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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 31 "Taylor Polynomial approximation" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ff:=t->1/t;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ff(2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "Taylor Polynomial degree one at x=1 (Straight line), and \+
degree two (Parabola)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "TP1:=t->1+(-1/1^2)*(t-1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "TP2:=t->1+(-1/1^2)*(t-1)+(2/
1^3)*(t-1)^2/2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 40 "Graphics showing the local approximation" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plo
t(\{ff(t),TP1(t),TP2(t)\}, t=0.5..3);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Interpolating Polynomials
: Lagrange Form:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 23 "Polynomials degree one." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P1:=t->(t-x[2])/(x[1]-x[2
])*f[1] + (t-x[1])/(x[2]-x[1])*f[2] ;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "x:=[1,2,4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "x[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=[1/x[1],1/
x[2],1/x[3]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "P1(4);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Polynomials degree two." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "
P2:=t->(t-x[2])*(t-x[3])/((x[1]-x[2])*(x[1]-x[3]))*f[1] + (t-x[1])*(t-
x[3])/((x[2]-x[1])*(x[2]-x[3]))*f[2] + (t-x[1])*(t-x[2])/((x[3]-x[1])*
(x[3]-x[2]))*f[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "P2(3);
" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Coefficients o
f the Lagrange Polynomial of second order." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "c1:=f[1]/((x[1]-x
[2])*(x[1]-x[3]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "c2:=f
[2]/((x[2]-x[1])*(x[2]-x[3]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 35 "c3:=f[3]/((x[3]-x[1])*(x[3]-x[2]));" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 94 "Therefore, the second order Lagrange Polynomial interpo
lating the previous points is given by " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "P2:=t->(t-x[2])*(t-x[3
])*c1 + (t-x[1])*(t-x[3])*c2 + (t-x[1])*(t-x[2])*c3;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Lagrange Polyno
mials and the actual function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(\{1/t,P1(t),P2(t)\}, t=
1..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "Comparing Lagrange and taylor Polynomials:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(
\{1/t,TP1(t),P1(t)\}, t=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "plot(\{1/t,TP2(t),P2(t)\}, t=1..3);" }}}{EXCHG {PARA 0 "> " 0 
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