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{SECT 0 {EXCHG {PARA 262 "" 0 "" {TEXT 256 26 "Tips for Maple Instruct
ors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 269 
"Posted with permission of Robert J. Lopez. Copyright 1996 by Robert J
. Lopez. All\nrights reserved. \n\nThis article has been published in \+
MapleTech, Vol 4, NO. 3, 1997. \n\nRobert J. Lopez\nDepartment of Math
ematics\nRose-Hulman Institute of Technology\nTerre Haute, IN 47803 " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 497 "In this, the third column of the series,
 we again illustrate some Maple V Release 4 functionality of significa
nce in the classroom.  In particular, we examine the role technology h
as in shaping pedagogy.  Inescapably, the available technology dictate
s the operable pedagogy.  In the pencil-and-paper world from which we \+
are emerging, the lack of graphics and convenient numerical calculatio
ns, for example, imposed an analytical style on didactics, overshadowi
ng experimentation and investigation." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 238 "In this issue's column, we consider \+
two major topics.  First, we examine the traditional approach to findi
ng the recursion relation for a power series solution of a differentia
l equation and compare that approach to one found in [1], the " }
{TEXT 322 22 "Maple V Flight Manual " }{TEXT -1 231 ".  Then, we explo
re the optimizing property of the Fourier series.  Each of these items
 reflects how the available technology influences the pedagogy.  In pa
rticular, each requires replacing infinite sums with finite sums in Ma
ple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 428 "
In a third section, we look at three Maple functionalities that affect
 the student's view of Maple in the classroom.  First, we consider an \+
error Maple makes in the midst of a lesson on removable singularities.
  Then, we contrast plotting a curve defined vectorially in two and th
ree dimensions.  Finally, we examine some of the difficulties branches
 pose for both students and Maple in the computation of curvature of a
 circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
611 "I continue to hope this column becomes a forum where common probl
ems can be identified and resolved, where useful hints and strategems \+
can be found, and where pedagogical insights can be shared.  Hence, I \+
urge all readers who have their own experiences with Maple in instruct
ion, successes and failures alike, to communicate with me by e-mail (r
.lopez@rose-hulman.edu), fax(812-877-3198), or letter (Math Dept., Ros
e-Hulman Institute of Technology, Terre Haute, IN 47803).  As much as \+
possible, I would like this column to address real issues, from real c
lasses.  Only the column's readers can make that happen." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Series so
lutions of ODEs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 391 "I recently suc
ceeded in getting Maple to replicate the steps I learned 35 years ago \+
for obtaining the recursion relation in a formal power series solution
 to an ordinary differential equation.  In comparison to a simpler tec
hnique appearing in [1], I conclude that the cherished classical forma
lism is probably inferior to a streamlined Maple solution.  I'll prese
nt both, and let you judge." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We write \+
an arbitrary second-order linear equation with variable coefficients.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "q := diff(y(x),x,x) + x^2*diff(y(x),x) + y(x) = 0;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG/,(-%%diffG6$-F(6$-%\"yG6#%\"xG
F/F/\"\"\"*&F/\"\"#F*F0F0F,F0\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Next, write a formal power seri
es as the solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "Y :=sum(a[n]*x^n,n=0..infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG-%$sumG6$*&&%\"aG6#%\"nG\"\"\")%
\"xGF,F-/F,;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 53 "Substitute the series into the differen
tial equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "q1 := eval(subs(y(x)=Y,q));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#q1G/,(-%$sumG6$,&**&%\"aG6#%\"nG\"\"\")%\"xGF/F0F/
\"\"#F2!\"#F0**F,F0F1F0F/F0F2F4!\"\"/F/;\"\"!%)infinityGF0*&F2F3-F(6$*
*F,F0F1F0F/F0F2F6F7F0F0-F(6$*&F,F0F1F0F7F0F9" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "All powers of " }{TEXT 
259 1 "x" }{TEXT -1 28 " submit to a combination of " }{TEXT 257 8 "si
mplify" }{TEXT -1 5 " and " }{TEXT 258 7 "combine" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "map(simplify@combine,q1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
$sumG6$,**(&%\"aG6#%\"nG\"\"\")%\"xG,&F,F-!\"#F-F-F,\"\"#F-*(F)F-F.F-F
,F-!\"\"*()F/,&F,F-F-F-F-F)F-F,F-F-*&F)F-)F/F,F-F-/F,;\"\"!%)infinityG
F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 60 "However, the indices on each separate series on the left of " }
{TEXT 307 2 "q1" }{TEXT -1 131 " must be shifted.  We have to access a
nd massage each such series separately.  Unfortunately, this requires \+
use of the \"low-level\" " }{TEXT 260 2 "op" }{TEXT -1 9 " command." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 65 "for j from 1 to 3 do\ny.j := simplify(combine(op(j,lhs(q1))));\n
od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G-%$sumG6$,&*(&%\"aG6#%\"n
G\"\"\")%\"xG,&F-F.!\"#F.F.F-\"\"#F.*(F*F.F/F.F-F.!\"\"/F-;\"\"!%)infi
nityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G-%$sumG6$*()%\"xG,&%\"n
G\"\"\"F-F-F-&%\"aG6#F,F-F,F-/F,;\"\"!%)infinityG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y3G-%$sumG6$*&&%\"aG6#%\"nG\"\"\")%\"xGF,F-/F,;\"\"!
%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 30 "Shift indices so the power of " }{TEXT 261 1 "x" }
{TEXT -1 48 " in the general term in each series is the same." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 84 "y4 := changevar(n-2=k,y1,k);\ny5 := changevar(n+1=k,y2,k);\ny6 :
= changevar(n=k,y3,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y4G-%$sum
G6$,&*(&%\"aG6#,&\"\"#\"\"\"%\"kGF/F/)%\"xGF0F/F-F.F/*(F*F/F1F/F-F/!\"
\"/F0;!\"#%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y5G-%$sumG
6$*(&%\"aG6#,&%\"kG\"\"\"!\"\"F.F.)%\"xGF-F.F,F./F-;F.%)infinityG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y6G-%$sumG6$*&&%\"aG6#%\"kG\"\"\")%
\"xGF,F-/F,;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 57 "Next, each sum must start at the same v
alue of the index " }{TEXT 263 1 "k" }{TEXT -1 22 ".  Here, we must ha
ve " }{TEXT 264 1 "k" }{TEXT -1 52 " = 1.  In the first sum, the terms
 corresponding to " }{TEXT 265 1 "k" }{TEXT -1 80 " = -2 and -1 are ze
ro.  In the first and third sums, the terms corresponding to " }{TEXT 
308 1 "k" }{TEXT -1 177 " = 0 must be extracted separately.  In Maple,
 however, it is difficult to manipulate the indices in a mathematicall
y meaningful way.  We settle for an artifice permitted by the " }
{TEXT 262 4 "subs" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "y4a := subs(-2=1,y4)
;\ny6a := subs(0=1,y6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y4aG-%$s
umG6$,&*(&%\"aG6#,&\"\"#\"\"\"%\"kGF/F/)%\"xGF0F/F-F.F/*(F*F/F1F/F-F/!
\"\"/F0;F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y6aG-%$sum
G6$*&&%\"aG6#%\"kG\"\"\")%\"xGF,F-/F,;F-%)infinityG" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "I suspect ther
e would be a problem if \"-2\" or \"-1\" appeared in a context other t
han just the summation limits.  Nonetheless, we are ready to combine t
he three transformed series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q2 := combine(y4a+y5+y6a);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G-%$sumG6$,**(&%\"aG6#,&\"\"#\"
\"\"%\"kGF/F/)%\"xGF0F/F-F.F/*(F*F/F1F/F-F/!\"\"*(&F+6#,&F0F/F4F/F/F1F
/F8F/F/*&&F+6#F0F/F1F/F//F0;F/%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We want a single coeffici
ent of " }{XPPEDIT 18 0 "x^k" ")%\"xG%\"kG" }{TEXT -1 9 ".  Hence," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "q3 := factor(q2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q3G-%$su
mG6$*&)%\"xG%\"kG\"\"\",.&%\"aG6#,&\"\"#F,F+F,F2*&F.F,F+F,\"\"$*&F.F,F
+F2F,*&&F/6#,&F+F,!\"\"F,F,F+F,F,F7F:&F/6#F+F,F,/F+;F,%)infinityG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The
 " }{TEXT 269 5 "coeff" }{TEXT -1 57 " command fails here, so we again
 resort to the low-level " }{TEXT 266 2 "op" }{TEXT -1 75 " command to
 access the summand of q3.  (Fortunately, Release 5 will have a " }
{TEXT 324 7 "summand" }{TEXT -1 25 " command to parallel the " }{TEXT 
271 9 "integrand" }{TEXT -1 16 " command of the " }{TEXT 325 7 "studen
t" }{TEXT -1 10 " package.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "q4 := op(1,q3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#q4G*&)%\"xG%\"kG\"\"\",.&%\"aG6#,&\"\"#F)F(F
)F/*&F+F)F(F)\"\"$*&F+F)F(F/F)*&&F,6#,&F(F)!\"\"F)F)F(F)F)F4F7&F,6#F(F
)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Now, " }{TEXT 270 5 "coeff" }{TEXT -1 10 " succeeds." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "q5 \+
:= coeff(q4,x^k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q5G,.&%\"aG6#,
&\"\"#\"\"\"%\"kGF+F**&F&F+F,F+\"\"$*&F&F+F,F*F+*&&F'6#,&F,F+!\"\"F+F+
F,F+F+F1F4&F'6#F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 22 "Finally, we solve for " }{XPPEDIT 18 0 "a[k+2]
" "&%\"aG6#,&%\"kG\"\"\"\"\"#F'" }{TEXT -1 13 " in terms of " }
{XPPEDIT 18 0 "a[k]" "&%\"aG6#%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "a[k-1]" "&%\"aG6#,&%\"kG\"\"\"\"\"\"!\"\"" }{TEXT -1 49 ", resultin
g in the sought-for recursion relation." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "isolate(q5,a[k+2]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,&\"\"#\"\"\"%\"kGF)*&,(*&&F
%6#,&F*F)!\"\"F)F)F*F)F1F.F)&F%6#F*F1F),(F(F)F*\"\"$*$F*F(F)F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Fi
nally, with yet another artifice, we attend to the " }{TEXT 309 1 "k" 
}{TEXT -1 25 " = 0 terms in the series " }{TEXT 267 2 "y4" }{TEXT -1 
5 " and " }{TEXT 268 2 "y6" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "value(subs(infini
ty=0,y4+y6)) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"aG6#\"\"#F
(&F&6#\"\"!\"\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 248 "Rather than continue with the rest of the clas
sical manipulations, we implement a device from [1] wherein a finite s
um replaces the infinite series Y.  The secret is to make the terms in
 the finite sum generic, and to span across the general term, " }
{XPPEDIT 18 0 "x^n" ")%\"xG%\"nG" }{TEXT -1 8 ".  Thus," }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Q := s
um(a[k]*x^k,k=n-1..n+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG,**&
&%\"aG6#,&%\"nG\"\"\"!\"\"F,F,)%\"xGF*F,F,*&&F(6#F+F,)F/F+F,F,*&&F(6#,
&F+F,F,F,F,)F/F7F,F,*&&F(6#,&\"\"#F,F+F,F,)F/F<F,F," }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Again, substitu
te into the differential equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q6 := eval(subs(y(x)=Q,q));
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#q6G/,<**&%\"aG6#,&%\"nG\"\"\"!
\"\"F-F-)%\"xGF+F-F+\"\"#F0!\"#F-**F(F-F/F-F+F-F0F2F.**&F)6#F,F-)F0F,F
-F,F1F0F2F-**F5F-F7F-F,F-F0F2F.**&F)6#,&F,F-F-F-F-)F0F<F-F<F1F0F2F-**F
:F-F=F-F<F-F0F2F.**&F)6#,&F1F-F,F-F-)F0FBF-FBF1F0F2F-**F@F-FCF-FBF-F0F
2F.*&F0F1,***F(F-F/F-F+F-F0F.F-**F5F-F7F-F,F-F0F.F-**F:F-F=F-F<F-F0F.F
-**F@F-FCF-FBF-F0F.F-F-F-*&F(F-F/F-F-*&F5F-F7F-F-*&F:F-F=F-F-*&F@F-FCF
-F-\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 15 "When in doubt, " }{TEXT 272 8 "simplify" }{TEXT -1 32 ". \+
 This will tame all powers of " }{TEXT 273 1 "x" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "q7 := simplify(q6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#q7G/,L
*&&%\"aG6#%\"nG\"\"\")%\"xGF+F,F,*&&F)6#,&F+F,F,F,F,)F.F2F,F,*&&F)6#,&
\"\"#F,F+F,F,)F.F7F,F,*()F.,&F+F,\"\"$F,F,F5F,F+F,F,*(F9F,F0F,F+F,F,*(
F3F,F(F,F+F,F,*(F-F,&F)6#,&F+F,!\"\"F,F,F+F,F,*&F-F,FAF,FD*(FAF,)F.,&F
+F,!\"$F,F,F+F,FI*&FAF,FGF,F8*(FAF,FGF,F+F8F,*&FAF,)F.FCF,F,*(F(F,)F.,
&F+F,!\"#F,F,F+F8F,*(F5F,F-F,F+F,F=*(F5F,F-F,F+F8F,*&F5F,F-F,F8*(F0F,F
MF,F+F,F,*(F0F,FMF,F+F8F,*(F(F,FOF,F+F,FD*&F;F,F5F,F8*&F9F,F0F,F,\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
32 "Free of the baggage encumbering " }{TEXT 274 3 "Sum" }{TEXT -1 2 "
, " }{TEXT 275 5 "coeff" }{TEXT -1 16 " now works.  We " }{TEXT 276 3 
"map" }{TEXT -1 36 " it onto both sides of the equation " }{TEXT 277 
2 "q7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q8 := map(coeff,q7,x^n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#q8G/,.&%\"aG6#%\"nG\"\"\"&F(6#,&F*F+!\"\"
F+F/&F(6#,&\"\"#F+F*F+F3*&F,F+F*F+F+*&F0F+F*F+\"\"$*&F0F+F*F3F+\"\"!" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 
"Finally, solving for " }{XPPEDIT 18 0 "a[n+2]" "&%\"aG6#,&%\"nG\"\"\"
\"\"#F'" }{TEXT -1 47 ", we get, in agreement with our earlier result,
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "isolate(q8,a[n+2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/&%\"aG6#,&\"\"#\"\"\"%\"nGF)*&,(*&&F%6#,&F*F)!\"\"F)F)F*F)F1&F%6#F*
F1F.F)F),(F(F)*$F*F(F)F*\"\"$F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 34 "Optimum property of Fourier series" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 53 "The discovery that for a function f(x) defined on [0
," }{XPPEDIT 18 0 "pi" "I#piG6\"" }{TEXT -1 59 "], the coefficients of
 its Fourier sine series are given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[n]=2/Pi" "/&%\"
bG6#%\"nG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(
f(x)*sin(n*x),x=0..Pi)" "-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$sinG6#*&%\"n
GF*F)F*F*/F);\"\"!%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 125 "traditionally involves manipulating a formal infini
te sum.  In fact, these coefficients are those which minimize the inte
gral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "q := Int((f(x) - Sum(s[n]*sin(n*x),n=1..infinity))^2,
x=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG-%$IntG6$*$,&-%\"fG
6#%\"xG\"\"\"-%$SumG6$*&&%\"sG6#%\"nGF.-%$sinG6#*&F6F.F-F.F./F6;F.%)in
finityG!\"\"\"\"#/F-;\"\"!%#PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 38 "In Maple, however, the differentiati
on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "diff(q,s[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
51 "fails, since Maple does not \"see\" the general term " }{XPPEDIT 
18 0 "s[n]" "&%\"sG6#%\"nG" }{TEXT -1 229 " in the data structure it u
ses to represent the infinite sum.  Thus, the traditional pedagogy use
d here must change if it is to be implemented in Maple.  Typically, th
is change amounts to working with an explicit finite sum as in" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 56 "q1 := Int((f(x) - sum(s[n]*sin(n*x),n=1..3))^2,x=0..Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G-%$IntG6$*$,*-%\"fG6#%\"xG\"\"\"
*&&%\"sG6#F.F.-%$sinGF,F.!\"\"*&&F16#\"\"#F.-F46#,$F-F9F.F5*&&F16#\"\"
$F.-F46#,$F-F@F.F5F9/F-;\"\"!%#PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Pay particular attention to the
 change from " }{TEXT 278 3 "Sum" }{TEXT -1 4 " to " }{TEXT 279 3 "sum
" }{TEXT -1 59 ".  Without that change the following differentiations \+
fail." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "for k from 1 to 3 do\neq.k := value(expand(diff(q1,s[
k]),sin)) = 0;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&-%$in
tG6$*&-%$sinG6#%\"xG\"\"\"-%\"fGF-F//F.;\"\"!%#PiG!\"#*&&%\"sG6#F/F/F5
F/F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,&-%$intG6$*&-%$sinG
6#,$%\"xG\"\"#\"\"\"-%\"fG6#F/F1/F/;\"\"!%#PiG!\"#*&&%\"sG6#F0F1F8F1F1
F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/,&-%$intG6$*&-%$sinG6#,$
%\"xG\"\"$\"\"\"-%\"fG6#F/F1/F/;\"\"!%#PiG!\"#*&&%\"sG6#F0F1F8F1F1F7" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 
"Solving each equation for its single Fourier coefficient yields" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "for k from 1 to 3 do\nisolate(eq.k, s[k]);\nod;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/&%\"sG6#\"\"\",$*&-%$intG6$*&-%$sinG6#%\"xGF'-%\"
fGF0F'/F1;\"\"!%#PiGF'F7!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/&%\"sG6#\"\"#,$*&-%$intG6$*&-%$sinG6#,$%\"xGF'\"\"\"-%\"fG6#F2F3/F2;
\"\"!%#PiGF3F:!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"sG6#\"\"
$,$*&-%$intG6$*&-%$sinG6#,$%\"xGF'\"\"\"-%\"fG6#F2F3/F2;\"\"!%#PiGF3F:
!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "from which we generalize to the familiar result stated ab
ove." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 230 "
Well, having to make this compromise left me feeling I had cheated my \+
students.  So, I thought I'd shift the same computations to a less fam
iliar domain.  I supposed a set of functions p0(x), p1(x), p2(x), ... \+
with two properties:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 9 "1)       " }{XPPEDIT 18 0 "Int(p[n](x)*p[m](x),x=-1..1)=
0,n<>m" "6$/-%$IntG6$*&-&%\"pG6#%\"nG6#%\"xG\"\"\"-&F*6#%\"mG6#F.F//F.
;,$\"\"\"!\"\"\"\"\"\"\"!0F,F3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "2)      " }{XPPEDIT 18 0 "Int(p[k](x)^2,x=
-1..1)=2/(2*k+1)" "/-%$IntG6$*$-&%\"pG6#%\"kG6#%\"xG\"\"#/F-;,$\"\"\"!
\"\"\"\"\"*&\"\"#\"\"\",&*&\"\"#F7F+F7F7\"\"\"F7F3" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "I asked \"Are these two \+
properties enough to reproduce the minimization property of the Fourie
r series?\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 125 "To tell Maple that the functions  p0(x), p1(x), p2(x) have the
se properties, define the following equations for Property (1)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 98 "q01:=Int(p0(x)*p1(x),x=-1..1)=0;\nq02:=Int(p0(x)*p2(x),x=-1..1)=
0;\nq12:=Int(p1(x)*p2(x),x=-1..1)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$q01G/-%$IntG6$*&-%#p0G6#%\"xG\"\"\"-%#p1GF,F./F-;!\"\"F.\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q02G/-%$IntG6$*&-%#p0G6#%\"xG\"\"\"
-%#p2GF,F./F-;!\"\"F.\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q12G/
-%$IntG6$*&-%#p1G6#%\"xG\"\"\"-%#p2GF,F./F-;!\"\"F.\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Then defi
ne the following equations for Property (2)." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "q00:=Int(p0(x)^2,
x=-1..1)=2;\nq11:=Int(p1(x)^2,x=-1..1)=2/3;\nq22:=Int(p2(x)^2,x=-1..1)
=2/5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q00G/-%$IntG6$*$-%#p0G6#%
\"xG\"\"#/F-;!\"\"\"\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$q11G/
-%$IntG6$*$-%#p1G6#%\"xG\"\"#/F-;!\"\"\"\"\"#F.\"\"$" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$q22G/-%$IntG6$*$-%#p2G6#%\"xG\"\"#/F-;!\"\"\"\"\"
#F.\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 179 "Now set up the same measure of performance as used for t
he Fourier series, namely, the integral of the square of the differenc
e between the function f(x) and an approximating sum." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Q := Int
((f(x)-s0*p0(x)-s1*p1(x)-s2*p2(x))^2,x=-1..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"QG-%$IntG6$*$,*-%\"fG6#%\"xG\"\"\"*&%#s0GF.-%#p0GF,
F.!\"\"*&%#s1GF.-%#p1GF,F.F3*&%#s2GF.-%#p2GF,F.F3\"\"#/F-;F3F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "D
ifferentiate the measure of performance with respect to each of the th
ree coefficients s0, s1, s2.  Set the derivatives equal to zero to det
ermine the values of the coefficients that minimize the measure of dev
iation Q." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 68 "eq1 := diff(Q,s0) = 0;\neq2 := diff(Q,s1) = 0;\neq3
 := diff(Q,s2) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/-%$IntG
6$,$*&,*-%\"fG6#%\"xG\"\"\"*&%#s0GF0-%#p0GF.F0!\"\"*&%#s1GF0-%#p1GF.F0
F5*&%#s2GF0-%#p2GF.F0F5F0F3F0!\"#/F/;F5F0\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$eq2G/-%$IntG6$,$*&,*-%\"fG6#%\"xG\"\"\"*&%#s0GF0-%#p
0GF.F0!\"\"*&%#s1GF0-%#p1GF.F0F5*&%#s2GF0-%#p2GF.F0F5F0F8F0!\"#/F/;F5F
0\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/-%$IntG6$,$*&,*-%\"f
G6#%\"xG\"\"\"*&%#s0GF0-%#p0GF.F0!\"\"*&%#s1GF0-%#p1GF.F0F5*&%#s2GF0-%
#p2GF.F0F5F0F<F0!\"#/F/;F5F0\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "Progress solving these equations \+
depends on simplifying them.  The parentheses need to be multiplied ou
t, and any possible integrations done." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq4 := expand(eq1);\neq
5 := expand(eq2);\neq6 := expand(eq3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$eq4G/,*-%$IntG6$*&-%#p0G6#%\"xG\"\"\"-%\"fGF-F//F.;!\"\"F/!\"
#*&%#s0GF/-F(6$*$F+\"\"#F2F/F;*&%#s1GF/-F(6$*&F+F/-%#p1GF-F/F2F/F;*&%#
s2GF/-F(6$*&F+F/-%#p2GF-F/F2F/F;\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$eq5G/,*-%$IntG6$*&-%#p1G6#%\"xG\"\"\"-%\"fGF-F//F.;!\"\"F/!\"#*&
%#s0GF/-F(6$*&-%#p0GF-F/F+F/F2F/\"\"#*&%#s1GF/-F(6$*$F+F=F2F/F=*&%#s2G
F/-F(6$*&F+F/-%#p2GF-F/F2F/F=\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$eq6G/,*-%$IntG6$*&-%#p2G6#%\"xG\"\"\"-%\"fGF-F//F.;!\"\"F/!\"#*&%#s
0GF/-F(6$*&-%#p0GF-F/F+F/F2F/\"\"#*&%#s1GF/-F(6$*&-%#p1GF-F/F+F/F2F/F=
*&%#s2GF/-F(6$*$F+F=F2F/F=\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 160 "No integrations have been done beca
use everything is symbolic.  Maple knows only properties (1) and (2) a
pply, but can't apply them until told.  Do this with a " }{TEXT 282 8 
"simplify" }{TEXT -1 76 " command containing the equations that define
 Property (1) and Property (2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "q0 := simplify(eq4,\{q00,q11
,q22,q01,q02,q12\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "q1 := simpl
ify(eq5,\{q00,q11,q22,q01,q02,q12\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "q2 := simplify(eq6,\{q00,q11,q22,q01,q02,q12\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#q0G/,&-%$IntG6$*&-%#p0G6#%\"xG\"\"\"-%\"fGF-F//
F.;!\"\"F/!\"#%#s0G\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1
G/,&-%$IntG6$*&-%#p1G6#%\"xG\"\"\"-%\"fGF-F//F.;!\"\"F/!\"#%#s1G#\"\"%
\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G/,&-%$IntG6$*&-%#p
2G6#%\"xG\"\"\"-%\"fGF-F//F.;!\"\"F/!\"#%#s2G#\"\"%\"\"&\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "So
lve each equation for the one coefficient it contains.  " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "isol
ate(q0,s0);\nisolate(q1,s1);\nisolate(q2,s2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#s0G,$-%$IntG6$*&-%#p0G6#%\"xG\"\"\"-%\"fGF,F./F-;!\"
\"F.#F.\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#s1G,$-%$IntG6$*&-%#
p1G6#%\"xG\"\"\"-%\"fGF,F./F-;!\"\"F.#\"\"$\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#s2G,$-%$IntG6$*&-%#p2G6#%\"xG\"\"\"-%\"fGF,F./F-;!\"
\"F.#\"\"&\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 50 "Generalize these definitions to a formula for the " 
}{TEXT 281 1 "n" }{TEXT -1 15 "th coefficient " }{XPPEDIT 18 0 "s[n]" 
"&%\"sG6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n]=[(2*k+1)/2]*Int(
f(x)*p[n](x),x=-1..1)" "/&%\"sG6#%\"nG*&7#*&,&*&\"\"#\"\"\"%\"kGF-F-\"
\"\"F-F-\"\"#!\"\"F--%$IntG6$*&-%\"fG6#%\"xGF--&%\"pG6#F&6#F9F-/F9;,$
\"\"\"F1\"\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 174 "Now, it was time to ask if such functions exist. \+
 Are there actually functions with properties (1) and (2)?  My student
s didn't know, be we surely do, that the the functions " }{XPPEDIT 18 
0 "p[n](x)" "-&%\"pG6#%\"nG6#%\"xG" }{TEXT -1 44 " are the Legendre po
lynomials, found in the " }{TEXT 280 9 "orthopoly" }{TEXT -1 9 " packa
ge." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "with(orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 76 "A convenient way to access the first
 five Legendre polynomials uses the loop" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for k from 0 to 4 do
\np.k:=P(k,x);\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0G\"\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G%\"xG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p2G,&*$%\"xG\"\"##\"\"$F(#!\"\"F(\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#p3G,&*$%\"xG\"\"$#\"\"&\"\"#F'#!\"$F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,(*$%\"xG\"\"%#\"#N\"\")*$F'\"\"
##!#:F(#\"\"$F+\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "Consider the function f(x) = sin(" }
{XPPEDIT 18 0 "Pi" "I#PiG6\"" }{TEXT -1 115 " x) on the interval [-1,1
], and compute the coefficients s0, s1, s2, s3, s4 by the integral for
mulas derived above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 66 "for k from 0 to 4 do\nA[k]:=(k+1/2)*int(s
in(Pi*x)*p.k,x=-1..1);\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG
6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"\",$*$%#PiG!
\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"#\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"$,$*&%#PiG!\"$,&*$F*\"\"#
\"\"\"!#:F/F/\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"%\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 74 "Form an approximating sum with these five coefficients.  Note h
ow the new " }{TEXT 283 3 "add" }{TEXT -1 44 " command lets us avoid t
he quotes needed by " }{TEXT 284 3 "sum" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s := a
dd(A[n]*p.n,n=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,&*&%#Pi
G!\"\"%\"xG\"\"\"\"\"$*(F'!\"$,&*$F'\"\"#F*!#:F*F*,&*$F)F+#\"\"&F0F)#F
-F0F*\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 69 "Plot the approximation and the function f(x) on the same \+
set of axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 56 "plot([sin(Pi*x),s],x=-1..1,linestyle=[2,1],color
=black);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7en7$$!\"\"\"\"!$
!1xB#QN1YA\"!#J7$$!1nmm;p0k&*!#;$!190Q#=y_O\"F17$$!1LL$3<XZ=*F1$!169Ff
*)GLDF17$$!1nmmT%p\"e()F1$!1/_QC.6.QF17$$!1mmm\"4m(G$)F1$!10.')Q4U7]F1
7$$!1ML$3i.9!zF1$!13DMpceDhF17$$!1nm;/R=0vF1$!1Rv;+FafqF17$$!1++]P8#\\
4(F1$!1HBz[!>8\"zF17$$!1nm;/siqmF1$!1c[3Qh-a')F17$$!1++](y$pZiF1$!1F?Z
5bcT#*F17$$!1LLL$yaE\"eF1$!10w/M\"oen*F17$$!1++]([j5i&F1$!1Z8[pzD5)*F1
7$$!1nmm\">s%HaF1$!1/9\\jt64**F17$$!1LL$3x&y8_F1$!1tzN\"Gau(**F17$$!1+
++]$*4)*\\F1$!0N<t@)******!#:7$$!1++++t_\"y%F1$!1/R!fFbk(**F17$$!1+++]
_&\\c%F1$!1MK#zGZn!**F17$$!1+++]zCcVF1$!1H(>Kl*='z*F17$$!1+++]1aZTF1$!
1'><)\\D`V'*F17$$!1nm;/#)[oPF1$!1msyOl'3E*F17$$!1MLL$=exJ$F1$!1$zgW2&o
N')F17$$!1MLLL2$f$HF1$!1W#[,/I-(zF17$$!1++]PYx\"\\#F1$!1=5bi?x_qF17$$!
1MLLL7i)4#F1$!1@O(oQZc7'F17$$!1++]P'psm\"F1$!1qJX7/k,]F17$$!1++]74_c7F
1$!1q%fK#HvXQF17$$!1JLL$3x%z#)!#<$!1v0DIQ%=d#F17$$!1MLL3s$QM%Fjs$!1X%R
*3]Ug8F17$$!1^omm;zr)*!#>$!1(H#HY)485$!#=7$$\"1<LLezw5VFjs$\"1n$f>&=8]
8F17$$\"1!****\\PQ#\\\")Fjs$\"1sP)R-$GKDF17$$\"1KLLe\"*[H7F1$\"1i90KNA
nPF17$$\"1*******pvxl\"F1$\"1XaI#)3zv\\F17$$\"1)****\\_qn2#F1$\"1zK4lU
CrgF17$$\"1)***\\i&p@[#F1$\"18uU3yMJqF17$$\"1)****\\2'HKHF1$\"12*)f%eG
L'zF17$$\"1lmmmZvOLF1$\"1MJ))3Mil')F17$$\"1+++]2goPF1$\"1ZW+Q***4E*F17
$$\"1KL$eR<*fTF1$\"1,nX%p[Pl*F17$$\"1mm\"HiBQP%F1$\"1xhUR682)*F17$$\"1
+++])Hxe%F1$\"1@bH#*>C;**F17$$\"1KLeR*)**)y%F1$\"1Nx7!fP!y**F17$$\"1lm
;H!o-*\\F1$\"12[!>E`*****F17$$\"1KL$3A_1?&F1$\"1&4OiTQ,)**F17$$\"1****
\\7k.6aF1$\"14))*e%=u;**F17$$\"1KLe9as;cF1$\"1lu+16*G\")*F17$$\"1mmm;W
TAeF1$\"1Co7J'z!o'*F17$$\"1****\\i!*3`iF1$\"1!3GZ>x]B*F17$$\"1MLLL*zym
'F1$\"1Fp(RhZ$e')F17$$\"1LLL3N1#4(F1$\"1$RKhH2o\"zF17$$\"1mm;HYt7vF1$
\"1+')ol?sUqF17$$\"1*******p(G**yF1$\"1[*HvbQ38'F17$$\"1mmmT6KU$)F1$\"
1<l5zp_v\\F17$$\"1LLLLbdQ()F1$\"1s\")z$eo*fQF17$$\"1++]i`1h\"*F1$\"1YO
EZQ=0EF17$$\"1++]P?Wl&*F1$\"1y\")49t'4O\"F17$$\"\"\"F*$\"1xB#QN1YA\"F-
-%*LINESTYLEG6#\"\"#-F$6$7hn7$F($\"1)R![OD7L?F17$$!1MLLe%G?y*F1$\"1X45
)R*fuwFjs7$F/$!1N]GG-duTFjs7$$!1++vV5Su$*F1$!1`Y!3PaTQ\"F17$F5$!1WMy'Q
pAH#F17$F:$!1cz]'))zo7%F17$F?$!1fRNI%RSp&F17$FD$!12v'o1=)))pF17$FI$!1g
3vv+(\\'zF17$FN$!1vcyX$R,w)F17$FS$!1Ue&p45WO*F17$FX$!1gFzQW4f(*F17$$!1
nmT&Gu,.'F1$!1Qn<f)[c))*F17$Fgn$!1XtL@Nji**F17$F\\o$!1(4SN*\\!3***F17$
Fao$!1OdDD)GJ)**F17$Ffo$!1z4Ewp6L**F17$F[p$!1\"\\8Cjl4%)*F17$Ffp$!1Ok!
4Lcc`*F17$F`q$!1_V1aM[+\"*F17$Feq$!1%e,Y8(>'f)F17$Fjq$!1<\\xtB*[(yF17$
F_r$!1eN&R:*frrF17$Fdr$!1<0gt!*fgiF17$Fir$!1?DVZ\"oCQ&F17$F^s$!1(4k'ei
daVF17$Fcs$!1\"Q*HjtZDLF17$Fhs$!1'*[()[Kk7AF17$F^t$!1[@m0Z6n6F17$Fct$!
1Z[U2BxdEFht7$Fjt$\"1qFn)Gl#e6F17$F_u$\"1xJ<(eU$y@F17$Fdu$\"1yTQF#GjD$
F17$Fiu$\"1$=/3K&HJVF17$F^v$\"1)>Vmn7>L&F17$Fcv$\"1A'pJ/+*RiF17$Fhv$\"
1n*=!p<`krF17$F]w$\"1/-^Vzw2zF17$Fbw$\"1Xvm$Hhjf)F17$Fgw$\"1\")yT&Gb_6
*F17$Fax$\"1dr0V*Qbb*F17$F[y$\"1!Gp>>\\o$)*F17$F`y$\"12W$f$HpG**F17$Fe
y$\"1\\>3[Xa!)**F17$Fjy$\"1(y#eEK-\"***F17$F_z$\"1tQfkl@g**F17$$\"1KLe
R<vPgF1$\"17t'H$R3#))*F17$Fdz$\"11WF(35`v*F17$Fiz$\"1ypCa&GwO*F17$F^[l
$\"1*RRP!o$\\w)F17$Fc[l$\"1'*\\8v_J[zF17$Fh[l$\"1AhbKPf%*pF17$F]\\l$\"
1!>y`@>([cF17$Fb\\l$\"1J3dF\\R/UF17$$\"1nm\"zW?)\\*)F1$\"1p&4!)G!pPLF1
7$Fg\\l$\"1`\"yYw)f,CF17$$\"1++++PDj$*F1$\"1V%ovBR\"R9F17$F\\]l$\"1jJ;
L]y,TFjs7$$\"1++v=5s#y*F1$!1l\\dY]_8xFjs7$Fa]l$!1)R![OD7L?F1-Ff]l6#Fb]
l-%+AXESLABELSG6$%\"xG%!G-%'COLOURG6&%$RGBGF*F*F*-%%VIEWG6$;F(Fa]l%(DE
FAULTG" 2 305 305 305 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -16706 -16706 0 0 0 0 0 0 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Miscellaneous Ma
ple observations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 391 "In this concl
uding section we consider three Maple characteristics of consequence f
or students whose mathematical instincts are still in the formative st
ages.  First, we consider an item from a discussion of removable singu
larities.  Then we consider plotting curves defined in vector notation
.  Finally, we look at the subtleties of branches in the calculation o
f the curvature of a circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Evaluate-at" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 14 "The function  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := sin(3*x)/sin(4*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&-%$sinG6#,$%\"xG\"\"$\"\"\"-F'
6#,$F*\"\"%!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 53 "is easily found to have a removable discontinuity \+
at " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 28 ".  Maple correct
ly evaluates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "q := Limit(f,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"qG-%&LimitG6$*&-%$sinG6#,$%\"xG\"\"$\"\"\"-F*6#,$F-\"\"%!\"
\"/F-\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "value(q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##
\"\"$\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "In an effort to show that there is indeed a discontinuity
 at " }{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 7 " we try" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "subs(x=0,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "a
 terribly erroneous result that steals all thunder from a lesson on re
movable discontinuities.  Clearly, Maple has calculated " }{XPPEDIT 
18 0 "sin(0)/sin(0)" "*&-%$sinG6#\"\"!\"\"\"-F$6#F&!\"\"" }{TEXT -1 
182 " and simplified that fraction to 1.  The \"error\" is that Maple \+
did not evaluate both numerator and denominator to 0.  However, observ
e that if the expression f had been entered as a " }{TEXT 285 8 "funct
ion" }{TEXT -1 38 ", Maple's behavior would be different." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F :=
 unapply(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG:6#%\"xG6\"6$%
)operatorG%&arrowGF(*&-%$sinG6#,$9$\"\"$\"\"\"-F.6#,$F1\"\"%!\"\"F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(0);" }}{PARA 8 "" 1 "
" {TEXT -1 30 "Error, (in F) division by zero" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "So, we observe, there
 is a difference between evaluating a function and substituting into a
n expression.  We see this difference again arising in the case of pie
cewise-defined functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g := piecewise(x<0,x, x<2,x^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%*PIECEWISEG6$7$%\"xG2F)\"\"!7$
*$F)\"\"#2F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=1,
g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$\"\"\"2F'\"\"!
7$F'2F'\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 86 "Substitution does not behave \"mathematically\" as \"pl
ug in and evaluate.\"  However, if " }{TEXT 287 1 "g" }{TEXT -1 19 " i
s redefined as a " }{TEXT 286 8 "function" }{TEXT -1 32 ", evaluation \+
occurs as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "G := unapply(g,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"GG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&
29$\"\"!F02F0\"\"#*$F0F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "G(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Mike Monagan al
erts me to a new paradigm coming in Release 5.  Since " }{TEXT 326 4 "
subs" }{TEXT -1 45 " is not the exact mathematical equivalent of " }
{TEXT 328 12 "substitution" }{TEXT -1 8 ", a new " }{TEXT 327 4 "eval
" }{TEXT -1 123 " command will support the notion of \"plug in and eva
luate at.\"  Both of the examples above yield to this new functionalit
y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Plotting cu
rves in vector form" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "The vector
 representation of a curve is a staple of a multivariable calculus cou
rse.  Hence, representing a helix via the radius vector " }{TEXT 288 
1 "R" }{TEXT -1 10 " = cos(t) " }{TEXT 289 1 "i" }{TEXT -1 10 " + sin(
t) " }{TEXT 290 1 "j" }{TEXT -1 5 " + t " }{TEXT 291 1 "k" }{TEXT -1 
45 " is most easily done in Maple with the syntax" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(linalg
):\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R := ve
ctor([cos(t),sin(t),t]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'M
ATRIXG6#7%7#-%$cosG6#%\"tG7#-%$sinGF,7#F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "A plot of the helix is si
mply obtained via the syntax " }{TEXT 299 10 "spacecurve" }{TEXT -1 
110 "(R, t = 0..4*Pi) but the analogous syntax for a plane curve will \+
fail.  For example, defining the plane curve " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r := vector
([cos(t),sin(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'MATRIXG
6#7$7#-%$cosG6#%\"tG7#-%$sinGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 21 "and using the syntax " }{TEXT 300 4 
"plot" }{TEXT -1 76 "(r, t = 0..2*Pi), fails to yield the expected cir
cle since Maple treats the " }{TEXT 293 6 "vector" }{TEXT -1 1 " " }
{TEXT 292 1 "r" }{TEXT -1 6 " as a " }{TEXT 294 4 "list" }{TEXT -1 
145 " of two functions and plots a sine and a cosine curve.  We are fo
rced to treat the curve parametrically and address the components of t
he vector " }{TEXT 295 1 "r" }{TEXT -1 53 ".  Hence, plotting a plane \+
curve defined as a vector " }{TEXT 302 1 "r" }{TEXT -1 10 " requires \+
" }{TEXT 301 4 "plot" }{TEXT -1 64 "([r[1], r[2], t = 0..2*Pi]).  Such
 blurring of the roles of the " }{TEXT 303 6 "vector" }{TEXT -1 5 " an
d " }{TEXT 304 4 "list" }{TEXT -1 119 " data-structures in Maple requi
res students to remember syntactical particulars that add to the instr
uctional overhead." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "C
urvature of a circle" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 315 "Another s
taple in a multivariable calculus course is the notion of curvature of
 a plane curve.  In fact, a standard shake-down of the definition of c
urvature is the verification that the curvature of a circle is a const
ant.  This calculuation, however, requires deft handling of branches o
f the square root funtion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 51 "We illustrate with a circle centered at the origi
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "q := x^2 + y^2 = a^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"qG/,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F**$%\"aGF)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "First, we
 obtain " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 85 " explici
tly, and compute the curvature on the upper and lower semicircles sepa
rately." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "qq := solve(q,y):\ny1 := qq[1];\ny2 := qq[2];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G*$,&*$%\"xG\"\"#!\"\"*$%\"aGF)\"
\"\"#F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G,$*$,&*$%\"xG\"\"#!
\"\"*$%\"aGF*\"\"\"#F.F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 22 "Defining curvature as " }{XPPEDIT 18 0 "k
appa=y^`\"`/[1+(y^`'`)^2]^(3/2)" "/%&kappaG*&)%\"yG%\"\"G\"\"\")7#,&\"
\"\"F(*$)F&%\"'G\"\"#F(*&\"\"$F(\"\"#!\"\"F4" }{TEXT -1 7 " we get" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 97 "kappa[1] := diff(y1,x,x)/(1+diff(y1,x)^2)^(3/2);\nkappa[2] := di
ff(y2,x,x)/(1+diff(y2,x)^2)^(3/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%&kappaG6#\"\"\"*&,&*&,&*$%\"xG\"\"#!\"\"*$%\"aGF.F'#!\"$F.F-F.F/*$F
+#F/F.F/F',&F'F'*&F+F/F-F.F'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&
kappaG6#\"\"#*&,&*&,&*$%\"xGF'!\"\"*$%\"aGF'\"\"\"#!\"$F'F-F'F1*$F+#F.
F'F1F1,&F1F1*&F+F.F-F'F1F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 59 "If we use simplification with respect to \+
the side relation " }{TEXT 296 1 "q" }{TEXT -1 11 ", we obtain" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "K1 := simplify(kappa[1],\{q\});\nK2 := simplify(kappa[2],\{q\});
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K1G,$*(%\"aG\"\"#,&*$%\"xGF(!\"
\"*$F'F(\"\"\"#!\"$F(,$*&F'F(,&F*F.F-F,F,F,F/F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#K2G*(%\"aG\"\"#,&*$%\"xGF'!\"\"*$F&F'\"\"\"#!\"$F',$
*&F&F',&F)F-F,F+F+F+F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "from which an application of the " }
{TEXT 329 8 "radsimp " }{TEXT -1 19 "command then yields" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rads
imp(K1);\nradsimp(K2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%\"aG!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$%\"aG!\"\"F&" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "The difference
 in signs is significant.  In contrast to newer caculus texts which de
fine curvature with an absolute value, older calculus texts such as [2
], define curvature so as to preserve  information about concavity con
tained in the sign of y\".  Hence, on the upper semicircle, " }
{XPPEDIT 18 0 "y(x)" "-%\"yG6#%\"xG" }{TEXT -1 136 " is concave downwa
rds so y\" is negative, whereas the opposite is true on the lower semi
circle.  Maple has yielded the correct results!  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "But we have violated Mona
gan's Prime Directive: " }{TEXT 310 4 "Thou" }{TEXT -1 1 " " }{TEXT 
311 5 "shalt" }{TEXT -1 1 " " }{TEXT 312 3 "not" }{TEXT -1 1 " " }
{TEXT 313 3 "use" }{TEXT -1 1 " " }{TEXT 314 7 "radsimp" }{TEXT -1 1 "
 " }{TEXT 315 5 "under" }{TEXT -1 1 " " }{TEXT 316 89 "any circumstanc
es!  That command exists for backwards compatibility only.  Thou shalt
 use" }{TEXT -1 1 " " }{TEXT 317 8 "simplify" }{TEXT -1 65 ".  It was \+
luck that got us the correct answers.  The weakness of " }{TEXT 318 7 
"radsimp" }{TEXT -1 150 ", and the correct way to deal with the branch
es, are illustrated by the following computation based on implicit dif
ferentiation as implemented by the " }{TEXT 319 12 "implicitdiff" }
{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "yx := implicitdiff(q,y,x);\nyxx := \+
implicitdiff(q,y,x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yxG,$*&%
\"xG\"\"\"%\"yG!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yxxG,$*&,
&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+F+F-!\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "kappa := yxx/(1+yx^2)^(3/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&kappaG,$*(,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+F+F-!\"$,&F
+F+*&F)F*F-!\"#F+#F.F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 34 "If we use the same combination of " }
{TEXT 297 8 "simplify" }{TEXT -1 5 " and " }{TEXT 298 7 "radsimp" }
{TEXT -1 18 " as before, we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "radsimp(simplify(kappa,\{q\}
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$%\"aG!\"\"F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Where exa
ctly did we lose the sign information inherent in the curvature of the
 branches?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 21 "Clearly, simplifying " }{XPPEDIT 18 0 "[y^2]^(3/2)" ")7#*$%\"yG
\"\"#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "y^3" 
"*$%\"yG\"\"$" }{TEXT -1 34 " is the culprit.  It should be |y|" }
{XPPEDIT 18 0 "``^3" "*$%!G\"\"$" }{TEXT -1 16 " so that [y/|y|]" }
{XPPEDIT 18 0 "``^3" "*$%!G\"\"$" }{TEXT -1 143 " is either 1 or -1, d
epending on the branch y represents.  The mathematically correct way (
and the proper Maple way) to proceed is to simplify " }{XPPEDIT 18 0 "
kappa" "I&kappaG6\"" }{TEXT -1 76 " with respect to the defining circl
e, then make assumptions on the signs of " }{TEXT 320 1 "a" }{TEXT -1 
6 ", and " }{TEXT 321 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q1 := simplify(ka
ppa,\{q\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G,$*(%\"aG\"\"#%\"
yG!\"$*&F'F(F)!\"##F*F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "assume(a>0);\ninterface(showassumed=0);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "(Incidentally, my stud
ents discovered before I did that the above " }{TEXT 323 9 "interface
" }{TEXT -1 314 " command builds into the worksheet the suppression of
 the tilde on assumed variables.  The alternative is to remember durin
g the lecture to use the Options menu.  And it is puzzling that the su
ppression of the tilde is not permanent.  Saving the worksheet and re-
opening it later finds the tildes have reappeared.)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Returning to the comput
ation at hand, on the upper semicircle, " }{XPPEDIT 18 0 "y(x)" "-%\"y
G6#%\"xG" }{TEXT -1 8 " > 0, so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(y>0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(q1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*$%#a|irG!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
33 "whereas on the lower semicircle, " }{XPPEDIT 18 0 "y(x)" "-%\"yG6#
%\"xG" }{TEXT -1 8 " < 0, so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(y<0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(q1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$%#a|irG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 59 "[1] Wade Ellis
, Eugene Johnson, Ed Lodi, and Dan Schwalbe, " }{TEXT 305 21 "Maple V \+
Flight Manual" }{TEXT -1 20 ", Brooks/Cole, 1992." }}{PARA 0 "" 0 "" 
{TEXT -1 77 "[2] William Anthony Granville, Percey F. Smith, and Willi
am Raymond Longley, " }{TEXT 306 50 "Elements of the Differential and \+
Integral Calculus" }{TEXT -1 25 ", Ginn and Company, 1934." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Biography
" }}{PARA 0 "" 0 "" {TEXT -1 718 "With a Purdue University Ph.D. (1970
) in Relativistic Cosmology, Robert J. Lopez is a classically trained \+
applied mathematician.  After a short stint at the University of Nebra
ska-Lincoln, he spent 12 years at Memorial University in St. John's, N
ewfoundland, Canada, an odyssey of cod fish, ice hockey, and long gray
 winters.  At Rose-Hulman Institute of Technology since 1985 where he \+
pioneered the use of Maple in the classroom, he has authored books and
 papers, represented Maple \"on the road\" for 30 months, and received
 his Institute's awards for both teaching excellence and distinguished
 scholarship.  He continues to promote technology as an active partner
 in undergraduate instruction and curriculum revision." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{PARA 269 "" 0 "" {HYPERLNK 17 "Part 1" 1 "tips1
.mws" "" }{TEXT -1 3 " - " }{HYPERLNK 17 "Part 2" 1 "tips2.mws" "" }}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }