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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 442 "MAPLE WORKSHEET FOR THE A
NIMATED DISPLAY OF TAYLOR POLYNOMIALS\n\nBased on a posting to sci.mat
h.symbolic on 15 February 1994 by Helmer Aslaksen (mathelmr@nusunix.nu
s.sg)\n\nModified by Douglas Meade (meade@math.scarolina.edu) to work \+
for an arbitrary function (not just sin).\n\nEXAMPLES:\n  taylor_anima
te( sin(x), x=0, x=-5..5, frames=10 );\n  taylor_animate( exp(x), x=2,
 x= 0..5, frames=5 );\n  taylor_animate( ln(x), x=1, x=0.1..3, frames=
10 ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 163 "Example 1. For the following power serie
s, (a) investigate what the interval of cnvergence will be? (b) what d
o you think the 'limit' of this power series will be?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=proc(x,n) sum(x^k/k!,k=0..n) end
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{f(x,3),f(x,5)\}
,x=-5..5,y=-2..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot
(\{f(x,3),exp(x)\},x=-5..5,y=-2..10);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "taylor(sin(x),x=1,5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "p5:=convert(%,polynom);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "p5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot
(\{sin(x),p5(x)\},x=0..3,y=-2..2,thickness=2);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "p:=proc(x,n) sum((x-3)^k/k,k=1..n) end;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p(x,3);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 60 "plot(\{p(x,5),p(x,10),p(x,15)\},x=1..5,y=-10.
.10,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 401 "taylo
r_animate := proc(FN::function, CENTER::name=numeric, DOMAIN::name=ran
ge, FRAMES::name=integer)\n  local VAR, t;\n  global X0, F, p, n;\n  X
0 := rhs(CENTER);\n  VAR := lhs(CENTER);\n  F := unapply(FN,VAR);\n  p
 := (n,x)->subs(X=x,convert(taylor(F(X),X=X0,n),polynom));\n  for n to
 rhs(FRAMES) do print( P[n](VAR)=p(n,VAR) ) od;\n  animate( \{F(x),'p(
trunc(t),x)'\}, DOMAIN, t=1..rhs(FRAMES), FRAMES );\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "taylor_animate(sin(x), x=0, x=-5..5
, frames=10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "taylor_ani
mate(exp(x), x=2, x=0..5, frames=5 );" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 266 "To close, here is an example for you to think about befo
re you execute the command. Some questions you might consider include:
 on what interval should the Taylor polynomials converge to the logari
thm? How will you be able to determine this graphically? Symbolically?
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "taylor_animate(ln(x), x=1, x=0.
1..2.5, frames=10);" }}}}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }