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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Evaluating the inverse sine funct
ion from its Maclaurin series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter
 Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:
  25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Graphing partial sums of
 the Maclaurin series for " }{XPPEDIT 18 0 "arcsin(x);" "6#-%'arcsinG6
#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " }{XPPEDIT 
18 0 "arcsin(x);" "6#-%'arcsinG6#%\"xG" }{TEXT -1 4 "  is" }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x) = x+``(1/2)*``(x^
3/3)+``(1/2)*``(3/4)*``(x^5/5)+``(1/2)*``(3/4)*``(5/6)*``(x^7/7)+` . .
 . `;" "6#/-%'arcsinG6#%\"xG,,F'\"\"\"*&-%!G6#*&F)F)\"\"#!\"\"F)-F,6#*
&F'\"\"$F4F0F)F)*(-F,6#*&F)F)F/F0F)-F,6#*&F4F)\"\"%F0F)-F,6#*&F'\"\"&F
@F0F)F)**-F,6#*&F)F)F/F0F)-F,6#*&F4F)F<F0F)-F,6#*&F@F)\"\"'F0F)-F,6#*&
F'\"\"(FOF0F)F)%(~.~.~.~GF)" }{TEXT -1 3 "   " }}{PARA 257 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "`` = x+Sum((2*n-1)!/(n!*(n-1)!*2^(2*n-
1))*x^(2*n+1)/(2*n+1),n = 1 .. infinity);" "6#/%!G,&%\"xG\"\"\"-%$SumG
6$**-%*factorialG6#,&*&\"\"#F'%\"nGF'F'F'!\"\"F'*(-F-6#F2F'-F-6#,&F2F'
F'F3F')F1,&*&F1F'F2F'F'F'F3F'F3)F&,&*&F1F'F2F'F'F'F'F',&*&F1F'F2F'F'F'
F'F3/F2;F'%)infinityGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can make a comparison of finite
 sums approximating " }{XPPEDIT 18 0 "arcsin(x);" "6#-%'arcsinG6#%\"xG
" }{TEXT -1 24 " graphically as follows." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "First we define a function " }
{TEXT 0 2 "AS" }{TEXT -1 84 " with two arguments or input parameters. \+
The degree of the polynomial approximation " }{TEXT 263 7 "AS(n,x)" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "arcsin(x);" "6#-%'arcsinG6#%\"xG" }
{TEXT -1 4 " is " }{XPPEDIT 18 0 "2*n+1;" "6#,&*&\"\"#\"\"\"%\"nGF&F&F
&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 144 "x := 'x': i := 'i': n := 'n':\nAS := (n,x)
 -> x+sum((2*i-1)!/(i!*(i-1)!*2^(2*i-1))*x^(2*i+1)/(2*i+1),i=1..n);\nA
S(1,x);\nAS(2,x);\nAS(3,x);\nAS(4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#ASGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF),&9%\"\"\"-%$sumG6$*.
-%*factorialG6#,&%\"iG\"\"#F/!\"\"F/-F56#F8F:-F56#,&F8F/F/F:F:)F9F7F:)
F.,&F8F9F/F/F/FBF:/F8;F/9$F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,&%\"xG\"\"\"*&#F%\"\"'F%)F$\"\"$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,(%\"xG\"\"\"*&#F%\"\"'F%)F$\"\"$F%F%*&#F*\"#SF%)F$\"\"&F%F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&#F%\"\"'F%)F$\"\"$F%F%*
&#F*\"#SF%)F$\"\"&F%F%*&#F/\"$7\"F%)F$\"\"(F%F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,,%\"xG\"\"\"*&#F%\"\"'F%)F$\"\"$F%F%*&#F*\"#SF%)F$\"\"
&F%F%*&#F/\"$7\"F%)F$\"\"(F%F%*&#\"#N\"%_6F%)F$\"\"*F%F%" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 25 "series(arcsin(x),x=0,10);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#+/%\"xG\"\"\"F%#F%\"\"'\"\"$#F(\"#S\"\"&#F+\"$7\"\"
\"(#\"#N\"%_6\"\"*-%\"OG6#F%\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(AS(12,0.5));\nevalf(ar
csin(0.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bx)fB&!#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+cx)fB&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Th
e degree 1, 3 and 5 approximations  " }{XPPEDIT 18 0 "y = x;" "6#/%\"y
G%\"xG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y = x+x^3/6;" "6#/%\"yG,&%\"
xG\"\"\"*&F&\"\"$\"\"'!\"\"F'" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "y \+
= x+x^3/6+3*x^5/40;" "6#/%\"yG,(%\"xG\"\"\"*&F&\"\"$\"\"'!\"\"F'*(F)F'
*$F&\"\"&F'\"#SF+F'" }{TEXT -1 28 "  can be plotted as follows." }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arctan(x);" "6#/%\"
yG-%'arctanG6#%\"xG" }{TEXT -1 33 " is also plotted for comparison. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "AS := (n,x) -> x+sum((2*i-1
)!/(i!*(i-1)!*2^(2*i-1))*x^(2*i+1)/(2*i+1),i=1..n):\nplot([arcsin(x),A
S(0,x),AS(1,x),AS(2,x)],x=-1.2..1.2,-1.6..1.6,\n        color=[black,r
ed,green,blue],linestyle=[2,1$3],\nnumpoints=100);" }}{PARA 13 "" 1 "
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214 "Successively higher degree Taylor polynomials produce progressive
ly better approximations over a fixed interval centred at the origin.
\nThe next picture shows the graphs of the degree 7, 9 and 11 Taylor p
olynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
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7$Fi]n$\"3jmz7qCbMLF*7$F^^n$\"3a\\S/yVG\"f$F*7$Fc^n$\"3%*RcPY6))\\QF*7
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n$\"30[#o/*yFO\\F*7$F\\`n$\"3?i3cEkr2_F*7$Fa`n$\"3s)ztIv[(*[&F*7$Ff`n$
\"3u*\\]\"oQj\"y&F*7$F[an$\"3i?&*>-T*[0'F*7$Fbbl$\"3c2%Rxd^>N'F*7$Fcan
$\"3!3(=^OyC`mF*7$Fhan$\"33p,0Al]$)pF*7$F]bn$\"3A_8ilma#G(F*7$Fbbn$\"3
@uyI;\"yuj(F*7$Fgbn$\"3qtnr$ebz'zF*7$F\\cn$\"3A#)QP!e5cI)F*7$Facn$\"3#
o\">63#**>o)F*7$Ffcn$\"3u(oF&o+ev!*F*7$F[dn$\"3U]Y!*H[7\\%*F*7$F`dn$\"
3]%3v([-3k)*F*7$Fedn$\"3t)GTyrg*G5F-7$Fjdn$\"3YU5JQ.rx5F-7$Fcfl$\"3$)[
1?aDSA6F-7$Fben$\"3/LU7By-x6F-7$Fgen$\"3i2*)4:i0K7F-7$F\\fn$\"3!yWbfs
\"f!H\"F-7$Fafn$\"3e'>D'y2z`8F-7$Fffn$\"3>%ohR:<PU\"F-7$F[gn$\"32(yZ9(
oz/:F-7$F`gn$\"3K8Oq%)RS'e\"F-7$Fegn$\"3ynib*G:[n\"F-7$Fjgn$\"3M1U%z)o
Dz<F-7$F_hn$\"3g/-'Hsq_*=F-7$Fdhn$\"3sI#o4\"*\\4,#F-7$Fihn$\"3K>0K.'yv
:#F--Fbil6&FdilFbinF_inF_inFcin-%+AXESLABELSG6$Q\"x6\"Q!F\\_q-%%VIEWG6
$;$!#7!\"\"$\"#7Fd_q;$!#;Fd_q$\"#;Fd_q" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "An adaptiv
e graph plotting procedure: " }{TEXT 0 5 "graph" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The procedure in this sec
tion is an alternative to the standard Maple function " }{TEXT 0 4 "pl
ot" }{TEXT -1 195 ", for plotting the graph of a single function. It a
llows control over how the plotting is performed, in a manner which is
 a bit different from the control one has over the standard Maple rout
ine " }{TEXT 0 4 "plot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 120 "In particular, the precision of the \+
floating point calculations is not changed internally, whereas, the Ma
ple procedure " }{TEXT 0 4 "plot" }{TEXT -1 137 " often uses hardware \+
floating point arithmetic whereby the effective precision is increased
 to the equivalent of about 15 decimal digits." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 12 "graph: usage" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 18 "Calling Sequenc
e:\n" }}{PARA 0 "" 0 "" {TEXT 272 4 "    " }{TEXT -1 17 "graph( f, xrn
g )\n" }{TEXT 273 1 "\n" }{TEXT -1 26 "    graph( f, xrng, yrng )" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 269 11 "Parame
ters:" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 3 
"   " }{TEXT 23 4 "f - " }{TEXT -1 83 "    an expression involving a s
ingle variable, say x, or a function x -> f(x)\n\n    " }{TEXT 23 7 "x
rng - " }{TEXT -1 79 "     horizontal plotting range in the form x=a..
b, when x is an expression in x" }}{PARA 0 "" 0 "" {TEXT -1 19 "      \+
             " }{TEXT 256 2 "OR" }{TEXT -1 48 "  in the form a..b when
 f is afunction x -> f(x)" }}{PARA 0 "" 0 "" {TEXT -1 12 "            \+
" }}{PARA 0 "" 0 "" {TEXT -1 4 "    " }{TEXT 23 7 "yrng - " }{TEXT -1 
92 "     vertical range (optional), which can be given in the form c..
d, or in the form y=c..d.\n" }}{PARA 256 "" 0 "" {TEXT 268 12 "Descrip
tion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 14 
"The procedure " }{TEXT 0 5 "graph" }{TEXT -1 22 " plots the graph of \+
a " }{TEXT 258 15 "single function" }{TEXT -1 26 " using an adaptive m
ethod." }}{PARA 15 "" 0 "" {TEXT -1 88 "An even spacing is aimed for a
long sections of the curve which are approximately linear." }}{PARA 
15 "" 0 "" {TEXT -1 79 "More points are plotted along sections of the \+
curve where the graph is bending." }}{PARA 15 "" 0 "" {TEXT -1 107 "An
y maximum or minimum points are located approximately by parabolic int
erpolation and added to the graph.\n" }}{PARA 256 "" 0 "" {TEXT 270 8 
"Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "Adaptive plot
ting will, where necessary, sub-divide the plotting interval in an att
empt to get a good graphical representation of the function. " }}
{PARA 0 "" 0 "" {TEXT -1 111 "By default, this option is set to \"true
\", but it can be turned off by setting the \"adaptive\" option to \"f
alse\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
312 "numpoints=n\nFor non-adaptive plotting the interval for the plot \+
is subdivided into a fixed number of sub-intervals of equal width by m
eans \"numpoints\" points. \nFor adaptive plotting \"numpoints\" contr
ols the spacing of points along the curve, that is, in the direction o
f the curve rather than just horizontally. " }}{PARA 0 "" 0 "" {TEXT 
-1 57 "The spacing between points is generally no greater than  " }
{XPPEDIT 18 0 "plotwidth/numpoints;" "6#*&%*plotwidthG\"\"\"%*numpoint
sG!\"\"" }{TEXT -1 97 ". Note that, in general, the number of points p
lotted could be vastly different from \"numpoints\"." }}{PARA 0 "" 0 "
" {TEXT -1 36 "The default value is \"numpoints=33\"." }}{PARA 0 "" 0 
"" {TEXT -1 12 "\nlinearity=n" }}{PARA 0 "" 0 "" {TEXT -1 82 "\"linear
ity\" controls the tolerance for the allowed deviation from a straight
 line." }}{PARA 0 "" 0 "" {TEXT -1 93 "Along any arc between two point
s on the graph, the curve will generally deviate no more than " }
{TEXT 274 16 "tol * plotheight" }{TEXT -1 62 " from the straight line \+
segment joining the two points, where " }{XPPEDIT 18 0 "tol = 10^(-4)/
linearity;" "6#/%$tolG*&)\"#5,$\"\"%!\"\"\"\"\"%*linearityGF*" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 98 "Thus increasing \"linearity\"
 reduces the tolerance, and so gives more points at bends in the graph
." }}{PARA 0 "" 0 "" {TEXT -1 36 "The default value is \"linearity=10
\"." }}{PARA 0 "" 0 "" {TEXT -1 12 "\nmaxpoints=n" }}{PARA 0 "" 0 "" 
{TEXT -1 108 "This option provides a cut-off for the adaptive subdivis
ion by specifying a minimum horizontal distance of  " }{XPPEDIT 18 0 "
plotwidth/maxpoints" "6#*&%*plotwidthG\"\"\"%*maxpointsG!\"\"" }{TEXT 
-1 65 " between points for the plot. \nThe default is \"maxpoints=1000
\".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 
"plotdata=true/false\nSetting the option \"plotdata\" to \"true\" caus
es the data points to be returned instead of the graph." }}{PARA 0 "" 
0 "" {TEXT -1 61 "This option can also be specified using the word \"p
lot_data\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
267 31 "Available standard plot options" }{TEXT -1 184 ":  color, line
style, thickness, scaling, xtickmarks, ytickmarks, tickmarks, labels, \+
style, symbol, symbolsize, title, axes, font, labelfont, titlefont, ax
esfont, view, labeldirections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a
ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ
e open the subsection, place the cursor anywhere after the prompt [ > \+
 and press [Enter].\nYou can then close up the subsection." }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 21 "graph: implementation" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "graph" {MPLTEXT 1 0 9031 "graph
 := proc(ff,rng)\n   local fx,x,y,t1,t2,eps,xL,yL,xR,yR,h,ymin,ymax,yM
in,yMax,\n      width,height,adaptdiv,xrange,yrange,Options,startoptio
ns,rs,\n      mxpts,lnrty,nmpts,delta,pdat,adptv,curve,dev,m,addmaxmin
,\n      fn,n1,n2,proctype,vars,y1,y2,testvals,aa,bb,i,opt;\n\n   if n
args<2 then\n      error \"at least 2 arguments are required; the basi
c syntax is: 'graph(f(x),x=a..b)'.\"\n   end if;\n\n   # Collect all t
he input data.\n   if type(ff,procedure) or \n       (op(0,ff)=`@@` an
d nops(ff)=2 and type(op(1,ff),procedure)) then\n      proctype := tru
e;\n      if type(rng,range) then\n         rs := rng;\n      else\n  \+
       error \"the 2nd argument, %1, is invalid .. it should have the \+
form 'a..b' to provide a horizontal range over which to plot the graph
 of %1\",rng,ff;\n      end if;\n   elif type(ff,algebraic) then \n   \+
   vars := indets(ff,name) minus indets(ff,realcons);\n      if nops(v
ars)<>1 then \n         if not type(ff,realcons) and not has(indets(ff
),\{Int,Sum\}) then\n            error \"the 1st argument, %1, is inva
lid .. it should be an expression which depends only on a single varia
ble\",ff;\n         end if;\n      else\n         x := op(1,vars);\n  \+
    end if;\n      if type(rng,name=range) then\n         proctype := \+
false;\n         x := op(1,rng);\n         if not member(x,vars) and n
ot type(ff,realcons) then\n            error \"the 1st argument, %1, i
s invalid .. it should be an expression which depends only on the vari
able %2\",ff,x;\n         end if;\n         rs := op(2,rng);\n      el
se\n         error \"the 2nd argument, %1, is invalid .. it should hav
e the form '%2=a..b' to provide a horizontal range over which to plot \+
the graph of %3\",rng,x,ff;\n      end if;\n   else\n      error \"the
 1st argument, %1, is invalid .. it should be an algebraic expression \+
in a single variable, or a numerical valued procedure with a single ar
gument\",ff;      \n   end if;\n   \n   xL := evalf(op(1,rs));\n   xR \+
:= evalf(op(2,rs));\n   if not type(xL,numeric) or not type(xR,numeric
) then\n      error \"each end point of the horizontal range %1 must e
valuate to a numeric\",rs;\n   end if;\n   if xL>=xR then\n      error
 \"2nd argument horizontal range is invalid\";\n   end if;\n   if proc
type then\n      xrange := xL..xR;\n   else\n      xrange := x=xL..xR;
\n   end if;\n\n   startoptions := 3;  \n   yrange := NULL;\n   yMax :
= infinity;\n   yMin := -infinity;\n   if nargs>2 then\n      if type(
args[3],range) or type(args[3],name=range) then\n         startoptions
 := 4;\n         if type(args[3],range) then\n            rs := args[3
]; \n         else\n            rs := op(2,args[3]);\n            y :=
 op(1,args[3]);\n         end if;\n         yMin := evalf(op(1,rs));\n
         yMax := evalf(op(2,rs));\n         if not type(yMin,numeric) \+
or not type(yMax,numeric) then\n            error \"each end point of \+
the vertical range %1 must evaluate to a numeric\",rs;\n         end i
f;\n         if yMin>=yMax then\n            error \"the 3rd argument \+
vertical range is invalid\";\n         end if;\n         if type(args[
3],range) then\n            yrange := yMin..yMax; \n         else\n   \+
         yrange := y=yMin..yMax;\n         end if;\n      end if;\n   \+
end if;\n\n   # Get the options, but first set default\n   nmpts := 33
;\n   lnrty := 10;\n   Options := [];\n   mxpts := 1000;\n   pdat := f
alse;\n   adptv := true;\n\n   if nargs>=startoptions then\n      Opti
ons := [args[startoptions..nargs]];\n      if not type(Options,list(eq
uation)) then\n         error \"each optional argument after the %-1 a
rgument must be an equation\",startoptions-1;\n      end if;\n      if
 hasoption(Options,'adaptive','adptv','Options') then\n         if not
 adptv=true then adptv := false end if;\n      end if;\n      if hasop
tion(Options,'numpoints','nmpts','Options') then\n         if not type
(nmpts,posint) or nmpts<2 then\n            error \"\\\"numpoints\\\" \+
must be an integer greater than 1\"\n         end if;\n      end if;\n
      if hasoption(Options,'linearity','lnrty','Options') then\n      \+
   if not type(lnrty,posint) then\n            error \"\\\"linearity\\
\" must be a positive integer\"\n         end if;\n      end if;\n    \+
  if hasoption(Options,'maxpoints','mxpts','Options') then\n         i
f not type(mxpts,posint) then\n            error \"\\\"maxpoints\\\" m
ust be a positive integer\"\n         end if;\n      end if;\n      if
 hasoption(Options,'plotdata','pdat','Options') or\n         hasoption
(Options,'plot_data','pdat','Options') then\n         if not pdat=true
 then pdat := false end if;\n      end if;\n      for i to nops(Option
s) do\n         opt := op(i,Options); \n         if not member(op(1,op
t),\n            \{'color','colour','linestyle','line_style','thicknes
s',\n            'scaling','xtickmarks','ytickmarks','tickmarks','labe
ls',\n            'style','symbol','symbolsize','title','axes','font',
\n            'labelfont','label_font','titlefont','title_font',\n    \+
        'axesfont','axes_font','view','labeldirections'\}) then\n     \+
       error \"unknown or invalid option: %1\",opt;\n         end if;
\n      end do;               \n   end if; \n\n   # Recursively define
d procedure to construct plotting data.\n   adaptdiv := proc(pL,pR)\n \+
     local x0,x1,x2,y0,y1,y2,p1,dx,dy,divL,divR;\n \n      x0 := pL[1]
;\n      x2 := pR[1];\n\n      x1 := (x0+x2)/2;\n      y1 := evalf(fn(
x1));\n\n      if y1<=yMax and y1>=yMin then\n         # Update estima
te of the height.\n         if y1>ymax then ymax := y1; height := ymax
-ymin end if;\n         if y1<ymin then ymin := y1; height := ymax-ymi
n end if;\n      end if;\n      y0 := pL[2];\n      y2 := pR[2];\n    \+
  p1 := [x1,y1];\n  \n      if not height=0 then\n         dy := (y2-y
0)/(height+height);\n         dev := ((y0+y2)/2-y1)/height;\n      els
e\n         dy := 0;\n         dev := (y0+y2)/2-y1;\n      end if;\n  \+
    dx := (x1-x0)/width;\n      m := dy/dx; # scaled derivative estima
te  \n\n      # Control deviation from being linear and spacing along \+
curve.\n      # As a last resort look at the horizontal spacing.\n    \+
  if (dev*dev<=eps*(1+m*m) and dx*dx+dy*dy<=h) or dx<delta then\n     \+
    return [pL,p1,pR]\n      else\n         divL := adaptdiv(pL,p1);\n
         divR := adaptdiv(p1,pR);\n         return [op(divL),op(2..nop
s(divR),divR)];\n      end if;\n   end proc; # of adaptdiv\n   \n   # \+
Add approximations for max and min points using parabolic interpolatio
n.\n   addmaxmin := proc(curve)\n      local i,x0,x1,x2,x3,y0,y1,y2,y3
,p3,dx1,dx2,dy1,p,q,r,numpoints,crv;\n   \n      crv := curve;\n      \+
numpoints := nops(crv);\n      i := 1;\n      while i<numpoints-2 do\n
         x0 := crv[i,1];\n         y0 := crv[i,2];\n         x1 := crv
[i+1,1];\n         y1 := crv[i+1,2];\n         x2 := crv[i+2,1];\n    \+
     y2 := crv[i+2,2];\n         dx1 := x1-x0;\n         dx2 := x2-x0;
\n         dy1 := y1-y0;\n\n         if dy1*(y2-y1)<0 then\n          \+
  dx2 := x2-x0;\n            r := dx2*dy1;\n            q := dx1*(y2-y
0);\n            p := dx2*r-dx1*q;\n            q := 2*(q-r);\n       \+
     if q > 0 then p := -p end if;\n            q := abs(q);\n        \+
    x3 := x0+p/q;\n            y3 := evalf(fn(x3));\n            if y3
<=yMax and y3>=yMin then\n               # Update estimate of the heig
ht.\n               if y3>ymax then ymax := y3; height := ymax-ymin en
d if;\n               if y3<ymin then ymin := y3; height := ymax-ymin \+
end if;\n            end if;\n            p3 := [x3,y3]; # max or min \+
pt estimate\n            if not x1=x3 then\n               if x1<x3 th
en\n                  crv := [op(1..i+1,crv),p3,op(i+2..numpoints,crv)
];\n               else\n                  crv := [op(1..i,crv),p3,op(
i+1..numpoints,crv)];\n               end if;\n               numpoint
s := numpoints+1;\n               i := i+1;\n            end if;\n    \+
     end if;\n         i := i+1;\n      end do;\n      return crv;\n  \+
 end proc: # of addmaxmin\n\n   if proctype then\n      fn := ff;\n   \+
else\n      # Evaluate any real constants in f\n      fn := unapply(ev
alf(ff),x);\n   end if;\n\n   yL := evalf(fn(xL));\n   yR := evalf(fn(
xR));\n   y1 := evalf(fn(.7101449275*xL+.2898550725*xR));\n   y2 := ev
alf(fn(.381966011*xL+.618033989*xR));\n   testvals := [yL,y1,y2,yR];\n
   if not type(testvals,list(numeric)) or nops(testvals)<>4 then\n    \+
  error \"1st argument %1 does not evaluate to a numeric at some point
, or points, in the plotting interval\",ff;\n   end if;\n\n   width :=
  xR-xL;\n\n   if adptv then   \n      ymin := max(yMin,min(yL,yR));\n
      ymax := min(yMax,max(yL,yR));\n\n      height := ymax-ymin;\n   \+
   eps := evalf(1/lnrty*0.0001);\n      delta := evalf(1/mxpts);\n    \+
  nmpts := iquo(nmpts,2);\n      h := evalf(1/nmpts)^2;\n      curve :
= addmaxmin(adaptdiv([xL,yL],[xR,yR]));\n      if pdat=true then\n    \+
     return curve;\n      else\n         return plot(curve,xrange,yran
ge,op(Options));\n      end if;\n   else\n      nmpts := nmpts-1;\n   \+
   h := width/nmpts;\n      n1 := iquo(nmpts,2);\n      n2 := nmpts-n1
-1;\n      curve := [[xL,yL],seq([xL+i*h,evalf(fn(xL+i*h))],i=1..n1),
\n               seq([xR+(i-n2)*h,evalf(fn(xR+(i-n2)*h))],i=0..n2-1),[
xR,yR]];\n      if pdat=true then\n         return curve;\n      else
\n         return plot(curve,xrange,yrange,op(Options));\n      end if
;\n   end if;\nend proc: # of graph" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Examples are given in the
 next section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 "A p
rocedure for evaluating the arcsine function from its Maclaurin series
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 696 "arcsin_series := proc(xx::realcons)\n   local x,z,fa
ct,pow,term,sum,eps,k,maxit;\n\n   maxit := Digits*20;\n   eps := Floa
t(2,-Digits);\n\n   x := evalf(xx);\n   if abs(x)>1 then\n      error \+
\"Maclaurin series only converges in the interval [-1,1]\"\n   end if;
\n\n   z := x*x;\n   term := x;\n   sum := term;\n   fact := 1.0;\n   \+
pow := x; \n   for k from 2 to maxit by 2 do\n      pow := pow*z;\n   \+
   fact := fact*(k-1)/k;\n      term := fact*pow/(k+1);\n      sum := \+
sum + term;\n      if abs(term)<=eps*abs(sum) then break end if;\n   e
nd do;\n  \n   if k>maxit then\n      print(`sum of`,k-1,`terms of ser
ies is`,sum);\n      error \"reached maximum iterations without conver
gence\"\n   end if;\n\n   evalf(sum);\nend proc:" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "When using the default pr
ecision of 10 digits, we can get about 10 digit accuracy in computing \+
" }{XPPEDIT 18 0 "arcsin(x);" "6#-%'arcsinG6#%\"xG" }{TEXT -1 27 " usi
ng this procedure with " }{XPPEDIT 18 0 "-4/5 <= x;" "6#1,$*&\"\"%\"\"
\"\"\"&!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= 4/5;" "6#1%!G*&\"\"%\"\"\"
\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xx := 0.8;\nevalf(arcsin_series(xx)
);\nevalf(arcsin(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\")
!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!=_HF*!#5" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+!=_HF*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 43 "We can check the accuracy of the procedur
e " }{TEXT 0 13 "arcsin_series" }{TEXT -1 12 " as follows." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot
(arcsin(x)-'evalf[10]@arcsin_series'(x),x=-0.84..0.84,color=blue);" }}
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.<)F*F_is7$$\"3]+++](=n=)F*F_is7$$\"3+++++]7.#)F*F_is7$$\"3^******\\7`
>#)F*Fc_l7$$\"38++++v$fB)F*Ffz7$$\"3k******\\PM_#)F*Fc_l7$$\"3E+++++vo
#)F*Fc_l7$$\"3y******\\i:&G)F*Ffft7$$\"3W+++7yD*G)F*Ffft7$$\"3S++++Dc,
$)F*F_is7$$\"3\"*******\\(ozJ)F*F_is7$$\"3%******RyP@L)F*F\\y7$$\"3`++
++]PM$)F*$\"3G+++++++!)F-7$$\"3/+++]7y]$)F*Ffz7$$\"3-+++%fLpN)F*Ffz7$$
\"3c*******\\(=n$)F*Fc_l7$$\"3=+++]Pf$Q)F*F\\y7$$\"3o*************R)F*
F_is-%'COLOURG6&%$RGBGF^rF^r$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fi^
u-%%VIEWG6$;$!#%)!\"#$\"#%)Fa_u%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 35 "Argument reduction to the interval " }
{XPPEDIT 18 0 "[0, 1/sqrt(2)];" "6#7$\"\"!*&\"\"\"F&-%%sqrtG6#\"\"#!\"
\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 56 "We can reduce the interval over which we \+
need to define " }{XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" }
{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[0, 1/sqrt(2)];" "6#7$
\"\"!*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 23 " by using the formu
la: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(-x)=-a
rcsin(x)" "6#/-%'arcsinG6#,$%\"xG!\"\",$-F%6#F(F)" }{TEXT -1 13 " ----
--- (i)," }}{PARA 257 "" 0 "" {TEXT -1 10 "          " }{TEXT 265 12 "
____________" }{TEXT -1 26 "                          " }}{PARA 258 "
" 0 "" {TEXT -1 10 " and, for " }{XPPEDIT 18 0 "x>=0" "6#1\"\"!%\"xG" 
}{TEXT -1 15 ", the formula  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x)=Pi/2-arcsin
(sqrt(1-x^2))" "6#/-%'arcsinG6#%\"xG,&*&%#PiG\"\"\"\"\"#!\"\"F+-F%6#-%
%sqrtG6#,&F+F+*$F'F,F-F-" }{TEXT -1 16 " ------- (ii).  " }}{PARA 257 
"" 0 "" {TEXT -1 1 " " }{TEXT 266 17 "_________________" }{TEXT -1 21 
"                     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 46 "To illustrate this idea, we define a function " }
{TEXT 0 5 "asin1" }{TEXT -1 48 " using Maple's arcsine function on the
 interval " }{XPPEDIT 18 0 "[0, 1/sqrt(2)];" "6#7$\"\"!*&\"\"\"F&-%%sq
rtG6#\"\"#!\"\"" }{TEXT -1 58 ", but ensure that it gives no value out
side this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "asin1 := \+
x -> if x>=0 and x<=0.7071067810 then arcsin(x) else FAIL end if;\nplo
t('asin1'(x),x=-0.2..1,thickness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&asin1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(@%31\"\"!9$1F0$\"+5y1rq
!#5-%'arcsinG6#F0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 
248 {PLOTDATA 2 "6&-%'CURVESG6$7M7$$\"3a+++g&zOb&!#@$\"3%)R-\\X)zOb&F*
7$$\"3'******>6o$[8!#?$\"3?tvd?&o$[8F07$$\"32+++o#o89#F0$\"3<2\"HX!*p8
9#F07$$\"3=+++D%oV$HF0$\"33n$)3OEPMHF07$$\"3S+++P(o._%F0$\"3/A*4=8%Q?X
F07$$\"3=+++]!pj5'F0$\"3X$QB%**pS1hF07$$\"3?+++v'p$y#*F0$\"3#*R!Gk!G]y
#*F07$$\"3%*******Hq.X7!#>$\"3!RlK\")>p]C\"FO7$$\"37+++brVz=FO$\"3Q&=)
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7Fho$\"3%=a@B6$Rk7Fho7$$\"39+++5cZ6:Fho$\"3o^tP$o!H<:Fho7$$\"3))*****z
22*Q<Fho$\"3%4X05-#zZ<Fho7$$\"36+++!4X$4?Fho$\"3'z3E0$z6B?Fho7$$\"3*)*
*****f:WQAFho$\"3%4\"yHr$pvD#Fho7$$\"32+++=_$\\]#Fho$\"3\"z%3Ke***=`#F
ho7$$\"3)*******fs#3u#Fho$\"3o%zW#RxNwFFho7$$\"3s*****z@Q'**HFho$\"3G'
pI=HZl/$Fho7$$\"3%******>X(3YKFho$\"3&**[/$Q/)fI$Fho7$$\"3$******\\PJK
]$Fho$\"39zog>3;zNFho7$$\"3()*****zwp$RPFho$\"3?FpEfL]KQFho7$$\"3A+++D
p2%*RFho$\"3/j*4nw1(3TFho7$$\"3;+++ygkeUFho$\"35:X'pjs\"*R%Fho7$$\"3++
++-V&*)[%Fho$\"3)3M)GK%)GbYFho7$$\"3#******\\\\$pPZFho$\"3(ROrlpgc$\\F
ho7$$\"37+++?am%*\\Fho$\"3iCa93!H)H_Fho7$$\"3k*****\\JigC&Fho$\"3?X$e'
>__AbFho7$$\"3w*****zt,$*[&Fho$\"3&[M%))4!Q3\"eFho7$$\"3/+++XwPfdFho$
\"3-B[Tn!3v8'Fho7$$\"30+++gG0-gFho$\"3(eLO*4udPkFho7$$\"3a******\\/;hi
Fho$\"3oz[<W.ilnFho7$$\"3%******zV]f\\'Fho$\"3W.@^[n^qqFho7$$\"3-+++5z
j_nFho$\"3%f+PQ*GA8uFho7$$\"3=+++k$*RtoFho$\"3Kg=<6P?yvFho7$$\"3K+++=3
;%*pFho$\"3ARJ')[<!eu(Fho7$$\"3/+++Z%=d-(Fho$\"338/wh=0!z(Fho7$$\"3m**
***\\2ws0(Fho$\"3wJ!HX9'\\MyFho7$$\"3#******H[l^1(Fho$\"3UT&>.'zjXyFho
7$%*undefinedGFhx-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FayF`y-%*THICKNESSG
6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fjy-%%VIEWG6$;$!\"#F_y$\"\"\"Fay%(DEFA
ULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now
 we define a function " }{TEXT 0 5 "asin2" }{TEXT -1 7 " using " }
{TEXT 0 5 "asin1" }{TEXT -1 31 " and the formulas (i) and (ii)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
149 "asin2 := proc(x)\n   if x < 0 then return -asin2(-x) end if;\n   \+
if x <=.7071067810 then asin1(x) else evalf(Pi/2)-asin1(sqrt(1-x^2)) e
nd if;\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&asin2Gf*6#%\"x
G6\"F(F(C$@$29$\"\"!O,$-F$6#,$F,!\"\"F3@%1F,$\"+5y1rq!#5-%&asin1G6#F,,
&-%&evalfG6#,$*&\"\"#F3%#PiG\"\"\"FDFD-F:6#-%%sqrtG6#,&FDFD*$)F,FBFDF3
F3F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xx := 0.85;\nevalf(ar
csin2(xx));\nevalf(arcsin(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
xxG$\"#&)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(arcsin2G6#$\"#&)!
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%H&)f,\"!\"*" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 37 "plot('asin2'(x),x=-1..1,thickness=2);" }}
{PARA 13 "" 1 "" {GLPLOT2D 305 296 296 {PLOTDATA 2 "6&-%'CURVESG6$7S7$
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er7$$\"3GKLLe\"*[H7F1$\"32=]A.!3EB\"F17$$\"3I*******pvxl\"F1$\"3aX*y?R
kam\"F17$$\"3#z****\\_qn2#F1$\"3WS([2O'*>4#F17$$\"3U)***\\i&p@[#F1$\"3
$4\">H?<R3DF17$$\"3B)****\\2'HKHF1$\"3G3,!)y>.wHF17$$\"3ElmmmZvOLF1$\"
3^#eBbI)*>S$F17$$\"3i******\\2goPF1$\"3cSm204/kQF17$$\"3UKL$eR<*fTF1$
\"37c!)[>6L!H%F17$$\"3m******\\)Hxe%F1$\"3c%RZb#z8mZF17$$\"3ckm;H!o-*
\\F1$\"3/d_t%)QvC_F17$$\"3y)***\\7k.6aF1$\"30L\"RJ@*[<dF17$$\"3#emmmT9
C#eF1$\"3ikNmY#H[@'F17$$\"33****\\i!*3`iF1$\"3+%yTN7t_v'F17$$\"3%QLLL$
*zym'F1$\"3!Q,M)[W!*)H(F17$$\"3wKLL3N1#4(F1$\"3'*)z)*)*G=P)yF17$$\"3Nm
m;HYt7vF1$\"3=`rt7[*)*\\)F17$$\"3Y*******p(G**yF1$\"3'G$zHt#Gp5*F17$$
\"3]mmmT6KU$)F1$\"3kf5^utQn)*F17$$\"3fKLLLbdQ()F1$\"3IOVz(*43j5F-7$$\"
3[++]i`1h\"*F1$\"3;<,Om\"f#e6F-7$$\"3W++]P?Wl&*F1$\"3i#\\m!Q/\"\\F\"F-
7$$\"\"\"F*$\"3c'*[zEjzq:F--%'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l-%*THICKN
ESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fj[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 
2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Further argument reduct
ion within the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }
{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 
0 "" 0 "" {TEXT -1 63 "Suppose that we decide to use the Maclaurin ser
ies to evaluate " }{XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" }
{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arc
sin(x) = x+``(1/2)*``(x^3/3)+``(1/2)*``(3/4)*``(x^5/5)+``(1/2)*``(3/4)
*``(5/6)*``(x^7/7)+` . . . `;" "6#/-%'arcsinG6#%\"xG,,F'\"\"\"*&-%!G6#
*&F)F)\"\"#!\"\"F)-F,6#*&F'\"\"$F4F0F)F)*(-F,6#*&F)F)F/F0F)-F,6#*&F4F)
\"\"%F0F)-F,6#*&F'\"\"&F@F0F)F)**-F,6#*&F)F)F/F0F)-F,6#*&F4F)F<F0F)-F,
6#*&F@F)\"\"'F0F)-F,6#*&F'\"\"(FOF0F)F)%(~.~.~.~GF)" }{TEXT -1 3 "   \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Sinc
e this series converges rather slowly, it is desirable to reduce the r
ange further within the interval " }{XPPEDIT 18 0 "[0, 1/sqrt(2)];" "6
#7$\"\"!*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 18 " to some interva
l " }{XPPEDIT 18 0 "[0,a]" "6#7$\"\"!%\"aG" }{TEXT -1 8 ", where " }
{TEXT 275 1 "a" }{TEXT -1 11 " less than " }{XPPEDIT 18 0 "1/sqrt(2)" 
"6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 38 "This is possible by using the formula " }}{PARA 257 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x) = arcsin(x[0])+arcsin(x*sq
rt(1-x[0]^2)-x[0]*sqrt(1-x^2));" "6#/-%'arcsinG6#%\"xG,&-F%6#&F'6#\"\"
!\"\"\"-F%6#,&*&F'F.-%%sqrtG6#,&F.F.*$&F'6#F-\"\"#!\"\"F.F.*&&F'6#F-F.
-F46#,&F.F.*$F'F:F;F.F;F." }{TEXT -1 15 " ------- (iii)." }}{PARA 257 
"" 0 "" {TEXT -1 1 " " }{TEXT 262 32 "________________________________
" }{TEXT -1 20 "                    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 45 "This formula comes from the addition fo
rmula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin(alpha-b
eta)=sin(alpha)*cos(beta)-cos(alpha)*sin(beta)" "6#/-%$sinG6#,&%&alpha
G\"\"\"%%betaG!\"\",&*&-F%6#F(F)-%$cosG6#F*F)F)*&-F16#F(F)-F%6#F*F)F+
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "
y = arcsin(x);" "6#/%\"yG-%'arcsinG6#%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y[0] = arcsin(x[0]);" "6#/&%\"yG6#\"\"!-%'arcsinG6#&%\"
xG6#F'" }{TEXT -1 7 ", then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "sin(y-y[0]) = sin(y)*cos(y[0])-cos(y)*sin(y[0]);" "6#/-
%$sinG6#,&%\"yG\"\"\"&F(6#\"\"!!\"\",&*&-F%6#F(F)-%$cosG6#&F(6#F,F)F)*
&-F36#F(F)-F%6#&F(6#F,F)F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "But, since " }{XPPEDIT 18 0 "
sin(y)=x, cos(y)=sqrt(1-x^2), sin(y[0])=x[0]" "6%/-%$sinG6#%\"yG%\"xG/
-%$cosG6#F'-%%sqrtG6#,&\"\"\"F1*$F(\"\"#!\"\"/-F%6#&F'6#\"\"!&F(6#F:" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(y[0])=sqrt(1-x[0]^2)" "6#/-%$c
osG6#&%\"yG6#\"\"!-%%sqrtG6#,&\"\"\"F/*$&%\"xG6#F*\"\"#!\"\"" }{TEXT 
-1 11 ", we have  " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"sin(y-y[0])=x*sqrt(1-x[0]^2) - x[0]*sqrt(1-x^2)" "6#/-%$sinG6#,&%\"yG
\"\"\"&F(6#\"\"!!\"\",&*&%\"xGF)-%%sqrtG6#,&F)F)*$&F06#F,\"\"#F-F)F)*&
&F06#F,F)-F26#,&F)F)*$F0F8F-F)F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "y-y[0]=arcsin(x*sqrt(1-x[0]^2)-x[0]*sqrt(1-x^2))" "6#/,&%\"yG\"\"\"
&F%6#\"\"!!\"\"-%'arcsinG6#,&*&%\"xGF&-%%sqrtG6#,&F&F&*$&F06#F)\"\"#F*
F&F&*&&F06#F)F&-F26#,&F&F&*$F0F8F*F&F*" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "arcsin(x) = arcsin(x[0])+arcsin(x*sqrt(1-x[0]^2)-x[0]*sqrt(1-x^2
))" "6#/-%'arcsinG6#%\"xG,&-F%6#&F'6#\"\"!\"\"\"-F%6#,&*&F'F.-%%sqrtG6
#,&F.F.*$&F'6#F-\"\"#!\"\"F.F.*&&F'6#F-F.-F46#,&F.F.*$F'F:F;F.F;F." }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "Divide the interval " }{XPPEDIT 18 0 "[0, 1/sqrt(2)];" "6
#7$\"\"!*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 20 " into the interv
als " }{XPPEDIT 18 0 "[0, sin(Pi/12)];" "6#7$\"\"!-%$sinG6#*&%#PiG\"\"
\"\"#7!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[sin(Pi/12), 1/sqrt(2
)];" "6#7$-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"*&F)F)-%%sqrtG6#\"\"#F+" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 84 "Once we have defined a (
restricted) function by the Maclaurin series on an interval " }
{XPPEDIT 18 0 "[0,a]" "6#7$\"\"!%\"aG" }{TEXT -1 57 ", we shall automa
tically have it defined on the interval " }{XPPEDIT 18 0 "[-a,a]" "6#7
$,$%\"aG!\"\"F%" }{TEXT -1 31 ", which is symmetrical about 0." }}
{PARA 0 "" 0 "" {TEXT -1 50 "It is now sufficient to find an approxima
tion for " }{XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" }{TEXT -1 
17 " on the interval " }{XPPEDIT 18 0 "[-sin(Pi/12), sin(Pi/12)];" "6#
7$,$-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 33 " o
r approximately [-0.259,0.259]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "evalf(sin(Pi/12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^/>)
e#!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Formula (iii) with  " 
}{XPPEDIT 18 0 "x[0] = 1/2;" "6#/&%\"xG6#\"\"!*&\"\"\"F)\"\"#!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "sqrt(1-x[0]^2) = sqrt(3)/2" "6#/-%%sqr
tG6#,&\"\"\"F(*$&%\"xG6#\"\"!\"\"#!\"\"*&-F%6#\"\"$F(F.F/" }{TEXT -1 
1 " " }{TEXT 264 1 "~" }{TEXT -1 11 " 0.866 and " }{XPPEDIT 18 0 "arcs
in(x[0]);" "6#-%'arcsinG6#&%\"xG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "arcsin(1/2) = Pi/6;" "6#/-%'arcsinG6#*&\"\"\"F(\"\"#!\"\"*&%#PiG
F(\"\"'F*" }{TEXT -1 26 " can be used to calculate " }{XPPEDIT 18 0 "a
rcsin(x)" "6#-%'arcsinG6#%\"xG" }{TEXT -1 18 " in the interval  " }
{XPPEDIT 18 0 "[sin(Pi/12), 1/sqrt(2)];" "6#7$-%$sinG6#*&%#PiG\"\"\"\"
#7!\"\"*&F)F)-%%sqrtG6#\"\"#F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, \+
we define a function " }{TEXT 0 5 "asin3" }{TEXT -1 49 " using Maple's
 arcsine function on the interval  " }{XPPEDIT 18 0 "[0, sin(Pi/12)];
" "6#7$\"\"!-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"" }{TEXT -1 58 ", but ensur
e that it gives no value outside this interval." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "asin3 := x \+
-> if abs(x) <=0.2588190451 then arcsin(x) else FAIL end if;\nplot('as
in3'(x),x=-0.4..0.4,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&asin3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(@%1-%$absG6#9$$\"+^/>)e#!#5
-%'arcsinGF0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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7$$\"3-+++fp'Rm\"F*$\"3w0H@_Gur;F*7$$\"3&******4%>4N=F*$\"3+4Txo3bX=F*
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LOURG6&%$RGBG$\"#5!\"\"$\"\"!FgwFfw-%*THICKNESSG6#\"\"#-%+AXESLABELSG6
$Q\"x6\"Q!F`x-%%VIEWG6$;$!\"%Few$\"\"%Few%(DEFAULTG" 1 2 0 1 10 2 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Now we define a function \+
" }{TEXT 0 5 "asin4" }{TEXT -1 7 " using " }{TEXT 0 5 "asin3" }{TEXT 
-1 64 " and the formula (iii) to extend the definition to the interval
 " }{XPPEDIT 18 0 "[-1, 1];" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 109 "The formuas (i) and (ii) of the previous
 section can be used to extend the definition to the whole real line.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 350 "asin4 := proc(xx)\n   local x,t;\n   x := evalf(xx);\n   if x
<0 then return -asin4(-x) end if;\n   if x<=0.7071067810 then \n      \+
if x>=0 and x<=0.2588190451 then return asin3(x)\n      else\n        \+
 t := evalf(sqrt(3)/2); \n         return evalf(Pi/6)+asin3(x*t-sqrt(1
-x^2)/2);\n      end if;\n   else return evalf(Pi/2)-asin4(sqrt(1-x^2)
) end if;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&asin4Gf*6#%
#xxG6$%\"xG%\"tG6\"F+C%>8$-%&evalfG6#9$@$2F.\"\"!O,$-F$6#,$F.!\"\"F;@%
1F.$\"+5y1rq!#5@%31F5F.1F.$\"+^/>)e#F@O-%&asin3G6#F.C$>8%-F06#,$*&#\"
\"\"\"\"#FS-%%sqrtG6#\"\"$FSFSO,&-F06#,$*&\"\"'F;%#PiGFSFSFS-FI6#,&*&F
MFSF.FSFS*&#FSFTFS-FV6#,&FSFS*$)F.FTFSF;FSF;FSO,&-F06#,$*&FTF;FjnFSFSF
S-F$6#FaoF;F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 92 "The following example shows how this works in the most co
mplicated case where the procedure " }{TEXT 0 5 "asin4" }{TEXT -1 73 "
 is called 3 times, each time with different argument, and the procedu
re " }{TEXT 0 5 "asin3" }{TEXT -1 17 " is called once. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xx := -
0.8;\nprintlevel := 50:\nevalf(asin4(xx));\nprintlevel := 1:\nevalf(ar
csin(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!\")!\"\"" }}
{PARA 9 "" 1 "" {TEXT -1 28 "\{--> enter asin4, args = -.8" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"xG$!\")!\"\"" }}{PARA 9 "" 1 "" {TEXT 
-1 27 "\{--> enter asin4, args = .8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"xG$\"\")!\"\"" }}{PARA 9 "" 1 "" {TEXT -1 37 "\{--> enter evalf/c
onstant/Pi, args = " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" 
}}{PARA 9 "" 1 "" {TEXT -1 56 "<-- exit evalf/constant/Pi (now in asin
4) = 3.141592654\}" }}{PARA 9 "" 1 "" {TEXT -1 27 "\{--> enter sqrt, a
rgs = .36" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++g!#5" }}{PARA 9 "
" 1 "" {TEXT -1 43 "<-- exit sqrt (now in asin4) = .6000000000\}" }}
{PARA 9 "" 1 "" {TEXT -1 36 "\{--> enter asin4, args = .6000000000" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+++++g!#5" }}{PARA 9 "" 1 "" 
{TEXT -1 25 "\{--> enter sqrt, args = 3" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"sG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"gG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"y
G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"$" }}{PARA 9 "" 
1 "" {TEXT -1 39 "<-- exit sqrt (now in asin4) = 3^(1/2)\}" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"tG$\"+SSDg')!#5" }}{PARA 9 "" 1 "" 
{TEXT -1 35 "\{--> enter sqrt, args = .6400000000" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+++++!)!#5" }}{PARA 9 "" 1 "" {TEXT -1 43 "<-- exit \+
sqrt (now in asin4) = .8000000000\}" }}{PARA 9 "" 1 "" {TEXT -1 36 "\{
--> enter asin3, args = .1196152424" }}{PARA 9 "" 1 "" {TEXT -1 37 "\{
--> enter arcsin, args = .1196152424" }}{PARA 9 "" 1 "" {TEXT -1 43 "
\{--> enter evalf/arcsin, args = .1196152424" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xG$\"+CC:'>\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#xrG$\"+CC:'>\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%absxG$\"+C
C:'>\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG$\"+LL-*>\"!#5" }}
{PARA 9 "" 1 "" {TEXT -1 52 "<-- exit evalf/arcsin (now in arcsin) = .
1199023333\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LL-*>\"!#5" }}
{PARA 9 "" 1 "" {TEXT -1 45 "<-- exit arcsin (now in asin3) = .1199023
333\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LL-*>\"!#5" }}{PARA 9 "" 
1 "" {TEXT -1 44 "<-- exit asin3 (now in asin4) = .1199023333\}" }}
{PARA 9 "" 1 "" {TEXT -1 44 "<-- exit asin4 (now in asin4) = .64350110
91\}" }}{PARA 9 "" 1 "" {TEXT -1 44 "<-- exit asin4 (now in asin4) = .
9272952179\}" }}{PARA 9 "" 1 "" {TEXT -1 49 "<-- exit asin4 (now at to
p level) = -.9272952179\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+z@&HF*
!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+!=_HF*!#5" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot('asin
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 72 "An arbitrary precision procedure to evaluate the inverse \+
sine function: " }{TEXT 0 6 "asinAP" }{TEXT -1 2 "  " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "
asinAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2568 "asinAP := proc(xx::realcons)\n   local x,z,s,t,save
Digits,eps,maxit,\n         isneg,flag,k,sum,term,pow,fact;\n  \n\011
\011 # Increase precision for the computation by a few digits\n\011
\011 saveDigits := Digits;\n   Digits := Digits+length(Digits)+1;\n   \+
x := evalf(xx);\n\n   if abs(x)>1 then\n      error \"argument must be
 in the interval [-1,1]\"\n   end if;\n\n   # Handle the argument redu
ction\n   if x<0 then\n      isneg := true;\n      x := -x;\n   else i
sneg := false end if;\n\n   if x<.25881904510252076235 then\n      fla
g := 1;\n   else\n      s := sqrt(1-x*x); # use Maple's square root\n \+
     if x>.96592582628906828675 then\n         flag := 4;\n         x \+
:= s;\n      else\n         t := evalf(root3)*0.5;    \n         if x<
.70710678118654752440 then\n            flag := 2;\n            x := x
*t-s*0.5;\n         else\n            flag := 3;\n            x := s*t
-x*0.5;\n         end if;\n      end if;\n   end if;\n   \n   # Initia
lisation for Maclaurin series loop.\n   eps := Float(1,-saveDigits);\n
   maxit := Digits*3;\n\n   z := x*x;\n   term := x;\n   sum := term;
\n   fact := 1;\n   pow := x; \n   for k from 2 to maxit by 2 do\n    \+
  pow := pow*z;\n      fact := fact*(k-1)/k;\n      term := fact*pow/(
k+1);\n      sum := sum + term;\n      if abs(term)<=eps*abs(sum) then
 break end if;\n   end do;\n  \n   if flag=2 then sum := evalf(Pi)/6+s
um\n   elif flag=3 then sum := evalf(Pi)/3-sum\n   elif flag=4 then su
m := evalf(Pi)/2-sum end if;\n   if isneg then sum := -sum end if;\n\n
   # Return arcsine rounded to the original precision\n   Digits := sa
veDigits;\n   return evalf(sum)\nend proc: # of asinAP\n\n`evalf/const
ant/root3` := proc()\nlocal d,r;\nglobal _root3;\n   if Digits<=55 the
n evalf(1.732050807568877293527446341505872366942805253810380628)\n   \+
elif Digits<=length(op(1,_root3)) then evalf(_root3)\n   else\n      d
 := length(op(1,_root3));\n      r := _root3;\n      while d<Digits do
\n         d := min(d+d-10,Digits+5);\n\011        r := evalf((3/r+r)/
2,d); # Newton iteration\n\011\011    end do;\n      _root3 := evalf(r
,Digits)\n   end if\nend proc:\n\n_root3 := 1.732050807568877293527446
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3055249016700235207111442886959909563657970871684980728994932964842830
20786408603988738697537582317317831:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 28 "Testing the procedure asinAP" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot('asinAP
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0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Di
gits := 300:\nsqrt(24)/5;\nxx:= evalf(%):\nasinAP(xx);\narcsin(xx);\nD
igits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"\"\"&!
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ae=.G@URh>wx#ec/gSQ%p8!$*H" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]l/H
m%yUuHPS^u(p*QUv`IS(\\_k;Z\")30b48,(R\\;#zni'=1VGjn#G%4'e2$4!3H>Y%eEMj
7EP)z)Q,&4S!3[[)[^3<h.CO^&o\"p6&)=&\\PB&p]CaU9(fe1e#[s\"Q*[ED9&G))H)p_
5AIU_C`G=mae=.G@URh>wx#ec/gSQ%p8!$*H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 79 "Digits := 300:\nsqrt(9/10);\nxx:= evalf(%):\nasinAP(xx);\narcs
in(xx);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"
\"\"\"#5!\"\"F'#F&\"\"#F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]l)G^
DlT5c!*31u8LA2$f*et.vPV\"Q#f&)f:Z(42$zDH98]a:AI?\")4ou-Q\\%\\U/jO9nE*)
3=tk.%=H2>3+fAw>UqGS8nl7SLz#z_<99^zQ(o1_&frT?.**3w&\\&RHPd'>:(zO#pY0>n
*)HTSHy2B,4\"Gxq\"*HeUa#)Rsd/\\7!$*H" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#$\"g]l)G^DlT5c!*31u8LA2$f*et.vPV\"Q#f&)f:Z(42$zDH98]a:AI?\")4ou-Q\\%
\\U/jO9nE*)3=tk.%=H2>3+fAw>UqGS8nl7SLz#z_<99^zQ(o1_&frT?.**3w&\\&RHPd'
>:(zO#pY0>n*)HTSHy2B,4\"Gxq\"*HeUa#)Rsd/\\7!$*H" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed comparisons " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " }
{TEXT 0 6 "arcsin" }{TEXT -1 16 " is faster than " }{TEXT 0 6 "asinAP
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 153 "st := time():\nfor i to 500 do asinAP(rand()*
Float(1,-12)) end do:\ntime()-st;\nst := time():\nfor i to 500 do arcs
in(rand()*Float(1,-12)) end do:\ntime()-st;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"%45!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$*>!\"
$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 181 "st := time():\nDigits := 50:\nfor i to 100 do asinAP
(rand()*Float(1,-12)) end do:\ntime()-st;\nst := time():\nfor i to 100
 do arcsin(rand()*Float(1,-12)) end do:\ntime()-st;\nDigits := 10:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%f5!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"$p$!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }