{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times
" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 
2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 {PARA 257 "" 0 "" {TEXT 258 48 "Calculus Exploration 12: Param
etric Curve Length" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 257 343 "In Calculus Exploration 11: Arc Length, we investigated
 arc length (curve length) problems defined by y = f(x) or x = g(y). H
owever, we did not have a means of doing this when x and y are both de
fined parametrically as x(t) and y(t). The Maple code below provides u
s with the tools needed to explore parametrically defined curves in th
e plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 23 "Parametric Curve Length" }{TEXT 256 0 "" }{TEXT -1 0 "" }
}{EXCHG {PARA 256 "" 0 "" {TEXT -1 48 "First Enter the five items desc
ribed below, and " }}{PARA 256 "" 0 "" {TEXT -1 45 "then click in the \+
red area and press [Enter]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4007 "re
start:\nwith(linalg):\nt0:=Pi/2:       # Enter t0 = the t parameter va
lue for which vel & acc lines are ploted.\ntstart:=0:      # Enter tst
art = the t parameter value where the plot begins. \ntend:=4*Pi:     #
 Enter tend = the t parameter value where the plot ends.\nx:=t->t*cos(
t): # Enter the x(t) function (change only the expression right of ' -
> ').\ny:=t->t*sin(t): # Enter the y(t) function (change only the expr
ession right of ' -> ').\nr:=t->[x(t),y(t)]:\n`r'`:=t->[D(x)(t),D(y)(t
)]:\n`r''`:=t->[(D@D)(x)(t),(D@D)(y)(t)]:\nmagr2:=t->dotprod(r(t),r(t)
,orthogonal):\nmagv2:=t->dotprod(`r'`(t),`r'`(t),orthogonal):\nmaga2:=
t->dotprod(`r''`(t),`r''`(t),orthogonal):\nmagr:=t->sqrt(magr2(t)):\nm
agv:=t->sqrt(magv2(t)):\nmaga:=t->sqrt(maga2(t)):\ndotva:=t->dotprod(`
r'`(t),`r''`(t),orthogonal):\nangleva:=t->arccos(dotva(t)/(magv(t)*mag
a(t))):\nTline:=(s,t)->`r'`(s)*t+r(s):\nNline:=(s,t)->(`r''`(s)-maga(s
)*cos(angleva(s))*(`r'`(s)/magv(s)))*t+r(s):\nTlinex:=(s,t)->op(1,simp
lify(expand(Tline(s,t)))):\nTliney:=(s,t)->op(2,simplify(expand(Tline(
s,t)))):\nNlinex:=(s,t)->op(1,simplify(expand(Nline(s,t)))):\nNliney:=
(s,t)->op(2,simplify(expand(Nline(s,t)))):\n\np1:=plot([x(t),y(t),t=ts
tart..tend],color=navy):\np2:=plot([Tlinex(t0,t),Tliney(t0,t),t=-tend.
.tend],color=blue):\np3:=plot([Nlinex(t0,t),Nliney(t0,t),t=-tend..tend
],color=red):\n#p2:=plot(g(x),x=xleft..xright,y=ybot..ytop,color=red):
\n#p3:=plot([[xleft,xleft],[xright,xright]],linestyle=2,color=navy):\n
t0str:=convert(evalf(t0,4),string):\nplotfcns:=plots[display](\n      \+
      [p1,p2,p3],\n             scaling=constrained,\n            labe
ls=[``,``],\n            titlefont=[HELVETICA,DEFAULT,14],\n          \+
  title=`2D Parametric Curve with Tan & Norm Lines at t = `||t0str):\n
plotfcns;\n`position:  `*'r(t)'=r(t),t=tstart..tend;\n` ||`*'r(t)'*`||
`^2=magr2(t);\n``=simplify(expand(magr2(t)));\n` ||`*'r(t)'*`||`=simpl
ify(expand(magr(t)));\n`velocity:  `*'`r '`(t)'=`r'`(t);\n` ||`*'`r '`
(t)'*`||`^2=magv2(t);\n``=simplify(expand(magv2(t)));\n` ||`*'`r '`'(t
)*`||`=simplify(expand(magv(t)));\n`acceleration:  `*'`r ''`(t)'=`r''`
(t);\n` ||`*'`r ''`(t)'*`||`^2=maga2(t);\n``=simplify(expand(maga2(t))
);\n` ||`*'`r ''`'(t)*`||`=simplify(expand(maga(t)));\n\n`position vec
tor at `*t=t0,'r'(t0)=r(t0);\n``=evalf(r(t0));\n`magnitude of position
 vector at `*t=t0,` ||`*'r'(t0)*`||`=simplify(expand(magr(t0)));\n``=e
valf(magr(t0));\n`direction of position vector at `*t=t0,'` r`'(t0)/(`
 ||`*'r'(t0)*`||`)=\nsimplify(expand(r(t0)/magr(t0)));\n``=evalf(r(t0)
/magr(t0));\n\n`velocity at `*t=t0,'`r '`'(t0)=`r'`(t0);\n``=evalf(`r'
`(t0));\n`magnitude of velocity (speed) at `*t=t0,` ||`*'` r '`'(t0)*`
||`=simplify(expand(magv(t0)));\n``=evalf(magv(t0));\n`direction of ve
locity at `*t=t0,'`r '`'(t0)/(` ||`*'` r '`'(t0)*`||`)=\nsimplify(expa
nd(`r'`(t0)/magv(t0)));\n``=evalf(`r'`(t0)/magv(t0));\n\n`acceleration
 at `*t=t0,'`r ''`'(t0)=`r''`(t0);\n``=evalf(`r''`(t0));\n`magnitude o
f acceleration at `*t=t0,` ||`*'` r ''`'(t0)*`||`=simplify(expand(maga
(t0)));\n``=evalf(maga(t0));\n`direction of acceleration at `*t=t0,'`r
 ''`'(t0)/(` ||`*'` r ''`'(t0)*`||`)=\nsimplify(expand(`r''`(t0)/maga(
t0)));\n``=evalf(`r''`(t0)/maga(t0));\n`angle between velocity & accel
eration at `*t=t0;\n'theta'=arccos(('`r '`'(t0)*` dot `*'`r ''`'(t0))/
((` ||`*'` r '`'(t0)*`|| `)*(`  ||`*'` r ''`'(t0)*`||  `)));\ndotvatxt
:=convert(evalf(dotva(t0)),string):\nmagvtxt:=convert(evalf(magv(t0)),
string):\nmagatxt:=convert(evalf(maga(t0)),string):\n\n'theta'='arccos
'(``||dotvatxt/(``||magvtxt*``||magatxt));\n'theta'=evalf(angleva(t0))
*`radians`;\n'theta'=evalf(angleva(t0)*(360)/(2*Pi))*`degrees`;\n`tang
ent line & normal line to the curve at `*t=t0;\n'Tanline(t)'=Tline(t0,
t);\n`` =evalf(Tline(t0,t));\n`` =[sort(evalf(Tlinex(t0,t))),sort(eval
f(Tliney(t0,t)))];\n'Normline(t)'=Nline(t0,t);\n`` =evalf(Nline(t0,t))
;\n`` =[sort(evalf(Nlinex(t0,t))),sort(evalf(Nliney(t0,t)))];\n`curve \+
arc length from `*t=tstart..tend;\nInt('` ||`*`r '`(t)*`||`',t=tstart.
.tend)=Int(magv(t),t=tstart..tend);\n``=int(magv(t),t=tstart..tend);\n
``=evalf(int(magv(t),t=tstart..tend));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Filename: ExploreCalc12.m
ws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Copyright 2005, All Rights Reserved
." }}{PARA 0 "" 0 "" {TEXT -1 44 "Permission is granted to use and mod
ify for " }}{PARA 0 "" 0 "" {TEXT -1 37 "academic and non-commercial p
urposes." }}{PARA 0 "" 0 "" {TEXT -1 13 "Dr. John Pais" }}{PARA 0 "" 
0 "" {TEXT -1 28 "Mathematics Department-MICDS" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "E-mail: pais@micds.org or pais@kinetigram.com" }}{PARA 0 
"" 0 "" {TEXT -1 33 "URL:  http://kinetigram.com/micds" }}{PARA 0 "" 
0 "" {TEXT -1 37 "_____________________________________" }}}{MARK "4" 
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }