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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "R-4 MCS 320 Wednesday 30 A
pril 2003" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "#6 test the fund
amental theorem of calculus" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "f := 1/x^3:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "int_f := int(f,x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&int_fG,$*(,&*$)%\"aG\"\"#\"\"\"F,*$)%\"bGF+F,!\"\"F,F/!\"#F*F
1#F0F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(a=-2,b=4,int
_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"#K" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 88 "This result is wrong, because of the origin which
 lies between a and b as a singularity." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(f,x=-2..4);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%*undefinedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "
An even better solution to this problem is to work with a function:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fun := \+
(a,b) -> int(1/x^3,x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$funG
f*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$intG6$*&\"\"\"F1*$)%\"xG\"
\"$F1!\"\"/F4;9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
"fun(-2,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 284 "# 2 The difference between unappl
y and the arrow operator is that the arrow operator is a fast and conv
enient way to define a small function (for example an anonymous functi
on), while the unapply turns a formula into a function.   We better us
e unapply when we have already a formula." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 53 "Here is an example how we can get around the unapply:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f := x \+
+ 3*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&%\"xG\"\"\"*&\"\"$
F')F&\"\"#F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Suppose we do n
ot have the unapply:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "ff := y -> subs(x=y,f);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#ffGf*6#%\"yG6\"6$%)operatorG%&arrowGF(-%%subsG6$/%\"xG9$%\"fG
F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ff(3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 
"The alternative with unapply is" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "fff := unapply(f,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$fffGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*&
\"\"$F.)F-\"\"#F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 
"fff(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 89 "If we would not have the arrow operator, we would \+
certainly miss the anonymous functions." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "h := matrix(2,3,(i,j) -> x^i+y^j);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG-%'matrixG6#7$7%,&%\"xG\"\"
\"%\"yGF,,&F+F,*$)F-\"\"#F,F,,&F+F,*$)F-\"\"$F,F,7%,&*$)F+F1F,F,F-F,,&
F8F,F/F,,&F8F,F3F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "#7 singular
ities usually make it hard for Maple to plot algebraic curves:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plots[i
mplicitplot](x^4 + x^2*y^2 - y^2,x=-10..10,y=-10..10): " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "polar_curve := subs(x=r*cos(t),y=r*
sin(t),x^4 + x^2*y^2 - y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,pol
ar_curveG,(*&)%\"rG\"\"%\"\"\")-%$cosG6#%\"tGF)F*F**(F'F*)F,\"\"#F*)-%
$sinGF.F2F*F**&)F(F2F*F3F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "s := solve(polar_curve,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"sG6&\"\"!F&,$*&,&\"\"\"F**$)-%$sinG6#%\"tG\"\"#F*!\"\"#F2F1F-F*F2F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plots[polarplot](s[3],
t=-Pi/2+0.2..Pi/2-0.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6$-%'CURVESG6$7in7$$\"3'p\"*>)ydm+)*!#=$!3IJ7<A-#[$[!#<7$
$\"3eb6Y5D*)p(*F*$!3]u1/&y&>vWF-7$$\"32YTY'oQpt*F*$!3#Hcmjc'zgTF-7$$\"
3v/y$=m6=q*F*$!3_is)zdcL)QF-7$$\"3#4lImF>Xm*F*$!3j=>A+B^OOF-7$$\"3=&)G
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*$!32nZGB1%)QiF*7$$\"3PearMezFjF*$!3#3)e3nz.r^F*7$$\"3wUNr@P:deF*$!3a'
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G[FDxa$!#?7$$\"35XDvi6A`8Fjt$!36NT)>>47$=!#B7$$!3v09o*[Yd!fFes$!3O7\"f
fV#)Q\\$Fjt7$$!3J(H=7osZ6\"F*$!3gz&zcf70D\"Fes7$$!3ab1xr;Tx;F*$!3UP\\D
n,:aGFes7$$!3OPPmhV'HD#F*$!3fh/Kn2z4_Fes7$$!3MQ1$enJ&3GF*$!3stU\\mWk=#
)Fes7$$!32\"Gz&=QFPLF*$!3E'zPJ*RZ\"=\"F*7$$!3VMi[b4@7RF*$!3M<7=eM4j;F*
7$$!3To>pa(yhT%F*$!3_GF9lUrt@F*7$$!33c;))3IDR\\F*$!3b=*)oh<w0GF*7$$!3'
4EwAYS$)R&F*$!3'\\wgcC%*>Y$F*7$$!3)Gc\"oN*GC)eF*$!31l`tm>#*yUF*7$$!3ds
b^B'z%>jF*$!3G/2)HVPH:&F*7$$!3A:E`=&Rdv'F*$!3/LxJ42A!>'F*7$$!3m!px&z3b
grF*$!3gh<=%R6`M(F*7$$!3_>G7m)Q*fvF*$!3&od$4Dc@J()F*7$$!3.6rc@/r>zF*$!
3=%RMBj&GF5F-7$$!3E^35Wn9h#)F*$!3#>#e;WW667F-7$$!3!fJ9R-\"=s&)F*$!3$*f
a;H$ppU\"F-7$$!3&fReZ:))G$))F*$!31(Qv@XVTm\"F-7$$!3z<2]?L:,\"*F*$!3CG?
G]V.**>F-7$$!3BGBa2'=.@*F*$!3E*[jA&4*z<#F-7$$!3IaKfv0p7$*F*$!3M&e7%*p:
/Q#F-7$$!3]95=gEF9%*F*$!3C-rh$HT#GEF-7$$!3pIfB]6'z]*F*$!3[f&*QL;*y\"HF
-7$$!3GbBmwr;!f*F*$!3!H8M])4'eC$F-7$$!3m*HaOy1]m*F*$!3)*)H4bVz%ROF-7$$
!3Tp6JBn:-(*F*$!3EAmB*zZe)QF-7$$!3Q=mtj\\:P(*F*$!3i>B$G9qE;%F-7$$!3y$[
$=VP**p(*F*$!3i:e18=EwWF-7$$!3'p\"*>)ydm+)*F*F+-%'COLOURG6&%$RGBG$\"#5
!\"\"$\"\"!F]_lF\\_l-%%VIEWG6$%(DEFAULTGFa_l" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "# 13 a polynomial in sin(x):" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "p := randpoly(sin(x),degree=
13,sparse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,.*$)-%$sinG6#%\"
xG\"#7\"\"\"\"#X*&\"\")F-)F(\"#5F-!\"\"*&\"#$*F-)F(F0F-F3*&\"##*F-)F(
\"\"%F-F-*&\"#VF-)F(\"\"$F-F-*&\"#iF-)F(\"\"#F-F3" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "q := subs(sin(x)=y,p);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"qG,.*$)%\"yG\"#7\"\"\"\"#X*&\"\")F*)F(\"#5F*!\"\"*&
\"#$*F*)F(F-F*F0*&\"##*F*)F(\"\"%F*F**&\"#VF*)F(\"\"$F*F**&\"#iF*)F(\"
\"#F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s := fsolve(q,y,
complex);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"sG6.$!+=>7:7!\"*^$$!+
Ne,9$)!#5$!+[zV[RF,^$F*$\"+[zV[RF,^$$!+3$=cR#F,$!+<`&4:\"F(^$F3$\"+<`&
4:\"F($\"\"!F;F:^$$\"+rDU(*HF,$!+1Nk(4\"F(^$F=$\"+1Nk(4\"F($\"+*>a2o'F
,^$$\"+1WuW5F($!+i$f'y;F,^$FG$\"+i$f'y;F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "For every solution in the sequence s we solve" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(s[2]=sin(x
),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!+e\"ez1)!#5$!+IXpNaF&" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(s[10]=sin(x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%)H>;t!#5" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 95 "Notice that this is a family of solutions: we may add \+
any multiple of 2*Pi to the answer above." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fsolve(p,x,complex);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "This last command excludes many interesting solutions and
 is not a good answer." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 90 "#8 we compute the common intersection points of two algeb
raic curves via a Groebner basis:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 41 "p := x^2 + y^2 - 1; q := 2*x*y + 2*x - 3;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*$)%\"xG\"\"#\"\"\"F**$)%
\"yGF)F*F*F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,(*&%\"xG\"
\"\"%\"yGF(\"\"#*&F*F(F'F(F(\"\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "grobner[gbasis]([p,q],[x,y],plex);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$,,*$)%\"yG\"\"$\"\"\"\"\"#*&F*F)F'F)!\"\"*&F(F)%\"x
GF)F)*&F*F))F'F*F)F)F*F,,*\"\"&F)*&\"\"%F))F'F4F)F)*&\"\")F)F&F)F)*&F7
F)F'F)F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 296 "With a Groebner basi
s, we can write any set or list of polynomials ( not polynomial equati
ons ) into a triangular form ( if we use pure lexicographical order ).
  We solve the last equation, which is an equation in one variable and
 substitute the values for that variable into the other equations." }
{MPLTEXT 1 0 0 "" }}}}{MARK "40 0 0" 296 }{VIEWOPTS 1 1 0 3 2 1804 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }