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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 19 "Taylor polynomials " }}
{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Taylor p
olynomials can be computed by the command  " }{MPLTEXT 1 0 6 "series" 
}{TEXT -1 113 ".    This command computes Taylor series of functions u
p to any given degree.  The  Maple output for the command " }{MPLTEXT 
1 0 6 "series" }{TEXT -1 312 "  has the data type of series.  In parti
cular it is not  a polynomial.  In order to be able to plot the graph \+
of the corresponding Taylor polynomial, we must convert the resulting \+
Taylor series into a polynomial.  That is done by the command convert.
  The following example illustrates the use of the commands.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 83 "f:=unapply(sin(x),x); # define the function for which we compute
 Taylor poylnomials" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "ser
ies(f(x),x=0);  #computes the Taylor series expansion and shows the Ta
ylor polynomial of degree 6 (default degree)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 102 "series(f(x),x=0,10); #computes the Taylor serie
s expansion and shows the Taylor polynomial of degree 9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 110 "The following line computes Taylor polynomials of order  3, 4,
 ..., 15 for the function f and gives the names " }{XPPEDIT 18 0 "P[3]
" "6#&%\"PG6#\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "P[4]" "6#&%\"PG6#
\"\"%" }{TEXT -1 8 ", ... , " }{XPPEDIT 18 0 "P[15]" "6#&%\"PG6#\"#:" 
}{TEXT -1 10 " for them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 
"for n from 3 to 15 do P[n]:=convert(series(f(x),x=0,n+1),polynom):od:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 78 "Let us next plot the function f  together with some \+
of its Taylor polyonmials." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
97 "plot([f(x),P[5],P[10],P[15]],x=-3*Pi..3*Pi,y=-5..5,\ncolor = [blac
k, green,red,blue],thickness=3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The conclusio
n is that " }{TEXT 256 76 "the higher order of a Taylor polynomial is,
 the better the approximation is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 76 "Compute and plot Taylor polynomials of degree 5, 10 an
d 15 for the function " }{XPPEDIT 18 0 "f(x) = cos(x)" "6#/-%\"fG6#%\"
xG-%$cosG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 76 "Compute and plot Taylor po
lynomials of degree 5, 10 and 15 for the function " }{XPPEDIT 18 0 "f(
x) = exp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
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