#F
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!#5$\"*76E-&!\")$\"*OD&RT!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"++
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{XPPMATH 20 "6%$\"+++]7y!#7$\")EVNa!\"($\"(c58\"!\"*" }}{PARA 11 "" 1 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i to 5 do S[i]:=(4*R[i+1]-R[i])
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{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+lzfOa!\"*$\"&4V$F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i to 4 do T[i]:=(8*S[i+1]-S[i])
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "lagrange:=proc(xn)local n,p
ol,i;   n:=nops(xn);   pol:=0;   for i to n do\npol:=pol+mul(x-xn[k],k
=1..i-1)*mul(x-xn[k],k=i+1..n)*y(xn[i])\n/mul(xn[i]-xn[k],k=1..i-1)/mu
l(xn[i]-xn[k],k=i+1..n) end do end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "pol:=lagrange([x0-h,x0,x0+h]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$polG,(*&#\"\"\"\"\"#F(**,&%\"xGF(%#x0G!\"\"F(,(F,F(F
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"> " 0 "" {MPLTEXT 1 0 31 "rule:=eval(diff(pol,x,x),x=x0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%ruleG,(*&-%\"yG6#,&%#x0G\"\"\"%\"hG!\"\"F
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "mtaylor(rule,h,8);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(---%#@@G6$%\"DG\"\"#6#%\"yG6#%#x0G\"\"\"*&#F/\"#
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{XPPMATH 20 "6%$\"++++]7!#5$\"*dD\"e;!\"($!)tk:F!\")" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%$\"++++]i!#6$\"*+5xj\"!\"($!(.4u'!\")" }}{PARA 11 "
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11 "" 1 "" {XPPMATH 20 "6%$\"+++]i:!#6$\"(!RJ;!\"&$!'.4U!\")" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%$\"+++]7y!#7$\"(x5j\"!\"&$!'.z5!\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i to 4 do S[i]:=(4*R[i+1
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20 "6$$\"+HX\"*H;!\")$!(oX0\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"
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\")$!%kTF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+*fp4j\"!\")$\"$-&F%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for i to 3 do T[i]:=(16
*S[i+1]-S[i])/15; print(T[i],T[i]-exct) end do:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+G3(4j\"!\")$\"%J<F%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$$\"+U*o4j\"!\")$!$b\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+5
*p4j\"!\")$\"$8)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for i
 to 2 do U[i]:=(64*T[i+1]-T[i])/63; print(U[i],U[i]-exct) end do:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+7*o4j\"!\")$!$&=F%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$$\"+D*p4j\"!\")$\"$G)F%" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 16 "Third derivative" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 200 "lagrange:=proc(xn)local n,pol,i;   n:=nops(xn);   po
l:=0;   for i to n do\npol:=pol+mul(x-xn[k],k=1..i-1)*mul(x-xn[k],k=i+
1..n)*y(xn[i])\n/mul(xn[i]-xn[k],k=1..i-1)/mul(xn[i]-xn[k],k=i+1..n) e
nd do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pol:=lagrange
([x0-2*h,x0-h,x0,x0+h,x0+2*h]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$
polG,,*&#\"\"\"\"#CF(*.,(%\"xGF(%#x0G!\"\"%\"hGF(F(,&F,F(F-F.F(,(F,F(F
-F.F/F.F(,(F,F(F-F.*&\"\"#F(F/F(F.F(-%\"yG6#,&F-F(*&F4F(F/F(F.F(F/!\"%
F(F(*&#F(\"\"'F(*.,(F,F(F-F.*&F4F(F/F(F(F(F0F(F1F(F2F(-F66#,&F-F(F/F.F
(F/F:F(F.*&#F(\"\"%F(*.F?F(F+F(F1F(F2F(-F66#F-F(F/F:F(F(*&#F(F=F(*.F?F
(F+F(F0F(F2F(-F66#,&F-F(F/F(F(F/F:F(F.*&F'F(*.F?F(F+F(F0F(F1F(-F66#,&F
-F(*&F4F(F/F(F(F(F/F:F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "rule:=eval(diff(pol,x,x,x),x=x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%ruleG,**&#\"\"\"\"\"#F(*&%\"hG!\"$-%\"yG6#,&%#x0GF(*&F)F(F+F(!\"
\"F(F(F3*&F+F,-F.6#,&F1F(F+F3F(F(*&F+F,-F.6#,&F1F(F+F(F(F3*&#F(F)F(*&F
+F,-F.6#,&F1F(*&F)F(F+F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "mtaylor(rule,h,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,(---%#@@G6$%\"DG\"\"$6#%\"yG6#%#x0G\"\"\"*&#F/\"\"%F/*&)%\"hG\"\"#F/
---F'6$F)\"\"&F+F-F/F/F/*&#F/\"#SF/*&)F5F2F/---F'6$F)\"\"(F+F-F/F/F/" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "exct:=evalf(eval(diff(exp
(x^2),x,x,x),x=1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exctG$\"+cOc
Oa!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x0:=1:   y:=x->ex
p(x^2.):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "h:=1/4.:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for i to 5 do R[i]:=rule; h:=h/2;  \+
print(h,rule,exct-rule) end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"
++++]7!#5$\")!4*yd!\"'$!*W`MU$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%
$\"++++]i!#6$\")[1?b!\"'$!)W6]$)!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6%$\"++++DJ!#6$\"(<tX&!\"&$!)WLv?!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6%$\"+++]i:!#6$\"'pTa!\"%$!(Wj7&!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6%$\"+++]7y!#7$\"&\"Qa!\"$$!(Wj`\"!\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "for i to 4 do S[i]:=(4*R[i+1]-R[i])/3.;print(S[i],S[i
]-exct) end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+o].)Q&!\")$!))e
G&[F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+)R$yLa!\")$!(e-y#F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+l5SOa!\")$!'\"fi\"F%" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$$\"+**4[Oa!\")$!&dE)F%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 70 "for i to 3 do T[i]:=(16*S[i+1]-S[i])/15; print(T
[i],T[i]-exct) end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+(GLoV&!
\")$\"'J'p#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+wbdOa!\")$\"&?>\"
F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+Gj[Oa!\")$!&Gt(F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for i to 2 do U[i]:=(64*T[i+
1]-T[i])/63; print(U[i],U[i]-exct) end do:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+'[rlV&!\")$\"%IyF%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$$\"+7\\[Oa!\")$!&W(yF%" }}}}{MARK "67" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }