{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 
11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 
0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "
Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dot:=proc(p,q);add(p
[i]*q[i],i=1..3)end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dotGf*6$%\"
pG%\"qG6\"F)F)-%$addG6$*&&9$6#%\"iG\"\"\"&9%F0F2/F1;F2\"\"$F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "cross:=proc(p,q);[p[2]*q[3]-
p[3]*q[2],p[3]*q[1]-p[1]*q[3],p[1]*q[2]-p[2]*q[1]]end;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&crossGf*6$%\"pG%\"qG6\"F)F)7%,&*&&9$6#\"\"#\"\"
\"&9%6#\"\"$F1F1*&&F.F4F1&F3F/F1!\"\",&*&F7F1&F36#F1F1F1*&&F.F=F1F2F1F
9,&*&F?F1F8F1F1*&F-F1F<F1F9F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "mag:=proc(p);sqrt(add(p[i]^2,i=1..3))end;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$magGf*6#%\"pG6\"F(F(-%%sqrtG6#-%$addG6$*$
)&9$6#%\"iG\"\"#\"\"\"/F4;F6\"\"$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "Set 8.5" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "Q26:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 2 "a)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "r:=[t,t^2,t^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%%\"tG*$
)F&\"\"#\"\"\"*$)F&\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "rd:=diff(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rdG7%\"\"\",$
*&\"\"#F&%\"tGF&F&,$*&\"\"$F&)F*F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "u:=evalm(rd/mag(rd));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"uG-%'vectorG6#7%*&\"\"\"F**$,(F*F**&\"\"%F*)%\"tG\"\"#F*F**&
\"\"*F*)F0F.F*F*#F*F1!\"\",$*(F1F*F,#F6F1F0F*F*,$*(\"\"$F*F,F9F0F1F*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "b)" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "rd:=eval(rd,t=1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#rdG7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 18 "evalm(rd/mag(rd));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,$*&\"#9!\"\"F)#\"\"\"\"\"#F,,$*&\"\"(F*F
)F+F,,$*(\"\"$F,F)F*F)F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "c)" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalm(
[1,1,1]+t*rd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&\"\"
\"F(%\"tGF(,&F(F(*&\"\"#F(F)F(F(,&F(F(*&\"\"$F(F)F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "with(plots): spacecurve(\{[t,t^2,t^
3],[1+t,1+2*t,1+3*t]\},t=-3..3,axes=BOXED);" }}{PARA 13 "" 1 "" 
{GLPLOT3D 494 494 494 {PLOTDATA 3 "6%-%'CURVESG6#7T7%$!\"$\"\"!$\"\"*F
*$!#FF*7%$!+?5bxG!\"*$\"+t)*H!G)F2$!+M&)p#Q#!\")7%$!+S?5bFF2$\"+^se!f(
F2$!+UUG\"4#F77%$!+gIlKEF2$\"+M@'3$pF2$!+TblC=F77%$!+!3/-^#F2$\"+BX7,j
F2$!+\\3r\"e\"F77%$!++^v(Q#F2$\"+=WP,dF2$!+%e[8O\"F77%$!+?hIlAF2$\"+<=
hJ^F2$!+mrYi6F77%$!+Sr&G9#F2$\"+An$=f%F2$!+1,lR)*F27%$!+g\"3/-#F2$\"+L
\"\\?3%F2$!+y`SZ#)F27%$!+!=fz*=F2$\"+\\!\\Ag$F2$!+^;#p$oF27%$!++-^v<F2
$\"+qkV_JF2$!+3J=(f&F27%$!+?71`;F2$\"+(R6Et#F2$!+IR<<XF27%$!+SAhI:F2$
\"+HQxUBF2$!+*Hyee$F27%$!+gK;39F2$\"+nP#H)>F2$!+'R!G#z#F27%$!+!G9dG\"F
2$\"+571`;F2$!+.WOD@F27%$!++`Ej6F2$\"+eh=`8F2$!+-X6u:F27%$!+?j\"3/\"F2
$\"+7')H$3\"F2$!+v[^F6F27%$!+/Mn$=*!#5$\"+>d)RV)F^s$!+Mq\\XxF^s7%$!+3O
=fzF^s$\"+r.'[L'F^s$!+#oJ?/&F^s7%$!+7QpMnF^s$\"+u+hNXF^s$!+5XfaIF^s7%$
!+;S?5bF^s$\"+I[BOIF^s$!+Nt-t;F^s7%$!+?Ur&G%F^s$\"+QYtO=F^s$!+b(><(y!#
67%$!+CWAhIF^s$\"+u\\4r$*F_u$!+\"[-(oGF_u7%$!+GYtO=F^s$\"+%4%ftLF_u$!+
$4(R'>'!#77%$!+?$[C7'F_u$\"+VtV[PF^v$!+\"Rh\\H#!#87%$\"+S'\\C7'F_u$\"+
f*Q%[PF^v$\"+wG'\\H#Ffv7%$\"+gZtO=F^s$\"+zXftLF_u$\"+H%)R'>'F^v7%$\"+c
XAhIF^s$\"+#y&4r$*F_u$\"+`GqoGF_u7%$\"+_Vr&G%F^s$\"+^ZtO=F^s$\"+#[?<(y
F_u7%$\"+[T?5bF^s$\"+v\\BOIF^s$\"+cu-t;F^s7%$\"+WRpMnF^s$\"+_-hNXF^s$
\"+!p%faIF^s7%$\"+SP=fzF^s$\"+\"eg[L'F^s$\"+L>.U]F^s7%$\"+ONn$=*F^s$\"
+if)RV)F^s$\"+ot\\XxF^s7%$\"+Lj\"3/\"F2$\"+R')H$3\"F2$\"+<\\^F6F27%$\"
+8`Ej6F2$\"+)='=`8F2$\"+bX6u:F27%$\"+$H9dG\"F2$\"+V71`;F2$\"+oWOD@F27%
$\"+tK;39F2$\"+.Q#H)>F2$\"+t/G#z#F27%$\"+`AhI:F2$\"+pQxUBF2$\"+!Ryee$F
27%$\"+L71`;F2$\"+S9hKFF2$\"+OS<<XF27%$\"+8-^v<F2$\"+;lV_JF2$\"+JK=(f&
F27%$\"+$>fz*=F2$\"+)4\\Ag$F2$\"+#z@p$oF27%$\"+t\"3/-#F2$\"+'=\\?3%F2$
\"+PbSZ#)F27%$\"+`r&G9#F2$\"+yn$=f%F2$\"+&G]'R)*F27%$\"+LhIlAF2$\"+w=h
J^F2$\"+'=nC;\"F77%$\"+8^v(Q#F2$\"+![u8q&F2$\"+1'[8O\"F77%$\"+$4/-^#F2
$\"+*eC6I'F2$\"+t3r\"e\"F77%$\"+tIlKEF2$\"+.A'3$pF2$\"+oblC=F77%$\"+`?
5bFF2$\"+Ate!f(F2$\"+sUG\"4#F77%$\"+L5bxGF2$\"+[**H!G)F2$\"+n&)p#Q#F77
%$\"+8+++IF2$\"+y+++!*F2$\"+N+++FF7-F$6#7T7%$!\"#F*$!\"&F*$F7F*7%$!+?5
bx=F2$!+S?5bZF2$!+gIlKwF27%$!+S?5b<F2$!+!3/-^%F2$!+?hIlsF27%$!+gIlK;F2
$!+?hIlUF2$!+!=fz*oF27%$!+!3/-^\"F2$!+g\"3/-%F2$!+SAhIlF27%$!++^v(Q\"F
2$!++-^vPF2$!++`EjhF27%$!+?hIl7F2$!+SAhINF2$!+g$=fz&F27%$!+Sr&G9\"F2$!
+!G9dG$F2$!+?9dGaF27%$!+g\"3/-\"F2$!+?j\"3/$F2$!+![C71&F27%$!++=fz*)F^
s$!+g$=fz#F2$!+Sv(Qp%F27%$!++?5bxF^s$!++/-^DF2$!++1`EVF27%$!++AhIlF^s$
!+SC71BF2$!+gO=fRF27%$!++C71`F^s$!+![C71#F2$!+?n$=f$F27%$!++Ej\"3%F^s$
!+?lK;=F2$!+!y*[CKF27%$!++G9dGF^s$!+g&G9d\"F2$!+SG9dGF27%$!++IlK;F^s$!
++1`E8F2$!++fz*[#F27%$!++?j\"3%F_u$!+SEj\"3\"F2$!+g*[C7#F27%$\"+gfEj\"
)F_u$!+3oMn$)F^s$!+@?5b<F27%$\"+#R;3/#F^s$!+;sO=fF^s$!+#3bxQ\"F27%$\"+
)=1`E$F^s$!+CwQpMF^s$!+W\"3/-\"F27%$\"+%)fz*[%F^s$!+K!3/-\"F^s$!+[?hIl
F^s7%$\"+!y&G9dF^s$\"+g:dG9F^s$!+gE9dGF^s7%$\"+wbxQpF^s$\"+_6bxQF^s$\"
+!GnK;)F_u7%$\"+s`Ej\")F^s$\"+W2`EjF^s$\"+;hz*[%F^s7%$\"+o^v(Q*F^s$\"+
O.^v()F^s$\"+/bEj\")F^s7%$\"+'\\C71\"F2$\"+$**[C7\"F2$\"+*[tO=\"F27%$
\"+wMn$=\"F2$\"+_pMn8F2$\"+G/-^:F27%$\"+cC718F2$\"+6\\C7;F2$\"+ntO=>F2
7%$\"+N9dG9F2$\"+qG9d=F2$\"+1Vr&G#F27%$\"+:/-^:F2$\"+I3/-@F2$\"+W71`EF
27%$\"+%RpMn\"F2$\"+*yQpM#F2$\"+$=3/-$F27%$\"+u$=fz\"F2$\"+[n$=f#F2$\"
+A^v(Q$F27%$\"+atO=>F2$\"+2ZtOGF2$\"+h?5bPF27%$\"+Lj\"3/#F2$\"+mEj\"3$
F2$\"+***[C7%F27%$\"+8`Ej@F2$\"+E1`ELF2$\"+Rfz*[%F27%$\"+$H9dG#F2$\"+'
eG9d$F2$\"+zG9d[F27%$\"+tK;3CF2$\"+YlK;QF2$\"+>)*[C_F27%$\"+`AhIDF2$\"
+1XAhSF2$\"+fn$=f&F27%$\"+L71`EF2$\"+mC71VF2$\"+*p$=ffF27%$\"+8-^vFF2$
\"+E/-^XF2$\"+R1`EjF27%$\"+$>fz*GF2$\"+'Q=fz%F2$\"+zv(Qp'F27%$\"+t\"3/
-$F2$\"+Yj\"3/&F2$\"+>XAhqF27%$\"+`r&G9$F2$\"+1Vr&G&F2$\"+f9dGuF27%$\"
+LhIlKF2$\"+mAhIbF2$\"+*R=fz(F27%$\"+8^v(Q$F2$\"+E-^vdF2$\"+R`Ej\")F27
%$\"+$4/-^$F2$\"+'=3/-'F2$\"+zAhI&)F27%$\"+tIlKOF2$\"+YhIliF2$\"+>#fz*
))F27%$\"+`?5bPF2$\"+1T?5lF2$\"+fhIl#*F27%$\"+L5bxQF2$\"+m?5bnF2$\"+*4
`Ej*F27%$\"+8+++SF2$\"+E+++qF2$\"+/+++5F7-%*AXESSTYLEG6#%$BOXG" 1 2 0 
1 10 0 2 1 1 2 2 1.000000 53.000000 -11.000000 0 0 "Curve 1" "Curve 2
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q30" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r:=[a*cos(t)^3,a*sin(t)^3,0]
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%*&%\"aG\"\"\")-%$cosG6#%
\"tG\"\"$F(*&F'F()-%$sinGF,F.F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "plot([cos(t)^3,sin(t)^3,t=0..2*Pi]);" }}{PARA 13 "" 
1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"\"\"\"\"
!$F*F*7$$\"3J$fMKJ4<s*!#=$\"3>2L+#Rd[a#!#?7$$\"3kF:eE2)H0*F/$\"3V2eH/;
vD;!#>7$$\"3QM[NsM!4\"zF/$\"3l9Uye\\o+bF87$$\"3shc;7>/zkF/$\"3wkOrl)Q$
f7F/7$$\"3bI%HoOb$Q\\F/$\"3#oi)eZ7\\)H#F/7$$\"3[#e7=a@Gb$F/$\"33TrF,WF
=NF/7$$\"3C^u.\\\"H#)G#F/$\"3!eopb.8;&\\F/7$$\"3g:Ap$3'4e7F/$\"3u^d8Iy
=\"['F/7$$\"3QfjhZ\"3\\d&F8$\"3o&R>nq+H*yF/7$$\"3xp@B^Ue5;F8$\"37^qi0B
ye!*F/7$$\"3#)eM8Z7)QV#F2$\"3/b<r![A)H(*F/7$$\"3q@l-*fyF-$!#@$\"3nUTpA
_^K**F/7$$!3NRsEmd!*G@!#F$\"2AM\\>l%******!#<7$$!31%oQS-ccA$Fco$\"3'f0
iH+K&H**F/7$$!3R'*Rlr_DHDF2$\"3d%o[SdUGs*F/7$$!3k')3gF3$H&=F8$\"3+rM\"
3Sl#o*)F/7$$!3q=!HDJI:P&F8$\"3'[p:'pqXUzF/7$$!3[pRE+e+#G\"F/$\"3!p,v!Q
q1SkF/7$$!3:VlCD_N5<F/$\"3%z2mI#G(\\v&F/7$$!3&3cWc%*HH?#F/$\"3-6`.5T0j
]F/7$$!3g\\Fc!)yV\\GF/$\"331u=!zq#pUF/7$$!3![HK4TxHc$F/$\"3Nt*e`Ah\"3N
F/7$$!3b,>D)4tuB%F/$\"3csSsbhIxGF/7$$!3`7e&Qsl#Q\\F/$\"3M\"fv!>2c)H#F/
7$$!3+VD%4MfI\\'F/$\"3%*3c!3rI7D\"F/7$$!3!*)Q^u#yplyF/$\"37P<lAwz(o&F8
7$$!3&)[;7kYVC!*F/$\"36H^\"*3a6,<F87$$!3ePWlZ]nB(*F/$\"3elq20]\"y^#F27
$$!3'H&z1q6;F**F/$\"33cMK!41+R$Fco7$$!31%fV&Gd&)****F/$\"37b=B+n(G)H!#
D7$$!33kE@h[]M**F/$!3iwA94X%**)GFco7$$!3u&y1*e9#ys*F/$!3qjV=%*f4hCF27$
$!3mqwdY,s`!*F/$!3qp,(zo:Qi\"F87$$!3+@E&[_A'[zF/$!3O(**ei`LkM&F87$$!34
-4+(>)eElF/$!3'G*=8N4$>B\"F/7$$!3ly$4mjYr,&F/$!38\"p')41hyB#F/7$$!3=Wv
)\\ClFH%F/$!3M)*)4c<\\*GGF/7$$!3E!frvl7^f$F/$!3UR\"3d&zGwMF/7$$!3[7@S[
0\")pGF/$!3SXn7mw+YUF/7$$!3gTI*oS**G@#F/$!3E%eHc98*\\]F/7$$!3?E&*yJR&=
p\"F/$!3yI!*G\"GpFy&F/7$$!3!*)>]9JHIC\"F/$!3R'y576zs]'F/7$$!3YDkvG>8q`
F8$!3;?\"z<J+G%zF/7$$!3M9j#fVK_x\"F8$!3#R34q&zz'**)F/7$$!3#pT**eJ?X:#F
2$!3Q(QgVsC3v*F/7$$!3M,\\88(ej!HFco$!3gZD))ptDM**F/7$$!3wT4,Fx\"z&GFbu
$!3#H:Ekyf)****F/7$$\"3C![H#pl()*\\#Fco$!3gmp$R^L0%**F/7$$\"3QpztZGMN@
F2$!33;K(*R$*H_(*F/7$$\"3K()pag8*zm\"F8$!3))*y'oZX#p.*F/7$$\"3O!*HrJTu
VcF8$!3aU-&4.)GwyF/7$$\"3+aJ]mqZ_7F/$!3o1p<6?!4\\'F/7$$\"3/-%3)eQE!G#F
/$!3AQf'*)GD>'\\F/7$$\"3C?jqy?0yNF/$!3qMTX&\\&=$\\$F/7$$\"3)y&y59SsI\\
F/$!3/&)Hb/%4WI#F/7$$\"3a0r,9*Hp_'F/$!3Fb%>A#\\tJ7F/7$$\"3kX\"G)*3N0&y
F/$!3)HR&zU^5^dF87$$\"3OXxb#zE%***)F/$!3g\"G-6!Q8o<F87$$\"3:t.ZMdXB(*F
/$!3&>kKo?F3_#F27$F($\"3GMDF^'H?_&!#X-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-
%+AXESLABELSG6$Q!6\"Fb_l-%%VIEWG6$%(DEFAULTGFg_l" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "simplify(int(mag(diff(r,t)),t=0..2*Pi));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"'\"\"\"-%%csgnG6#%\"aGF&F*F&F&
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Set 8.6" }{MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q8a" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r:=[2*cos(t)+cos(2*t),2*sin(t)-sin(
2*t),0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%,&*&\"\"#\"\"\"-%$
cosG6#%\"tGF)F)-F+6#,$*&F(F)F-F)F)F),&*&F(F)-%$sinGF,F)F)-F5F/!\"\"\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([r[1],r[2],t=0.
.2*Pi]);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6%-%'C
URVESG6$7ao7$$\"\"$\"\"!$F*F*7$$\"3>foZ%y#*R%H!#<$\"3cw#[z:Gob#!#?7$$
\"3)=^32O4k!GF/$\"3]$*))Q$32Fl\"!#>7$$\"3?I;\"HLW/c#F/$\"32-0'4/<ar&F8
7$$\"3&4$RF5p7GAF/$\"3')*H;h5v-N\"!#=7$$\"30+#f0T%RI=F/$\"3ud+3Vn`nDFC
7$$\"3:sUW=h(*G;F/$\"3Q@S4$[u@H$FC7$$\"3L<M\"3(>x>9F/$\"3?%QNmWQ\">TFC
7$$\"35\"[w\"*4[t>\"F/$\"3EQb!*)H933&FC7$$\"3E0_(>n<^r*FC$\"3640wE:zWh
FC7$$\"37I84!\\E.P(FC$\"3//ej(fE[M(FC7$$\"3XpZ^I%4J/&FC$\"3g]7!Gn#QN')
FC7$$\"3!p!yV,X%pw#FC$\"3Yx^1j$f%)***FC7$$\"3_Sz)HqE(*e&F8$\"3JY&fkoJA
9\"F/7$$!3^OTD,K\"ph\"FC$\"3!*3(QM3_JH\"F/7$$!3G@:tyArtOFC$\"3C>$\\\\`
nkW\"F/7$$!3OkHT6uoq`FC$\"3?pEyQWl\"e\"F/7$$!3MJ(e/uZy%pFC$\"3O\"*p$)*
4T_r\"F/7$$!3`B8)3]$>,5F/$\"3)R90s&Q>,?F/7$$!329ME3$p`B\"F/$\"3]&zge#[
I^AF/7$$!3![091$3>*Q\"F/$\"3T`qz$Qx!RCF/7$$!3-?$*4mZ*)p9F/$\"3A>(y=Z=5
b#F/7$$!3-^_xCPC$\\\"F/$\"3'[J6Q&f%pe#F/7$$!37lBN)=k**\\\"F/$\"3I$Hf%
\\R,)f#F/7$$!3KSh<\\0$R\\\"F/$\"3!enrs@=qe#F/7$$!3#)**3dK#*Qy9F/$\"3/'
[+lTonb#F/7$$!3LuG1W^p79F/$\"3fH[Ad5a5CF/7$$!3'Q,\\'e>JJ8F/$\"3S<&)>Lo
\\$>#F/7$$!3/\"ezO[*=K7F/$\"3[8$f?&Gam=F/7$$!3nkJ0lN)>9\"F/$\"3=D_^'e[
\"z9F/7$$!3i.2D:B,l5F/$\"3')>2ju`V65F/7$$!3`-#p'>5UQ5F/$\"3OR=kP>R-yFC
7$$!3S73xF6U=5F/$\"3Z-t%HG/kT&FC7$$!3qOOQ7f&[+\"F/$\"3]GeE$f!o&y#FC7$$
!3)Q4(4='4++\"F/$\"3><Ir\"z?0C\"F87$$!3s9JgNjO/5F/$!3)p\\&[rboTEFC7$$!
3q#3FVsW\"=5F/$!3WG'[_F.eP&FC7$$!3S]wfrcXP5F/$!3\"f^S'fgq/xFC7$$!3#obc
,CiI1\"F/$!3k#f\")fTgS'**FC7$$!31t%eg!>\\O6F/$!3UpU!fH!Q^9F/7$$!3TMRLD
r+I7F/$!3aG1O.pPe=F/7$$!3)[^SPp8kK\"F/$!3Y_+;A#*4z@F/7$$!3A'z.&G-\"4T
\"F/$!3AA.I2o?1CF/7$$!3Ia9(*H$4!y9F/$!3%3=dAM.gb#F/7$$!3[Vr5<^O%\\\"F/
$!3-#4e1]Hye#F/7$$!3%z!*))eE)***\\\"F/$!3AG&o-@t!)f#F/7$$!3'ppp*>Fg#\\
\"F/$!3M;zI%R<fe#F/7$$!3MUy;%py)p9F/$!3AAUbpX*4b#F/7$$!3=&4eRRIcQ\"F/$
!3frpEyxTMCF/7$$!3p-,Ng/&\\A\"F/$!3ObO[s\")RRAF/7$$!3Vn=nrg415F/$!3y7U
m!R0h+#F/7$$!3g/l9ds!H4(FC$!30\"GLv)z%zs\"F/7$$!3?Rmn0[w2aFC$!3cAy4dKq
%e\"F/7$$!3MB3%H0qUe$FC$!36)pc@1p&R9F/7$$!3SooEZ@*Ha\"FC$!3Cd)zmUTyG\"
F/7$$\"3'*GYX?imUhF8$!3,C&\\ZbP&Q6F/7$$\"3u?*f$\\%35y#FC$!3Ukq_*R=(*)*
*FC7$$\"3oj4:$[/K,&FC$!3K'QAN5KEl)FC7$$\"3))*\\D`W4%RtFC$!3+ZJDJBJhtFC
7$$\"3oV>sfK`$o*FC$!3UH@ZDYIghFC7$$\"37E(p2Y;**>\"F/$!3e#efXlB#p]FC7$$
\"3>-&RoRhyU\"F/$!39pI(=Bkd3%FC7$$\"3SR\">dm'zJ;F/$!3#Qad<2S:G$FC7$$\"
3'e'pz^?HG=F/$!3'e$[5^MtuDFC7$$\"3kTY9I,vRAF/$!3F@n#*))Q<>8FC7$$\"3'p#
)zB/8qa#F/$!3ED)H^$R%H)fF87$$\"3]\"\\uq6)>&z#F/$!3W757bz>*z\"F87$$\"3s
'4DqYYV%HF/$!3M!Hw8Z4E`#F2F'-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLAB
ELSG6$Q!6\"Fcal-%%VIEWG6$%(DEFAULTGFhal" 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 "
" {MPLTEXT 1 0 13 "v:=diff(r,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"vG7%,&*&\"\"#\"\"\"-%$sinG6#%\"tGF)!\"\"*&F(F)-F+6#,$*&F(F)F-F)F)F)F
.,&*&F(F)-%$cosGF,F)F)*&F(F)-F7F1F)F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "speed:=simplify(mag(v));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&speedG,$*(\"\"#\"\"\"F'#F(F',$*&,&-%$cosG6#%\"tGF(F(
!\"\"F(),&F(F(*&F'F(F-F(F(F'F(F1F)F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "a:=diff(v,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
aG7%,&*&\"\"#\"\"\"-%$cosG6#%\"tGF)!\"\"*&\"\"%F)-F+6#,$*&F(F)F-F)F)F)
F.,&*&F(F)-%$sinGF,F)F.*&F0F)-F8F2F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 44 "atang:=simplify(evalm(dot(a,v)/dot(v,v)*v));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&atangG-%'vectorG6#7%,(*&\"\"'\"\"\"
)-%$cosG6#%\"tG\"\"#F,!\"\"*&\"\"$F,F.F,F3F5F,,$*(F5F,,&*&F2F,F.F,F,F,
F3F,-%$sinGF0F,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a
norm:=simplify(evalm(a-atang));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
anormG-%'vectorG6#7%,(-%$cosG6#%\"tG\"\"\"*&\"\"#F.)F*F0F.!\"\"F.F.*&-
%$sinGF,F.,&F.F.*&F0F.F*F.F.F.\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 3 "Q8c" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "r:=[cos(t),sin(2*t),cos(2*t)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "spacecurve(r,t=0..2*Pi,axes=BOXED);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG7%-%$cosG6#%\"tG-%$sinG6#,$*&\"\"#\"\"\"F)F0F0-F'
F," }}{PARA 13 "" 1 "" {GLPLOT3D 494 494 494 {PLOTDATA 3 "6$-%'CURVESG
6#7T7%$\"\"\"\"\"!$F*F*F(7%$\"+Q,!z\"**!#5$\"+SeaODF/$\"+I'[Hn*F/7%F2$
\"+@b<2\\F/$\"+Tq=8()F/7%$\"+tv;p#*F/$\"+2b#o&pF/$\"++N\\$=(F/7%F7$\"+
JwU^&)F/$\"+\"oDR=&F/7%$\"+=i89!)F/$\"+J&yme*F/$\"+keFXGF/7%F>$\"+i@'[
***F/$\"+Fx:0K!#67%$\"+<!)*[B'F/$\"+@\"z#\\(*F/$!+V$4_A#F/7%$\"+!oDR=&
F/$\"+gI*f'))F/$!+4HQDYF/7%$\"+CM$y/%F/$\"+g*zFS(F/$!+:*3Is'F/7%$\"+ae
FXGF/$\"+\"**[`X&F/$!+j5)3Q)F/7%$\"+M*)*ff\"F/$\"+[@3^JF/$!+\"[d0\\*F/
7%$\"+Zv:0KFQ$\"+z:-2kFQ$!+IRXz**F/7%$!+UGI-'*FQ$!+Pje6>F/$!+g:f:)*F/7
%$!+o$4_A#F/$!+Wu$)QVF/$!+`')o4!*F/7%$!+w0l`MF/$!+/SG#['F/$!+S&fWh(F/7
%$!+OHQDYF/$!+*eA<?)F/$!+Rl;@dF/7%$!+Om;@dF/$!+]Uo%Q*F/$!+l/l`MF/7%$!+
P*3Is'F/$!+Q6z`**F/$!+l;I-'*FQ7%$!+<'fWh(F/$!+=y\"=()*F/$\"+]!**ff\"F/
7%$!+z5)3Q)F/$!+'=ET9*F/$\"+KN$y/%F/7%$!+0()o4!*F/$!+^ZJ=yF/$\"+6\")*[
B'F/7%$!+!\\d0\\*F/$!+._5\")fF/$\"+&HOT,)F/7%$!+$e\"f:)*F/$!+B*pEv$F/$
\"+Cw;p#*F/7%$!+KRXz**F/$!+u9xy7F/$\"+d,!z\"**F/7%$!+BRXz**F/$\"+o<xy7
F/$\"+>,!z\"**F/7%$!+b:f:)*F/$\"+)>qEv$F/$\"+8v;p#*F/7%$!+Vub!\\*F/$\"
+Ta5\")fF/$\"+<h89!)F/7%$!+S')o4!*F/$\"+O\\J=yF/$\"+zy*[B'F/7%$!+)*4)3
Q)F/$\"+1j7W\"*F/$\"+hK$y/%F/7%$!+@&fWh(F/$\"+ly\"=()*F/$\"+e()*ff\"F/
7%$!+F)3Is'F/$\"+56z`**F/$!+;YI-'*FQ7%$!+9l;@dF/$\"+ZTo%Q*F/$!+V2l`MF/
7%$!+0GQDYF/$\"+?Cs,#)F/$!+#ym6s&F/7%$!+P/l`MF/$\"+zPG#['F/$!+K(fWh(F/
7%$!+C#4_A#F/$\"+xr$)QVF/$!+#y)o4!*F/7%$!+n8I-'*FQ$\"+Yge6>F/$!+<;f:)*
F/7%$\"+H!f^?$FQ$!+PX-2kFQ$!+6RXz**F/7%$\"+!3**ff\"F/$!+IC3^JF/$!+(Qd0
\\*F/7%$\"+'*fFXGF/$!+S#\\`X&F/$!+,4)3Q)F/7%$\"+fN$y/%F/$!+f,y-uF/$!+&
p3Is'F/7%$\"+1e#R=&F/$!+(>$*f'))F/$!+ZEQDYF/7%$\"+N\")*[B'F/$!+)=z#\\(
*F/$!+]!4_A#F/7%$\"+2O\\$=(F/$!+_@'[***F/$\"+53;0KFQ7%$\"+8j89!)F/$!+T
%yme*F/$\"+nhFXGF/7%$\"+?r=8()F/$!+juU^&)F/$\"+ff#R=&F/7%$\"+Ow;p#*F/$
!+o_#o&pF/$\"+JP\\$=(F/7%$\"+t'[Hn*F/$!+C_<2\\F/$\"+3s=8()F/7%$\"+g,!z
\"**F/$!+.baODF/$\"+>([Hn*F/7%$\"+++++5!\"*$\"+pE3kN!#<Fc`l-%*AXESSTYL
EG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v:=diff(r,t);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%,$-%$sinG6#%\"tG!\"\",$*&\"\"
#\"\"\"-%$cosG6#,$*&F.F/F*F/F/F/F/,$*&F.F/-F(F2F/F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "speed:=simplify(mag(v));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&speedG*$,&\"\"&\"\"\"*$)-%$cosG6#%\"tG\"\"#F(!
\"\"#F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=diff(v,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7%,$-%$cosG6#%\"tG!\"\",$*&
\"\"%\"\"\"-%$sinG6#,$*&\"\"#F/F*F/F/F/F+,$*&F.F/-F(F2F/F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "atang:=simplify(evalm(dot(a,v)/dot(
v,v)*v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&atangG-%'vectorG6#7%,$
*(-%$cosG6#%\"tG\"\"\",&F/!\"\"*$)F+\"\"#F/F/F/,&\"\"&F1F2F/F1F1,$*,F4
F/F+F/-%$sinGF-F/,&*&F4F/F3F/F/F/F1F/F5F1F1,$**\"\"%F/F+F4F0F/F5F1F1" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "anorm:=simplify(evalm(a-a
tang));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&anormG-%'vectorG6#7%,$*(
\"\"%\"\"\"-%$cosG6#%\"tGF,,&\"\"&!\"\"*$)F-\"\"#F,F,F3F,,$*,F6F,F-F,-
%$sinGF/F,,&\"#>F3*&F6F,F5F,F,F,F1F3F3,$*(F+F,,(*&\"#5F,F5F,F3F2F,*$)F
-F+F,F,F,F1F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Set 8.7" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q8" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r:=[cosh(t),sinh
(t),0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%-%%coshG6#%\"tG-%%s
inhGF(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot([r[1],r
[2],t=-1..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "
6%-%'CURVESG6$7S7$$\"3rV_\"[j!3V:!#<$!3Q,Qk$>,_<\"F*7$$\"3/:`CR))H$\\
\"F*$!3!3!o!>CF!46F*7$$\"3cY*ePV'H_9F*$!3AdHv%**oJ0\"F*7$$\"3=G%3Qsy'3
9F*$!33Eoc!=r:#**!#=7$$\"358uW#)4On8F*$!3_Y)35PUcK*F<7$$\"3')zL@4MuG8F
*$!3IV\\h?Ji\\()F<7$$\"3))p%o(R(4^H\"F*$!34!4L'Q*z)H#)F<7$$\"3!)\\g)yB
ECE\"F*$!3Ar,:vGK0xF<7$$\"31`?')4(f3B\"F*$!3g^9V8*ek<(F<7$$\"3]KMl<+g,
7F*$!3%yv$RA8:imF<7$$\"35IR#\\;XP<\"F*$!3-J!)y%*)[b9'F<7$$\"3aBJDKD0^6
F*$!3*)HQx#Q#>+dF<7$$\"34BTfVp_F6F*$!3^oevE*4)3_F<7$$\"3;92x)4;g5\"F*$
!3:vsrWD;DZF<7$$\"3-(o!fD0D(3\"F*$!3'48r7CyuE%F<7$$\"3c3HX\\=&=2\"F*$!
3?G%=@r?$eQF<7$$\"3?!*G/4Vab5F*$!3+\">Pl?h*yLF<7$$\"3KcSc;*3M/\"F*$!3)
Hy!H04HyHF<7$$\"3-\\y(fm07.\"F*$!3`8d.v-k<DF<7$$\"3$oL'H&*>5A5F*$!35cJ
n4)fS6#F<7$$\"3k*QDo;JR,\"F*$!3a#>^x![+v;F<7$$\"3\"[jZfh/z+\"F*$!3#oaJ
G%*H)f7F<7$$\"3C#oIaWHM+\"F*$!3%=o5GfR*)G)!#>7$$\"3-vBW%fV4+\"F*$!3U%)
[&fR._M%Ffr7$$\"3,$Rhs[+++\"F*$!3Y'yZ+F$zr)*!#@7$$\"3xK2\"*z#H4+\"F*$
\"3z79)*yJ57VFfr7$$\"3\\siLUBK.5F*$\"3%H+!*3Bh#e\")Ffr7$$\"3co3'Qunv+
\"F*$\"3e'GRF3*eK7F<7$$\"3?HY%)*fsP,\"F*$\"31V&f?Oz`m\"F<7$$\"3&zN<z\\
U;-\"F*$\"3/%oUh=J<4#F<7$$\"3/ihQ?V'4.\"F*$\"3ZGi5Fmt2DF<7$$\"3A>TyM2I
V5F*$\"3gp2-n')\\uHF<7$$\"3OUK=,\")=c5F*$\"3Ws&f30>!*R$F<7$$\"3/!****=
>c=2\"F*$\"3K&RA[MT%eQF<7$$\"3W$42M`zx3\"F*$\"3'y'HmO\"Q4G%F<7$$\"3d.m
#\\5&426F*$\"3uCU.&*QO]ZF<7$$\"3D&G*oq$>r7\"F*$\"3+[1)y\\\")**>&F<7$$
\"35Fb'Gh.+:\"F*$\"3Kl1J2:)*ycF<7$$\"3'fPcq]XV<\"F*$\"3Q-X(e92q:'F<7$$
\"3_iC%Rjf>?\"F*$\"3)Q5')[DN'omF<7$$\"3NGla*fi1B\"F*$\"3E+'H#zw2trF<7$
$\"3++4Z%31AE\"F*$\"3v`<B\"R:<q(F<7$$\"3K\"*GiC:t&H\"F*$\"3b+97.8mR#)F
<7$$\"3Bw#eQIe&G8F*$\"3-fyf+=\"ou)F<7$$\"31B7^Mjio8F*$\"3N^*41f&=W$*F<
7$$\"3+l96.ut19F*$\"3%R)ej'o))R*)*F<7$$\"3)QO\">>m!)\\9F*$\"3#)zc:MHt
\\5F*7$$\"3crWCnCX$\\\"F*$\"3T+%zC4M#46F*7$F($\"3Q,Qk$>,_<\"F*-%'COLOU
RG6&%$RGBG$\"#5!\"\"$\"\"!F`[lF_[l-%+AXESLABELSG6$Q!6\"Fd[l-%%VIEWG6$%
(DEFAULTGFi[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 "" {MPLTEXT 1 0 31 "rd:=diff(r,t
): rdd:=diff(rd,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "sqrt
(dot(rd,rd)*dot(rdd,rdd)-dot(rd,rdd)^2)/dot(rd,rd)^(3/2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&\"\"\"F$*$),&*&\"\"#F$)-%%coshG6#%\"tGF)F$F$F$!\"\"#
\"\"$F)F$F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q14" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:='a';" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"aGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "r:=[a*cos(t),a*sin(t),c*t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"rG7%*&%\"aG\"\"\"-%$cosG6#%\"tGF(*&F'F(-%$sinGF+F(*&%\"cGF(F,F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rd:=diff(r,t):  rdd:=diff
(rd,t):  rddd:=diff(rdd,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
64 "dot(rd,cross(rdd,rddd))/(dot(rd,rd)*dot(rdd,rdd)-dot(rd,rdd)^2):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#*&%\"cG\"\"\",&*$)%\"aG\"\"#F%F%*$)F$F*F%F%!\"
\"" }}}}{MARK "14 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }